Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks

Aldo Battista and Rémi Monasson

Laboratory of Physics of the Ecole Normale Supérieure, CNRS UMR 8023 & PSL Research, Paris, France (Dated: January 8, 2020)

Recurrent neural networks (RNN) are powerful tools to explain how attractors may emerge from noisy, high-dimensional dynamics. We study here how to learn the ∼ N 2 pairwise interactions in a RNN with N neurons to embed L manifolds of dimension D N . We show that the capacity, i.e. the maximal ratio L/N , decreases as | log | -D , where is the error on the position encoded by the neural activity along each manifold. Hence, RNN are flexible memory devices capable of storing a large number of manifolds at high spatial resolution. Our results rely on a combination of analytical tools from statistical mechanics and random matrix theory, extending Gardner's classical theory of learning to the case of patterns with strong spatial correlations.

How sensory information is encoded and processed by neuronal circuits is a central question in computational neuroscience. In many brain areas, the activity of neurons, σ, is found to depend strongly on some continuous sensory correlate r; examples include simple cells in the V1 area of the visual cortex coding for the orientation of a bar presented to the retina, and head direction cells in the subiculum or place cells in the hippocampus, whose activities depend, respectively, on the orientation of the head and the position of an animal in the physical space. Over the past decades, Continuous Attractor (CA) neural networks have emerged as an appealing concept to explain such findings, more precisely, how a large and noisy neural population can reliably encode 'positions' in lowdimensional sensory manifolds, σ = Φ(r), and continuously update their values over time according to input stimuli [1][2][START_REF] Ben-Yishai | Proc. Natl. Acad. Sci[END_REF][START_REF] Fung | [END_REF][5].

Models for the embedding of a CA in Recurrent Neural Network (RNN) generally assume that, after a Hebbianlike learning phase, the connection W ij between the neurons i, j having their place fields centered in positions r i and r j , takes value

W ij = w |r i -r j | , (1)
where | • | denotes the distance in the sensory space. If w is sufficiently excitatory at short distances and inhibitory at long ones, a bump state spontaneously emerges, in which active neurons tend to code for nearby positions in the sensory space. Weak external inputs suffice to move the bump and span the D-dimensional manifold of all possible positions r (Fig. 1(a)). This mechanism was observed in the ellipsoid body of the fly, where a bump of activity points towards the heading direction [6]. Indirect evidences for the presence of CA have been reported, e.g. in the grid-cell system [7] and in the prefontal cortex [8].

Hebbian connections (1) can be modified to embed in the same network of N neurons multiple, unrelated CAs (Fig. 1(a)), such as multiple hippocampal spatial maps corresponding to different environments [START_REF] Alme | Proc. Natl. Acad. Sci[END_REF] or contextual situations [START_REF] Jezek | [END_REF]. Assuming each one of the L maps con- FIG. 1: (a) A recurrent network with N neurons and connectivity matrix W (top left) generates high-dimensional activity configurations attracted to multiple low-dimensional manifolds (right); on each manifold, we require to memorize p points (bottom left, red crosses), whose separation defines the spatial resolution . (b) Place fields (PF) of N = 5 neurons in two maps, out of the three shown in panel (a). Each color identifies one neuron; the corresponding PF define the regions (with periodic boundary conditions) in the maps in which the neuron is active. The table lists, for each map, p = 3 activity patterns corresponding to the marked points.

tributes equally to the learning process, connections take the form [11]

W ij = L =1 w |r i -r j | , (2)
where r i is the center of the place field (PF) of neuron i in environment (Fig. 1

(b)). Theoretical calculations

show that a bump state can exist (in any map) as long as L < α c N , where α c defines the critical capacity that can be sustained by the network [12,13].

There are, however, serious practical and conceptual issues with the current theoretical understanding of multiple CAs based on (2). First, as soon as L ≥ 2, the activity bump gets stuck in some preferred locations in the retrieved map due to the interferences coming from the other L -1 non-retrieved maps [14]. In other words, rule (2) does not define truly CAs, as large barriers oppose the motion of the bump along the map [15]. The spatial error with which the environment is encoded, defined as the average discrepancy between any initial position r for the bump and the closest stable position in which it finally settles after neural relaxation dynamics, becomes quite large as L increases (Fig. 2(a)). The issue of spatial resolution is also unclear from a theoretical point of view. Capacity calculations [12,13] require that a bump can form in any of the L maps, in at least one position: they offer no guarantee about the existence of other memorized positions, and, more generally, about the value of . Secondly, the values of the critical capacity α c with rule (2) are generally quite low. It is reasonable to expect that the optimal storage capacity could be much higher: a ∼ 15-fold increase was found from the Hebb-rule critical capacity, 0.14 [17], to the optimal capacity, α c = 2 [18] in the case of 0-dimensional attractors, corresponding to the Hopfield model [START_REF] Hopfield | Proc. Nat. Acad. Sci[END_REF]. Optimal learning could also provide detailed insights on the statistical structure of the neural couplings W ij , which could be compared to the physiological distribution of synaptic connections [START_REF] Brunel | [END_REF].

In this Letter, we present a theory of optimal storage of multiple quasi-continuous maps with prescribed spatial resolution in a RNN with N binary neurons (σ i = 0, 1) and real-valued, oriented connections W ij . A map in this context is defined through the set of the input (place) fields of the N neurons, each covering a volume fraction φ 0 of the D-dimensional cube (Fig. 1(b)). In practice, the centers r i of the PFs are uniformly drawn at random in the cube, independently of each other, in all = 1...L maps. For each map , we draw uniformly at random p positions r ,µ , µ = 1...p, and collect the p corresponding patterns of activity: the neuron i is active (σ ,µ In order to learn these patterns we use Support Vector Machines (SVM) with linear kernels and hard margin classification (SM, Sec. I.B) [START_REF] Scholkopf | Learning with kernels: support vector machines, regularization, optimization, and beyond[END_REF] [23] [24]. We train N SVM, one for every row i in the coupling matrix W ij , in which we consider the neuron i as the output and the other N -1 neurons j( = i) as the inputs [START_REF] Chung | The case of hetero-associative classification of manifolds was recently studied[END_REF]. The training set {σ ,µ i } is common to all SVM. Once learning is complete, we normalize each row of the coupling matrix to j( =i) W 2 ij = 1. SVM find the coupling matrix W maximizing the stability of the stored patterns,

κ = max W min {i=1...N, =1...L,µ=1...p} 2σ ,µ i -1 j( =i) W ij σ ,µ j .
(3) SVM couplings share some qualitative features with their Hebbian counterparts. First, the couplings W ij are correlated with the distances d ij = |r i -r j | between the PF centers of the neurons i and j in the different maps , see SM Sec. I.C. Secondly, when simulating the trained network with simple rules for updating the neuron activities (SM, Sec. I.E), the activity bump forms and diffuses within a map, and occasionally jumps to other maps [11,15,16]. However, with the maximal-stability learning rule, the spatial error can be tuned at will by varying p, see Fig. 2(a). For a fixed p, remains remarkably stable as the load increases until its critical value is reached. This is in sharp contradistinction with the Hebb rule case, for which quickly increases with the number of maps. The p patterns form a discrete approximation of the map, with average spatial error scaling as = p -1/D , i.e. as the typical distance between neighboring points (Figs. 1(a)&2(b)).

The optimal stability κ (3) is shown in Fig. 3 as a function of the load α = L/N and of the number p of prescribed fixed points; it is much higher than the maximal stability achievable with rule (2) after optimization over the interaction kernel w, see SM, Sec. I.D. As expected, κ(α, p) is a decreasing function of α and p: increasing the number of maps or enforcing finer spatial resolution reduces the stability. The value of the load at which κ(α, p) vanishes defines the critical capacity α c (p), that is, the maximal load sustainable by the network as a function of the required spatial resolution. Figure 3(inset) shows that α c (p) decreases proportionally to 1/p at low p, and then much more slowly as p grows. For small p, all L × p patterns are roughly independent, and we have α c (p) αc (1) p , where α c (1) is the capacity of the perceptron with independent, biased patterns having a fraction φ 0 of active neurons [18]. As p gets large, substantial redundancies between the p patterns within a map appear, as nearby positions define similar patterns (Fig. 1(b)), and the capacity is expected to decrease less quickly with p. The cross-over takes place at p c.o. ∼ 1/φ 0 (SM, Sec. I.D). The non-trivial behavior of α c (p) when p p c.o. will be characterized in the theoretical study below.

Gardner's framework [18] can, in principle, be applied to the optimal couplings corresponding to maximal stability κ (3). Following standard calculations (SM, Sec. II.A [START_REF] Mézard | Spin glass theory and beyond: An Introduction to the Replica Method and Its Applications[END_REF]), we find that the maximal load at fixed κ and p is given by

α c (κ, p) = 1/ min m E p ( R, Z, m; κ) R,Z , (4)
where the minimum is taken over m = φ 0 j( =i) W ij . In the formula above, • denotes the average over the vectors R = (r 1 , ..., rp ) of p positions rµ drawn uniformly at random in the D-dimensional cube, and Z = (z 1 , ...z p ) drawn from the multivariate centered Gaussian distribution with R-dependent covariance matrix

Γ µ,ν ( R) = Γ |r µ -rν | -φ 2 0 .
(5) Here, Γ(d) is the overlapping volume between two PFs, whose centers are at distance d from one another; hence, Γ(0) = φ 0 . Function E p in ( 4) is defined through

E p ( R, Z, m; κ) = min {tµ≥κ+m,µ=1...p} p µ,ν=1 t µ -z µ -2 m Φ(|r µ |) Γ -1 µ,ν ( R) t ν -z ν -2 m Φ(|r ν |) , (6)
where Φ(d) = 1 if d < r c , 0 otherwise; r c is the radius of the PF, i.e. the smallest number such that Γ(2 r c ) = 0. In practice, computing α c (κ, p) from ( 4) is quite involved from a numerical point of view, as it requires to solve the p-dimensional semi-definite quadratic optimization problem in (6), as well as to average over the random vectors R and Z. This can be accurately done for small enough p, with results in excellent agreement with the SVM simulations, see Fig. 3. Notice that, for p = 1, our calculation reproduces Gardner's critical capacity α c (1) for independent and biased patterns (SM, Sec. II.B). This is expected as spatial correlations between patterns within a map appear when p ≥ 2. Formula (4) seems, unfortunately, intractable for large p. The intricate dependence on p, e.g. showing up through the Gaussian correlations between the p random fields z µ in (6), stems from the average (in each map ) over the N PF centers, {r i }, at fixed positions {r ,µ }. To avoid introducing these correlations and have an explicit dependence on the parameter p, we consider an alternative calculation scheme, where the p positions in each map are averaged out, while keeping the L × N centers quenched. To further simplify the calculation we neglect in the effective action all terms of order ≥ 3 in the couplings W ij [26]; this Gaussian approximation is expected to be exact in the large-p limit. Details about the calculation can be found in SM, Sec. II.C. Within our quenched PF theory the optimal load α c (κ, p) is the root of F defined through

F (α; m, q, U, V, T ) = V q + U - m 2 1 -4 g(U ) + 4U (7) +T 1 + U g(U ) -1 V -αp(q -m 2 ) ∞ x dz e -z 2 2 √ 2π (z -x) 2 with x = m-κ √ q-m 2 . In (7), m = j( =i) (2 C ij -φ 0 )W ij , q = j,k( =i) W ij C jk W ik ,
and the Lagrange multipliers U, V, T enforcing, respectively, the normalization of W and the definition of the order parameters, are all chosen to optimize F . C denotes the N × N multi-space Euclidean Random Matrix (ERM)

C jk {r i } = 1 L L =1 Γ |r j -r k | , (8)
with resolvent g(U ) = 1 N Trace (U Id+C) -1 . While ERM have been intensively studied in the literature [27], superimpositions of ERM mixing up different spaces have not been considered so far to our knowledge. The resolvent g(U ) can nevertheless be computed using tools from Random Matrix Theory [START_REF] Livan | Introduction to random matrices: theory and practice[END_REF] [START_REF] Battista | On the spectrum of multi-space Euclidean random matrices[END_REF], and shown to be solution of the implicit equation

U = 1 g(U ) - k =0 α Γ(k) α + g(U ) Γ(k) , (9)
where the Γ(k)'s are the components of the Fourier transform of Γ on the D-dimensional infinite reciprocal cube.

Resolution of these equations gives access to κ(α, p), in very good agreement with the numerical results obtained with SVM (Fig. 3). Small deviations can, however, be noticed and diminish with increasing p as expected. The order parameters q and m are shown as functions of p in Fig. 4, in good agreement with SVM results for large p ( p c.o. ). The value of p at which the confluence between the results from the quenched theory and SVM takes place is a decreasing function of the PF size φ 0 (SM, Sec. II.D [37] [38]) and of the map dimension D (SM, Sec. I.D).

Due to the explicit dependence of F on p in (7) the asymptotic behaviour of the critical capacity can be analytically determined in the large-p limit:

α c (p) ∼ A(D) φ -(D-1) 0 (log p) D (p → ∞) , (10)
where the constant A is made explicit in SM, Sec. II.C. Equation ( 10) is our main result. Informally speaking, the very slow decay of the critical capacity with p (Fig. 3, inset) means that recurrent neural nets can efficiently store multiple spatial maps, even at high spatial resolution. More precisely, enforcing a strong reduction of the spatial error, such as → 2 , results in a moderate drop of the maximal sustainable load, α c → α c /2 D . In addition, the capacity is predicted to be a decreasing function of the PF size in dimensions D = 2, 3, but not in dimension D = 1. This asymptotic statement is qualitatively corroborated by SVM results, even for moderate values of p (SM, Sec. I.D). Many extensions of the current work can be contemplated. First, our theory can be easily generalized to the FIG. 4: Order parameters m (left) and q (right) vs. p. Dots: SVM results (N = 2500), averaged over 50 samples; Dasheddotted lines: quenched PF theory (7). Parameters: D = 2, φ0 = .3, α = .02 (top) and .05 (bottom), for which up to, respectively, pc 2500 and pc 250 points can be memorized.

case of spatial resolutions varying from map to map, by substituting p with its average value over the maps in [START_REF] Jezek | [END_REF]. This suggests that the fraction of maps with finest spatial resolution should not exceed ∼ D when → 0, in order not to affect too much the critical capacity.

Secondly, it would be very interesting to understand how much the scaling of α c in ( 10) is robust against the choice of the parametrization Φ(r) of the manifolds. We have shown that reducing the number of active neurons in each map and allowing for variations in the sizes of the PFs from neuron to neuron do not affect this scaling [START_REF] Battista | On the spectrum of multi-space Euclidean random matrices[END_REF]. While we have assumed here for the sake of simplicity that the distribution of points was statistically uniform across space, this need not be the case in practice. Experiments have shown that spatial representations of environments are enriched in place fields close to spots of interests (such as water pots [START_REF] Hollup | [END_REF] or objects [31]) with respect to void regions. Numerical simulations reported in SM, Sec. I.D&E show that increasing the density of prescribed positions in regions of the physical space allows us to carve specific attractors in the neural activity space, representing preferentially those regions. This result is compatible with recent studies establishing the link between PF distribution and behavioral place preference [32]. Interestingly, our quenched PF theory can be applied to any particular set of PF, not necessarily homogeneously distributed over space; knowledge of the PF characteristics, e.g. from experimental measurements, allows us to determine the multispace correlation matrix C in (8) and to make specific predictions. A proof of principle of this approach is shown in SM, Sec. II.D, where we compare the couplings found with SVM and with our quenched PF theory on synthetic data.

Thirdly, the biological implications of our work remain to be worked out. Several improvements should be first brought in terms of biological plausibility. In particular one should consider continuous rather than binary neurons, explicitly distinguish excitatory and inhibitory neurons and impose Dale's law on the associated synapses, and take into account the sparse nature of synapses [33] and of place-cell activity [START_REF] Alme | Proc. Natl. Acad. Sci[END_REF] observed in CA3. Border effects, known to be important for hippocampal maps [34], should also be considered instead of the simple periodic boundary conditions assumed here. Finally, it would be very interesting to study the dynamics of learning. As a preliminary attempt, we consider in SM, Sec. I.F, an online version of the SVM learning algorithm [36], and show that the number of presentations of the patterns needed to stabilize a map is approximately proportional to p. Studying plausible learning rules could ultimately elucidate how the network progressively maturates to account for more and more fixed points and eventually defines a quasi-continuous attractor, as seems to be the case during the first weeks of development in rodents [35].

FIG. 2 :

 2 FIG. 2: (a) Spatial error vs. number L of two-dimensional maps in a network of N = 1000 neurons. Black: rule (2), with w(d) = e -d/.01 + w0, where w0 < 0 enforces a fraction φ0 = .3 of active cells. Colors: SVM results for different numbers p of prescribed positions. Line widths show the error bars, see SM Sec. I.E for details about the calculation of . (b) Spatial error vs. number p of positions in a network of N = 1000 neurons storing L = 5 maps, in dimensions D = 1, 2, 3. Lines show the expected scalings ∼ p -1/D in log-log scale.

i = 1 FIG. 3 :

 13 FIG. 3: Optimal stability κ as a function of the load α and the number p of positions. Dots: SVM results; Dashed lines: Gardner's theory (4); Dashed-dotted lines: quenched PF theory (7). Parameter values: D = 2, φ0 = .3, N = 1000 for SVM. Inset: αc(p) decreases proportionally to 1/p (straight line) at low p, and much more slowly for large p. Dots indicate results from SVM (N = 5000), averaged over 50 samples, see SM Sec. I.D for details on the estimation of αc(p); the dot size indicates the maximal error bar. The dashed-dotted line shows the predictions from the quenched PF theory.

Acknowledgements. We are grateful to A. Treves for useful discussions. This work was funded by the HFSP RGP0057/2016 project.


Birgit Griese
email: birgit.griese@math.upb.de

Analyzing attitude towards learning and teaching mathematics in members of professional learning communities: A case study

Keywords: Teacher professional development, professional learning community, constructivism

This paper presents a first step in a larger study in the context of professional learning communities following a five-day professional development course in stochastics at upper secondary level. Its focus is the analysis of one teacher's attitude towards learning and teaching mathematics in connection with his first teaching sequence on inference statistics after taking part in a professional learning community. The aim is to explore his views in reference to learning as transmission of knowledge actuated by the instructor or as a constructive activity by the learner. This is realized on the basis of the account the teacher gives on his teaching, on his observations in class, and on his reasons for the decisions involved. Additionally, a Q-sorting with statements for transmissive and constructivist views on learning was employed. Together, our investigations of the data reveal a nuanced picture of the teacher's attitudes and notions, of the diverse learning support he offered, and of perspectives of his further professional development.

Introduction

In the course of their active careers, teachers need to regularly adapt their work due to shifts such as changes in curriculum guidelines, an altered structure of their classes, or technological development. In Germany, various innovations constitute challenges, e.g. competence-orientation, digitalization, and inclusive settings. The increased emphasis on statistics and probability theory is particularly relevant for teachers at upper secondary level. It has created a high demand for mathematics teacher professional development (PD), and a majority of teachers report that they would like to have more PD than they receive (OECD, 2009, Figure 3.5). However, after decades of neglecting stochastics, "the more challenging a reform is to teachers' existing beliefs and practices, […] the more it may need […] ongoing professional development to achieve depth" (Coburn, 2003, p. 9). Theoretically corroborated PD courses, designed e.g. by researchers and teacher educators in the German Center for Mathematics Teacher Education (Deutsches Zentrum für Lehrerbildung Mathematik, DZLM) aim to fulfil the demand for sustainable PD that impacts not only at the surface but effects a real change towards learning based on understanding. Professional learning communities (PLCs) which allow teachers to reflect and discuss their teaching stand the chance to contribute substantially to this goal. Nevertheless, mid-or long-term professional growth fostered by the activities connected with a PD course remains under-researched.

In order to describe individual PD appropriately, it is advisable to first classify the underlying attitudes towards the learning and teaching of mathematics (as in [START_REF] Grigutsch | Einstellungen gegenüber Mathematik bei Mathematiklehrern[END_REF] because "the affective cannot be separated from the cognitive" (Brown & Coles, 2011, p. 864): "A teacher's belief about how students engage in mathematical activity and learn are critical factors in the ability and tendency to design and carry out inquiry-based instruction" (Lloyd, 2002, p. 150). To be particular, mathematics educators and academic researchers tend to take the view that mathematical knowledge needs to be (re)constructed by the learner in order to be understood properly, and teachers often feel constrained by the necessity to also adequately prepare their students for tests and examinations presumed to stress routines and algorithmseven if the majority of teachers report constructivist views on learning (OECD, 2009, Figure 4.2). Thus, there is the possibility, if and when these two positions clash, that the potential benefit of a PD course suffers. Thus it is a meaningful first step in the evaluation of the impact of a PD course and the PLCs that emerged in its wake to a) investigate the attitude towards the learning and teaching of mathematics held by the participants of the course. Later steps include exploring b) what the participating teachers relate and reflect about their lessons, c) what tasks and tests they use, d) the correlations between their attitudes, their reflections, and their tasks, and finally e) theorizing about possible reasons for the connections.

Theoretical background

There are numerous epistemological theories on the evolvement of mathematical knowledge (e.g. APOS by Dubinsky, cf. [START_REF] Arnon | APOS Theory: A framework for research and curriculum development in mathematics education[END_REF]Abstraction in Context, cf. Dreyfus, 2012; or the theory on advanced mathematical thinking found in [START_REF] Tall | Advanced mathematical thinking[END_REF], and many share the basic belief that learners need to actively (re)construct mathematical knowledge in order to comprehend it. The corresponding view on how mathematics should be taught is described as relativistic, meaning an instructional style "based on […] student understanding" (Wilson & Cooney, 2002, p. 132). These attitudes can be summarized under the label constructivism, viewing mathematics as a process, seeing learning as an independent and discursive activity pursued by the learner, and trusting in students' eagerness and ability to learn.

However, particularly concerning beliefs on how mathematics should be taught, another, transmissive (or dualistic, according to [START_REF] Wilson | Mathematics teacher change and development[END_REF], view gains importance: of mathematics as a toolbox containing recipes and algorithms that solve tasks, of definite solutions, of learning by following examples and instructions.

There is consensus that epistemological beliefs on how mathematical knowledge is constructed and beliefs on how mathematics should be taught are not perfectly congruent: Studies [START_REF] Chan | Relational analysis of personal epistemology and conceptions about teaching and learning[END_REF] found correlations, though mostly both traits are found in individuals. Moreover, it remains an open question if transmissive and constructionist views are the extremes of one dimension or two separate dimensions (Voss, Kleickmann, Kunter, & Hachfeldt, 2011). It is to be expected that an individual teacher may exhibit attitudes for both transmissive and constructivist positions, e.g. that he or she may seem convinced that acquiring knowledge is an active process demanding discussion and exchange of ideas, but at the same time prefer teaching methods that rely on guided step-bystep instructions or an instructor demonstrating the only correct way to solve a task very early on. This ostensible contradiction may allow deeper understanding of teacher practices in combination with their convictions and therefore an important gain of insight for research in mathematics teacher PD.

What is more, studies have identified mathematics as the subject that is most often taught in a prestructured way (OECD, 2009, Figure 4.5), so this may be particularly true for the subject we are interested in. Previous research has shown that, not only for mathematics, "short-term professional development initiatives often remain at the surface and do not effect the desired change" [START_REF] Roesken | Beliefs and beyond: Affect and the teaching and learning of mathematics[END_REF], which is ascribed to the fact that the underlying beliefs are not easily changed and the beliefs on how mathematical knowledge is built sustainably influence the way teachers teach mathematics. PLCs present a chance to address these beliefs and allow access to participant teachers' views and maybe even classrooms.

The relevance of teachers' transmissive or constructionist views towards teaching and learning is most often seen in connection with students' progression and performance in tests. This is not our main focus here; we intend to understand teachers' attitudes towards learning and teaching first, with a later perspective on exploring their individual PD. Our research questions for this paper are:

(RQ1) What attitude towards learning and teaching mathematics does a teacher show in connection with having taught stochastics after partaking in professional development on this topic? (RQ2) In how far can this attitude be categorized as transmissive or as constructivist?

Circumstances of the study

The data collected for this study must be seen in connection with an approved five-day PD course on stochastics at upper secondary level conceptualized by DZLM mathematics educators and researchers [START_REF] Oesterhaus | Designing and implementing an alternative teaching concept within a continuous professional development course for German secondary teachers[END_REF]. The course stretched out over several months and covered a broad range of stochastic content, strictly following the principle of promoting understanding. Some participants of this course came forward to partake in subsequent work in a professional learning community (PLC) with other teachers from their respective schools. For this, selected contents of the PD course were presented, discussed, and adapted. This allowed a unique insight into teaching practice in connection with a radically constructivist PD approach. Changes in the curriculum had involved teaching in-depth stochastics, which traditionally is often avoided.

Consequently, the PD course was well-frequented, and participants were motivated.

Methodology

Interviews with individual teachers were conducted after he or she had taught a sequence on stochastics, usually covering probability, tree diagrams, conditional probability, binomial distributions, and in advanced courses also hypothesis testing. The interviews were conceptualized to serve the overall research purposes and thus followed a guideline touching on the teaching sequence (underpinned by the course register, worksheets etc.) with specifics about didactic decisions, students' reactions, PLC meetings and the individual's benefit for professional growth connected to them, and cooperation among teachers. For the research focus presented here, interviewees' utterances were examined for passages revealing indications for the respective teacher's attitude towards learning and teaching, and wherever possible, these passages were categorized as referring to a transmissive or a constructivist view. In the examples given below, Vic 1 recounts a classroom situation typical of a constructivist teaching attitude, and Joe reports that his students opposed his approach to establish understanding and demanded a more transmissive method.

Vic: I ventured to say, frequently: I'm lost here, I have to think about this again. Or: You are right, as far as I can see at the moment. And the next time, I could say: I thought about this, and it occurred to me that … (Vic_1, 01:05:25)

Joe: Subsequently, they [the students] reproached me: Why did we start with a task that is not one-to-one relevant for the exams? (Joe_1, 00:02:55)

In this case study, we concentrate on Vic, who is an experienced teacher with hardly any history of teaching stochastics himself, but who addressed teaching this content in an advanced course. Figure 1: Q-method score sheet by Vic, statements (abbreviated here) placed in given fields, from "1: less important" to "7: superimportant"

The Q-method [START_REF] Fluckinger | Positive reactions to a Q sort for personality assessment[END_REF]) is a popular tool for assessing personality. Subjects (in our case, teachers participating in a PLC at their school, after one of them had attended the five-day PD course) are asked not to rate statements (like in Likert-scale surveys) but to rank them by placing them on a score sheet of a given shape (see Figure 1, where statements are condensed to keywords). This way, the common case is avoided that the vast majority of statements are rated as equally "very important". In a Q-survey the number of fields on the "important" right side of the score sheet is limited. The Q-sorting can take place with statements printed on cards to be physically placed on the score sheet, or with digital alternatives (e.g. FlashQ by Christian Hackert, http://www.hackert.biz/flashq) that can be sent via e-mail; we use both. The 16 statements used for our investigation of attitude towards the learning of mathematics, the socalled Q-sample, are taken from Jaschke (2017). Eight statements each represent the transmissive or the constructivist view. For example, Learners are to develop routines by drill and repeated practice (bottom statement in column 3 in Figure 1) stands for a transmissive perception of learning, and Learners are to work on mathematical problems in an autonomous and self-regulated way (top statement in column 6 in Figure 1) describes a constructivist notion. The completed score sheet allows the calculation of an index: Transmissive statements are equipped with a negative algebraic sign to their respective position; constructivist statements retain their respective position with a positive algebraic sign; the index is calculated by adding up all values (Figure 2 shows an example). In our survey, this results in index values between -20 (clearly transmissive) and +20 (clearly constructivist).

+ + + - - + + + + - - - - + - - -1 -4 +3 +8 +5 +0 -7

Results

Vic's utterances in the interview reveal his differentiated perspective on the learning and teaching of mathematics, containing elements both of transmissive and of constructivist attitude.

Interview utterances categorized as transmissive

Vic reports that at the beginning of the teaching unit, he opted for a guided approach ("I set that [the null hypothesis in the introductory example], […] and then I defined the errors of the first and second kind", Vic_1, 00:19:58) with clear-cut step-by-step tasks for the students ("Then I kept to the worksheets from [name of publisher]: […] very well preprocessed, for filling in, for working with", Vic_1, 00:21:33). He continues to secure the essence of the lessons and calculations by scanning and printing important summaries from his preferred publisher's material in color on his home printer (Vic_1, 00:21:55), thus choosing scripted synopses over students' individual résumés.

Interview utterances categorized as constructivist

On the other hand, Vic sees the narrow solutions provided for the marking of central exams critically (Vic_1, 00:31:59). He relates that his students "had to try out [their approaches at calculating the probability of an error]" (Vic_1, 00:20:36), which is typical of a constructivist view. He very much appreciates the fact that his students liked to take part in discussions, and acknowledges that both himself and the students profited from these interactions (Vic_1, 00:58:25).

Vic appreciates the creative strategies some of his students use (e.g. when attempting to solve a third degree polynomial equation by trying systematically, Vic_1, 00:35:56), although he is aware that in exams, such strategies can result in an inconvenient loss of credits (Vic_1, 00:36:31). His concerns about his students' performance also become clear when he talks about the case of a student who did not abide to "what is to be done at school, [he] always reinvented the wheel" (Vic_1, 00:38:48) and thus was graded lower than his presumed potentialuntil in the final exams, when he came out with full credits. This experience boils down to the fact that a constructivist approach to learning bears the danger of demerits. Still, Vic likes to offer a choice (Vic_1, 00:28:04) of different topics or complexities. He also appreciates it when understanding is promoted (like in double tree diagrams, where a higher base rate results in a higher number of false-positive diagnoses, Vic_1, 00:25:59, 00:26:14, or in a responsive spreadsheet that enables the observation that with a higher sample size, the standard deviation of a random variable decreases, but the mean stays the same, Vic_1, 00:53:37).

Interview utterances referring to affective aspects

Vic also keeps an eye on students' motivation ("And they found that interesting, I think", when referring to the latter of the above examples, Vic_1, 00:53:55), particularly as he has learned that many students in this course possess a pronounced dislike against probability and stochastics ("the term stochastics brought horror to some students'about a third of the course -faces at first", Vic_1, 00:56:12). He does not lose hope of imparting some of the fascination he feels to his students and underpins this with anecdotic evidence of a student who was rather successful, in spite of her initial strict dislike (Vic_1, 00:57:25). He is particularly aware of offering alternative procedures to weaker students (e.g. using absolute numbers when calculating conditional probabilities, Vic_1, 00:26:35) and values their potential ("when I was able to reconstruct it, I gave full credits", Vic_1, 00:26:51). Vic mentions more than one incident of perceiving distinct learning difficulties, e.g. when he recounts that some students were at a loss when having to transfer routines to new contexts (Vic_1, 00:33:33).

Vic's vision of ideal teaching

As a prospect, when considering how to teach the content next time, Vic's ideal of constructivist instruction based on solid content knowledge becomes clear; he phrases his goals as "feeling more at home" (Vic_1, 00:47:00) in inference statistics, as "enhancing my collection of tasks" (Vic_1, 00:47:59). His vision of the teaching he would like to conduct in the future encompasses extended phases of reflection (Vic_1, 00:50:00), which he substantiates with the observation that what students find out themselves stays in their minds, even if it is wrong (Vic_1, 00:50:00), thus displaying his awareness of the pitfalls of self-guided learning and his own responsibility for his students' learning goals and their examination performance.

The Q-sorting Vic's Q-sorting produces the Q-index +4, a moderately constructivist view. We found that particularly his positioning of the transmissive statements are worthwhile interpreting. One of the transmissive statements considered most important reveals Vic's concern with his students' learning outcome: Learners are to master mathematical terms and procedures reliably and accurately (column 6). The other one describes what you could call a general goal for mathematics lessons: Learners are to think strictly logically and precisely (column 7). The transmissive statements rated least important by Vic also tell a story (see columns 1 and 2 in Figure 1): They refer to rote-learning without considering if the content is understood. The overall picture presented by this Q-sorting can be summarized as rejecting mindless routines, emphasizing autonomous exploring, but simultaneously keeping an eye on learning goals.

Discussion

Vic's data indicates his constructivist belief of sustainable learning as an active process that needs time to explore, to discuss, and to reflect in order to be effective, based on his personal experience. Nevertheless, his responsibility to promote all students' learning success makes him avoid very open approaches to new content that might throw some students off track, and he makes sure that learning outcomes are clearly communicated and accessible for everyone. This illustrates one of [START_REF] Clarke | Ten key principles from research for the professional development of mathematics teachers[END_REF] important principles of PD, that "changes in teachers' beliefs about teaching and learning are derived largely from classroom practice" (p. 38). Vic's teaching decisions are also influenced by the specific students in his course. There are some students keen to debate and discuss, some rattled by the prospect of having to come to grips with the unpopular content, and others with limited mathematical abilities. As a consequence, Vic does not follow only one approach in his teaching, but switches between more guided and more open activities. The reasons he gives do not stem from general scientific results but from current observations of this specific group of learners and previous experiences that form his treasure trove and enable him to draw on his collection of reactions, explanations, examples, and tasks that can support a variety of learners to overcome their unique learning obstacles. His aim of augmenting this trove with more tasks in order to further improve his teaching metaphorically sums up this perspective. Vic's attitude towards the learning and teaching of mathematics cannot be seen as one place on a line with the poles transmissive and constructivist view, although the Q-index +4 gives an orientation and acts as a first classification. Vic's reports on his teaching as flexible with respect to individual learners and variable teaching aims (such as introducing a topic, or corroborating a procedure in preparation of a centralized exam). When analyzing his teaching decisions in more detail, e.g. his choice of task or method later in our ensuing studies, we need to consider these variables, too.

Further research perspectives

As indicated above, this paper presents a first step in a larger study aimed at investigating the impact of a PD course and its subsequent PLCs on the professional growth of the members of these PLCs. The next steps could be as follows:

 Broaden the investigation of attitude towards the learning and teaching of mathematics to more members of the PLCs.  Analyze PLC members' reflections in interviews in terms of evidence for professional growth.  Explore PLC members' tasks and tests used when teaching selected content.  Look for correlations between teachers' attitudes, evidence for professional growth, and their tasks and test, and search for qualitative and theoretical corroboration that (some of) these correlations are causal.

In discussions at CERME, other lines of research were suggested for consideration, such as taking a closer look at the interactions among the PLC, or concentrating on affective aspects, e.g. specificities of the content, or teachers' expectations of what they should be able to achieve.

Altogether, these lines of research raise the hope of generating interesting results.

Figure 2 :

 2 Figure 2: Calculation of Q-index -1-4+3+8+5+0-7=+4, transmissive statements with negative, constructivist statements with positive algebraic sign, absolute values by position

All names given are aliases. Teachers' statements were translated from German by the author.


François Bremond

Remi Trichet
email: remi.trichet@inria.fr

Francois Bremond
email: francois.bremond@inria.fr

How to train your dragon: best practices in pedestrian classifier training

Keywords: pedestrian detection, real-time, data selection, parameter estimation, big data, dataset training

The present year witnesses another milestone in Pedestrian detection's journey: it has achieved remarkable progresses in the course of the past 15 years, and experts foresee an everyday use of numerous stemming applications within the next 15. Standing on the tipping point between yesterday and tomorrow pushes to the field's retrospect. Features, context, the combination of approaches, and feature data are responsible for most of the breakthroughs in pedestrian detection. If the first three elements cover the bulk of the literature, much less efforts have been dedicated to feature data. In many aspects, the construction of a training set remains similar to what it was at the birth of the domain, some related problems are not well studied, and sometimes still tackled empirically. This paper gets down to the study of pedestrian classifier training conditions. More than a survey of existing training classifiers or features, our goal is to highlight impactful parameters, potential new research directions, and combination dilemmas. Our findings are experimentally verified on two major datasets: the INRIA and Caltech-USA datasets.

I. INTRODUCTION

Accurately detecting human beings in an image is a task that an infant can perform at the age of one, but a computer algorithm still can't reliably achieve. Nevertheless, a wide range of applications such as congestion analysis, automotive, abnormal event detection, retail data mining, robotics, human gait characterization have boosted the field's progresses which is now at the core of many computer vision advances.

In a recent study [START_REF] Benenson | Ten years of pedestrian detection, what have we learned? ECCV[END_REF], features, context, feature data, and the combination of approaches have been identified as the key performance elements for pedestrian detection. Roughly 30% of approaches focus on developing, combining or adapting features, and this direction has lead to most of the breakthroughs over the past years, such as channel features [START_REF] Dollar | Integral channel features[END_REF], or convolutional network [START_REF] Ouyang | Joint deep learning for pedestrian detection[END_REF]. Context adaptation [START_REF] Galleguillos | Context based object categorization: A critical survey[END_REF], harnessing frequent geometry [START_REF] Hoiem | Putting objects in perspective[END_REF] [START_REF] Park | Multiresolution models for object detection[END_REF] or environment patterns [START_REF] Ouyang | Single-pedestrian detection aided by multipedestrian detection[END_REF] [START_REF] Yan | Robust multi-resolution pedestrian detection in traffic scenes[END_REF], also successfully received attention from the community, but this algorithm adaptation is often application or even dataset dependent.

However, less progress has been stated concerning other areas such as feature data, threshold estimation, or nonmaximum suppression. Training sets are still generated by applying the same techniques than a decade ago: Increasing the data variability is done by blindly mixing datasets [START_REF] Benenson | Ten years of pedestrian detection, what have we learned? ECCV[END_REF], and Bootstrapping [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] [START_REF] Felzenszwalb | Cascade object detection with deformable part models[END_REF] remains among the preferred choices for training set generation. To the best of our knowledge, alternatives to greedy non-maximum suppression are never considered, and the majority of training parameters are overlooked in short-length papers.

Consequently, lots of question have never been clearly answered and the related approaches are often empirical. How to augment the number of positive examples in my training set ? Is a soft-cascade always working ? What are the alternatives to non-maximum suppression ? Should we always consider data cleaning before training ? What is the best strategy to set up the classifier cascade thresholds ? These details are often understated in short-length papers. Hence, by conducting thorough parameter evaluation and comprehensive method variation tests, this detail oriented paper strives to determine what are the best practices to train a pedestrian classifier. To the best of our knowledge, these experiments are new to the community. They bring a new light on some overlooked parameters, and precise or disprove some of the community empirical findings that were never experimentally validated before.

More specifically, our contributions are manyfold. We provide extensive experiments and analysis in the domains of model size, instance selection, data cleaning, data generation, as well as near real-time classifier choice and thresholding, soft-cascading, and non-maximum suppression.

In order to study underexplored state-of-the-art areas and assess parameters impact independently, we first define a baseline that performs on par with common pedestrian detectors while remaining generic enough to test on the most common competitors. To validate our experiments, we provide tests with two types of features: HoG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] and LBP [START_REF] Ojala | A comparative study of texture measures with classification based on feature distributions[END_REF]; and on two major datasets, INRIA [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] and Caltech USA [START_REF] Dollr | Pedestrian detection: A benchmark[END_REF].

The rest of this paper is organized as follows. Section 2 reviews the related state-of-the-art. Section 3 describes the reference classifier training methodology along with the variants that we will utilize throughout this study. Section 4 covers the large set of experiments we have undertaken and Section 5 sums up our findings and discusses possible future directions.

II. RELATED WORK

Classifier optimization always has been a major concern in computer vision. Techniques ranging from active learning [START_REF] Liu | Early active learning with pairwise constraint for person re-identification[END_REF] to feature augmentation [START_REF] Chang | Feature interaction augmented sparse learning for fast kinect motion detection[END_REF] [START_REF] Liu | Unified discriminating feature analysis for visual category recognition[END_REF] have been employed to fine-tune the classifiers on tasks as various as person reidentification [START_REF] Ma | Joint attributes and event analysis for multimedia event detection[END_REF][62] [START_REF] Zhiqiang | Twostream multi-rate recurrent neural network for video-based pedestrian reidentification[END_REF] or event recognition [START_REF] Chang | Semantic pooling for complex event analysis in untrimmed videos[END_REF] [START_REF] Luo | An adaptive semi-supervised feature analysis for video semantic recognition[END_REF].

More specifically, data selection from imbalanced datasets has been a concern for pedestrian detection since the birth of the field. [START_REF] Japkowicz | Learning from imbalanced data sets: a comparison of various strategies[END_REF] proved that disproportioned datasets degrade SVMs prediction accuracy, especially for non-linearly separable data. Subsequent research on these experiments [START_REF] Yan | On predicting rare classes with svm ensemble in scene classification[END_REF] showed that best performance was obtained for approximately comparable class cardinalities when over-sampling the minority set.

Limited work carries out these issues. Bootstrapping [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF][25] is probably the most common adaptation to the problem. This data training mechanism improves accuracy thanks to successive training set updates focusing on samples difficult to classify. It is first initialized on uniformly drawn negative samples. At each iteration, the negative training set is augmented with the false positives of the previous run. The process keeps repeating until the performance converges or a memory threshold is reached. [START_REF] Walk | New features and insights for pedestrian detection[END_REF] further showed that 2 iterations lead to optimal performance. [27] adopted a different practice for random forest. They first train t trees with uniformly-drawn samples, use them to obtain harder positive and negative sample to train the next t trees, and iteratively repeat this process. Each tree group bears its own bias, therefore improving the overall forest performance. [START_REF] Felzenszwalb | Object detection with discriminatively trained part-based models[END_REF] incorporated latent values to optimise hard negative mining.

Generative techniques consider over-sampling the minority class through artificial data generation. This type of approach suffers from both, over-generalization and the risk to create erroneous data. The SMOTE algorithm [START_REF] Wang | Imbalanced data set learning with synthetic samples[END_REF] is probably the most renowned one. It creates new data as the linear combination of a randomly selected data point and one of its (same class) k nearest neighbors. Borderline-SMOTE [START_REF] Han | Borderline-smote: A new oversampling method in imbalanced data sets learning[END_REF] subsequently proposed to improve SMOTE through clever selection and strengthening of weak pairs. ADASYN [START_REF] He | Adasyn: Adaptive synthetic sampling approach for imbalanced learning[END_REF] considered limiting the over-generalization through density-based data generation. In the same vein, [START_REF] Jo | Class imbalances versus small disjuncts[END_REF] introduced clusterbased oversampling, simultaneously tackling the intra-and inter-class imbalance issue. Finally, [START_REF] Bengio | Curriculum learning. ICML[END_REF] proposed curriculum learning to progressively incorporate into the training set data ordered according to their generative power (i.e. from easy samples to hard ones).

Cost sensitive learning methods [START_REF] Elkan | The foundations of cost-sensitive learning[END_REF][53] are an alternative to generative techniques that learn the cost of misclassifying each data sample. They optimize the classification performance by up-weighting important data and provide a natural way to enhance the minority class.

Finally, [START_REF] Hwang | A two phase approach for pedestrian detection[END_REF] highlighted and tackled the imbalance issue with a two classifier cascade: The first one aims for a high recall whereas the second one enhances precision.

An alternate strategy to deal with the excess of negative data is to successively prune out easy-to-classify instances via the use of several consecutive classifiers. This cascade-of-rejectors [69][7] can also be considered to speed up the detection. The technique, inspired by Viola & Jones face detector [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF], builds up a classifier cascade that consists of successive rejection stages that get progressively more complex, therefore rejecting more difficult candidates as the classifiers get more specialized. [START_REF] Souded | Optimized cascade of classifiers for people detection using covariance features[END_REF] further upgraded the principle from features to data by performing hierarchical clustering on the dataset, leading to improved classification results. However, it requires, for each new datum, to find the corresponding set of clusters it belongs to. See [START_REF] He | Learning from imbalanced data[END_REF] for more details on learning from imbalanced datasets.

III. REFERENCE TRAINING METHODOLOGY

The definition of a reference method is a difficult task. On one hand, it needs to be generic enough to allow the evaluation of a wide spectrum of common techniques and its reproducibility under various application environments. But on the other hand, results competitive with the state-of-the-art are required for the conclusions to be usable with up-to-date algorithms. Bearing this compromise in mind, we chose the following reference method. For the sake of conciseness, we restricted it to real-time or near real-time techniques, but in most cases, the extension to more compute-intensive strategies remain possible.

Our training methodology, illustrated in figure 1, unfolds as follows. It decomposes in two distinct parts: The initial training set generation and the classifier training. The initial training set generation carefully selects data from a set of images while balancing negative and positive sample cardinalities. We then train a cascade of 1 to n classifiers. This cascade could include a cascade-of-rejectors [START_REF] Costea | Word channel based multiscale pedestrian detection without image resizing and using only one classifier[END_REF][69][7] [START_REF] Souded | Optimized cascade of classifiers for people detection using covariance features[END_REF][56], a soft cascade [START_REF] Bourdev | Robust object detection via soft cascade[END_REF], or both. In addition, each independent classifier is learnt through bootstrapping [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF][25] to improve performance. One key aspect is to seek balanced positive and negative sets at all time. Hence, all along the cascade, the minority class is oversampled to create balanced positive and negative sets.

The body of this section describes each of these components independently while putting the emphasis on important aspects or parameters that will be analysed in the next section dedicated to experiments.

A. Minority class oversampling

As the studies on the topic prove that class imbalance leads to deteriorated performance, one may seek class balance before any training. Two options exist. The first one consists in undersampling the majority class. This solution doesn't satisfy our needs as a large training set is preferable to fully represent the data variability and avoid overfitting during the training process. The remaining solution oversamples the minority class. This process implies the creation of extra data that will not match any existing instance in the dataset images. Nevertheless, these newly created instances have to credibly represented a hypothetical data. The risk of introducing spurious data in the dataset should be minimized, especially in our case, where we may generate over 90% of one class instances. Consequently, the large majority of data generation algorithms create new data by fusing the values of existing samples. In order to respect the credibility constraint, the parent data often are feature space neighbors, leading to the new data to be created in their feature space local vicinity. This is the overgeneralization issue that cripples most generative techniques, like SMOTE [START_REF] Wang | Imbalanced data set learning with synthetic samples[END_REF]: Instead of exploring the emptiest areas of the feature space, these algorithms keep populating the denser areas, thus reinforcing the dataset biases.

In order to free the process from this issue, [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF] takes advantage of the spatial localization of most pedestrian feature values. Indeed, due to their block or region decomposition, geometrically structured descriptors, such as HoG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF], LBP [START_REF] Ahonen | Face description with local binary patterns: Application to face recognition[END_REF], DPM [START_REF] Felzenszwalb | Object detection with discriminatively trained part-based models[END_REF], or Haar-inspired strategies [START_REF] Dollar | Integral channel features[END_REF], each of their feature values have a precise localization within the proposal window. The new data generation function f (.) unfolds as follows. We randomly select 2 instances x 1 = [x 11 , ...x 1n ] and x 2 = [x 21 , ...x 2n ] from the same class S i and create a new sample with their lower and upper body features, while avoiding doublons. Uniform sampling with replacement is employed. More formally:

∀x 1 , x 2 ∈ S i f (x 1 , x 2 ) = [x 11 , ...x 1 n 2 , x 2 n 2 +1 , ...x 2n ] (1
) So, a minority class of n samples can spawn a maximum of n × (n -1) new instances. Not normalizing the final vector gives better results than a re-normalization.

In their approach, [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF] solely used the horizontal dichotomy in order to guarantee the creation of valid data. In this paper, we choose to explore the limits of this strategy by relaxing this constraint in the experiments section.

B. Training Set Generation

This first step aims to form the initial training set by selecting insightful instances from the swarm of features extracted from the training images I. Let D be the desired training set cardinality, with training set positive and negative instances cardinality P and N so that D = P + N . To get a representative negative subset of the training set, the common approach is to randomly extract n = D/|I| data from each image. This typically leads to a large imbalanced training set with few positives P and a large subset N of negative instances. Data generation techniques [START_REF] Wang | Imbalanced data set learning with synthetic samples[END_REF][35] can then be employed to limit the class bias. Positive instances horizontal flipping is also frequently used.

To get a better control over the selected instances, we borrowed the strategy from [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF]. In this approach, ground truth positive examples are augmented with near positives, defined as negative data having a spatial overlap with the ground truth positive samples higher than 90%. Let P i be the set of already selected positive data after the processing of i images. When processing image i + 1, negative samples are selected as follows:

1) Data negative samples are sorted according to their similarity with positive samples P i . 2) Select the subset of the n 0 most similar negative samples (based on euclidean distance), with n 0 ≥ n. 3) n = D/|I| negative instances are randomly selected among the remaining subset n 0 . Finally, the positive class set is oversampled to reach a balance with the negative instances.

n 0 represents the overall similarity between the selected negative N and positive data P. Setting n 0 low restricts the selected negative data to those similar to positive ones, therefore building a classifier that focusses on the often misclassified data that lie on the border between the two classes. Inversely, n 0 =100% comes back to the common approach for instance selection where the full training set is represented. Figure 3 depicts the idea. Only ground truth positive and the same amount of original negative samples are fed to the training set, the rest of the original data and the augmented data are directed to the validation set.

C. Data cleaning

The purpose of data cleaning is to spruce the dataset off data that could potentially mislead the classifier training. In this paper, we experimented with two data cleaning methods with distinct spirits: Tomek links [START_REF] Tomek | Two modifications of cnn[END_REF] focusses on mixed classes neighborhoods while DBSCAN cleans overpopulated areas [START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF]. Tomek links are defined by pairs of opposite classes data that are respective nearest neighbors. More formally, given an instance pair (x i , x j ), where x i and x j are respectively positive and negative instances, and d(x i , x j ) is the distance between x i and x j , the d(x i , x j ) pair is called a Tomek link if there is no instance

x k , such that d(x i , x k ) < d(x i , x j ) or d(x j , x k ) < d(x i , x j ).
Therefore, Tomek links are drawn either from noise, or on classes border. So they are a powerful tool, for data clean-up, especially after synthetic sampling. Tomek links can then be removed until all nearest neighbor pairs belong to the same class. In our case, we employ euclidian distance and restrict the deletion to negative data to spare the scarce minority class. DBSCAN cleaning makes use of the renowned density-based clustering algorithm [START_REF] Ester | A density-based algorithm for discovering clusters in large spatial databases with noise[END_REF]. It basically performs a density correction, removing data from the mostly populated areas of the feature space. The algorithm unfolds as follows. With number of neighbors present within distance d associated to each point, the data points are ranked according to their neighbor cardinality. The algorithm then iteratively removes the sample in the most populated neighborhood, updates the cardinalities accordingly,and stops when all data samples have less than Nmin points in their vincinity. We used the euclidian distance, d=0.8 and Nmin=5 for all our experiments.

D. Classifier

The two employed classifiers in this work are typical Adaboost [START_REF] Freund | A decision-theoric generalization of online learning and an application to boosting[END_REF] and random forest [START_REF] Breiman | Random forests[END_REF]. After specifying our random forest setting, we will remind bootstrapping, the cascade-of-rejectors and soft cascade methods in the next subsections.

In our paper, the tree construction is grounded on typical entropy optimization S:

S = 1 ∑ Ci=0 -p Ci log(p Ci ) (2)
And the final confidence score for instance x is obtained by voting:

P (C i (x)) = 1 F F ∑ t=0 c t (C i ) (3)
where F is the number of trees, c t (C i ) is the count for category C i at the leaf node L t of the t th decision tree.

E. Classifier bootstrapping

We use bootstrapping [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF][25] to augment the training set. At each iteration, we incorporate the most badly classified data from the validation set into the training set and retrain the classifier. We aim to repeat this process until the training set reaches its desired initial size T .

However, a maximum of 2 iterations is advised to avoid overfitting [START_REF] Walk | New features and insights for pedestrian detection[END_REF]. Therefore, we employ [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF] method to deal with this issue, the idea being to introduce new data to prevent overfitting. Instead of simply selecting data from the validation set, we generate it the same way as detailed in the minority class oversampling section. Each new datum is generated based on 2 data samples selected among the misclassified data. Weighted sampling is utilized in this case, the probability of an instance to be drawn being set in relation to the extent of the error. The training set reduction algorithm, illustrated in figure 4, iterates until the desired training set size is reached.

F. Cascade-of-Rejectors

The cascade-of-rejectors is an efficient strategy for sieving out a large proportion of false positives, and has been applied several times in the context of pedestrian detection [START_REF] Costea | Word channel based multiscale pedestrian detection without image resizing and using only one classifier[END_REF][69][7][52] [START_REF] Viola | Rapid object detection using a boosted cascade of simple features[END_REF]. However, tuning this cascade of classifiers remains problematic. First, the classifiers are prone to errors that propagate along the cascade. Second, most authors employing this technique solely reject negative data after each classification step to avoid the creation of classifiers likely to reject positive data during the subsequent stages. However, this leads to progressively more imbalanced datasets. Fine tuning of the original positive versus negative instances ratio is usually necessary to obtain satisfactory results (a 2 to 4 times bigger negative set is the common practice). But this strategy is still under-optimal as balanced datasets are proved [START_REF] Yan | On predicting rare classes with svm ensemble in scene classification[END_REF] to be the best performing ones.

To deal with these issues, we propose a new cascade-ofrejectors. Our approach borrows [START_REF] Hwang | A two phase approach for pedestrian detection[END_REF]'s main idea with a twostage classification: The first one aims for a high recall whereas the second one focuses on precision.

The first stage embodies a cascade of n -1 rejectors. To avoid increasing the false negative rate, these n -1 initial classifiers enforce a high recall, moderately optimizing the performance on the precision. The metric formalizes as:

M = (T N × W T N + T P )/(P + N ) (4)
with P, N, T P, T N , and W T N being respectively positive, negative, true positive, true negative instances, and the weight associated to T N . W T N enforces the recall. Note that W T N = 0 comes back to optimizing for recall. When W T N ≤ 0.5, its parametrization has little impact on the results. We used W T N = 0.5 for all our experiments. Negative data, with a confidence below 0.5, are rejected at each stage. Negative data over-sampling is then undertaken to avoid emptying the validation set. When insufficient to match the positive instances cardinality, random positive data under-sampling is then considered. We employ Adaboost as classifiers for this part. These decision trees have demonstrated competitive performance whilst retaining computational efficiency [61][44]. During testing, typically over 95% of the data are safely discarded after 4-5 iterations. This cascade-of-rejectors guaranties a high recall, increased performance induced by balancing the datasets at each iteration, and few parameters to tune.

The second stage performs a finer classification, aiming for high precision results. Since the bulk of the data have been removed, more demanding computation can be performed at this stage without slowing down the detector. We employ a dense forest classifier, that has shown [START_REF] Benenson | Ten years of pedestrian detection, what have we learned? ECCV[END_REF] giving slightly better performance than its counterparts on pedestrian detection. We optimize for the log-average miss rate [START_REF] Wojek | Pedestrian detection: An evaluation of the state of the art[END_REF]. Figure 5 illustrates the cascade effect on a toy dataset.

The only parameters that require careful tuning are the rejection thresholds t(C i ), with C i the i t h classifier of the cascade. During i t h stage of the testing cascade, any data d j with confidence C i (d j ) < t(C i ) will be rejected. Since the method optimizes for the recall, the large majority of data around the separating border between the two classes are negative ones. It is then sensible to set t(C i ) > 0.5. See the experiment section for a thorough evaluation of this parameter impact on performance.

G. Non-Maximum Suppression

Objects of interest generating lots of detections in their neighborhood, non-maximum suppression is a method that keeps the maxima while deleting other proposals. Its purpose is two-fold. First, aggregating the information of nearby detections. It also works as an overlap control mechanism, balancing overlapping false detections and detections over self-occluding objects. Figure 6 illustrates these difficulties.

In this study, we use 3 different implementations of the threshold non-maximum suppression. First, Greedy nonmaximum suppression is the systematically employed technique that compares the bounding boxes that overlap over a pre-defined threshold and keeps the best scoring one. We set the overlap threshold to 0.6 for all our experiments. We also question its supremacy by testing out to other methods. Threshold non-maximum suppression (t-NMS) [START_REF] Buil | Non-maximum suppression[END_REF] groups the detection according to the bounding boxes overlap with the group best candidate, keeps the d candidates with highest confidence for each group, and builds the final candidate position with the group mean border positions. Despite its simplicity, t-NMS doesn't require heavy parametrization and combines well with the cascade algorithms. Even though the complexity is O(n 2 ), it is quite fast in practice since most of the negatives have already been suppressed by the cascade-ofrejectors. We set the overlap threshold to 0.4 and d = 2 for all our experiments. We also considered Scale non-maximum suppression (s-NMS) [START_REF] Buil | Non-maximum suppression[END_REF] that differs from t-NMS by treating each scale separately.

In the occurrence of nested detections, our parametrization always favors the biggest one. Threshold non-maximum suppression is utilized by default.

IV. EXPERIMENTS

This section, dedicated to our experimental validation, breaks down into several sub-parts. After presenting the datasets and the experimental setup details, we run individual experiments on each part main parameters or overlooked variations. The last part compares our work to the state-ofthe-art.

A. Datasets

We experimented on the INRIA and Caltech-USA datasets.

The INRIA [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] features 1832 training and 741 testing high resolution pictures. This is among the most widely used dataset for person detection. Despite its small size compared to more recent benchmarks, the INRIA dataset boasts high quality annotations and a large variety of angles, scenes, and backgrounds. The Caltech-USA [START_REF] Dollr | Pedestrian detection: A benchmark[END_REF] dataset totals 350000 images. Despite some annotation errors [START_REF] Zhang | How far are we from solving pedestrian detection[END_REF], its large size along with crowded environments, tiny pedestrians, and numerous occlusions probably make the Caltech-USA dataset the most widely used one. We experimented on the "reasonable set" that restricts algorithms to pedestrians over 50 pixels in size and a maximum of 35% occlusion. With respectively a large variability and tiny occluded detections, everyday pictures and automotive application, these two datasets offer complementary settings for our experiments.

B. Experimental setup

We experimented on the INRIA [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] and Caltech-USA (reasonable set) [START_REF] Dollr | Pedestrian detection: A benchmark[END_REF] datasets with HoG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF] and Haar-LBP [START_REF] Corvee | Haar like and lbp based features for face, head and people detection in video sequences[END_REF] features. HoGs are configurated with 12 × 6 cells, 2 × 2 blocks and 12 angle orientations, for a total of 2640 values. Block as well as full histogram normalisation are performed. The LBP descriptor follows the same structure, features a value per channel for each cell, and non-uniform patterns are pruned out [START_REF] Ojala | Multiresolution gray-scale and rotation invariant texture classification with local binary patterns[END_REF]. Candidates are selected using a multi-scale sliding window approach [START_REF] Zhu | Fast human detection using a cascade of histograms of oriented gradients[END_REF] with a stride of 4 pixels. Approximately 60% are loosely filtered according to "edgeness" and symmetry. No background subtraction or motion features are employed on the Caltech dataset. The training set size is up-bounded at 12K samples. Increasing this threshold doesn't improve performance.

No generative resampling is performed. The model height is set to 100 pixels for the INRIA dataset, 50 pixels for Caltech.

We also implemented a cascade-of-rejectors baseline with the commonly used settings: Rejector optimisation is done according to the log-average miss rate metric. We tested this baseline with a typical initial validation set containing 4 times more negative samples than positive ones. To compare with the common multi-dataset training technique, we also used an augmented version with PETS2009 [START_REF] Ferryman | An overview of the pets2009 challenge[END_REF] positive samples. This set gathers 50K positive instances from both datasets and 110K I summarizes the default methods and parameters that will be utilized for the different components alternatives.

We utilised the openCV implementation for the Adaboost and random forest classifiers. Random forest parameter set includes the number of decision trees, the number of sampled feature dimensions and the max tree depth. They were selected by measuring out-of-bag errors (OOB) [START_REF] Breiman | Random forests[END_REF]. It was computed as the average of prediction errors for each decision tree, using the non-selected training data. Adaboost is initialised with a tree depth of 2 and 256 weak classifiers. Each extra run adds 64 weak classifiers. This is a low number of weak classifiers compared to typical settings (i.e. [1024,2048]). However, in practice, increasing this value leads us to lower performance and speed.

We use a variant of the threshold non-maximum suppression (t-NMS) [START_REF] Buil | Non-maximum suppression[END_REF] that groups the detections according to the bounding boxes overlap with the group d best candidate, keeps the d candidates with highest confidence for each group, and builds the final candidate position with the group mean border positions. We set the overlap threshold to 0.6 and d = 3 for all our experiments. Log-Average Miss Rate (LAMR) [START_REF] Wojek | Pedestrian detection: An evaluation of the state of the art[END_REF] is employed as metric for all runs.

C. Model size

state-of-the-art detectors empirically set the model size [START_REF] Wojek | Pedestrian detection: An evaluation of the state of the art[END_REF]. In this subsection, we conducted thorough experimentation to determine what should be the optimal model size given the initial conditions and what is its actual impact on performance. We varied the model height size in relation to the pedestrian minimal size observed in the dataset. We identified the smallest detections to be 50 and 60 pixels in height for respectively the Caltech (reasonable set) and INRIA datasets. And, as shown in table II, the best corresponding model sizes for these 2 benchmarks are 70 and 100 pixels in height.

The overall rule should be "the bigger, the better" as implicitly stated in [START_REF] Wojek | Pedestrian detection: An evaluation of the state of the art[END_REF]. Indeed, a bigger model will harness more information. However, the quality of the detection deteriorates when the model height size exceeds 150% of the smallest dataset pedestrian. The performance gain is then a factor of small detection frequency. As shown in [START_REF] Dollr | Pedestrian detection: A benchmark[END_REF], in Caltech's reasonable set, small detections represents the bulk of occurring pedestrians whereas they remain pretty rare in INRIA, which explains the discrepancy between the two datasets. the size m within the testset. When the frequency is unknown, we advise setting it up to m × 150%.

D. Instances selection

Experiments on the training set D=P+N size have been conducted in [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF], with the performance reaching a plateau around 200K data instances. Therefore we will restrict our tests to the parameter n 0 tuning the similarity of the selected negative instances to the positive ones. We implemented 3 strategies for this purpose.

1) Percentage: Select the n 0 percent most similar data.

2) Threshold: Select the data with similarity inferior to a threshold n 0 . 3) Amount: Select the n 0 most similar data. In every case, a minimum of n = D/m data are selected to insure the same training set cardinality. Figure 7 compares these variants performance.

Despite the rather simple strategies employed here, data selection leads to a significant classifier performance improvement. The best strategy clearly consists in selecting the p data most similar to positives. The other two strategies, exploring parts of the feature space further away from positive instances lead to less discriminative classifiers. We explain the lower amount of data preselected for the Caltech dataset by the lower data variability of this dataset. Indeed, the angle and commonly encountered background objects are similar over the frames.

lesson learned: data selection and adaptation is an important area that deserves more attention from the community. As shown in these experiments, careful initial instances selection is crucial, and can lead to major performance boost. The selected negatives similarity with positive data must be set in accordance to the dataset variability.

E. Data cleaning

Data cleaning is a rarely used pre-processing step. Is it overlooked or just dispensable? To answer this question, we evaluated its true importance for pedestrian training through the 2 proposed data cleaning methods. Since our training set generation method is not deterministic, each result is averaged over a total of 6 runs on the two datasets. Approximately 0.01 to 0.2% of the negative data are removed. No significant dataset discrepancy has been observed. Results are detailed in table III. The use of data cleaning shows a small but steady mean improvement. Tomek links outperform DBSCAN cleaning on this experiment. We assume that the DBSCAN cleaning may remove important data on the fuzzy class borders that fine tuned training sets tend to densely populate in order to improve the classifier discriminative power. Also, its use typically counterbalances the over-generalization problem which does not impair the training set construction presented in this study (see section III-A). We also observe that the data cleaning efficiency is inversely correlated to the training set size. We assume that the repercussions of the few spurious data instances weaken in accordance to the population density in their neighbourhood.

lesson learned: If cleaning the data could be dispensable, especially for big fine tuned training sets, it never harms the classification. Counter-intuitively, the popular DBSCAN does not perform best for pedestrian detection.

F. Additional data generation

Let's define a credible datum as a generated datum that visually looks like the object it represents. Note that the credibility of newly generated data can only be checked visually, so, the overall credibility of a generated set can hardly be quantified. In their original work, [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF] restricted their data generation methodology to horizontal cuts, arguing that the newly generated instance credibility was paramount. In this experiment we test the limits of this hypothesis through the implementation of the following variations:

1) Horizontal cuts in 2 parts of equal areas. The original strategy proposed in [START_REF] Trichet | Dataset optimisation for real-time pedestrian detection[END_REF]. See figure 2. 2) Vertical cuts in 2 parts of equal areas. Note that cutting along a vertical axis doesn't equal to an instance horizontal flip since the two parent instances are likely to be different data. 3) Random cuts in 2 parts of equal areas. This implies the cut passing through the centre of the bounding box. Each new data will be generated according to a different cut. 4) Random walks. The first half of the blocks are selected with the random walk algorithm [START_REF] Doyle | Random walks and electric networks[END_REF] from the first sample. The other half of the blocks are selected from the second sample. Each new data will be generated according to a different walk. Using random walks guaranties that, at least, one of the 2 sets will be fully connected. 5) Random walks and horizontal cut combination. This strategy generates negative data utilizing the random walks and positive data with horizontal cuts. The methods are ranked according to their generative power and the loss of credibility of the newly generated data. For instance, with the same initial set, random walks will generate a much greater variety of new data than horizontal cuts, leading to a better exploration of the feature space. However, its likelihood to create spurious data values, that doesn't correspond to a legitimate pedestrian image is also much higher. Results are provided in table IV.

With performance inversely decreasing with the loss of newly produced data credibility, this experiment confirms that the credibility is paramount. But counter-intuitively, the same rule also applies for negative example generation that do not have this restriction. Indeed, the combination of strategies 1 and 4 doesn't lead to any improvement. This implies that the negative samples structural geometry (i.e. a window frame, a car cabin, or tree branches) also cannot be tampered with.

lesson learned: Data generation credibility is essential and should not be jeopardized for both positive and negative instances. Horizontal cuts is the best performing method.

G. Classifier type

The general predominance of Random forest classifiers over Adaboost for pedestrian recognition have already been observed in the past [START_REF] Benenson | Ten years of pedestrian detection, what have we learned? ECCV[END_REF]. In this study, We aim to push the experiment further by evaluating its actual impact within a cascade-of-rejectors. For this purpose, we tune the number of Adaboost and Random forest classifiers, keeping to 6 their total amount. Adaboost classifiers always come first in the cascade in order to benefit from their higher speed-up. This experiment is only presented for HoG features and the INRIA dataset as a similar behaviour is observed with other datasets and features.

Results are reported in table V. The only noticeable improvement is observed when the 4 or 5 last classifiers are random forests. This is sensible as over 90% of the data have been removed after these stages and the remaining ones are mostly grouped around the pedestrians. Using random forest as first classifier doesn't increase the results. We assume that a more discriminative classifier is less important during this first stage that focusses on negative data broadly different from the positive ones. We also state that random forest only brings in a marginal improvement of 0.42% compared to Adaboost classifiers, contrary to what is usually reported [START_REF] Benenson | Ten years of pedestrian detection, what have we learned? ECCV[END_REF] [START_REF] Gall | Class-specific hough forests for object detection[END_REF]. We assume that the Adaboost and random forest performance converge with the overall result improvement. lesson learned: Powerful classifiers perform best when used in all but first classifiers of the cascade. general progresses in the field also call back into question the random forest classifier predominance [START_REF] Benenson | Ten years of pedestrian detection, what have we learned? ECCV[END_REF].

H. Soft cascade

We tested our pipeline with several variations of the famous soft-cascade [START_REF] Bourdev | Robust object detection via soft cascade[END_REF], ranked according to their speed-up and their loss of discriminative power:

1) soft-cascade: The standard soft cascade [START_REF] Bourdev | Robust object detection via soft cascade[END_REF] that adds one feature value at each step of the cascade. Standard soft cascade provides results on par with the stateof-the-art [START_REF] Bourdev | Robust object detection via soft cascade[END_REF], which are significantly worse than our baseline. This makes sense as classifiers trained on low feature dimensionality are far less discriminative than full-length descriptors. The preprocessing classifier based on reverse engineering doesn't degrade the results but provides only a 2-fold speedup.

lesson learned: These results may sound the death knell of soft-cascade. If a soft-casacade provides a huge speed boost, its classifiers are independently not discriminative enough to compete with a the most recent classifiers trained on finely selected instances.

I. Classifiers threshold setting

Training various classifiers on different data, with varying features of a classifier cascade invariably leads to a set of disparate classifiers. In addition to that, variations between train and test sets may induce more discrepancies. The use of any classifier cascade is pointless without a mechanism to cope with these variations. To deal with it, the actual strategy estimates and sets a different threshold t(C i ) for each classifier C i . Setting up a set of cascade thresholds is not an easy task and the methods that tackle this particular pedestrian detection sub-problem are, at best, empirical. During testing, the optimal threshold setting is supposed to replicate the training phase dataset reduction across the cascade. In other words, each step should progressively eliminate harder negative examples while the recall should remain close or equal to 1.

In order to compare the efficacy of various commonly employed thresholding technique, we first needed to find the optimal setting. We manually looked for the threshold set leading to the best performance, exhaustively trying out all possibilities with a 0.01 step. Then, we tested the following methods: This experiment revealed several interesting points. First of all, while looking for the optimal thresholds, we stated that they are not data but feature dependent. In other words, the best threshold varies a lot when the feature dimensionality changes but remain quite similar otherwise. Second, the most common threshold estimation strategy (3), along with the removal of a predefined amount of data at each step (4) perform poorly when used with these classifiers settings. The best performing thresholding technique uses a constant threshold (1). This technique leads to an average deletion of respectively 80% and 35% of the data during the first 2 steps of the cascade, and 5-10% during the remaining stages. Hence, we assume that the thresholding technique (3) and (4) might also be dependent to classifier sets displaying a rather regular deletion behaviour across the cascade steps. Finally, all the tested techniques, are, on average performing 4.48% worse than the optimal. This elects threshold estimation as a potentially rewarding research direction. lesson learned: Counter-intuitively, the variability between the threshold values is not much caused by the data, but mostly by the features. None of existing approaches are close to the optimal and there is still space for improvement in that particular direction.

J. Non maximum suppression

Table IX covers our evaluation the 3 existing non-maximum suppression algorithms previously described in section III-G. The best results for both HoG and LBP features are always obtained by the threshold non-maximum suppression with an average 2% boost compared to the common greedy nonmaximum suppression.

lesson learned: Surprisingly, the choice of the non maximum suppression algorithm significantly impacts the over- all detector performance. The over-employed greedy nonmaximum suppression, that underperforms in every case, shouldn't be blindly chosen.

K. Comparison with the state-of-the-art

Table VI shows our method's ranking compared to the stateof-the-art, in terms of performance and speed. Table I parameters and random forest were employed for this experiment. T =0.58 is used for these runs. This work compares favorably to the state-of-the-art. We used using Adaboost classifiers and soft coding reverse engineering for this experiment.While not being the best detector on the market, with respectively 17% and 46% log-average miss rate on the INRIA and Caltech datasets, our approach still provides a very competitive performance/speed ratio. For instance, it scores better than the famous DPM [START_REF] Felzenszwalb | Object detection with discriminatively trained part-based models[END_REF] and similarly to the integral channel features [START_REF] Dollar | The fastest pedestrian detector in the west[END_REF] while running 7 times faster. VeryFast [START_REF] Benenson | Pedestrian detection at 100 frames per second[END_REF] and the crosstalk cascades provide a viable alternative in terms of speed while displaying lower performance. LBP is the best descriptor on the INRIA dataset, while HoGs show the best performance on Caltech. We assume that the numerous little pedestrians on the latter impact the texture descriptor more strongly than the gradients. The HoG and LBP fusion yields little improvement. Packed crowds, teensy pedestrians and awkward poses remain the main failure cases. Finally, Deep learning technique [START_REF] Ke | Pedestrian detection via pca filters based convolutional channel features[END_REF], [START_REF] Ren | Faster r-cnn: Towards real-time object detection with region proposal networks[END_REF] perform worse and are slower, [START_REF] Liliang | Is faster r-cnn doing well for pedestrian detection? ECCV[END_REF] performs much better but is slower, while [START_REF] Liu | Ssd: Single shot multibox detector[END_REF] is slightly more efficient and faster. It is worth stating that careful machine learning choices and parametrisation can bring standard features close to state-of-the-art deep learning performance. The end-to-end system is trained in 4-14hours on one core for 10-200K data and processes 3-4 fps on an Intel Xeon 2.1GHz CPU, calculated over frames of 640×480 pixels in size. Further speed up is possible when using a GPU, leading our detector to perform in real-time. This makes this detector one of the best performing real-time pedestrian detector to date.

V. CONCLUSION

This paper studied through extensive experimentation underexplored mechanisms at various stages of a pedestrian classifier training, with an emphasis on data selection, dataset generation at the initial stage and during the classifier cascade. Our study highlighted data selection and non-maximum suppression as underexploited components for high performance, and soft-cascade thresholding assessment as potential research direction. We also disproved a few empirically received ideas, namely the predominance of the random forest classifier (with high performing detectors) or the use of the DBSCAN algorithm for data cleaning. Besides extra experiments in domains such as active learning or part-based models, the application of the optimized Fair-Train pipeline to deep learning is an obvious possible next step for this work.

VI. ACKNOWLEDGMENT

To appear in camera ready version.

Fig. 1 .

 1 Fig. 1. Training pipeline. The initial training set generation selects data while balancing negative and positive sample cardinalities (Top). A cascade classifiers is then trained on it (Left), each independent classifier being learnt through bootstrapping (right). balanced positive and negative sets is sought all along the cascade. cl:classifier Fig. 2. data generation example with horizontal cut. the Feature values corresponding to upper and lower body parts are concatenated to form a new data sample.

Fig. 3 .

 3 Fig. 3. Representation of the parameter n 0 for instances selection. Positive samples are in red, selected negative ones are in blue, unselected ones in black. The rough class boundaries are depicted with dashed lines. From left to right, n 0 =100%, n 0 =50%, n 0 =10%

Fig. 4 .

 4 Fig. 4. Set cardinality balancing pipeline. '+' and '-' signs represent the positive and negative classes. Best viewed in color.

Fig. 5 .

 5 Fig. 5. Toy example of a cascade-of-rejectors training for binary classification. Blue and red colors represent the classes. The line depicts a (hypothetical) separating hyperplane. The first iterations optimize on recall, the final one optimizes for precision. Minority and majority classes are respectively underand over-sampled between each iteration to learn on balanced classes. Best viewed in color.

Fig. 6 .

 6 Fig. 6. Difficulties that non-maximum suppression tackles. Detections are in green. Best viewed in color.

Fig. 7 .

 7 Fig. 7. Comparison of the instance selection strategy variants, setting the log-average miss rate performance in relation to the parametrization. The INRIA dataset is in red, Caltech in blue.

TABLE V LAMR

 V PERFORMANCE AND SPEED OF A HOG-BASED CASCADE ON THE INRIA DATASET, WHILE VARYING THE CASCADE CLASSIFIER TYPES (ADABOOST OR RANDOM FOREST). BEST RESULTS IN BOLD.

  2) block-cascade: A more discriminative but slower version of the soft cascade that adds one feature block at each step of the cascade. 3) Reverse engineering : This method Learns first the classifier based on the full length descriptor and recursively removes the less discriminative half until only 2 blocks are left. It can be considered as a preprocessing step that quickly prunes out most of the negatives before running the standard cascade. Results are presented in table VII.

1 )

 1 Constant: Using the same threshold t(C i ) = T for each cascade classifier t(C i ); i = 0...1 2) Increasing: Monotonously increasing the classifiers threshold. t(C i ) = T + c × 0.01. 3) Recall: Estimating thresholds during training such that recall equals to r. This is the method most commonly used in the literature. 4) Regular: Using the same purposely low threshold t(C i ) = T for each cascade classifier t(C i ) and insuring that d% of the remaining data are removed at each step of the cascade. the parameter T, and d are manually set to the best performing value. However, the same value is employed for all datasets. T=0.58 is utilized for experiment 1, T=0.55 for experiment 2, and d=40% for experiment 4. Results are displayed in table in table VIII.

  OF-THE-ART. NEAR REAL-TIME METHODS ARE SEPARATED FROM OTHERS. Ours IS IN BOLD. DEEP LEARNING TECHNIQUES ARE IN RED. COMPUTATION TIMES ARE CALCULATED ACCORDING TO 640×480 RESOLUTION FRAMES. THE USED METRIC IS THE LOG-AVERAGE MISS RATE (THE LOWER THE BETTER).

	Method			INRIA	Speed(CPU/GPU)	Method	CALETCH Speed(CPU/GPU)
	HoG [16]			46%		0.5fps	HoG [16]	69%	0.5fps
	HoG-LBP [59]		39%	Not provided	DPM [24]	63.26%	< 1fps
	MultiFeatures [60]		36%		< 1fps	FeatSynth [2]	60.16%	< 1fps
	FeatSynth [2]		31%		< 1fps	MultiFeatures+CSS [57]	60.89%	No
	MultiFeatures+CSS [57]	25%		No	FPDW [17]	57.4%	2-5fps
	Channel Features [18]		21%		0.5fps	Channel Features [18]	56.34%	0.5fps
	FPDW [17]			21%		2-5fps	Roerei [4]	48.35%	1 fps
	DPM [24]			20%		< 1fps	MOCO [13]	45.5%	< 1fps
	RF local experts [43]		15.4%		3fps	JointDeep [47]	39.32%	< 1fps
	PCA-CNN [36]		14.24%		< 0.1fps	InformedHaar [66]	34.6%	< 0.63fps
	Ours -HoG			19.01%		4/60fps	katamari-v1 [5]	22.49%	< 1fps
	VeryFast [3]			18%		8/135fps	FRCNN [51]	56%	7fps
	WordChannels [15]		17%		0.5/8fps	CrossTalk cascades [19]	53.88%	30-60fps
	crossTalk cascades [19]		17%		30-60fps	Ours -LBP	51.12%	3.6/54fps
	Ours -LBP			17.28%		3.7/54fps	Ours -HoG	45.4%	3.9/58fps
	Ours -HoG+LBP		17.08%		2.6/39fps	Ours -HoG+LBP	45.2%	1.8/27fps
	SSD [38]			15%		56fps	WordChannels [15]	42.3%	0.5/8fps
	FRCNN [51]			13%		7fps	SSD [38]	34%	56fps
	RPN+PF [37]		7%		6fps	RPN+PF [37]	10%	6fps
						TABLE VI
	COMPARISON WITH THE STATE-Soft cascade type Feature		INRIA		Caltech
	No soft-cascade	HoG		20.31% / 2.1fps	46.25% / 1.9fps
	soft-cascade	HoG	32.21% / 18.1fps 58.26% / 17.8fps
	block soft-cascade	HoG		27.86% / 9.9fps	53.97% / 10.1fps
	reverse engineering	HoG		20.32% / 4 fps	46.23% / 3.9fps
	No soft-cascade	LBP		18.42% / 1.9fps	51.36% / 1.7fps
	soft-cascade	LBP	30.87% / 17.8fps 61.32% / 17.1fps
	block soft-cascade	LBP		28.33% / 9.8fps	55.45% / 9.5fps
	reverse engineering	LBP		18.41% / 3.7fps	51.33% / 3.6fps

TABLE VII LAMR

 VII PERFORMANCE AND CPU SPEED IN FRAME-PER-SECOND OF DIFFERENT SOFT-CASCADE STRATEGIES. SEE TEXT FOR THE DESCRIPTION OF THE METHODS. BEST RESULTS IN BOLD.

TABLE VIII THRESHOLD

 VIII ESTIMATION TECHNIQUES EVALUATION ON INRIA AND CALTECH BENCHMARKS. SEE TEXT FOR THE DESCRIPTION OF THE METHODS. BEST RESULTS IN BOLD.

	NMS type	Feature	INRIA	Caltech
	Greedy-NMS	HoG	22.91%	49.62%
	t-NMS	HoG	20.23%	46.25%
	s-NMS	HoG	21.47% 46.98%
	Greedy-NMS	LBP	20.21%	52.32%
	t-NMS	LBP	18.63% 51.26%
	s-NMS	LBP	19.87%	52.65%

TABLE IX NON

 IX -MAXIMUM SUPPRESSION ALGORITHM EVALUATION ON INRIA AND CALTECH BENCHMARKS. BEST RESULTS IN BOLD.