Bertrand Moulin

Olivier Barge

Hydrological modeling and management of water ressources in the Syrian steppe during the Early Bronze Age, the case of the fortified city of Al Rawda

During the end of the third millennium, populations settled in the Syrian steppe amid the arid margins of the Fertile Crescent. The establishment of a series of settlements, often fortified, raises the question of how these settlements subsisted in an area of marked aridity.

An investigation planned over 5 years, centered around the sedentary site beginning at the end of the Early Bronze Age, Al Rawda, located in a fayda, is aimed to chart precisely the vestiges present in this pioneer territory. Initial results have made it possible to identify several hydro-agricultural installations.

A geoarcheologic survey was initiated and coupled with a space analysis using GIS, which aims to specify the functionality of installations. The construction of a DEM using statements in mesh using a differential GPS Trimble Pro XRS made it possible to identify clearly, through detailed hydrological analysis, the hydrographic network as well as the contributive surfaces of the basin's slopes. Intial results show:

• Stock management of water was probably articulated around a hydraulic installation system, making it possible alternatively to use the flows available from the main wadi and those of its secondary tributaries. The hydraulic system, of which remains only a palimpsest, made it possible to optimise all flows available and to direct them, using a distributor, in a differentiated way towards zones of agriculture where water needs temporally varied.

• Stock management of water may have determined for a great part the structural organization of space around the site.

Lastly, one measurement made following an observation of a pluviometric event, by the intermediary of the DEM, allowed the quantification and subsequent modeling of the contributions of possible water and the measurement of various flooded surfaces.

Introduction

In the far eastern Syrian steppe, well below the 200 mm isohyet, sedentary populations established territories at the turn of the second millennium BC. This occupation was relatively brief -just a few centuries -and the reasons for its existence are still unknown. We can however study the natural environment of this small region, and through the vestiges of this period attempt to understand how this occupation and organization established and existed.

The climate of the area is not favourable to agriculture. If it can be established that these populations were herders, their sedentary way of life would imply that they practised other modes of subsistence. It is 1

Table of Content

therefore necessary to study the local resources in terms of crop cultivation, which in this arid area would need a rudimentary form of irrigation.

Our investigation is organized on the study of environments with a geo-archaeological approach, closely linked to the archaeological evidence; the goal is to understand the manmade elements in their environmental context, in accordance with stratigraphic evidence. Another goal of the study is to clarify the horizontal relationships of proximity and distance of the vestiges of occupation. For this, all the field dataarchaeological data and environmental data -are located with a differential GPS and recorded in a Geographic Information System (GIS). The GIS enables organisation of the inventories and production of multiple maps (Barge, Sanz, Mouraille in press). This tool enables us to analyse the main results of the survey. Results were acquired during the first two field seasons (2002 and 2003), each lasting two weeks.

We will present the results of a spatial analysis, conducted with the GIS, a calculation of water flow lines, which will enable us to confirm the hypothesis of cultivation in immediate proximity to the town of al-Rawda.

Presentation of the study area

), directed by Bernard Geyer between 1995 and 2002, which had as an objective an exhaustive inventory of archaeological sites for all periods. One of the important results of this research was the revelation of the ebb and flow of human populations in the region. The western part of the study area lies in the Fertile Crescent, which has been continuously occupied, whereas the arid eastern part has only been occupied by sedentary people periodically. This was the case at the end of the Early Bronze Age when these lands were first conquered [START_REF] Geyer | Les steppes arides de la Syrie du Nord au Bronze ancien ou « la première conquête de l'Est », Conquête de la steppe et appropriation des terres sur les marges du croissant fertile[END_REF]. The limits of sedentary occupation in the east (Fig. 1) were only exceeded in the Byzantine period; even today, semi-nomadic populations occupy the region only in winter. There are many Early Bronze Age sites, which densely occupied the territory; their eastern limits are marked by the Wadi al-Amur, which provides water to al-Rawda. The sub-region under study is thus a window of detailed investigations in the heart of the study zone of the previous survey, astride the edge of sedentary occupation defined by that survey (Fig. 1).

The recent excavations at al Rawda have shown that this is a fortified urban site, measuring some fifteen hectares of which 12 are intra muros, dating to the end of the Early Bronze Age [START_REF] Castel | Rapport préliminaire sur les activités de la première mission archéologique franco-syrienne dans la microrégion d'al-Rawda (Syrie intérieure) : la campagne de 2002[END_REF].

The Franco-Syrian archaeological survey, directed by Corinne Castel, delineated a sub-region of 100 km 2 centred on this site in order to study its territory. The region is not totally inhospitable, which is the case farther south and east, but has high aridity with very irregular precipitation [START_REF] Jaubert | Utilisation du territoire et couvert végétal, Régions arides et semi-arides des provinces d'Alep et Hama (Syrie)[END_REF]. It is a marginal zone.

In the center of this sub-region lays the site of al-Rawda in a faydas. The faydas are alluvial depressions, where downstream channel narrowing has led to high sedimentation rates. The clayey-silty soils are deep, well drained and receive water from the wadis that cross them. Thus, these are favourable environments, which contrast strongly with the interfluves that are covered with a calcareous crust; dry environments where sparse vegetation grows on thin strips of soil.

Remnants of cultivation in the sub-region are few; dry farming remained difficult because of the high aridity, even in favourable environments such the valley beds or the faydas. Nevertheless, the archaeobotanical results obtained in stratigraphy on the site of al-Rawda [START_REF] Herveux | Étude archéobotanique préliminaire de tell al-Rawda, site de la fin du Bronze ancien en Syrie intérieure[END_REF]) identified ruderal vegetation indicative of irrigation, as well as the cultivation of olives, vines and legumes, which strongly suggests there was at least minimal irrigation. Moreover, around the tell of al-Rawda, beyond the urban context, are a series of structurs which are generally poorly preserved but whose general plan suggests an organized system of water flow (Fig. 2). Although visible on the ground, their presence and their organization were revealed from air photographs taken from a kite (Sanz, Barge in press).

The GIS have verified the hypothesis of the existence of relatively crude hydraulic constructions, enabling the use of natural water flows for agriculture, hydrographically.

Methodology

A Digital Elevation Model (DEM), a digital description of the relief where each pixel of the image carries an altitude value, was obtained by interpolation of points measured in the field using a differential GPS, over a 3 km x 3 km surface area centred on al-Rawda, with a regular grid of 100 m.

In order to develop the DEM, we used a Trimble Differential GPS Pro XRS and a cross-country vehicle. The altitude point precision recorded using the DGPS is decimetric. Further, we used points provided by topographers using theodolites, which are reliable and adjacent to the site.

The more than 4000 topographic points recorded allowed for the calculation of a digital elevation model, or DEM, aided by the topogrid control mechanism in ArcInfo (ESRI). We used this DEM, to test our hydrological hypotheses.

Results and Discussion

Description of the presence of a system of river flows control and management :

An analysis of the theoretical water flows using the DEM delineates the present course of the Wadi al-Amur, which passes east of the tell of al-Rawda, as well as the small tributary wadis that originate locally (Fig 3a). Just upriver from the tell are partly destroyed structures, consisting of an in place dike as well as several adjoining walls. Some appear to have been longer, but their outline in the course of the wadi has been destroyed. One of these walls has been reconstituted virtually in the DEM. A new analysis of theoretical water flows shows a diversion of the main wadi towards the west of the tell (Fig. 3b). This structure could have been used to divert water when the stream swelled and conduct it right or left of the tell according to necessity, towards plots requiring water. Similarly, a sluice gate was simulated in the structure, and a new diversion appeared. The water flows to a second dike in place, which allows flow to be directed to the easternmost part of the fayda (Fig. 3c).

Thus, there probably existed a hydro-agricultural system, of which there are only remnants but from which the main orientations can be detected. Moreover, the interfluve between the Wadi al-Amur and a small tributary from the left bank ends with a low ridge, which adjoins the dike. It is possible given the low relief that the dike originally continued westward to harness the waters of this wadi (Fig. 3d). This structure would allow the exploitation of this water source; temporal water availability would increase as the two water courses may not have flow at the same time. The small wadi could flow as a result of a local rainstorm while the main wadi remained dry.

Estimate of water volume :

A rainfall event (18 mm) that occurred during the 2003 survey enabled the evaluation of the volumes that pass through this system from this tributary wadi (Fig. 4). The theoretical threshold of 10 % was retained to calculate the infiltration value. The potential in terms of arable land of the fayda could have been captured by hydro-agricultural structures that enabled the exploitation of local water flows or those coming from the main wadi. The low relief of the area required only modest structures to utlilize rainfall to the best advantage. The simulations correspond to earth embankments of no more than 80 cm. One can imagine that the fayda as a whole was adapted in a manner similar to that of the villagers in the 1950s: earth embankments set perpendicular to the flow accumulated the water and sediments behind them. We know that these techniques were known in this period, and similarly used near al-Rawda (Braemer, [START_REF] Echallier | Eléments d'appréciation de l'évolution du milieu[END_REF].

Conclusion

There is strong evidence that a system of irrigation existed on the Al Rawda site. This system of clever gravitating irrigation was well adapted to the specific hydrological temporality at the fayda of Al Rawda junction.

Although the practice of irrigation agriculture has been revealed, it seems not to have extended beyond the immediate environs of the ancient town. At the end of the third field season, the survey was able to identify new indications of hydro-agricultural structures, but their function remains unclear. They usually affect only small surfaces, not necessarily associated with habitation sites, which is the case at al-Rawda.

Two sectors with this type of construction situated in the fayda of Tarmaz in the Wadi Qastal are candidates for future study.

Fig. 3 -

 3 Fig. 3 -Theoretical water flows and simulations of hydraulic works

Fig. 4 -

 4 Fig. 4 -Calculation of the volumes of water infiltrated during the rainfall event of 3/10/2003


Jérémy Matias
email: jeremy.matias@uca.fr

Jean-Charles Quinton

Michèle Colomb

Marie Izaute

Laetitia Silvert

Attentional capture by reward-distractors under perceptual load

come

INTRODUCTION

RESULTS

DISCUSSION

Value-Driven Attentional Capture (Anderson, 2013)  Recent research revealed that visual distractors associated with or signaling reward outcomes capture attention in spite of being neither salient nor relevant for the ongoing task.

 If attentional capture by reward distractors is automatic, therefore, interference by reward distractors should be immune to perceptual load manipulations.

 In situation of low perceptual load spare capacities from relevant processing "spills over" to the irrelevant items, resulting in distractor interference.

 Attentional capture by irrelevant distractors can be prevented in situations of high perceptual load that exhaust all available capacities in the perception of relevant stimuli.

Perceptual Load Theory (Lavie, 2010) Perceptual Load and Reward ?

Load manipulation:

High Load

In an adaptation of a go/nogo task, participants had to identify the target letter (H or S) depending on the cue displayed, while ignoring the presence of a distractor color letter that could be either identical to (compatible) or different from (incompatible) the target.  Valence effect (with reward as positive events and fearful faces as negative events) might explain those discrepant results.

METHOD

 However, the persistence of the interference effect for high reward distractors may be due to the ability of those stimuli to capture attention, but also to hold attention once it has been captured, or both. • Growing body of studies demonstrating an advantage for positive over negative emotional information in attracting and capturing attention (e.g., Srivastava & Srinivasan, 2010).

EEG study in progress…  Distraction by irrelevant fearful faces is reduced under high load.

 Processing of irrelevant negative emotional stimuli depends on available attentional resources.

Automatic attentional capture

Feature



  Compatibility Effect: Reward (Low; High) x Load (Low-load; High-load) • Reward: F(1, 29) = 1.23, p = .28 • Load: F(1, 29) = 5.09, p < .05 Errors: Reward (2) x Load (2) • Reward p < .01 • Load p < .001 confirming load manipulation  Reward x Load p < .High perceptual load reduces low but not high-reward peripheral distractor interference.  High-reward distractor interference was observed despite low level of free residual attentional resources (i.e., high perceptual load )  At odds with what is observed for fearful faces distractors.

  reward distractor interference whereas high-reward distractor interference seems « immune » to load manipulation. Automatic attentional capture ? -Anderson, B. A. (2013). A value-driven mechanism of attentional selection. Journal of Vision, 13, 1-16. -Chen, Z., & Cave, K. R. (2016). Zooming in on the cause of the perceptual load effect in the go/no-go paradigm. Journal of Experimental Psychology. Human Perceptionand Performance, 42(8), 1072-1087. -Gupta, R., Hur, Y.-J., & Lavie, N. (2016). Distracted by Pleasure: Effects of Positive Versus Negative Valence on Emotional Capture Under Load. Emotion, 16(3), 328-337. -Lavie, N. (2010). Attention, Distraction, and Cognitive Control Under Load. Current Directions in Psychological Science, 19(3), 143-148. Perceptual Load and Emotion (e.g., Silvert et al., 2007) -Silvert, L., Lepsien, J., Fragopanagos, N., Goolsby, B., Kiss, M., Taylor, J. G., … Nobre, A. C. (2007). Influence of attentional demands on the processing of emotional facial expressions in the amygdala. NeuroImage, 38(2), 357-366. -Srivastava, P., & Srinivasan, N. (2010). Time course of visual attention with emotional faces. Attention, Perception, & Psychophysics, 72, 369-377.  Reward x Load: F(1, 29) = 4.37, p < .05• Interference by centrally presented distractor of negative but not positive valence is reduced under high perceptual load(Gupta et al., 2016).

Attentional capture by reward-This work has been sponsored by the French government research program "Investissements d'Avenir" through the IMobS3 Laboratory of Excellence (ANR-10-LABX-16-01) and the RobotEx Equipment of Excellence (ANR-10-EQPX-44), by the European Union through the Regional Competitiveness and Employment program -2014-2020-(ERDF -AURA region) and by the AURA région.


Ahmet Iscen

Yannis Avrithis

Giorgos Tolias

Teddy Furon
email: teddy.furon@inria.fr

Ondřej Chum
email: chum@cmp.felk.cvut.czioannis.avrithis

Fast Spectral Ranking for Similarity Search

Despite the success of deep learning on representing images for particular object retrieval, recent studies show that the learned representations still lie on manifolds in a high dimensional space. This makes the Euclidean nearest neighbor search biased for this task. Exploring the manifolds online remains expensive even if a nearest neighbor graph has been computed offline.

This work introduces an explicit embedding reducing manifold search to Euclidean search followed by dot product similarity search. This is equivalent to linear graph filtering of a sparse signal in the frequency domain. To speed up online search, we compute an approximate Fourier basis of the graph offline. We improve the state of art on particular object retrieval datasets including the challenging Instre dataset containing small objects. At a scale of 10 5 images, the offline cost is only a few hours, while query time is comparable to standard similarity search.

1. We cast image retrieval as linear filtering over a graph, efficiently performed in the frequency domain.

Introduction

Image retrieval based on deep learned features has recently achieved near perfect performance on all standard datasets [START_REF] Radenović | CNN image retrieval learns from bow: Unsupervised fine-tuning with hard examples[END_REF]14,[START_REF] Gordo | End-to-end learning of deep visual representations for image retrieval[END_REF]. It requires fine-tuning on a properly designed image matching task involving little or no human supervision. Yet, retrieving particular small objects is a common failure case. Representing an image with several regions rather than a global descriptor is indispensable in this respect [START_REF] Razavian | Visual instance retrieval with deep convolutional networks[END_REF][START_REF] Tolias | Particular object retrieval with integral max-pooling of cnn activations[END_REF]. A recent study [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] uses a particularly challenging dataset [START_REF] Wang | Instre: a new benchmark for instance-level object retrieval and recognition[END_REF] to investigate graph-based query expansion and re-ranking on regional search.

Query expansion [7] explores the image manifold by recursive Euclidean or similarity search on the nearest neighbors (NN) at increased online cost. Graph-based methods [44,[START_REF] Shen | Spatially-constrained similarity measure for large-scale object retrieval[END_REF] help reducing this cost by computing a k-NN graph offline. Given this graph, random walk 1 processes [START_REF] Page | The PageRank citation ranking: bringing order to the web[END_REF][START_REF] Zhou | Ranking on data manifolds[END_REF] provide a principled means of ranking. Iscen 1 We avoid the term diffusion [11,[START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] in this work. The low-pass filtering of an impulse over the real line (top) and a graph (bottom). In a weighted undirected graph the information "flows" in all directions, controlled by edge weights. In retrieval, the impulse in red is the query, and the output x is its similarity to all samples. et al. [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] transform the problem into finding a solution x of a linear system Ax = y for a large sparse dataset-dependent matrix A and a sparse query-dependent vector y. Such a solution can be found efficiently on-the-fly with conjugate gradients (CG). Even for an efficient solver, the query times are still in the order of one second at large scale.

In this work, we shift more computation offline: we exploit a low-rank spectral decomposition A ≈ U ΛU ⊤ and express the solution in closed form as x = U Λ -1 U ⊤ y. We thus treat the query as a signal y to be smoothed over the graph, connecting query expansion to graph signal processing [START_REF] Sandryhaila | Discrete signal processing on graphs[END_REF]. Figure 1 depicts 1d and graph miniatures of this interpretation. We then generalize, improve and interpret this spectral ranking idea on large-scale image retrieval. In particular, we make the following contributions: 3. We reduce manifold search to a two-stage similarity search thanks to an explicit embedding. 4. A rich set of interpretations connects to different fields.

The text is structured as follows. Section 2 describes the addressed problem while Sections 3 and 4 present a description and an analysis of our method respectively. Section 5 gives a number of interpretations and connections to different fields. Section 6 discusses our contributions against related work. We report experimental findings in Section 8 and draw conclusions in Section 9.

Problem

In this section we state the problem addressed by this paper in detail. We closely follow the formulation of [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF].

Representation

A set of n descriptor vectors V = {v 1 , . . . , v n }, with each v i associated to vertex v i of a weighted undirected graph G is given as an input. The graph G with n vertices V = {v 1 , . . . , v n } and ℓ edges is represented by its n × n symmetric nonnegative adjacency matrix W . Graph G contains no self-loops, i.e. W has zero diagonal. We assume W is sparse with 2ℓ ≪ n(n -1) nonzero elements.

We define the n × n degree matrix D : = diag(W 1) where 1 is the all-ones vector, and the symmetrically normalized adjacency matrix W : = D -1/2 W D -1/2 with the convention 0/0 = 0. We also define the Laplacian and normalized Laplacian of G as L : = D -W and L : = D -1/2 LD -1/2 = I -W, respectively. Both are singular and positive-semidefinite; the eigenvalues of L are in the interval [0, 2] [START_REF] Chung | Spectral graph theory[END_REF]. Hence, if λ 1 , . . . , λ n are the eigenvalues of W, its spectral radius ̺(W) : = max i |λ i | is 1. Each eigenvector u of L associated to eigenvalue 0 is constant within connected components (e.g., L1 = D1 -W 1 = 0), while the corresponding eigenvector of L is D 1/2 u.

Transfer function

We define the n × n matrices L α : = β -1 (D -αW ) and

L α : = D -1/2 L α D -1/2 = β -1 (I -αW), where α ∈ [0, 1)
and β := 1α. Both are positive-definite. Given the n × 1 sparse observation vector y online, [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] computes the n × 1 ranking vector x as the solution of the linear system

L α x = y. (1)
We can write the solution as h α (W)y, where

h α (W) : = (1 -α)(I -αW) -1 (2)
for a matrix W such that I -αW is nonsingular; indeed,

L -1 α = h α (W).
Here we generalize this problem by considering any given transfer function h : S → S, where S is the set of real symmetric matrices including scalars, R. The general problem is then to compute

x * : = h(W)y (3)
efficiently, in the sense that h(W) is never explicitly computed or stored: W is given in advance and we are allowed to pre-process it offline, while both y and h are given online.

For h α in particular, we look for a more efficient solution than solving linear system (1).

Retrieval application

The descriptors V are generated by extracting image descriptors from either whole images, or from multiple sampled rectangular image regions, which can be optionally reduced by a Gaussian mixture model as in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. Note that the global descriptor is a special case of the regional one, using a single region per image. In the paper, we use CNN-based descriptors [START_REF] Radenović | CNN image retrieval learns from bow: Unsupervised fine-tuning with hard examples[END_REF].

The undirected graph G is a k-NN similarity graph constructed as follows. Given two descriptors v, z in R d , their similarity is measured as

s(v, z) = [v ⊤ z] γ
+ , where exponent γ > 0 is a parameter. We denote by s(v i |z) the similarity s(v i , z) if v i is a k-NN of z in V and zero otherwise. The symmetric adjacency matrix W is defined as w ij : = min(s(v i |v j ), s(v j |v i )), representing mutual neighborhoods. Online, given a query image represented by descriptors {q 1 , . . . , q m } ⊂ R d , the observation vector y ∈ R n is formed with elements y i : = m j=1 s(v i |q j ) by pooling over query regions. We make y sparse by keeping the k largest entries and dropping the rest.

Method

This section presents our fast spectral ranking (FSR) algorithm in abstract form first, then with concrete choices.

Algorithm

We describe our algorithm given an arbitrary n × n matrix A ∈ S instead of W. Our solution is based on a sparse low-rank approximation of A computed offline such that online, x ≈ h(A)y is reduced to a sequence of sparse matrix-vector multiplications. The approximation is based on a randomized algorithm [START_REF] Rokhlin | A randomized algorithm for principal component analysis[END_REF] that is similar to Nyström sampling [START_REF] Drineas | On the Nyström method for approximating a gram matrix for improved kernel-based learning[END_REF] but comes with performance guarantees [START_REF] Halko | Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions[END_REF][START_REF] Witten | Randomized algorithms for low-rank matrix factorizations: Sharp performance bounds[END_REF]. In the following, r ≪ n, p < r, q and τ are given parameters, and r = r + p.

1. (Offline) Using simultaneous iteration [62, §28], compute an n × r matrix Q with orthonormal columns that represents an approximate basis for the range of A, i.e. QQ ⊤ A ≈ A. In particular, this is done as follows [18, §4.5]: randomly draw an n × r standard Gaussian matrix B (0) and repeat for t = 0, . . . , q -1:

(a) Compute QR factorization Q (t) R (t) = B (t) . (b) Define the n × r matrix B (t+1) : = AQ (t) .
Finally, set Q : = Q (q-1) , B : = B (q) = AQ. 2. (Offline-Fourier basis) Compute a rank-r eigenvalue decomposition U ΛU ⊤ ≈ A, where n × r matrix U has orthonormal columns and r × r matrix Λ is diagonal.

In particular, roughly following [18, §5.3]: 3. (Offline) Make U sparse by keeping its τ largest entries and dropping the rest. 4. (Online) Given y and h, compute

x : = U h(Λ)U ⊤ y. (4)
Observe that U ⊤ projects y onto R r . With Λ being diagonal, h(Λ) is computed element-wise. Finally, multiplying by U and ranking x amounts to dot product similarity search in R r . The online stage is very fast, provided U only contains few leading eigenvectors and y is sparse. We consider the following variants:

• FSR.SPARSE: This is the complete algorithm. To see why FSR.EXACT works, consider the case of h α (W). Let W ≃ U ΛU ⊤ . It follows that h α (W)/β = (I -αW) -1 ≃ U (I -αΛ) -1 U ⊤ , where (I -αΛ) -1 is computed element-wise. Then, x * ≃ βU (I -αΛ) -1 U ⊤ y. The general case is discussed in section 4.

Retrieval application

Returning to the retrieval problem, we compute the ranking vector x ∈ R n by (4), containing the ranking score x i of each dataset region v i . To obtain a score per image, we perform a linear pooling operation [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] represented as x : = Σx where Σ is a sparse N × n pooling matrix. The N × r matrix U : = ΣU is indeed computed offline so that we directly compute x = U h(Λ)U ⊤ y online.

Computing y involves Euclidean search in R d , which happens to be dot product because vectors are ℓ 2normalized. Applying U and ranking x amounts to a dot product similarity search in R r . We thus reduce manifold search to Euclidean followed by dot product search. The number of nonzero elements of y and rows of U , whence the cost, are the same for global or regional search.

Analysis

We derive the asymptotic space and time complexity of different algorithm variants and derive necessary condition for correctness and error bounds of approximate variants.

Complexity

The offline complexity is mainly determined by the number of columns r of matrix Q: Stage 1 reduces the size of the problem from n 2 down to nr. The online complexity is determined by the number of nonzero entries in matrix U . A straightforward analysis leads to the following:

• FSR.APPROX: The offline complexity is O(qn(k + r)r) time and O(nr) space; its online (time and space) complexity is O(nr). • FSR.SPARSE: The offline complexity is O(qn(k + r)r + τ log τ ) time and O(nr) space; its online complexity is O(τ ).

Stage 1 is "embarrassingly parallelizable" meaning that it is dramatically accelerated on parallel and distributed platforms. Since the online stage 4 amounts to NN search, any approximate method applies, making it sublinear in n.

Correctness

We derive here the conditions on h and A under which our algorithm is correct under no truncation, i.e., FSR.EXACT(y|A, h) = h(A)y. We also show, that h α and W satisfy these conditions, which is an alternative proof of correctness to the one in Section 3.1.

Starting from the fact a real symmetric matrix A is diagonalizable, there exists an exact eigenvalue decomposition U ΛU ⊤ = A, where U is orthogonal. According to [1, §9.14,9.2], we have h

(A) = U h(Λ)U ⊤ = U diag(h(λ 1 ), . . . , h(λ n ))U ⊤ if

and only if there exists a series expansion of h converging for this specific

A: h(A) = ∞ t=0 c t A t .
(5)

This holds in particular for h α admitting the geometric progression expansion

h α (A) : = β(I -αA) -1 = β ∞ t=0 (αA) t , (6)
which converges absolutely if ̺(αA) < 1 [1, §9.6,9.19]. This holds for A = W because α < 1 and ̺(W) = 1.

Error bound

We present main ideas for bounding the approximation error of FSR.RANK-r and FSR.APPROX coming from literature, and we derive another condition on h under which [61, §3.3]. Therefore, our method makes sense only when the restriction of h to scalars is nondecreasing. This is the case for h α .

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.2 0.4 0.6 0.8 1 h α (x) = 1 -α 1 -αx 1 -x h α (x) α = 0.99 α = 0.9 α = 0.7 Figure 2: Function h α (2)
Since A is symmetric, A ≈ QQ ⊤ AQQ ⊤ = QCQ ⊤ ≈ QV ΛV ⊤ Q ⊤ = U ΛU ⊤ . The latter approximation C ≈ V ΛV ⊤ is essen- tially a best rank-r approximation of C = Q ⊤ AQ.

Interpretation

Our work is connected to studies in different fields with a long history. Here we give a number of interpretations both in general and in the particular case h = h α .

Graph signal processing

In signal processing [START_REF] Oppenheim | Discrete-Time Signal Processing: Pearson New International Edition[END_REF], a discrete-time signal of period n is a vector s ∈ R n where indices are represented by integers modulo n, that is, sī : = s (i mod n)+1 for i ∈ Z. A shift (or translation, or delay) of s by one sample is the mapping sī → s i-1 . If we define the n × n circulant matrix C n : = (e 2 e 3 . . . e n e 1 ) 2 , a shift can be represented by s → C n s [START_REF] Sandryhaila | Discrete signal processing on graphs[END_REF]. A linear, time (or shift) invariant filter is the mapping s → Hs where H is an n × n matrix with a series representation

H : = h(C n ) = ∞ t=0 h t C t n . Matrix C n has the eigenvalue decomposition U ΛU ⊤ where U ⊤ is the n × n discrete Fourier transform matrix F. If the series h(C n ) converges, filtering s → Hs is written as s → F -1 h(Λ)Fs. (7)
That is, s is mapped to the frequency domain, scaled element-wise, and mapped back to the time domain.

2 Observe that Cn is the adjacency matrix of the directed graph of Figure 1 after adding an edge from the rightmost to the leftmost vertex.

Graph signal processing [START_REF] Sandryhaila | Discrete signal processing on graphs[END_REF]54] generalizes the above concepts to graphs by replacing C n by W, an appropriately normalized adjacency matrix of an arbitrary graph. If U ΛU ⊤ is the eigenvalue decomposition of W, we realize that (4) treats y as a (sparse) signal and filters it in the frequency domain via transfer function h to obtain x. Function h α in particular is a low-pass filter, as illustrated in Figure 2. By varying α from 0 to 1, the frequency response varies from all-pass to sharp low-pass.

Random walks

Consider the iterating process: for t = 1, 2, . . .

x (t) : = αAx (t-1) + (1 -α)y. ( 8
)
If A is a stochastic transition matrix and x (0) , y are distributions over vertices, this specifies a random walk on a (directed) graph: at each iteration a particle moves to a neighboring vertex with probability α or jumps to a vertex according to distribution y with probability 1α. This is called a Markov chain with restart [2] or random walk with restart [START_REF] Pan | Automatic multimedia cross-modal correlation discovery[END_REF]. State x (t) converges to x * = h α (A)y as t → ∞ provided ̺(αA) < 1 [START_REF] Zhou | Learning with local and global consistency[END_REF]. In fact, ( 8) is equivalent to Jacobi solver [17] on linear system (1) [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF].

If y = e i , the i-th canonical vector, then x * is used to rank the vertices of G, expressing a measure of "similarity" to v i [START_REF] Zhou | Ranking on data manifolds[END_REF]. Parameter α controls how much x * is affected by boundary condition y [64]: x * equals y for α = 0, while in the limit α → 1, x * tends to a dominant eigenvector of A. Indeed, for α = 1, (8) becomes a power iteration.

Random fields

Given a positive-definite n × n precision matrix A ∈ S and a mean vector µ ∈ R n , a Gaussian Markov random field (GMRF) [49] with respect to an undirected graph G is a random vector x ∈ R n with normal density p(x) : = N (x|µ, A -1 ) iff A has the same nonzero offdiagonal entries as the adjacency matrix of G. Its canonical parametrization p(x) ∝ e -E(x|b,A) where E(x|b, A) : =

1 2 x ⊤ Ax -b ⊤ x is a quadratic energy. Its expectation µ = A -1 b is the minimizer of this energy. Now, x * = L -1 α y (1)
is the expectation of a GMRF with energy

f α (x) : = E(x|y, L α ) = 1 2 x ⊤ L α x -y ⊤ x. (9)
A mean field method on this GMRF is equivalent to Jacobi or Gauss-Seidel solvers on (1) [66]. Yet, conjugate gradients (CG) [START_REF] Nocedal | Numerical optimization[END_REF] is minimizing f α (x) more efficiently [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]5].

If we expand f α (x) using βL α = αL + (1α)I, we find that it has the same minimizer as

α i,j w ij xi -xj 2 + (1 -α) x -y 2 , (10)
where x : = D -1/2 x. The pairwise smoothness term encourages x to vary little across edges with large weight whereas the unary fitness term to stay close to observation y [START_REF] Zhou | Learning with local and global consistency[END_REF]. Again, α controls the trade-off: x * equals y for α = 0, while for α → 1, x * tends to be constant over connected components like dominant eigenvectors of W.

Regularization and kernels

The first term of ( 9) is interpreted as a regularization operator related to a kernel K = L -1 α [START_REF] Smola | The connection between regularization operators and support vector kernels[END_REF]57,[START_REF] Kondor | Diffusion kernels[END_REF]. In a finite graph, a kernel can be seen either as an n × n matrix K or a function κ : V 2 → R operating on pairs of vertices. More generally, if h(x) > 0 for x ∈ R, which holds for h α , then K : = h(W) is positive-definite and there is an

n × n matrix Φ such that K = Φ ⊤ Φ, or κ(v i , v j ) = φ(v i ) ⊤ φ(v j )
where feature map φ : V → R n is given by φ(v i ) : = Φe i . A particular choice for Φ is

Φ : = h(Λ) 1/2 U ⊤ (11)
where U ΛU ⊤ is the eigenvalue decomposition of W. If we choose a rank-r approximation instead, then Φ is an r × n matrix and φ is a low-dimensional embedding onto R r . The goal of out-of-sample extension is to compute a "similarity" κ(z 1 , z 2 ) between two unseen vectors z 1 , z 2 ∈ R d not pertaining to the graph. Here we define

κ(z 1 , z 2 ) : = ψ(z 1 ) ⊤ Φ ⊤ Φψ(z 2 ) (12)
given any mapping ψ : R d → R n , e.g. ψ(z) i : = s(v i |z) discussed in section 2. This extended kernel is also positivedefinite and its embedding φ(z) = Φψ(z) is a linear combination of the dataset embeddings. For r ≪ n, our method allows rapid computation of κ or κ for any given function h, without any dense n × n matrix involved.

Paths on graphs

Many nonlinear dimension reduction methods replace Euclidean distance with an approximate geodesic distance, assuming the data lie on a manifold [START_REF] Lee | Nonlinear dimensionality reduction[END_REF]. This involves the all-pairs shortest path (APSP) problem and Dijkstra's algorithm is a common choice. Yet, it is instructive to consider a naïve algorithm [9, §25.1]. We are given a distance matrix where missing edges are represented by ∞ and define similarity weight w ij = e -dij . A path weight is a now a product of similarities and "shortest" means "of maximum weight". Defining matrix power A ⊗t as A t with + replaced by max, the algorithm is reduced to computing max t W ⊗t (element-wise). Element i, j of W ⊗t is the weight of the shortest path of length t between v i , v j .

Besides their complexity, shortest paths are sensitive to changes in the graph. An alternative is the sum 3 of weights 3 In fact, similar to softmax due to the exponential and normalization. over paths of length t, recovering the ordinary matrix power W t , and the weighted sum over all lengths ∞ t=0 c t W t , where coefficients (c t ) t∈N allow for convergence [64], [52, §9.4]. This justifies (5) and reveals that coefficients control the contribution of paths depending on length. A common choice is c t = βα t with β = 1α and α ∈ [0, 1) being a damping factor [64], which justifies function h α (6).

Related work

The history of the particular case h = h α is the subject of the excellent study of spectral ranking [64]. The fundamental contributions originate in the social sciences and include the eigenvector formulation by Seeley [START_REF] Seeley | The net of reciprocal influence. a problem in treating sociometric data[END_REF], damping by α (6) by Katz [START_REF] Katz | A new status index derived from sociometric analysis[END_REF] and the boundary condition y (1) by Hubbell [22]. The most well-known follower is PageRank [START_REF] Page | The PageRank citation ranking: bringing order to the web[END_REF]. In machine learning, h α has been referred to as the von Neumann [27,[START_REF] Shawe-Taylor | Kernel methods for pattern analysis[END_REF] or regularized Laplacian kernel [57]. Along with the diffusion kernel [START_REF] Boldi | Graph fibrations, graph isomorphism, and PageRank[END_REF][START_REF] Kondor | Diffusion kernels[END_REF], it has been studied in connection to regularization [START_REF] Smola | The connection between regularization operators and support vector kernels[END_REF]57].

Random fields are routinely used for low-level vision tasks where one is promoting smoothness while respecting a noisy observation, like in denoising or segmentation, where both the graph and the observation originate from a single image [59, 5]. A similar mechanism appears in semi-supervised learning [START_REF] Zhou | Learning with local and global consistency[END_REF][START_REF] Zhu | Semi-supervised learning: From Gaussian fields to Gaussian processes[END_REF][START_REF] Zhu | Semi-supervised learning using Gaussian fields and harmonic functions[END_REF][START_REF] Chapelle | Cluster kernels for semisupervised learning[END_REF] or interactive segmentation [START_REF] Grady | Random walks for image segmentation[END_REF]30] where the observation is composed of labels over a number of samples or pixels. In our retrieval scenario, the observation is formed by the neighbors in the graph of an external query image (or its regions).

The random walk or random walk with restart (RWR) formulation [START_REF] Zhou | Ranking on data manifolds[END_REF][START_REF] Zhou | Learning with local and global consistency[END_REF][START_REF] Pan | Automatic multimedia cross-modal correlation discovery[END_REF] is an alternative interpretation to retrieval [11]. Yet, directly solving a linear system is superior [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. Offline matrix decomposition has been studied for RWR [START_REF] Tong | Fast random walk with restart and its applications[END_REF]13,26]. All three methods are limited to h α while sparse LU decomposition [13,26] assumes an uneven distribution of vertex degrees [START_REF] Kang | Beyond 'caveman communities': Hubs and spokes for graph compression and mining[END_REF], which is not the case for k-NN graphs. In addition, we reduce manifold search to two-stage Euclidean search via an explicit embedding, which is data dependent through the kernel K = L -1 α . In the general case, the spectral formulation (4) has been known in machine learning [START_REF] Chapelle | Cluster kernels for semisupervised learning[END_REF][START_REF] Shawe-Taylor | Kernel methods for pattern analysis[END_REF][START_REF] Nadler | Diffusion maps, spectral clustering and eigenfunctions of fokker-planck operators[END_REF][START_REF] Zhu | Graph kernels by spectral transforms[END_REF][START_REF] Vishwanathan | Graph kernels[END_REF] and in graph signal processing [START_REF] Sandryhaila | Discrete signal processing on graphs[END_REF]54,[START_REF] Hammond | Wavelets on graphs via spectral graph theory[END_REF]. The latter is becoming popular in the form of graph-based convolution in deep learning [4,[START_REF] Henaff | Deep convolutional networks on graph-structured data[END_REF][START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF]3,[START_REF] Monti | Geometric deep learning on graphs and manifolds using mixture model cnns[END_REF]43]. However, with few exceptions [4,[START_REF] Henaff | Deep convolutional networks on graph-structured data[END_REF], which rely on an expensive decomposition, there is nothing spectral when it comes to actual computation. It is rather preferred to work with finite polynomial approximations of the graph filter [START_REF] Defferrard | Convolutional neural networks on graphs with fast localized spectral filtering[END_REF]3] using Chebyshev polynomials [START_REF] Hammond | Wavelets on graphs via spectral graph theory[END_REF]55] or translation-invariant neighborhood templates in the spatial domain [START_REF] Monti | Geometric deep learning on graphs and manifolds using mixture model cnns[END_REF]43].

We cast retrieval as graph filtering by constructing an appropriate observation vector. We actually perform the computation in the frequency domain via a scalable so-lution. Comparing to other applications, retrieval conveniently allows offline computation of the graph Fourier basis and online reuse to embed query vectors. An alternative is to use random projections [START_REF] Tremblay | Graph wavelets for multiscale community mining[END_REF]48]. This roughly corresponds to a single iteration of our step 1. Our solution is thus more accurate, while h is specified online.

Practical considerations

Block diagonal case. Each connected component of G has a maximal eigenvalue 1. These maxima of small components dominate the eigenvalues of the few (or one) "giant" component that contain the vast majority of data [START_REF] Kang | Beyond 'caveman communities': Hubs and spokes for graph compression and mining[END_REF]. For this reason we find the connected components with the union-find algorithm [START_REF] Cormen | Introduction to algorithms[END_REF] and reorder vertices such that A is block diagonal: A = diag(A 1 , . . . , A c ). For each n l × n l matrix A l , we apply offline stages 1-3 to obtain an approximate rank-r l eigenvalue decomposition Ûl Λl Û⊤ l ≈ A l with r l = max(ρ, ⌈rn l /n⌉) if n l > ρ, otherwise we compute an exact decomposition. Integer ρ is a given parameter. We form (U l , Λ l ) by keeping up to ρ slices from each pair ( Ûl , Λl ) and complete with up to r slices in total, associated to the largest eigenvalues of the entire set diag( Λ1 , . . . , Λc ). Online, we partition (y 1 ; . . . ; y c ) = y, compute each x l from y l by (4) and form back vector x = (x 1 ; . . . ; x c ). Sparse neighborhoods. Denote by η i the ℓ 2 -norm of the i-th row of U . FSR.EXACT yields η = 1 but this is not the case for FSR.RANK-r. Larger (smaller) values appear to correspond to densely (sparsely) populated parts of the graph. For small rank r, norms η i are more severely affected for uncommon vectors in the dataset. We propose replacing each element x i of (4) by

x ′ i = x i + (1 -η i )v ⊤ i q, (13)
for global descriptors, with a straightforward extension for regional ones. This is referred to as FSRw and is a weighted combination of manifold search and Euclidean search. It approaches the former for common elements and to the latter for uncommon ones. Our experiments show that this is essential at large scale.

Experiments

This section introduces our experimental setup, investigates the performance and behavior of the proposed method and its application to large-scale image retrieval.

Experimental Setup

Datasets. We use three image retrieval benchmarks: Oxford Buildings (Oxford5k) [START_REF] Philbin | Object retrieval with large vocabularies and fast spatial matching[END_REF], Paris (Paris6k) [START_REF] Philbin | Lost in quantization: Improving particular object retrieval in large scale image databases[END_REF] and Instre [START_REF] Wang | Instre: a new benchmark for instance-level object retrieval and recognition[END_REF], with the evaluation protocol introduced in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] for the latter. We conduct large-scale experiments by following a standard protocol of adding 100k distractor images from Flickr [START_REF] Philbin | Object retrieval with large vocabularies and fast spatial matching[END_REF] to Oxford5k and Paris6k, forming the so called Oxford105k and Paris106k. Mean average precision (mAP) evaluates the retrieval performance in all datasets. Image Descriptors. We apply our method on the same global and regional image descriptors as in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. In particular, we work with d-dimensional vectors extracted from VGG [START_REF] Simonyan | Very deep convolutional networks for large-scale image recognition[END_REF] (d = 512) and ResNet101 [20] (d = 2, 048) networks fine-tuned specifically for image retrieval [START_REF] Radenović | CNN image retrieval learns from bow: Unsupervised fine-tuning with hard examples[END_REF][START_REF] Gordo | End-to-end learning of deep visual representations for image retrieval[END_REF]. Global description is R-MAC with 3 different scales [START_REF] Tolias | Particular object retrieval with integral max-pooling of cnn activations[END_REF], including the full image as a separate region. Regional descriptors consist of the same regions as those involved in R-MAC but without sum pooling, resulting in 21 vectors per image on average. Global and regional descriptors are processed by supervised whitening [START_REF] Radenović | CNN image retrieval learns from bow: Unsupervised fine-tuning with hard examples[END_REF]. Implementation. We adopt the same parameters for graph construction and search as in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. The pairwise descriptor similarity is defined as s(v, z) = [v ⊤ z] 3 + . We use α = 0.99, and keep the top k = 50 and k = 200 mutual neighbors in the graph for global and regional vectors, respectively. These choices make our experiments directly comparable to prior results on manifold search for image retrieval with CNN-based descriptors [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. In all our FSR.APPROX experiments, we limit the algorithm within the largest connected component only, while each element x i for vertex v i in any other component is just copied from y i . This choice works well because the largest component holds nearly all data in practice. Following [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF], generalized max-pooling [35,[START_REF] Iscen | Memory vectors for similarity search in high-dimensional spaces[END_REF] is used to pool regional diffusion scores per image. Reported search times exclude the construction of the observation vector y, since this task is common to all baseline and our methods. Time measurements are reported with a 4-core Intel Xeon 2.00GHz CPU.

Retrieval Performance

Rank-r. We evaluate the performance of FSR.RANK-r for varying rank r, which affects the quality of the approximation and defines the dimensionality of the embedding space. As shown in Figure 4, the effect of r depends on the dataset. In all cases the optimal performance is already reached at r = 1k. On Paris6k in particular, this happens as soon as r = 100. Compared to FSR.EXACT as implemented in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF], it achieves the same mAP but 150 times faster on Oxford5k and Paris6k and 300 times faster on Instre. Global search demonstrates a similar behavior.

We achieve 97.0 mAP on Paris6k, which is near-perfect. Figure 3 shows the two queries with the lowest AP and their top-ranked negative images. In most cases the ground-truth is incorrect, as these images have visual overlap with the query bounding box. The first correct negative image for "La Défense" appears at rank 126, where buildings from the surroundings are retrieved due to "topic drift". The same happens with "Pyramide du Louvre", where the first correct negative image is at rank 108. Figure 4: Performance of regional search with FSR.RANKr. Runtimes are reported in text labels. ⋄ refers to FSR.EXACT performed with conjugate gradients as in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] Regional search performs better than global [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] at the cost of more memory and slower query. We unlock this bottleneck thanks to the offline pooling U = ΣU . Indeed, global and regional search on Instre take 0.040s and 0.042s respectively with our method, while the corresponding times for FSR.EXACT are 0.055s and 3s.

Approximate eigendecomposition keeps the off-line stage tractable at large scale. With 570k regional descriptors on Instre, FSR.RANK-5000 and FSR.APPROX yield a mAP of 89.5 and 89.2 respectively, with offline cost 60 and 3 hours respectively, using 16-core Intel Xeon 2.00GHz CPU. This is important at large scale because the off-line complexity of FSR.RANK-r is polynomial.

When new images are added, one can express them according to existing ones, as in [START_REF] Drineas | On the Nyström method for approximating a gram matrix for improved kernel-based learning[END_REF]. We evaluate such extension by constructing the graph on a random subset of 100%, 90%, 70%, 50%, 30% and 10% of Instre, yielding 80.5, 80.1, 78.3, 75.8, 70.2 and 40.6 mAP respectively on the entire dataset, with global search. The drop is graceful until 30%; beyond that, the graph needs to be updated.

Large-scale experiments

We now apply our approach to a larger scale by using only 5 descriptors per image using GMM reduction [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. This choice improves scalability while minimizing the accuracy loss. FSRw.APPROX becomes crucial, especially at large scale, because vectors of sparsely populated parts of the graph are not well represented. Figure 5 shows the comparison between FSRw.APPROX and FSR.APPROX. We achieve 90.2 and 94.2 with FSR.APPROX and FSRw.APPROX respectively, with r = 10k and Resnet101 descriptors. We further report the performance separately for each of the 11 queries of Oxford105k dataset. Results are shown in Figure 6. Low values of r penalize sparsely populated parts of the graph, i.e. landmarks with less similar instances in the dataset. FSRw.APPROX partially solves this issue.

The search time is 0.14s and 0.3s per query for r = 5k and r = 10k respectively on Oxford105k. It is two orders of magnitude faster than FSR.EXACT: The implementation of [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] requires about 14s per query, which is reduced to 1s with dataset truncation: manifold search is a re-ranking only applied to top-ranked images. We do not use any truncation. This improves the mAP by 3% and our method is still one order of magnitude faster.

Sparse embeddings. Most descriptors belong only to few manifolds and each embedding vector has high energy in the corresponding components. Setting r = 10k, large enough to avoid compromising accuracy, Figure 7 shows the effect of sparsifying the embeddings with FSRw.SPARSE on Oxford105k. Remarkably, we can make up to 90% memory savings with only %2 drop of mAP.

Quantized descriptors. Construction of the observation vector requires storing the initial descriptors. We further all souls (78) ashmolean [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF] balliol ( 12) bodleian [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] christ church (78) cornmarket ( 9) hertford ( 54) keble (7) magdalen (54) pittrivers ( 6) radcliffe (221) use product quantization (PQ) [START_REF] Jégou | Product quantization for nearest neighbor search[END_REF] to compress them. Using FSRw.APPROX on Oxford105k, mAP drops from 94.4 with uncompressed descriptors to 94.2 and 91.1 with 256and 64-byte PQ codes, respectively.

Comparison to other methods

Table 1 compares our method with the state-of-the-art. We report results for r = 5k, FSR.RANK-r for global description, FSR.APPROX for regional description, and FSRw.APPROX in large-scale (with 100k distractors) and regional experiments. GMM reduces the number of regions per image from 21 to 5 [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. We do not experiment at large-scale without GMM since there is not much improvement and it is less scalable. Our method reaches performance similar to that of FSR.EXACT as evaluated with CG [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. Our benefit comes from the dramatic speed-up. For the first time, manifold search runs almost as fast as Euclidean search. Consequently, dataset truncation is no longer needed and this improves the mAP.

Discussion

This work reproduces the excellent results of online linear system solution [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] at fraction of query time. We even improve performance by avoiding to truncate the graph online. The offline stage is linear in the dataset size, embarrassingly parallelizable and takes a few hours in practice for the large scale datasets of our experiments. The approximation quality is arbitrarily close to the optimal one at a 1: Performance comparison to the baseline methods and to the state of the art on manifold search [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF]. Points at 512D are extracted with VGG [START_REF] Radenović | CNN image retrieval learns from bow: Unsupervised fine-tuning with hard examples[END_REF] and at 2048D with ResNet101 [START_REF] Gordo | End-to-end learning of deep visual representations for image retrieval[END_REF]. Regional representation with m = 5 descriptors per image uses GMM. Large-scale regional experiments use the FSRw.APPROX variant. Dataset truncation is used in [START_REF] Iscen | Efficient diffusion on region manifolds: Recovering small objects with compact cnn representations[END_REF] at large scale.

given embedding dimensionality. The required dimensionality for good performance is large but in practice the embedded vectors are very sparse. This resembles an encoding based on a large vocabulary, searched via an inverted index.

Our method is generic and may be used for problems other than search, including clustering and unsupervised or semisupervised learning.

Figure 1 :

 1 Figure1: The low-pass filtering of an impulse over the real line (top) and a graph (bottom). In a weighted undirected graph the information "flows" in all directions, controlled by edge weights. In retrieval, the impulse in red is the query, and the output x is its similarity to all samples.

( a )

 a Form the r × r matrix C : = Q ⊤ B = Q ⊤ AQ. (b) Compute its eigendecomposition V Λ V⊤ = C. (c) Form (V, Λ) by keeping from ( V , Λ) the slices (rows/columns) corresponding to the r largest eigenvalues. (d) Define the matrix U : = QV .

  • FSR.APPROX: Drop sparsification stage 3. • FSR.RANK-r: Drop approximation stage 1 and sparsification stage 3. Set r = n, Q = I, B = A in stage 2. • FSR.EXACT: same as FSR.RANK-r for r = n.

Figure 3 :

 3 Figure 3: Two queries with the lowest AP from Paris6k (left) and the corresponding top-ranked negative images based on the ground-truth, with their rank underneath. Ranks are marked in blue for incorrectly labeled images, and red otherwise.

  Figure 5: mAP vs. r on Oxford105k with FSR.APPROX and FSRw.APPROX, using Resnet101(R) and VGG(V).

Figure 6 :

 6 Figure 6: mAP vs. rank r separately per landmark in Oxford105k with FSR.APPROX (left) and FSRw.APPROX (right). Number of positive images per landmark is shown in the legend.

  Figure 7: mAP vs. sparsity of U by keeping its τ largest values and varying τ with FSRw.SPARSE on Oxford105k, Resnet101 descriptors and rank r = 10k.

  is a 'low-pass filter'; 1x represents eigenvalues of L, where 0 is the DC component. our algorithm is valid under truncation. The approximation QQ ⊤ A ≈ A of stage 1 is studied in [18, §9.3,10.4]: an average-case bound on A -QQ ⊤ A decays exponentially fast in the number of iterations q to |λ r+1 |. Stage 2 yields an approximate eigenvalue decomposition of A:

Table

	Method	m × d INSTRE Oxf5k Oxf105k Par6k Par106k
		Global descriptors -Euclidean search
	R-MAC [45]	512 47.7	77.7	70.1 84.1 76.8
	R-MAC [15]	2,048 62.6	83.9	80.8 93.8 89.9
		Global descriptors -Manifold search
	Diffusion [24]	512 70.3	85.7	82.7 94.1 92.5
	FSR.RANK-r	512 70.3	85.8	85.0 93.8 92.4
	Diffusion [24]	2,048 80.5	87.1	87.4 96.5 95.4
	FSR.RANK-r	2,048 80.5	87.5	87.9 96.4 95.3
	Regional descriptors -Euclidean search
	R-match [46]	21×512 55.5	81.5	76.5 86.1 79.9
	R-match [46] 21×2,048 71.0	88.1	85.7 94.9 91.3
	Regional descriptors -Manifold search
	Diffusion [24]	5×512 77.5	91.5	84.7 95.6 93.0
	FSR.APPROX	5×512 78.4	91.6	86.5 95.6 92.4
	Diffusion [24] 21×512 80.0	93.2	90.3 96.5 92.6
	FSR.APPROX 21×512 80.4	93.0	-	96.5	-
	Diffusion [24] 5×2,048 88.4	95.0	90.0 96.4 95.8
	FSR.APPROX 5×2,048 88.5	95.1	93.0 96.5 95.2
	Diffusion [24] 21×2,048 89.6	95.8	94.2 96.9 95.3
	FSR.APPROX 21×2,048 89.2	95.8	-	97.0	-

We provide a truly scalable solution to computing an approximate Fourier basis of the graph offline, accompanied by performance bounds.

Acknowledgments The authors were supported by the MSMT LL1303 ERC-CZ grant. The Tesla K40 used for this research was donated by the NVIDIA Corporation. The authors would like to thank James Pritts for fruitful discussions during this work.


Arnaud Regazzi

Stéphane Corn
email: stephane.corn@mines-ales.fr

Patrick Ienny

Jean-Charles Bénézet

Anne Bergeret

Reversible and irreversible changes in physical and mechanical properties of biocomposites during hydrothermal aging

Keywords:

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The growing demand for sustainable policies from public opinion drives industry toward the development of bio-based materials. In the field of thermoplastic composites, poly(lactic acid) reinforced with plant fibers meets a growing success for industrial application because of their good specific mechanical properties, availability, and similar processing properties than traditional polymers [START_REF] Bocz | Flax fibre reinforced PLA/TPS biocomposites flame retarded with multifunctional additive system[END_REF]. A limited number of products can already be found on the market targeting various fields of application such as automotive, mobile phones, or plant pots [START_REF] Graupner | Natural and man-made cellulose fibre-reinforced poly(lactic acid) (PLA) composites: an overview about mechanical characteristics and application areas[END_REF][START_REF] Summerscales | Manufacturing methods for natural fibre composites[END_REF]. Such products are fully biodegradable and offer the advantage to be either recycled, composted or incinerated at their end of life [START_REF] Le Moigne | Study of the interface in natural fibres reinforced poly(lactic acid) biocomposites modified by optimized organosilane treatments[END_REF]. However the intrinsic nature of these materials results in a high sensitivity to the presence of water and temperature (Le [START_REF] Duigou | Seawater ageing of flax/poly(lactic acid) biocomposites[END_REF]. Indeed several papers highlight the specific problems encountered in the presence of a hydrothermal environment [START_REF] Islam | Influence of accelerated ageing on the physico-mechanical properties of alkali-treated industrial hemp fibre reinforced poly(lactic acid) (PLA) composites[END_REF] resulting in a decrease of mechanical properties. Though they do not rule out the idea of using such materials for outdoor applica-tions [START_REF] Cheung | Natural fibre-reinforced composites for bioengineering and environmental engineering applications[END_REF][START_REF] Dittenber | Critical review of recent publications on use of natural composites in infrastructure[END_REF]. As a result, the success of the commercial use of biobased composites lies in the control of the evolution of their mechanical properties in real-life conditions. This evolution of properties (physical and mechanical), which are the effects of aging, can be either temporary (reversible) or permanent (irreversible). In order to control aging, the first step consists in identifying the aging factors susceptible to result in a change of these properties. Among them, bacterial degradation, UV radiations and mechanical loadings are not negligible but temperature and humidity are the major causes for short-term aging [START_REF] Kumar | Biodegradation of flax fiber reinforced poly lactic acid[END_REF][START_REF] Van Den Oever | Agrofibre reinforced poly(lactic acid) composites: effect of moisture on degradation and mechanical properties[END_REF]. The processes induced by the aging factors and their combination are well identified since they have been studied for years for usual polymers. However, their extension to biobased polymers requires additional research efforts because of the numerous processes at stake, their interactions, and the heterogeneity of both materials and aging processes. In addition to these difficulties, aging effects may be reversible, like plasticizing for instance, or, in most cases, irreversible [START_REF] Mercier | Characterisation and modelling of aging of composites[END_REF]Pochiraju et al., 2012;[START_REF] Weitsman | Fluid Effects in Polymers and Polymeric Composites[END_REF]. This distinction is essential to predict the evolution of properties, especially when aging factors are expected to evolve during the materials lifetime. Indeed, for any outdoor application, aging is cycled: day/night, sun/rain, seasons, indoor storage/outdoor use. In order to properly assess the sensitivity of materials to aging in real-life conditions within a time period of several years, a complete understanding of the impact of both reversible and irreversible effects of aging is necessary. The purpose of this study is to assess the consequences of both the reversible and irreversible effects of a hydrothermal aging on the properties of a plant fiber reinforced poly(lactic acid) in order to offer new perspectives for predicting the lifetime of natural fibers reinforced composites.

Materials and methods

Methodology

Poly(lactic acid)/short flax fiber biocomposites have been selected for the present study. As described in Fig. 1, the analysis was decomposed into two steps:

1. Firstly the impact of hydrothermal aging was assessed on composites with various fiber contents. In this case, reversible and irreversible effects of aging occurred simultaneously, and only their combined effects were evaluated. 2. In a second step, materials were brought to the same hydrothermal conditions as before aging by removing water by desiccation. This procedure allowed the assessment of potential damage, and thus the distinction between reversible and irreversible effects of aging.

It is worth to mention that, in this study, the properties of the materials after an initial desiccation effected prior to aging constitutes the reference for the analysis of their changes. The total change of a property due to aging corresponds to the variation between its value after immersion and its reference value. Accordingly, the irreversible change of a property is defined by its variation at final desiccation. Then, the reversible part of a property is assessed from the difference between its total change and its irreversible change.

Materials

2.2.1. Poly(lactic acid) matrix PLA Ingeo TM 7000D resin was obtained from NatureWorks ® LLC (Blair, NE, USA). These granules were designed for injection stretch blow molded applications. This grade of PLA had a density of 1.24 g/cm 3 , a glass transition temperature between 55 and 60 • C and a melting temperature between 155 and 165 • C (NatureWorks, 2005).

Flax fibers

The short flax fibers (Linum usitatissimum) FIBRA-S ® 6A used for this study were provided by Fibres Recherches Développement ® (Troyes, France). According to the technical data sheet [START_REF] Frd | Fibres et préformes végétales -Solutions composites[END_REF], fibers (bundles) were 6 mm long with a diameter of 260 ± 150 m and their density was between 1.4 and 1.5 g/cm 3 . Concerning quasi-static mechanical properties, Young's modulus of bundles was 36 ± 13 GPa, maximum stress was 750 ± 490 MPa and strain at break was 3.0 ± 1.9%.

Experimental techniques 2.3.1. Processing conditions

Various fiber weight contents were used: 0% (neat PLA) hereafter named PLA, 10% hereafter named PLA-F10, and 30% hereafter named PLA-F30. Polylactic acid granules and flax fibers were dried at 80 • C for at least 24 h and under vacuum at 120 • C for 4 h respectively. Composite granules were obtained with a corotative twin-screw extruder (Clextral BC21, screw length = 900 mm; temperature profile along the screw and at the die = 180 • C). After a second drying step under vacuum at 80 • C during 24 h, compounded granules were molded with an injection molding machine (Krauss Maffei KM50-180CX) into dog-bone samples according to the standard ISO 527-2 1BA. The temperature profile was increasing up to 200 • C and the mold was kept at 25 • C. After processing, samples were stored at room temperature and 2%rh (relative humidity) before characterization or aging. This equilibrium state constitutes the reference for evaluating the effects of aging on the materials.

Aging experiments

Three series of isothermal water immersion experiments were conducted for each material (PLA, PLA-F10 and PLA-F30) at 20 • C, 35 • C, and 50 • C. For each temperature and material, 10 samples were immersed in water, slightly wiped, characterized then re-immersed, up to 51 days. Afterward, samples were desiccated at room temperature and 2%rh until their mass reached an equilibrium. This desiccation was supposed not to imply other mechanisms than those previously caused by hydrothermal aging. This equilibrium state constitutes the final condition for evaluating the irreversible effects of aging.

Weight, volume, and density measurements

Samples mass was measured using a weighing scale Mettler-Toledo AT200 with a precision of 0.1 mg. Thickness and width were determined with a 1 m-accurate palmer and length with a 10 maccurate caliper. A mean value performed on each sample was determined. The volume of the sample was approximately evaluated on the basis of dimensional measurements. As a result, the evaluation of density was based on water uptake and volume gain measurements. The reproducibility was evaluated on 10 measurements.

Scanning electron microscope observations

The morphology of the samples after several aging durations was analyzed using an environmental scanning electron microscope (ESEM) FEI TM Quanta 200 FEG. They were freeze-fractured in the middle part, and the surface was carbon-coated with Balzers CED030 sputter-coating device.

Size exclusion chromatography

The molecular mass of PLA was evaluated by size exclusion chromatography (SEC) with Optilab ® rEX TM of Wyatt Technology (CIRAD, UMR 1208 (IATE), Université de Montpellier, France). 90 mg of each aged material was diluted in tetrahydrofuran stabilized with butylated hydroxytoluene, and then kept at 30 • C during 40 h in a water bath. After a 45 m-filtration, each solution was injected in the column for measurement. The reproducibility was evaluated on 3 samples.

Differential scanning calorimetry

A Perkin-Elmer ® Diamond DSC was used in order to assess the crystallinity of PLA. Samples mass ranged from 5 to 10 mg and ramp temperature was set at 10 • C/min. The enthalpy of a 100% crystalline PLA was assumed to be 93.7 J/g in order to determine crystallinity from thermograms [START_REF] Garlotta | A literature review of poly(lactic acid)[END_REF].

Vibration analysis

From a general point of view, vibration techniques aim to study the dynamic behavior of a structure using its fundamental natural mode of vibration [START_REF] Corn | Vibration analysis techniques for detecting filler-matrix decohesion in composites[END_REF][START_REF] Hwang | Determination of elastic constants of materials by vibration testing[END_REF]. In the present study, this technique was expected to evaluate the intrinsic viscoelastic properties of the materials and more particularly their dynamic elastic modulus (storage modulus) and damping coefficient [START_REF] Adams | The dynamic properties of fibre-reinforced polymers exposed to hot, wet conditions[END_REF].

Samples were set in cantilever position (one end clamped, the other end free) with a free length of 60 ± 0.2 mm. The free end was stimulated by a short impulse leading the sample to vibrate. The displacement during vibration was recorded by a laser sensor (SunX HL-C203F). The time response was then converted to a frequency response by means of a Fast Fourier Transform (FFT) [START_REF] Gibson | Modal vibration response measurements for characterization of composite materials and structures[END_REF]. The eigenfrequency and the damping factor of the first fundamental natural mode of vibration were determined using an interpolation of the Nyquist plot of the frequency response function. This mode corresponded to the first flexural mode. As no simple analytical solution existed for such "dog-bone"-shaped samples, a finite element analysis was conducted in order to extract the dynamic elastic modulus from the eigenfrequency.

The main advantages of this vibration technique are its nondestructive nature and its easy implementation, so that it was well suited for monitoring materials properties during aging. However, it requires the knowledge of the density and the dimensions of the sample which varied during aging. All vibration tests were carried out at room temperature whatever the sample aging conditions. The reproducibility was validated on 5 samples.

Uniaxial tensile tests

Ultimate tensile properties were assessed with a testing machine Zwick B Z010 according the ISO 527 standard. Samples were loaded at 5 mm/min at room temperature. The reproducibility was evaluated on 3 samples. [START_REF] Crank | The Mathematics of Diffusion[END_REF]. Other sorption curves (except PLA-F30 at 50 • C) displayed a gradual increase at long times. This diffusion is also called "pseudo-Fick" diffusion [START_REF] Weitsman | Effects of fluids on polymeric composites -a review[END_REF] since only the beginning of the sorption follows Fick's model. However at 50 • C, PLA-F30 exhibited a weight loss beyond 144 h (or 12 √ h). The volume and density variations were also assessed during these experiments. Density is plotted as a function of immersion time in Fig. 3. Since the density of flax fibers was higher than the density of PLA, the initial density (unaged samples) of composites increased with the fiber content -from 1.238 ± 0.005 g/cm 3 for PLA to 1.254 ± 0.005 g/cm 3 for PLA-F10 and 1.285 ± 0.005 g/cm 3 for PLA-F30. Density turned out to decrease with immersion time with or without the presence of fibers. However this decrease was amplified by both temperature and fiber content. Before aging, an increase in the initial DEM with the fiber content was observed: +24% and +95% for PLA-F10 and PLA-F30 respectively, compared to PLA. These results are in accordance with literature where mechanical properties were assessed with quasistatic measurements [START_REF] Bax | Impact and tensile properties of PLA/Cordenka and PLA/flax composites[END_REF][START_REF] Duigou | Effect of recycling on mechanical behaviour of biocompostable flax/poly(l-lactide) composites[END_REF][START_REF] Oksman | Natural fibres as reinforcement in polylactic acid (PLA) composites[END_REF][START_REF] Van Den Oever | Agrofibre reinforced poly(lactic acid) composites: effect of moisture on degradation and mechanical properties[END_REF].

Results

Physical properties during aging

Mechanical properties during aging

During aging, the DEM of all samples decreased with immersion time. The amount of the decrease increased with aging temperature and fiber content.

• For aging at 20 • C, PLA DEM remained rather constant for the duration of our tests. However the DEM of PLA-F10 and PLA-F30 decreased and reached equilibrium at long immersion times (>400 h or 20 √ h). Nevertheless, at the end of the experiment, composites remained stiffer than unaged PLA.

Table 1

Irreversible and total changes of ultimate tensile strength (UTS) and strain at break (SB) of PLA and flax/PLA composites after 144 h of aging.

Aging temperature 20 • For aging at 35 • C, the decrease in DEM was more significant, even for PLA. The DEM of composites was approximately the same as PLA at the end of the experiment.

• C 35 • C 50 • C Sample PLA PLA-F10 PLA-F30 PLA PLA-F10 PLA-F30 PLA PLA-F10 PLA-F30 Total change of UTS (%) -11 ± 2 -20 ± 3 -31 ± 2 -29 ± 2 -44 ± 3 -53 ± 2 - - - Irreversible change of UTS (%) -1 ± 3 -4 ± 3 -6 ± 6 -2 ± 3 -13 ± 3 -28 ± 6 3 ± 3 -16 ± 3 -41
• For aging at 50 • C, the assessment of mechanical properties of PLA and PLA-F10 became impossible respectively after 144 h and 750 h because of a very high brittleness. But PLA-F30 was strong enough for characterization up to the end of the experiment. At long immersion times, it exhibited a DEM much lower than unaged PLA.

The evolution of ultimate tensile properties during immersion was evaluated as well (cf. Table 1). Before aging, the ultimate tensile strength of PLA, PLA-F10, and PLA-F30 was 67 ± 1, 62 ± 2, and 66 ± 4 MPa, respectively. During immersion, ultimate tensile strength decreased for all materials but more significantly as temperature and fiber content increased. Characterization tests could not be performed at 50 • C due to the extreme ductility of the samples. Prior to aging, the strain at break of PLA, PLA-F10, and PLA-F30 was 5.5 ± 0.3, 3.1 ± 0.2, and 1.6 ± 0.2%, respectively. During immersion, ductility was significantly improved regardless of the fiber content. To sum up, results showed that, during aging, strain at break increased whereas ultimate tensile strength and dynamic elastic modulus decreased, which was a characteristic of softening materials. Fig. 5 presents the evolution of damping of the first fundamental natural mode of vibration for the three materials at each aging temperature. Damping is controlled by dissipative factors such as friction at micro-and meso-scale [START_REF] Meyers | Mechanical Behavior of Materials[END_REF]. Before aging, it appeared that energy dissipation slightly increased with fiber content. During aging, the evolution of damping with aging temperature and fiber content was similar to the evolution of water uptake (cf. Fig. 2).

Reversibility of physical properties

All the results presented so far were the combination of reversible effects (e.g. plasticizing) and irreversible effects (e.g. damage). The measurement of the properties after desiccation suppressed the reversible effects and enabled to quantify the irreversible effects of aging. Reversibility/irreversibility was only evaluated over the first 144 h of immersion because results presented previously showed that transient phenomena were mostly occurring in this duration. Another reason was the very high brittleness of some samples beyond this time which rendered characterization impossible.

Results for 144 h of immersion are gathered in Table 2. The weight of samples after water removal was approximately the same as their initial weight whatever the fiber content and the aging temperature. The volume and the density were completely recovered after desiccation for PLA immersed at 20 • C and 35 • C as well as PLA-F10 at 20 • C. In all other cases, the recovery was only partial and was less and less significant for higher temperatures and larger fiber contents. This partial recovery indicated the presence of damage.

Reversibility of mechanical properties

The evolution of the DEM for different immersion times after desiccation is presented in Fig. 6. The changes were different for pure PLA and PLA/flax composites. For PLA, a slight increase in the DEM was observed as a function of aging time. As shown in Table 3, this result was more significant at higher temperatures (1.6%, 3.0% and 5.2% at 20 • C, 35 • C and 50 • C respectively for 144 h of aging). The ultimate tensile stress exhibited a similar increase for aging at 50 • C (cf. Table 1). On the contrary, the strain at break drastically decreased.

For composites, desiccation only allowed partial recovering of the DEM (cf. Fig. 6). The DEM dropped with aging temperature (-9.6%, -28.4% and -38.6% at 20 • C, 35 • C and 50 • C respectively for PLA-F30 after 144 h of aging) and with fiber content (-13% and -39% for PLA-F10 and PLA-F30 respectively after 144 h of aging at 50 • C). The ultimate tensile strength also dropped due to irreversible effects of aging and in a similar manner as DEM according to results in Table 1. Also, the increase in strain at break observed Fig. 6. Dynamic elastic modulus of PLA and flax/PLA composites after immersion in water at different aging temperatures and complete desiccation; the legend is the same as in Fig. 2. during aging turned out to be partially irreversible, especially when fiber content was significant. Except for PLA at 50 • C, the irreversible effects of aging led to an improvement of the ductility.

As shown in Fig. 7, the reversibility of damping was fully complete whatever the fiber content and the aging temperature.

Discussion

Reversible and irreversible mechanisms

Results showed that the strong sensitivity of PLA/flax composites to temperature and water was responsible for important changes in physical and mechanical behavior.

First of all, an increase in water content and temperature both led to an increase in polymer chain mobility [START_REF] Weitsman | Effects of fluids on polymeric composites -a review[END_REF]. This mobility facilitated the penetration of water into the polymer matrix. As a result, it enhanced both water storage capacity and diffusion rate [START_REF] Badia | Hygrothermal ageing of reprocessed polylactide[END_REF]. Besides, in composites, these quantities were even further increased by the high affinity of lignocellulosic fibers to water [START_REF] Ndazi | Characterization of hydrolytic degradation of polylactic acid/rice hulls composites in water at different temperatures[END_REF]. In the case of flax, the water content can reach up to 50% for a loose fiber [START_REF] Müssig | Industrial Applications of Natural Fibres[END_REF] which is much more than PLA. As a result, both temperature and fiber content led to the increase of water content and diffusion kinetics. However the weight loss of PLA-F30 at 50 • C was the expression of non-reversible damage [START_REF] Weitsman | Effects of fluids on polymeric composites -a review[END_REF].

The absorption of water resulted in the samples swelling because of the physical presence of water molecules causing the deviation of chains. As water is less dense than PLA and flax fibers (cf. Section 2.2), the overall density tended to decrease as water penetrated the materials. Since flax fibers absorb more water than PLA, their swelling was more significant (up to 30% for a loose fiber) [START_REF] Müssig | Industrial Applications of Natural Fibres[END_REF]. As a result, the decrease in density was more significant as well. Similarly the increase in water uptake with temperature led to a more significant swelling and a more significant decrease in density.

As shown by characterization tests after desiccation, the presence of water induced modifications of the physical properties, which remained after complete desiccation. Such irreversible changes affected all areas of composites and can be explained as follows:

(i) Matrix: PLA was prone to undergo chemical (hydrolysis, cavitation) and mechanical (cracks, relaxations) damage under the combination of water and temperature. In the presence of water, when temperature was close to the glass transition temperature, the high mobility of polymeric chains led to hydrolysis [START_REF] Zhang | Morphological behaviour of poly(lactic acid) during hydrolytic degradation[END_REF]. The loss in number-average molecular mass in Fig. 8 indicates the chain scissions caused by hydrolysis. Besides, the mobility induced by both swelling (physi-crystallization) and chain scissions (chemicrystallization) caused chains to reorganize in a more stable conformation. This crystallization of PLA is shown by DSC measurements. Indeed, PLA, which was completely amorphous after processing, reached a crystallinity of 6 ± 2% after 144 h in immersion at 50 • C. Finally, swelling was prone to induce internal stresses which, in the presence of water, led to a mechanism called environmental stress cracking [START_REF] Arnold | The influence of liquid uptake on environmental stress cracking of glassy polymers[END_REF]. Fig. 9 shows the presence of numerous cracks in PLA immersed at 50 • C, suggesting that irreversible swelling (or density loss) in these samples was caused by swelling induced cracks.

(ii) Fibers: the hydrophilic behavior of plant fibers influences significantly the moisture sorption capacity of the composite. It is caused by their composition (polar groups prone to establish hydrogen bonds with water molecules) and their structure (porosity and high exchange surface) [START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF]. As a result, when increasing the fiber content, the composites absorbs more water and the change of its properties is more likely to be irreversible. The dimensional stability of plant fibers is known to be highly dependent to moisture [START_REF] Madsen | Hemp yarn reinforced composites -III. Moisture content and dimensional changes[END_REF]. But their swelling is also highly anisotropic.

For flax fibers, it can reach 25% in their transverse direction but rarely exceeds 0.2% in their longitudinal direction [START_REF] Müssig | Industrial Applications of Natural Fibres[END_REF]. Hygroscopic strains are generally reversible but when they become substantial the micro-structure of cellulosic fibers undergo damage [START_REF] Lee | Longitudinal and concurrent dimensional changes of cellulose aggregate fibrils during sorption stages[END_REF]. Consequently, with such a significant macroscopic strain, it is very likely that flax fibers were damaged during the sorption and/or desorption process. (iii) Interface fibers/matrix: the substantial swelling in the transverse direction of lignocellulosic fibers induced hoop stresses in the matrix, leading to plastic strain and micro-cracks (cf. Fig. 10). The occurrence of micro-cracks has also been reported in several different nature of matrix [START_REF] Azwa | A review on the degradability of polymeric composites based on natural fibres[END_REF][START_REF] Bismarck | Surface characterization of flax, hemp and cellulose fibers; surface properties and the water uptake behavior[END_REF][START_REF] Chow | Moisture absorption studies of sisal fibre reinforced polypropylene composites[END_REF][START_REF] Hu | Moisture absorption, tensile strength and microstructure evolution of short jute fiber/polylactide composite in hygrothermal environment[END_REF]. The diffusion of water molecules was increased through the bulk matrix as well as along the fiber/matrix interfaces [START_REF] Célino | The hygroscopic behavior of plant fibers: a review[END_REF]. Besides, water molecules are supposed to create hydrogen bonds with the fibers, which should have led to a reduction of fiber/matrix interactions [START_REF] Dhakal | Effect of water absorption on the mechanical properties of hemp fibre reinforced unsaturated polyester composites[END_REF]. After drying, the shrinkage of fibers resulted in a gap between fibers and matrix visible in SEM pictures (cf. Fig. 9). The increase in the irreversible swelling with fiber content in Table 2 is mainly the consequence of the void created by fibers shrinking during drying (le [START_REF] Le Duigou | Improving the interfacial properties between flax fibres and PLLA by a water fibre treatment and drying cycle[END_REF]. Besides, an extensive swelling of PLA led to shear stresses at the interfaces, due to the almost nonexistent swelling of fibers in their longitudinal direction. Such shear was likely to contribute to fiber/matrix decohesion (Le [START_REF] Duigou | In-situ evaluation of flax fibre degradation during water ageing[END_REF]. To sum up, the lack of dimensional stability of natural fibers to moisture is supposed to be the most decisive factors in the irreversibility of aging at the interface between fibers and matrix.

All these damage mechanisms are expected to lead to a drop of the mechanical properties. The likely consequences of these phenomena are assessed in the next part.

Reversibility/irreversibility of mechanical properties

The changes of physical properties during aging of materials indubitably affected their mechanical properties. Indeed, the increase in polymeric chains mobility caused by the diffusion of water between polymer chains induces a softening of polymeric materials (both fibers and matrix) called plasticizing [START_REF] Weitsman | Fluid Effects in Polymers and Polymeric Composites[END_REF]. By comparing mechanical properties immediately before and after desiccation (cf. Tables 1 and3), it was possible to evaluate their reversibility, since plasticizing is the only reversible process occurring. Plasticizing turned out to be the main reason for the drop in DEM during immersion (except PLA-F30 at 35 and 50 • C). Surprisingly, it turned out to be the only reason of the increase in damping during immersion, since the reversibility was complete (cf. Fig. 7).

While reversible phenomena were quite dependent of the sole water content, the influence of irreversible mechanisms seemed to be more complex.

For PLA, the irreversible effects of aging were responsible for a slight increase in DEM with aging temperature. This observation was attributed to the increase in crystallinity measured by DSC and described previously. While aging did not affect the irreversibility of the ultimate tensile strength, it increased the irreversibility of the strain at break, probably due to stress relaxation induced by the forming process. However when aging temperature was close to the glass transition temperature, the irreversibility of the strain at break dropped. In this case, hydrolysis-caused damage prevailed (cf. Fig. 8) which quickly led to a very high brittleness (immediate breaking). As a result, hydrolysis was essentially responsible for the irreversible loss of mechanical properties and its consequences occurred quite rapidly in these conditions.

For composites, the irreversible effects of aging should theoretically be enhanced compared to PLA due to the additional damaging mechanisms occurring in the presence of fibers. This hypothesis was mainly verified by the growing influence of irreversible mechanisms with fiber content. Damage induced by the presence of fibers became so important that irreversibility prevailed over reversibility in the drop of the DEM of PLA-F30 beyond 35 • C.

Yet fibers did not always induce a more significant irreversibility. Indeed, composites immersed during 144 h at 50 • C exhibited an improvement in ductility while hydrolysis drastically deteriorated PLA. In addition, for longer aging times, the lifetime of materials was significantly extended by the presence of fibers which postponed the failure of PLA. Consequently, the damage engendered by the swelling of fibers in composites did not seem to be the main reason of the irreversible loss of properties (and thus of the failure mechanisms), but rather the larger amount of water (cf. Fig. 2) which led to an advanced hydrolysis (cf. Fig. 8). On the contrary, the presence of fibers significantly limited the propagation of cracks in PLA [START_REF] Bourmaud | Observation of the structure of a composite polypropylene/flax and damage mechanisms under stress[END_REF].

Conclusions

This study exposed the effects of hydrothermal aging on PLA/flax biocomposites processed by extrusion and injection molding. It presented a methodology to distinguish the consequences of reversible and irreversible phenomena induced by hydrothermal aging of these biocomposites.

On the one hand, the reversible effects of aging were characterized by a limited swelling and a plasticizing of both PLA and flax fibers. On the other hand, the irreversible effects were characterized by a drop of PLA average molecular mass caused by hydrolysis which led to an increase in its crystallinity and the formation of cracks. In composites, it was accompanied by a substantial swelling of fibers leading to hoop stresses in the fibers vicinity and eventually decohesion from the PLA matrix.

Generally, the changes in mechanical properties were demonstrated to be primarily reversible. Damping turned out to be exclusively affected by the amount of water.

The irreversible effects of aging were strongly influenced by both the amount of fiber and the aging temperature. It became the primary cause for the drop of mechanical properties for the composite with the highest fiber content. Besides, the irreversible and reversible effects of aging did not always have the same effects on mechanical properties, i.e. while crystallizing induced a slight increase in elastic modulus, plasticizing induced a decrease. The use of the term "damage" implies some ambiguities, since such modification in the microstructure can lead to the increase of some mechanical properties (e.g. rigidity and ductility). Finally, even though the use of flax fibers in PLA led, for short-term aging, to a more significant irreversible decrease of several properties, it enabled to extend drastically the durability of such materials.

Fig. 1 .

 1 Fig. 1. Schematic representation of the methodology to evaluate the irreversible changes in the characterized properties during hydrothermal aging.

Fig. 2

 2 Fig.2shows experimental results for water uptake of the three materials at 20 • C, 35 • C and 50 • C. It can be observed that both water content and diffusion kinetics were enhanced by both temperature and fiber content. At 20 • C and 35 • C, PLA seemed to obey to a Fick diffusion since water uptake reached an equilibrium for long immersion times[START_REF] Crank | The Mathematics of Diffusion[END_REF]. Other sorption curves (except PLA-F30 at 50 • C) displayed a gradual increase at long times. This diffusion is also called "pseudo-Fick" diffusion[START_REF] Weitsman | Effects of fluids on polymeric composites -a review[END_REF] since only the beginning of the sorption follows Fick's model. However at 50 • C, PLA-F30 exhibited a weight loss beyond 144 h (or 12 √ h).

Fig. 2 .

 2 Fig. 2. Water uptake of PLA and flax/PLA composites during immersion in water at different aging temperatures.

Fig. 3 .

 3 Fig. 3. Density of PLA and flax/PLA composites during immersion in water at different aging temperatures; the legend is the same as in Fig. 2.

Fig. 4

 4 Fig. 4 shows the evolution of the dynamic elastic modulus (DEM) as a function of immersion time for the three materials during immersion in water at 20, 35, and 50 • C.Before aging, an increase in the initial DEM with the fiber content was observed: +24% and +95% for PLA-F10 and PLA-F30 respectively, compared to PLA. These results are in accordance with literature where mechanical properties were assessed with quasistatic measurements[START_REF] Bax | Impact and tensile properties of PLA/Cordenka and PLA/flax composites[END_REF][START_REF] Duigou | Effect of recycling on mechanical behaviour of biocompostable flax/poly(l-lactide) composites[END_REF][START_REF] Oksman | Natural fibres as reinforcement in polylactic acid (PLA) composites[END_REF][START_REF] Van Den Oever | Agrofibre reinforced poly(lactic acid) composites: effect of moisture on degradation and mechanical properties[END_REF].During aging, the DEM of all samples decreased with immersion time. The amount of the decrease increased with aging temperature and fiber content.

Fig. 4 .

 4 Fig. 4. Dynamic elastic modulus of PLA and flax/PLA composites during immersion in water at different aging temperatures; the legend is the same as in Fig. 2.

Fig. 5 .

 5 Fig. 5. Damping (measured from the first fundamental natural mode of vibration) of PLA and flax/PLA composites during immersion in water at different aging temperatures; the legend is the same as in Fig. 2.

Fig. 7 .

 7 Fig. 7. Damping (measured from the first fundamental natural mode of vibration) of PLA and flax/PLA composites after immersion in water at different aging temperatures and complete desiccation; the legend is the same as in Fig. 2.

Fig. 8 .

 8 Fig. 8. Number average molecular mass of PLA and flax/PLA composites during immersion in water at different aging temperatures; the legend is the same as in Fig. 2.

Fig. 9 .

 9 Fig. 9. Freeze-fracturing surfaces of unaged (left) and aged (right) PLA/flax 10%wt composites. The aged sample was immersed in water at 50 • C during 1225 h (magnification: ×2000).

Fig. 10 .

 10 Fig. 10. Damage mechanisms likely to occur in lignocellulosic composites during hydrothermal aging (adapted from (Azwa et al., 2013)) -(a) water diffusion into the composite, (b) swelling of fibers leading to matrix microcracks, (c) leaching of water soluble substances from fibers, (d) decohesion of fiber/matrix interface during drying.

Table 2

 2 Irreversible changes of morphological properties (mass, volume and density changes) of PLA and flax/PLA composites after 144 h of aging.

	Aging	20 • C			35 • C			50 • C
	temperature
	Sample	PLA	PLA-F10	PLA-F30	PLA	PLA-F10	PLA-F30	PLA	PLA-F10	PLA-F30
	Irreversible	0.00 ± 0.01	0.00 ± 0.01	0.01 ± 0.01	0.00 ± 0.01	0.01 ± 0.01	0.05 ± 0.04	0.04 ± 0.01	0.04 ± 0.01	0.04 ± 0.04
	change of
	mass (%)
	Irreversible	-0.42 ± 1.40	0.10 ± 0.54	0.25 ± 0.26	0.52 ± 1.00	1.46 ± 0.42	4.36 ± 0.54	2.67 ± 1.79	5.49 ± 0.63	10.06 ± 0.27
	change of
	volume (%)
	Irreversible	0.01 ± 0.28	-0.05 ± 0.10	-0.26 ± 0.22	0.16 ± 0.47	-1.20 ± 0.14	-4.01 ± 0.31	-2.42 ± 0.37	-5.06 ± 0.22	-9.42 ± 0.12
	change of
	density (%)

Table 3

 3 Reversible and irreversible changes of dynamic elastic modulus of PLA and flax/PLA composites after 144 h of aging.

	Aging temperature	20 • C			3 5 • C			5 0 • C
	Sample	PLA	PLA-F10	PLA-F30	PLA	PLA-F10	PLA-F30	PLA	PLA-F10	PLA-F30
	Total change (%)	-1.7	-8.1	-26.0	-5.9	-20.0	-45.1	-3.4	-28.1	-60.3
	Irreversibility (%)	1.6	-2.3	-9.6	3.0	-7.8	-28.4	5.2	-13.3	-38.6
	Reversibility (%)	-3.3	-5.9	-16.4	-8.9	-12.2	-16.8	-8.7	-14.8	-21.7

Acknowledgments

The authors would like to thank Frédéric Bonfils from CIRAD, UMR 1208 (IATE), Montpellier for his assistance with the size exclusion chromatography experiments.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.indcrop.2016.01. 052.