Nicolas P Rougier

Yann Boniface
email: yann.boniface@loria.fr

Motivated Self-Organization

We present in this paper a variation of the selforganizing map algorithm where the original time-dependent (learning rate and neighborhood) learning function is replaced by a time-invariant one. The resulting self-organization does not fit the magnification law and the final vector density is not directly proportional to the density of the distribution. This lead us to introduce the notion of motivated self-organization where the self-organization is biased toward some data thanks to a supplementary signal. From a behavioral point of view, this signal may be understood as a motivational signal allowing a finer tuning of the final self-organization where needed. We illustrate this behavior through a simple robotic arm setup.

I. INTRODUCTION

We introduced in [START_REF] Rougier | Dynamic self-organizing map[END_REF] the Dynamic Self-Organized Map (DSOM) architecture that is a variation of the self-organizing map algorithm [START_REF] Kohonen | Self-organized formation of topologically correct feature maps[END_REF] where the original time-dependent (learning rate and neighborhood) learning function has been replaced by a time-invariant learning rule. This modification of the learning rule yields several interesting new properties. First, and because of the time invariance, it is possible for the network to support life-long learning. This means that the network can be fed continuously with new input and the network is able to self-organize itself around the whole set of data (using a set of hypothesis detailed in [START_REF] Rougier | Dynamic self-organizing map[END_REF]). This kind of property cannot be easily achieved using SOM-like algorithms because they generally and explicitly depend on a time decreasing learning rate and/or neighborhood function (SOM, NG, GNG) that requires to know beforehand the number of data to be processed. The second property deals with the magnification law as introduced by [START_REF] Villman | Magnification control in self-organizing maps and neural gas[END_REF]. Most vector quantization (VQ) algorithms try to match the density through the density of their code book: high density regions of the distribution tend to have more associated prototypes than low density regions. This generally allows to minimize the loss of information (or distortion) as measured by the mean quadratic error. However, in the case of DSOM, the magnification law is not fit and the density of the code book is uncorrelated with the density of the data, hence leading to a regular quantification a priori of the underlying probability density function. This second property could be easily considered as a serious drawback for vector quantization if one wants, for example, to estimate data density function. However, from a more behavioral point of view, we argue in this paper that this may be a desirable property provided that we can modulate learning using a dedicated signal, i.e. a motivation to concentrate learning on what is relevant to the task in order to have finely tune representations where necessary.

II. MODEL

We'll use here the definitions that we first proposed in [START_REF] Rougier | Dynamic self-organizing map[END_REF].

A. Definitions

Let us consider a probability density function f (x) on a compact manifold Ω ∈ R d . A vector quantization (VQ) is a function Φ from Ω to a finite subset of n code words {w i ∈ R d } 1≤i≤n that form the code book. A cluster is defined as

C i def = {x ∈ Ω|Φ(x) = w i },
which forms a partition of Ω and the distortion of the VQ is measured by the mean quadratic error

ξ = n i=1 Ci x -w i 2 f (x)dx. (1)
If the function f is unknown and a finite set {x i } of p non biased observations is available, the distortion error may be empirically estimated by

ξ = 1 p n i=1 xj ∈Ci x j -w i 2 . ( 2
)
In the following, we will use definitions and notations introduced by [START_REF] Villman | Magnification control in self-organizing maps and neural gas[END_REF] where a neural map is defined as the projection from a manifold Ω ⊂ R d onto a set N of n neurons which is formally written as Φ : Ω → N . Each neuron i is associated with a code word w i ∈ R d , all of which established the set {w i } i∈N that is referred as the code book. The mapping from Ω to N is a closest-neighbor winner-take-all rule such that any vector v ∈ Ω is mapped to a neuron i with the code w v being closest to the actual presented stimulus vector v,

Φ : v → arg min i∈N ( v -w i ). (3)
The neuron w v is called the winning element and the set

C i = {x ∈ Ω|Φ(x) = w i } is called the receptive field of the neuron i.
The geometry corresponds to a Voronoï diagram of the space with w i as the center.

B. Self-Organizing Maps (SOM)

SOM is a neural map equipped with a structure (usually a hypercube or hexagonal lattice) and each element i is assigned a fixed position p i in R q where q is the dimension of the lattice (usually 1 or 2). The learning process is an iterative process between time t = 0 and time t = t f ∈ N + where vectors v ∈ Ω are sequentially presented to the map with respect to the probability density function f . For each presented vector v at time t, a winner s ∈ N is determined according to equation [START_REF] Villman | Magnification control in self-organizing maps and neural gas[END_REF]. All codes w i from the code book are shifted towards v according to

∆w i = ε(t) h σ (t, i, s) (v -w i ) (4)
with h σ (t, i, j) being a neighborhood function of the form

h σ (t, i, j) = e - p i -p j 2 2σ(t) 2 . ( 5
)
where ε(t) ∈ R is the learning rate and σ(t) ∈ R is the width of the neighborhood defined as

σ(t) = σ i σ f σ i t/t f , with ε(t) = ε i ε f ε i t/t f , (6)
while σ i and σ f are respectively the initial and final neighborhood width and ε i and ε f are respectively the initial and final learning rate. We usually have σ f σ i and ε f ε i .

C. Dynamic Self-Organizing Map (DSOM)

DSOM is a neural map equipped with a structure (a hypercube or hexagonal lattice) and each neuron i is assigned a fixed position p i in R q where q is the dimension of the lattice (usually 1 or 2). The learning process is an iterative process where vectors v ∈ Ω are sequentially presented to the map with respect to the probability density function f . For each presented vector v, a winner s ∈ N is determined according to equation (3). All codes w i from the code book W are shifted towards v according to

∆w i = ε v -w i Ω h η (i, s, v) (v -w i ) (7)
with ε being a constant learning rate and h η (i, s, v) being a neighborhood function of the form

h η (i, s, v) = e -1 η 2 p i -ps 2 v-ws 2 Ω ( 8
)
where η is the elasticity or plasticity parameter. If v = w s , then h η (i, s, v) = 0

D. Standard distributions

The DSOM algorithm reflects two main ideas:

• If a neuron is close enough to the data, there is no need for others to learn anything: the winner can represent the data.

• If there is no neuron close enough to the data, any neuron learns the data according to its own distance to the data.

The closeness of the winner to the data is controlled using the elasticity parameter as illustrated on figure 1. and for uniform distributions, DSOM with proper elasticity is comparable to SOM as illustrated on figure 2 and detailed in [START_REF] Rougier | Dynamic self-organizing map[END_REF].

III. EXPERIMENTAL RESULTS

As we explained in the introduction, the dynamic nature of the DSOM algorithm leads to a regular self-organization, that is, a self organization that does not fit the magnification law and consequently, code-vectors are evenly spread on the distribution support. Even from a behavioral point of view, this may not be satisfactory since we may want to have finer representations in some region of the input space that are of some interest and more generic representations in some other parts of the input space.

1 L 1 θ 2 L 2
Fig. 3. The robotic arm is made of two segments of respective size L 1 and L 2 whose positions are given by angles θ 1 and θ 2 . The gray area represents reachable positions in the case L 1 = L 2 and the dotted disc area corresponding to the region of interest (arbitrary defined) .

A. Experimental setup

Let us consider the case of a simple robotic arm made of two linked segments L 1 and L 2 . Segment L 1 can rotate freely around its base in the range [-π/2, +π/2] and segment L 2 can rotate around L 1 endpoint in the range [-π/2, +π/2] (see figure 3). The goal of the experiment is to learn the correspondence between {θ 1 , θ 2 } and the Cartesian coordinates {x, y} of the end point of the arm with finer representations in the dotted disc area. Since we do not want to use the inverse model of the arm, the only way to generate data is then to draw {θ 1 , θ 2 } from their respective domain and to compute the corresponding {x, y} end position of the arm.

B. Results

The mapping from {θ 1 , θ 2 } to {x, y} is not linear and leads to a larger density on the frontier of the domain. If we use such a distribution, we obtain the self-organization drawn on left part of figure 4. This self-organization spreads almost evenly on the whole reachable region with some noticeable and useless representations outside the region. However, as we explained we would like to have finer representations within the dotted gray area and consequently, we built a modulation signal defined as the distance of the end-point to the center of the disc area. This modulation is associated to any sample given to the model and this allow us to modify the original neighborhood equation ( 8) as follows:

h η (i, s, v) = e -α η 2 p i -ps 2 v-ws 2 Ω (9)
with α being the modulation. This α modulation modifies the overall elasticity on a per sample basis and allows to have a finer tuning of the self-organization. This is illustrated on the left part of the figure 4 where self-organization has concentrated code vectors in the region of interest.

IV. CONCLUSION

Since the early work of [START_REF] Hubel | The period of susceptibility to the physiological effects of unilateral eye closure in kittens[END_REF], [START_REF] Daw | Mechanisms of plasticity in the visual cortex[END_REF], the idea of a critical period in the early years of development, where most sensory Sample data are generated by drawing uniformly θ 1 and θ 2 in [-π/2, +π/2] and computing the end point position. This leads to a nonuniform distribution where higher density is found on the periphery of the distribution support. The left figure displays resulting self-organization of a DSOM with an elasticity of 2.5 and no modulation while the right figure displays the same DSOM algorithm where each sample is modulated according to its distance to the center of the figure .   or motor properties are acquired and stabilized have been widely accepted. In such context, the original SOM algorithm gives a fair account of such development. However, cortical representations are not fixed entities, but rather, are dynamic and are continuously modified by experience as explained by [START_REF] Buonomano | Cortical plasticity: From synapses to maps[END_REF] and the capacity of the cortex to re-organize itself in face of lesions, deficits or change in the environment [START_REF] Rita | Brain Mechanisms in Sensory Substitution[END_REF], [START_REF] Ramachandran | Perceptual correlates of massive cortical reorganization[END_REF] cannot be easily explained using SOM-like algorithm since after learning period, the resulting self-organization is frozen and cannot be easily changed.

Thanks to its dynamic nature, the DSOM algorithm may help to solve this dilemma by ensuring a tight coupling between representations and the environment with a code book that does not fit data density and cover the whole data support evenly. Since we may need nonetheless to have more representations in some region of the input space, we proposed to modulate learning by providing the algorithm with an explicit signal giving the importance of the data. In a more general framework, we could expect a cerebral structure (e.g. basal ganglia, amygdala) to compute such signal: if some regions of the perceptive space is judged behaviorally relevant, model could develop precise representations in this region. This has been illustrated through a very simple robotic experiment where the motivation signal is computed according to the distance of the arm end-point to the center of the distribution. If learning was to be driven solely by data density (like in most VQ), such modulation would certainly be strongly attenuated or not possible at all.

Fig. 1 .

 1 Fig.1. Three DSOM with respective elasticity equal to 1, 1.5 and 2 have been trained for 20 000 iterations on a normal distribution using a regular grid covering the [0, 1] 2 segment as initialization. Low elasticity leads to loose coupling between neurons while higher elasticity results in a tight coupling between neurons.

Fig. 2 .

 2 Fig. 2. Side by side comparison of the SOM and DSOM algorithms on very simple and uniform distributions. For each figure, network of size 8 × 8 has been trained for 20 000 epochs using 10 000 samples. Initialization has been done by placing initial code vectors randomly over the [0, 1] 2 area.

  θ

  Fig. 4.Sample data are generated by drawing uniformly θ 1 and θ 2 in [-π/2, +π/2] and computing the end point position. This leads to a nonuniform distribution where higher density is found on the periphery of the distribution support. The left figure displays resulting self-organization of a DSOM with an elasticity of 2.5 and no modulation while the right figure displays the same DSOM algorithm where each sample is modulated according to its distance to the center of the figure.

APPENDIX

  Ω : a compact manifold of R d where d ∈ N + f (x) : a probability density function (pdf) Ω → R {x i } : a set of p non-biased observations of f . N : a set of n elements, n ∈ N + . Φ : a function defined from Ω → N w i ∈ R d : code word associated to an element i of N {w i } : code book associated to N C i : cluster associated to element i such that C i = {x ∈ Ω|Φ(x) = w i } x : euclidean norm defined over R d x Ω : normalized euclidean norm defined over Ω as x →

		x
	ξ ξ	maxy,z∈Ω( y-z ) : distortion error defined as 2 f (x)dx w i : estimated distortion error defined as n i=1 Ci x -1 p n i=1 xj ∈Ci x j -w i 2
	ε(t)	: learning rate at time t
	λ(t) or σ(t) : neighborhood width at time t
	η	: elasticity or plasticity
	%


1 2 + 1 p s -1 = ζ(s) ( 1
)
where s ∈ C, and p ∈ P.

We can observe the perfect zeta function, valid on the whole complex plane, with a convergent series represented by the following polar. We can also define the Riemann zeta function in many different ways. Proof.

∞ k=1 s 2 (p k -1) = ζ(s) = ∞ n=0 1 2 p n (-1)s ( 2
)
where p k denotes the k n-th prime.

It is specially true for all cases for which the Riemann zeta function appear to be an infinite holographic function. Indeed, we assume, under suitable conditions

∞ p ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 2 + 1 p ⎛ ⎜ ⎝ 1 2 + 1 p z -1 ⎞ ⎟ ⎠ -1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ = ζ(s) ( 3
)
where

s = 1 2 + 1 p z -1 , z = 1 2 + 1 p z1 -1 and z n = 1 2 + 1 p zn+1 -1 .
We can illustrate the situation by the following plot :

THE NEXT PRIME

If p > 3 is prime then there exists x and O integers such that

2 p -2 x ≡0 (mod 2) = O (4)
and p divides O.

SOWING THE SEEDS OF PRIMES

To build an infinite tree of primes we need to consider the following sequence for any given integer n = pq and α as any factor of n such that

2α 2 = prime (5)
where β ≡ 5 (mod 10) corresponds to the gear. We can also replace n by any prime and α by 1 such that for p = 3 we have

3 2 + 5 2 2 = 17 (6)
For example with n = 551 we have the following constellation of primes Which means that every prime number has its own satellite which is also prime. Likewise the distribution of the prime numbers depends on the last digit. We can illustrate the situation as follows:

THE DNA OF PRIMES

As we saw it previously, the distribution of the primes depends on the last digit. Then, there exists four classical basis as well as for the DNA. We can illustrate the situation as follows:

Indeed, there exists a special zeta function for which the distribution of the primes follows only some specifics values. The special zeta function is on the form:

sf = p + ( 7
)
where f = 5 6 corresponds to the constant of the light and p is prime. This function is an elegant proof of the Einstein's theory on the light bends. The modular distribution of the primes can be expressed as follows:

If the last digit of the prime is 1 then = 2 3 and so on... The first gates of primes can be illustrate as follows:

FACTORING IN O (log (n))

The RSA set n = pq can be expressed as follows: 6 Searching p and q for the large numbers, it's like looking for a needle in a haystack but, for the crafty mathematicians, the situation is not all doom and gloom. In the other hand, we know that the sinews of war is about cryptography.

Exercice 1

n = x 2 + x -2xy 6 (8)
and

p = n + p (y -1) 2 x (9) Algorithm 3. Code Mathematica # " ! n=RSA-220 FindInstance[\[Sqrt]((n + (p (y -1))) / (2ˆ363)) == p && y > 2ˆ728 && Mod[y, 2] == 0 && y < n && n == p q && p > 2ˆ365 && p < 2ˆ375 && Element[p,Primes] && Element[q,Primes], {p, q, y}, Integers, 1] Exercice 2 n -1 2 ± 2 a -2 b c = p (10)
c < 2 α and 1 < α < 11.

n -z q = 2 x (11) Algorithm 5. Code Mathematica # " ! n=RSA-220 FindInstance[((n -(z)) / q) == 2ˆ98 && Mod[z, q] == 0 && n == p q && Element[q,Primes] && Element[p,Primes] && q > 0 && z > 0 && z < n, {p, q, z}, Integers, 1] Exercice 4 q (x -q -20) n = 2 3 (12)
and

q (x -20) -2 3 n = q (13) Algorithm 6. Code Mathematica n=RSA-1024 FindInstance[((q (x -q -20) )/ n) == 2 3 && n == p q && Element[q,Primes] && Element[p,Primes] && x > 0 && x < n && q > 0, {p, q, x}, Integers, 1] Exercice 5 n 2 x q = 9 16 - y 2 x (

FACTORING AND THE ZETA FUNCTION

If n = pq then there exists a complex number s on the form:

n 2 x p + n 2 x q = y 2 x-2 = √ s ( 16
)
which means that the simplest expression of the Riemann zeta function is:

Theorem 2. ∞ √ s = ζ(s). ( 17

RIEMANN AND THE MODERN PHYSICS

The most important contribution of the Riemann Hypothesis to the Modern Physics may be thought as the deep connections between the Riemann zeta function and the complex structure of a Black Hole. We can illustrate the situation as follows:

Algorithm 1 .

 1 Code MathematicaPolarPlot[{{Cos[(tˆt + 1) / (2 tˆt -2)]}, Sin[(tˆt + 1) / (2 tˆt -2)]}, {t, -2π, 2π}, PlotStyle->{Red,Directive[Dashed,Green,Orange]}, PlotRange-> All]

  ((n -1) / 2) -(2ˆa -2ˆb)) / c == p && c > -20 && c < 20 && n == p q && Element[p,Primes] && Element[q,Primes] && a >= 0 && a <= 700 && b >= 0 && b <= 700, {p, q, a, b, c}, Integers, 1] Exercice 3

  2n+2)ˆ((nˆ2+1)/4),{n,1,Infinity}]

Algorithm 8 .

 8 Code Mathematica © SphericalPlot3D[1+Cos[φˆφ] / (2 φˆφ -2),{θ,0,π},{φ,0,2π}, PlotStyle-> Directive[Black,Opacity[0.7], Specularity[White,10]], Mesh->None, PlotPoints->30] 10 π DECODED? There are finitely many primes a, b, c, d and e satisfying

Algorithm 2. Code Mathematica

	©
	ListContourPlot3D[Table[nˆy + z / 2 nˆy -2 z,{n,-2,2,0.2},{y,-2,2,0.2},{z,-2,2,0.2}],
	Contours->5, Mesh->None, ContourStyle->{Black,Diamond,Red}]


Inria scientific information policy in concrete terms IFIP: editorial management

  • Serial crisis • Big deals • Green and gold OA Diving in the (cold waters of a) new world • Coordinating scholarly publication management • Digital content management licencing, persitent identifiers, access statistics • Drafts, reports, publication, theses, and data …no management of physical content … nor of open access (it's in the genes) High-Low  low stewardship? • Research and learning materialA FAVOURABLE CONTEXTInria -a research organisation with a visionVision for a scientific information policy • Maximising the dissemination of our scientific assets (visibility and swift dissemination of knowledge), for a reasonable price • Constitution of a reliable and sovereign institutional corpus (documentation, preservation, access), with clear public governance principles • Contribution to shaping the scientific communication landscape in terms of editorial processes and usage made of scientific productions

--•

  Development limitations of pdf "Converting PDF to XML is a bit like converting hamburgers into cows. You may be best off printing it and then scanning the result through a decent OCR package." Michael Kay (http://lists.xml.org/archives/xml-dev/200607/msg00509.html) Inspired from: Duncan Hull Structuring content • GROBID: information extraction from PDF documents • Meta-data: title, authors, affiliations, abstracts • Bibliographical references (with crossref consolidation) • Standards-based representation • TEI (Text Encoding Initiative) as a reference format • State of the art performance(CRF models) • Cf. M. Lipinski, et al., 2013 • Used at EPO/ResearchGate/Mendeley/CERN/NASA • Integrated in HAL • Automatic meta-data extraction for author's deposit Sciencesconf-a conference management tool Services • Online conference management tool • Abstracts, full paper, peer review, conference program, registration Technologies • Modular services • Unique authentication system • Connection to HAL authorities • Upload to HAL Library support • Supervises the creation of a conference instance • Create conference series • Moderates the integration of the papers as a collection in HAL épisciences One of the major scholarly society in Information & Communications -Organized in technical committees and working groups • Objective: setting up a sustainable digital library -All volumes publishes as IFIP conferences -Visibility, trust, technical facilities• Publication agreement with Springer -Systematic publication in the AICT, LNBIP and LNCS series -3-year embargo before free online dissemination -Provision of meta-data (XML Springer) and author post-review manuscripts (PDF)• Note: XML Springer formats often contains full-text • For Inria: an experiment in ingesting and aggregating Standardized ingestion workflow • Towards a unified TEI-based format for scholarly papers -Cf. EU Peer project (massive ingestion of preprints) -Istex, OpenEdition, HAL, EPO -More flexible and extensible than JATS (e.g. full text)

•--

  Objectives Designing a scholarly dashboard • Scientific profiles: publications and authors -Experimenting various data extraction mecanisms • Reference content from the HAL publication repositories -Meta-data: title, periods, authors, affiliations -Full-text: full-text indexing, conceptual search • Technical background -NERD and Grobid full text • Librarians are shaping the service with developpers -Identifying measured quantities in scientific documents -Value, unit, measured entity -Multiple search scenarios in a scholarly context Example (example 2, streptomycin) Plugging additional services -2 • Re-publishing content -Grobid-TEI as a pivot format in the publishing environment -Generation of multiple derived format • HTML • ePub • Braille • How far are we from this? -Improving the performances of Grobid (e.g. book) -Partially implemented in various initiatives • Revues.org • Istex • There again, a central role for the library staff -Identifying user expectations -Maintaining formats and transforms Conclusion • Digital sovereignty -Mastering scholarly content at all stages of the publication process -Mastering the whole scholarly process • Understanding where and when we need to resort to the private sector -Mastering interoperability by means of open formats -A general culture of openness • Focusing our efforts -More brain, budget etc. in shaping this digital turn -Stop loosing time in "negotiating" open access: let's just do it • Dedicated research activity on scholarly information management -TDM breakthroughs at the service of scientific information -Dealing with the quickly evolving digital context -Coupling service units with adequate research teams • Training future digital librarians -Formats, standards and related technologies -Data mining and visualisation

Mapping

The mapping is a set of domain specific wrappers transforming the input/output from/to the internal data model. The internal data model is composed of:

HAL authors

Local attributes: surname, last name, title, email, etc Comparison using distances depending upon the type of the attribute: e. g. Person Name is computed using Dice Sorensen, Cosine similarity and Jaro Winkler.

Relation attributes: co-authorship, affiliations, popularity*, years of activity*, etc. Relations are computed using affiliation information Preliminary results are promising, F-1 score of 0.964