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1 Geometric Methods for Learning and Memory A thesis presented by Dimitri Nowicki To Universite Paul Sabatier in partial fulfillment for the degree of Doctor es Science in the subject of Applied Mathematics

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2 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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3 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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4 Models and Algorithms requiring Geometric Approach Kalman–like filters Blind Signal Separation Feed-Forward Neural Networks Independent Component Analysis

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5 Introduction Riemannian spaces Lie groups and homogeneous spaces Metric spaces without any Riemannian structure Spaces emerging in learning problems

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6 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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7 Outline Some facts from Riemannian geometry Optimization algorithms –Smooth –Nonsmooth Implementation –The case of Submanifolds –Computing exponential maps –Computing Hessian etc.

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8 Some concepts from Riemannian Geometry Geodesics

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9 Exponential map

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10 Parallel transport Computing parallel transport using an exponential map Where u such that

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11 Newton Method for Geometric optimization The modified Newton operator

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12 Wolfe condition for Riemannian manifolds

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13 Global convergence of modified Newton method

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14 Nonsmooth methods The subgradient:

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15 The r -algorithm. Here

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16 Problem of constrained optimization Equality constraints

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17 Classical (extrinsic) methods The Lagrangian Newton-Lagrange method Sequential quadratic programming

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18 Classical methods Penalty functions and the augmented Lagrangian

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19 Advantages of Geometric methods Dimension of the manifold is n-m against n+m in the case of Lagrangian-based methods We may have convex function in the manifold even if the Lagrangian is non- convex Geometric Hessian may be positive- definite even if the classical one is not

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20 Implementation: The case of Submanifolds

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21 Hamilton Equations for the Geodesics The Lagrangian: The Hamiltonian:

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22 Hamilton Equations for the Geodesics

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23 Lagrange equation are also constrained Hamiltonian We can rewrite Lagrange equations in the form:

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24 Symplectic Numerical Integration A transformation is called symplectic if it preserves following differential 2-form:

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25 Implicit Runge-Kutta Integrators The IRK method is called symplectic if associated transformation preserves y=(x,p)

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26 The Gauss method of order 4 i=1i=2 j=11/4 j=2 1/4 1/2

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27 Backward error analysis

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28 Covariant Derivative on the Submanifold

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29 Computing the constrained Hessian Direct computation Mixed computation where

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30 Example of geometric iterations

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31 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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32 Neural Associative memory Hopfield-type auto-associative memory. Memorized vectors are bipolar: vk {-1, 1} n, k=1…m. Suppose these vectors are columns of n m matrix V. Then synaptic matrix C of the memory is given by: Associative recall is performed using following procedure: the input vector x 0 is a starting point of the iterations: where f is a monotonic odd function such that

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33 Attraction radius We will call the stable fixed point of this discrete-time dynamical system an attractor. The maximum Hamming distance between x 0 and a memorized pattern v k such that the examination procedure still converges to v k is called an attraction radius.

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34 Problem statement

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35 Generalized averaging on the manifold argmin

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36 Computing generalized average on the Grassmann manifold Generalized averaging as an optimization problem Transforming objective function:

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37 Statistical estimation

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38 Statistical estimation

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39 Experimental results: the simulated data n=256– for all experiments Nature of the data

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40 Experimental results: simulated data

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41 Experimental results: simulated data Frequencies of attractors of associative clustering network for different m, p=8

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42 Experimental results: simulated data Frequencies of attractors of associative clustering network for different p, and m=p

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43 Experimental results: simulated data Distinction coefficients of attractors of associative clustering network for different p, and m=p

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44 The MNIST database: data description Gray-scale images 28 28 10 classes: digits from 0 to 9 Training sample: 60000 images Test sample:10000 images Before entering to the network images were tresholded to obtain 784- dimensional bipolar vectors

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45 Experimental results: the MNIST database Example of handwritten digits from MNIST database

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46 Experimental results: the MNIST database Generalized images of digits found by the network

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47 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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48 Kernel AM The main algorithm

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49 Kernel AM The Basic Algorithm (Continued)

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50 Algorithm Scheme

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51 Experimental Results Gaussian Kernel

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52 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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53 Model of Signal

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54 Signal Trajectories in the phase space

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55 The Manifold

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57 Example of Signal Processing

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58 Outline Introduction Geodesics, Newton Method and Geometric Optimization Generalized averaging over RM and Associative memories Kernel Machines and AM Quotient spaces for Signal Processing Application: Electronic Nose

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59 Application for Real-Life Problem Electronic Nose: QCM Setup overview Variance Distribution between principal Components

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60 Chemical images in space spanned by first 3 PCs

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