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Learning Riemannian metrics for motion classification Fabio Cuzzolin INRIA Rhone-Alpes Computational Imaging Group, Pompeu Fabra University, Barcellona 25/1/2007

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Myself gesture recognition Masters thesis on gesture recognition at the University of Padova Visiting student, ESSRL, Washington University in St. Louis theory of belief functions Ph.D. thesis on the theory of belief functions Young researcher in Milan with the Image and Sound Processing group Post-doc at UCLA in the Vision Lab Marie Curie fellowship, INRIA Rhone-Alpes

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My research research Discrete mathematics linear independence on lattices Belief functions and imprecise probabilities geometric approach algebraic analysis combinatorial analysis Computer vision object and body tracking data association gesture and action recognition identity recognition

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Todays talk Motion classification Motion classification is one of most popular vision problems Applications: surveillance, biometric, human- computer interaction Issue: choice of distance function Learning Riemannian metrics for motion classification

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Riemannian metrics for classification Distances between dynamical models Learning a metric from a training set Pullback metrics Spaces of linear systems and Fisher metric Experiments on scalar models

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Distances between dynamical models Problem: motion classification linear dynamical model Approach: representing each movement as a linear dynamical model for instance, each image sequence can be mapped to an ARMA, or AR linear model distance function in the space of dynamical models Classification is then reduced to find a suitable distance function in the space of dynamical models We can then use this distance in any distance-based classification scheme: k-NN, SVM, etc.

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A review of the literature Some distances have been proposed Fisher information matrix a family of probability distributions depending on a n- dimensional parameter can be regarded in fact as an n- dimensional manifold, with Fisher information matrix [Amari] Kullback-Leibler divergence Kullback-Leibler divergence Gap metric Gap metric [Zames,El-Sakkary]: compares graphs associated with linear systems thought of as input-output maps Cepstrum norm Cepstrum norm [Martin] Subspace angles Subspace angles between column spaces of the observability matrices

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Riemannian metrics for classification Distances between dynamical models Learning a metric from a training set Pullback metrics Spaces of linear systems and Fisher metric Experiments on scalar models

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Learning metrics from a training set All those metrics are task-specific Besides, it makes no sense to choose a single distance for all possible classification problems as… Labels can be assigned arbitrarily to dynamical systems, no matter what the underlying structure is When some a-priori info is available (training set).... we can learn in a supervised fashion the best metric for the classification problem!.. we can learn in a supervised fashion the best metric for the classification problem! volume minimization of A feasible approach: volume minimization of pullback metrics pullback metrics

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Learning distances Of course many unsupervised algorithms take an input dataset and embed it in some other space, implicitly learning a metric (LLE, Laplacian Eigenmaps, etc.) they fail to learn a full metric for the whole input space, but only images of a set of samples optimal Mahalanobis distance [Xing, Jordan]: maximizes classification performance for linear maps y=A 1/2 x > optimal Mahalanobis distance reduces to convex optimization relevant component analysis [Shental et al]: relevant component analysis – changes the feature space by a global linear transformation which assigns large weights to relevant dimensions" and low weights to irrelevant dimensions

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Riemannian metrics for classification Distances between dynamical models Learning a metric from a training set Pullback metrics Spaces of linear systems and Fisher metric Experiments on scalar models

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Learning pullback metrics Some notions of differential geometry give us a tool to build a parameterized family of metrics pullback metrics The diffeomorphism F induces on M a family of pullback metrics geodesics The geodesics of the pullback metric are the liftings of the geodesics associated with the original metric Consider than a family of diffeomorphisms F between the original space M and a metric space N M F N D

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Pullback metrics - detail Diffeomorphism Diffeomorphism on M: Push-forward Push-forward map: Given a metric on M, g:TM TM, the pullback metric pullback metric is

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Inverse volume Inverse volume: Inverse volume maximization The natural criterion would be to optimize the classification performance In a nonlinear setup this is hard to formulate and solve Reasonable to choose a different but related objective function Effect: finding the manifold which better interpolates the data (i.e. forcing the geodesics to pass through crowded regions)

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Riemannian metrics for classification Distances between dynamical models Learning a metric from a training set Pullback metrics Spaces of linear systems and Fisher metric Experiments on scalar models

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Space of AR(2) models Given an input sequence, we can identify the parameters of the linear model which better describes it We chose the class of autoregressive models of order 2 AR(2) Fisher metric on AR(2) to get a distance: compute the geodesics of the pullback metric on M

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Under stability (|a|<1) and minimality (b 0) this family forms a manifold Space of M(1,1,1) models Consider instead the class of stable discrete-time linear systems of order 1 After choosing a canonical setting c = 1 the transfer function becomes h(z) = b/(z a) Fisher tensor:

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Families of diffeomorphisms We chose two different families of diffeomorphisms For AR(2) systems: For M(1,1,1) systems:

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Riemannian metrics for classification Distances between dynamical models Learning a metric from a training set Pullback metrics Spaces of linear systems and Fisher metric Experiments on scalar models

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6 cameras Mobo database: 25 people performing 4 different walking actions, from 6 cameras action, id, view Each sequence has three labels: action, id, view MOBO database

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Classification of scalar models recognition of actions and identities from image sequences scalar feature, AR(2) and M(1,1,1) models compared performance of all known distances, with pullback Fisher metric built the geodesic distance used NN algorithm to classify new sequences

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Results - action Action recognition performance, all views considered – second best distance function Action recognition performance, all views considered – pullback Fisher metric Action recognition, view 5 only – difference between classification rates pullback metric – second best

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Results – action 2 Recognition performance of the second-best distance (blue) and the optimal pull-back metric (red), increasing size of training set View 1View 5 View 3View 6

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Effect of the training set The size of the training set obviously affects the recognition rate Systems of the class M(1,1,1) Increasing size of the training set on the abscissae All views considered View 2 only

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Conclusions Movements can be represented as dynamical systems Motion classification then reduces to finding a distance between dynamical model having a training set of such models we can learn the best metric for a given classification problem… … and use it to classify new sequences Pullback metrics induced by the Fisher metric structure on linear models is a possible choice Design of a family of diffeomorphisms Future: multidimensional observations, better objective function

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