Manifold learning for multi-dimensional auto-regressive dynamical models

Manifold learning [2, 5, 28, 32, 36, 12] has become a popular topic in machine learning and computer vision in the last few years, as many objects of interests (like natural images, or sequences representing walking persons), in spite of their apparent high dimensionality, live in a non-linear space of usually limited dimension. Many unsupervised algorithms (e.g. locally linear embedding [27]) take an input dataset and embed it into some other space, implicitly learning a metric. Extensions learning a full metric for the whole input space have been recently formulated [4]. In particular, videos or image sequences are often represented as realizations of some sort of dynamical model, either stochastic (e.g. HMMs) or deterministic (e.g. ARMA). Such an approach has proven to be effective in problems such as video coding (e.g. dynamic textures [11]), action recognition [7], or identity recognition from gait [31]. Several metrics or distance functions on linear systems have been introduced [6] in the context of system identification [21, 35, 30]. A vast literature also exists on dissimilarity measures between Markov models [10, 33], mainly concerning variants of the Kullback-Leibler divergence [18]. Consider, though, the problem of classifying a dynamical model (as the representative of an input image sequence). Since models (or sequences) can be endowed with different labeling, while maintaining the same geometrical structure, no single distance function can possibly outperform all the others in each and every classification problem. A reasonable approach, when possessing some a-priori information, consists therefore in trying to learn in a supervised fashion the “best” distance function for a specific classification problem [2, 5, 28, 32, 36, 12]. A natural optimization criterion seeks to maximize the classification performance achieved by means of the learnt metric. Efforts have been done in this sense in the linear case [29, 34].