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Convex low-rank models: from matrices to tensors

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In this talk I will present low-rank models in two domains and how they can be set-up as convex optimization problems. The fist domain is the so-called brain-computer interface. The problem of learning a set of discriminative spatio-temporal filters for the P300 speller system is formulated as a low-rank matrix learning problem. Using the trace norm regularization combined with an appropriately defined likelihood function I obtained both state-of-the art classification performance and highly interpretable spatio-temporal filters. The second domain is audio-signal separation. I propose positive semidefinite tensor factorization (PSDTF), which is a generalization of nonnegative matrix factorization (NMF), that decomposes a collection of PSD matrices into linear combinations of a small number of basis PSD matrices. I propose a non-parametric Bayesian model for PSDTF that automatically infers the number of basis PSD matrices. Finally, I present an ongoing work on convex relaxation of tensor (multilinear)-rank. I propose generalizations of trace norm for tensors and analyze their statistical performance. The wide gap between the performance that can be obtained by tractable algorithms and that can only be obtained by intractable algorithms points to an optimization-statistics tradeoff that opens up many future direction. Includes joint work with Klaus-Robert Müller, Kazuyoshi Yoshii, Daichi Mochihashi, Masataka Goto, Taiji Suzuki, Kohei Hayashi, and Hisashi Kashima.

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