Tuesday, December 17, 2013

Sample Complexity of Dictionary Learning and other Matrix Factorizations

Here is something that touches on the several advanced matrix factorization techniques that have sprung up in the past few years:

Dear Igor, 
We have just finalized a paper which considers the sample complexity of DL and other matrix factorizations. The topic might be of interest for the Nuit Blanche readers and we would be delighted if you could advertise our preprint, which is available here

Thanks a lot and best wishes,

Prof. Dr. Martin Kleinsteuber
Geometric Optimization & Machine Learning Group
Cluster CoTeSys www.cotesys.de
TU München  www.gol.ei.tum.de

Many modern tools in machine learning and signal processing, such as sparse dictionary learning, principal component analysis (PCA), non-negative matrix factorization (NMF), K-means clustering, etc., rely on the factorization of a matrix obtained by concatenating high-dimensional vectors from a training collection. While the idealized task would be to optimize the expected quality of the factors over the underlying distribution of training vectors, it is achieved in practice by minimizing an empirical average over the considered collection. The focus of this paper is to provide sample complexity estimates to uniformly control how much the empirical average deviates from the expected cost function. Standard arguments imply that the performance of the empirical predictor also exhibit such guarantees. The level of genericity of the approach encompasses several possible constraints on the factors (tensor product structure, shift-invariance, sparsity \ldots), thus providing a unified perspective on the sample complexity of several widely used matrix factorization schemes.

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