  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. The derived generalization bounds behave proportional to
$\sqrt{\log(n)/n}$ w.r.t.\ the number of samples $n$ for the considered matrix
factorization techniques.
