The Generalization Error of Dictionary Learning with Moreau Envelopes

This is a theoretical study on the sample complexity of dictionary learning with a general type of reconstruction loss. The goal is to estimate a $m \times d$ matrix $D$ of unit-norm columns when the only available information is a set of training samples. Points $x$ in $\mathbb{R}^m$ are subsequently approximated by the linear combination $Da$ after solving the problem $\min_{a \in \mathbb{R}^d}~ \Phi(x - Da) + g(a)$; function $g:\mathbb{R}^d \to [0,+\infty)$ is either an indicator function or a sparsity promoting regularizer. Here is considered the case where $ \Phi(x) = \inf_{z \in \mathbb{R}^m}~{ ||x-z||_2^2 + h(||z||_2)}$ and $h$ is an even and univariate function on the real line. Connections are drawn between $\Phi$ and the Moreau envelope of $h$. A new sample complexity result concerning the $k$-sparse dictionary problem removes the spurious condition on the coherence of $D$ appearing in previous works. Finally, comments are made on the approximation error of certain families of losses. The derived generalization bounds are of order $\mathcal{O}(\sqrt{\log n /n})$ and valid without any further restrictions on the set of dictionaries with unit-norm columns.