Dictionary Identification - Sparse Matrix-Factorisation via $\ell_1$-Minimisation

TL;DR

This paper investigates conditions under which a dictionary and sparse coefficients can be uniquely recovered via $ ext{l}_1$-minimisation, showing that incoherent bases can be identified with a near-linear number of samples relative to the signal dimension.

Contribution

It provides algebraic conditions for local identifiability of dictionaries and demonstrates that incoherent bases are recoverable with high probability using a nearly linear number of samples.

Findings

Algebraic conditions for local identifiability are derived.

Incoherent bases are identifiable with high probability.

Sample complexity grows logarithmically with signal dimension.

Abstract

This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via $\ell_1$-minimisation. The problem can also be seen as factorising a $\ddim \times \nsig$ matrix $Y=(y_1 >... y_\nsig), y_n\in \R^\ddim$ of training signals into a $\ddim \times \natoms$ dictionary matrix $\dico$ and a $\natoms \times \nsig$ coefficient matrix $\X=(x_1... x_\nsig), x_n \in \R^\natoms$, which is sparse. The exact question studied here is when a dictionary coefficient pair $(\dico,\X)$ can be recovered as local minimum of a (nonconvex) $\ell_1$-criterion with input $Y=\dico \X$. First, for general dictionaries and coefficient matrices, algebraic conditions ensuring local identifiability are derived, which are then specialised to the case when the dictionary is a basis. Finally, assuming a random Bernoulli-Gaussian sparse model on the coefficient matrix, it is shown that sufficiently incoherent bases are locally identifiable with high probability. The perhaps surprising result is that the typically sufficient number of training samples $\nsig$ grows up to a logarithmic factor only linearly with the signal dimension, i.e. $\nsig \approx C \natoms \log \natoms$, in contrast to previous approaches requiring combinatorially many samples.