Dictionary Identification - Sparse Matrix-Factorisation via $\ell_1$-Minimisation - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Journal Articles IEEE Transactions on Information Theory Year : 2010

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

Rémi Gribonval
Karin Schnass
• Function : Author
• PersonId : 883985

#### 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.

### Dates and versions

hal-00541297 , version 1 (30-11-2010)

### Identifiers

• HAL Id : hal-00541297 , version 1
• DOI :

### Cite

Rémi Gribonval, Karin Schnass. Dictionary Identification - Sparse Matrix-Factorisation via $\ell_1$-Minimisation. IEEE Transactions on Information Theory, 2010, 56 (7), pp.3523--3539. ⟨10.1109/TIT.2010.2048466⟩. ⟨hal-00541297⟩

### Export

BibTeX XML-TEI Dublin Core DC Terms EndNote DataCite

513 View