Stable recovery of the factors from a deep matrix product

François Malgouyres (1) , Joseph Landsberg (2)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 Department of Mathematics [Texas A&M University]

We study a deep matrix factorization problem. It takes as input the matrix $X$ obtained by multiplying $K$ matrices (called factors) and aims at recovering the factors. When $K=1$, this is the usual compressed sensing framework; $K=2$: Examples of applications are dictionary learning, blind deconvolution, self-calibration; $K\geq 3$: can be applied to many fast transforms (such as the FFT). In particular, we apply the theorems to deep convolutional network. Using a Lifting, we provide : a necessary and sufficient conditions for the identifiability of the factors (up to a scale indeterminacy); - an analogue of the Null-Space-Property, called the Deep-Null-Space-Property which is necessary and sufficient to guarantee the stable recovery of the factors.

Mots clés: matrix factorization; convolutional network; stable recovery

Domaines: Optimisation et contrôle [math.OC]

François Malgouyres, Joseph Landsberg. Stable recovery of the factors from a deep matrix product. Signal Processing with Adaptive Sparse Structured Representations (SPARS) , 2017, Lisbonne, Portugal. ⟨hal-01417943v2⟩