2012/10/02 by Rodolphe Jenatton, Rémi Gribonval, Jenatton, Rodolphe +3
Engineering · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Photoacoustic and Ultrasonic Imaging #Sparse and Compressive Sensing Techniques #Ultrasonics and Acoustic Wave Propagation
paper · pdf · doi:10.48550/arxiv.1210.0685
openalex publication_date 2012/10/02 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
A popular approach within the signal processing and machine learning communities consists in modelling signals as sparse linear combinations of atoms selected from a learned dictionary. While this paradigm has led to numerous empirical successes in various fields ranging from image to audio processing, there have only been a few theoretical arguments supporting these evidences. In particular, sparse coding, or sparse dictionary learning, relies on a non-convex procedure whose local minima have not been fully analyzed yet. In this paper, we consider a probabilistic model of sparse signals, and show that, with high probability, sparse coding admits a local minimum around the reference dictionary generating the signals. Our study takes into account the case of over-complete dictionaries and noisy signals, thus extending previous work limited to noiseless settings and/or under-complete dictionaries. The analysis we conduct is non-asymptotic and makes it possible to understand how the key quantities of the problem, such as the coherence or the level of noise, can scale with respect to the dimension of the signals, the number of atoms, the sparsity and the number of observations.