arXiv subjects

Simon Ruetz

Publications and source records attributed to Simon Ruetz.

4 recordsLinked to original sources

arXiv2023

Convergence of alternating minimisation algorithms for dictionary learning

In this paper we derive sufficient conditions for the convergence of two popular alternating minimisation algorithms for dictionary learning - the Method of Optimal Directions (MOD) and Online Dictionary Learning (ODL), which can also be thought of as approximative K-SVD. We show that given a well-behaved initialisation that is either within distance at most $1/\log(K)$ to the generating dictionary or has a special structure ensuring that each element of the initialisation only points to one generating element, both algorithms will converge with geometric convergence rate to the generating dictionary. This is done even for data models with non-uniform distributions on the supports of the sparse coefficients. These allow the appearance frequency of the dictionary elements to vary heavily and thus model real data more closely.

math.OC↗

arXiv2022

Bounds for matrices of inclusion probabilities in rejective sampling

Motivated by a desire to model n-sparse signals more realistically, we study matrices of inclusion probabilities for conditional Poisson or rejective sampling in the non-asymptotic regime. In particular, we provide bounds in the semi-definite ordering for matrices that collect first and second order inclusion probabilities as their diagonal and off-diagonal entries respectively, and operator norm bounds of their Hadamard products with other matrices. Finally, we demonstrate the usefulness on these bounds, which only involve first order inclusion probabilities, in a toy application from dictionary learning.

math.PR↗

arXiv2022

Adapted variable density subsampling for compressed sensing

Recent results in compressed sensing showed that the optimal subsampling strategy should take into account the sparsity pattern of the signal at hand. This oracle-like knowledge, even though desirable, nevertheless remains elusive in most practical application. We try to close this gap by showing how the sparsity patterns can instead be characterised via a probability distribution on the supports of the sparse signals allowing us to again derive optimal subsampling strategies. This probability distribution can be easily estimated from signals of the same signal class, achieving state of the art performance in numerical experiments. Our approach also extends to structured acquisition, where instead of isolated measurements, blocks of measurements are taken.

cs.IT↗

arXiv2020

Submatrices with non-uniformly selected random supports and insights into sparse approximation

In this paper we derive tail bounds on the norms of random submatrices with non-uniformly distributed supports. We apply these results to sparse approximation and conduct an analysis of the average case performance of thresholding, Orthogonal Matching Pursuit and Basis Pursuit. As an application of these results we characterise sensing dictionaries to improve average performance in the non-uniform case and test their performance numerically.

math.PR↗