SOME RECENT PREPRINTS

Title (No specific order)

• Compressed sensing: perturbations of the measurement matrices and the dictionaries
  
• On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering
  

Compressed sensing: perturbations of the measurement matrices and the dictionaries

Akram Aldroubi, Xuemei Chen, and Alex Powell

The compressed sensing problem for redundant dictionaries aims to use a small number of linear measurements to represent signals that are sparse with respect to a general dictionary. Under an appropriate restricted isometry property for a dictionary, reconstruction methods based on $\ell^q$ minimization are known to provide an effective signal recovery tool in this setting. We show that $\ell^q$ minimization is jointly stable with respect to imprecise knowledge of the measurement matrix $A$ and the dictionary $D$ when $A$ satisfies the restricted isometry property. We also show that if $A$ satisfies an appropriate null space property (a weaker condition), then the $\ell^q$ minimization produces solutions that are also robust to noise and stable with respect to compressible signals and perturbations of $A$ and $D$.

On the Existence of Optimal Unions of Subspaces for Data Modeling and Clustering

Akram Aldroubi, and Romain Tessera

Given a set of vectors $\F=\{f_1,\dots,f_m\}$ in a Hilbert space $\HH$, and given a family $\CC$ of closed subspaces of $\HH$, the {\it subspace clustering problem} consists in finding a union of subspaces in $\CC$ that best approximates (nearest to) the data $\F$. This problem has applications and connections to many areas of mathematics, computer science and engineering such as Generalized Principle Component Analysis (GPCA), learning theory, compressed sensing, and sampling with finite rate of innovation. In this paper, we characterize families of subspaces $\CC$ for which such a best approximation exists. In finite dimensions the characterization is in terms of the convex hull of an augmented set $\CC^+$. In infinite dimensions however, the characterization is in terms of a new but related notion of contact half-spaces. As an application, the existence of best approximations from $\pi(G)$-invariant families $\CC$ of unitary representations of abelian groups is derived.