Robust analysis ℓ1\ell_1-recovery from Gaussian measurements and total variation minimization

Analysis $\ell_1$-recovery refers to a technique of recovering a signal that is sparse in some transform domain from incomplete corrupted measurements. This includes total variation minimization as an important special case when the transform domain is generated by a difference operator. In the present paper we provide a bound on the number of Gaussian measurements required for successful recovery for total variation and for the case that the analysis operator is a frame. The bounds are particularly suitable when the sparsity of the analysis representation of the signal is not very small

Tempered Radon Measures

A tempered Radon measure is a σ-finite Radon measure in Rn which generates a tempered distribution. We prove the following assertions. A Radon measure μ is tempered if, and only if, there is a real number βsuch that ……. finite. A Radon measure is finite if, and only if, it belongs to the positive cone…….. (equivalent norms).A tempered Radon measure is a σ-finite Radon measure in Rn which generates a tempered distribution. We prove the following assertions. A Radon measure μ is tempered if, and only if, there is a real number βsuch that ……. finite. A Radon measure is finite if, and only if, it belongs to the positive cone…….. (equivalent norms)

Stable low-rank matrix recovery via null space properties

The problem of recovering a matrix of low rank from an incomplete and possibly noisy set of linear measurements arises in a number of areas. In order to derive rigorous recovery results, the measurement map is usually modeled probabilistically. We derive sufficient conditions on the minimal amount of measurements ensuring recovery via convex optimization. We establish our results via certain properties of the null space of the measurement map. In the setting where the measurements are realized as Frobenius inner products with independent standard Gaussian random matrices we show that $10 r (n_1 + n_2)$ measurements are enough to uniformly and stably recover an $n_1 \times n_2$ matrix of rank at most $r$. We then significantly generalize this result by only requiring independent mean-zero, variance one entries with four finite moments at the cost of replacing $10$ by some universal constant. We also study the case of recovering Hermitian rank-$r$ matrices from measurement matrices proportional to rank-one projectors. For $m \geq C r n$ rank-one projective measurements onto independent standard Gaussian vectors, we show that nuclear norm minimization uniformly and stably reconstructs Hermitian rank-$r$ matrices with high probability. Next, we partially de-randomize this by establishing an analogous statement for projectors onto independent elements of a complex projective 4-designs at the cost of a slightly higher sampling rate $m \geq C rn \log n$. Moreover, if the Hermitian matrix to be recovered is known to be positive semidefinite, then we show that the nuclear norm minimization approach may be replaced by minimizing the $\ell_2$-norm of the residual subject to the positive semidefinite constraint. Then no estimate of the noise level is required a priori. We discuss applications in quantum physics and the phase retrieval problem.Comment: 26 page

Besov spaces on fractals and tempered Radon measures

We study Besov spaces on d-sets and provide their characterization by means of Hölder-continuous atoms, wavelets and counterparts of Faber-Schauder functions. We follow the connection between isotropic Besov spaces on d-sets, which are obtained as a cartesian product of bizarre fractal curves, and anisotropic Besov spaces on the unit cube. We also clarify the relation between Radon measure, tempered distributions and weighted Besov spaces

Analysis ℓ1-recovery with frames and Gaussian measurements

This paper provides novel results for the recovery of signals from undersampled measurements based on analysis ℓ1-minimization, when the analysis operator is given by a frame. We both provide so-called uniform and nonuniform recovery guarantees for cosparse (analysissparse) signals using Gaussian random measurement matrices. The nonuniform result relies on a recovery condition via tangent cones and the uniform recovery guarantee is based on an analysis version of the null space property. Examining these conditions for Gaussian random matrices leads to precise bounds on the number of measurements required for successful recovery. In the special case of standard sparsity, our result improves a bound due to Rudelson and Vershynin concerning the exact reconstruction of sparse signals from Gaussian measurements with respect to the constant and extends it to stability under passing to approximately sparse signals and to robustness under noise on the measurements