Over the past decade, compressed sensing has delivered significant advances in the theory and application of measuring and compressing data. Consider capturing a 10 mega pixel image with a digital camera. Emailing an image of this size requires an unnecessary amount of storage space and bandwidth. Instead, users employ a standard digital compression scheme, such as JPEG, to represent the image ...

This paper considers the problem of recovering an unknown sparse p× p matrix X from an m ×m matrix Y = AXBT , where A and B are known m × p matrices with m p. The main result shows that there exist constructions of the “sketching” matrices A and B so that even if X has O(p) non-zeros, it can be recovered exactly and efficiently using a convex program as long as these non-zeros are not concentra...

Recall the setup in compressive sensing. There is an unknown signal z ∈ R, and we can only glean information about z through linear measurements. We choose m linear measurements a1, . . . , am ∈ R. “Nature” then chooses a signal z, and we receive the results b1 = 〈a1, z〉, . . . , bm = 〈am, z〉 of our measurements, when applied to z. The goal is then to recover z from b. Last lecture culminated i...

In this paper we introduce a nonuniform sparsity model and analyze the performance of an optimized weighted `1 minimization over that sparsity model. In particular, we focus on a model where the entries of the unknown vector fall into two sets, with entries of each set having a specific probability of being nonzero. We propose a weighted `1 minimization recovery algorithm and analyze its perfor...

In this chapter, we introduce a unjfied rugh-dimensional geometric framework for analyzing the phase transition phenomenon of (1 minimization in compressive sensing. This framework connects srudying the phase transitions of ( 1 minimization with computing the Grassmann angles in high-dimensional convex geometry. We demonstrate the broad applications of this Grassmann angle framework by giving s...

In [1], we proved the asymptotic achievability of the Cramér-Rao bound in the compressive sensing setting in the linear sparsity regime. In the proof, we used an erroneous closed-form expression of ασ for the genie-aided Cramér-Rao bound σTr(AIAI) −1 from Lemma 3.5, which appears in Eqs. (20) and (29). The proof, however, holds if one avoids replacing σTr(AIAI) −1 by the expression of Lemma 3.5...

Corresponding author: Vasilis Ntziachristos Ingoldstädter Landstraße 1, 85764 ,Neuherberg, Germany; E-mail: [email protected]

In traditional compressed sensing MRI methods, single sparsifying transform limits the reconstruction quality because it cannot sparsely represent all types of image features. Based on the principle of basis pursuit, a method that combines sparsifying transforms to improve the sparsity of images is proposed. Simulation results demonstrate that the proposed method can well recover different type...

F. Krzakala , M. Mézard , F. Sausset , Y. F. Sun and L. Zdeborová 4 1 CNRS and ESPCI ParisTech, 10 rue Vauquelin, UMR 7083 Gulliver, Paris 75005, France. 2 Univ. Paris-Sud & CNRS, LPTMS, UMR8626, Bât. 100, 91405 Orsay, France. 3 LMIB and School of Mathematics and Systems Science, Beihang University, 100191 Beijing, China. 4 Institut de Physique Théorique, IPhT, CEA Saclay, and URA 2306, CNRS, 9...

Is it possible to obliviously construct a set of hyperplanes H such that you can approximate a unit vector x when you are given the side on which the vector lies with respect to every h ∈ H? In the sparse recovery literature, where x is approximately k-sparse, this problem is called onebit compressed sensing and has received a fair amount of attention the last decade. In this paper we obtain th...