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We consider two theorems from the theory of compressive sensing. Mainly a theorem concerning uniform recovery of random sampling matrices, where the number of samples needed in order to recover an $s$-sparse signal from linear measurements…
This paper investigates total variation minimization in one spatial dimension for the recovery of gradient-sparse signals from undersampled Gaussian measurements. Recently established bounds for the required sampling rate state that uniform…
Motivated by recent results in the statistical physics of spin glasses, we study the recovery of a sparse vector $\mathbf{x}_0\in \mathbb{S}^{n-1}$, $\|\mathbf{x}_0\|_{\ell_0} = k<n$, from $m$ quadratic measurements of the form $…
One of the key issues in the acquisition of sparse data by means of compressed sensing (CS) is the design of the measurement matrix. Gaussian matrices have been proven to be information-theoretically optimal in terms of minimizing the…
This paper demonstrates that if the restricted isometry constant $\delta_{K+1}$ of the measurement matrix $A$ satisfies $$ \delta_{K+1} < \frac{1}{\sqrt{K}+1}, $$ then a greedy algorithm called Orthogonal Matching Pursuit (OMP) can recover…
Uniform bounds on sketched inner products of vectors or matrices underpin several important computational and statistical results in machine learning and randomized algorithms, including the Johnson-Lindenstrauss (J-L) lemma, the Restricted…
The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an…
The problem of 1-bit compressive sampling is addressed in this paper. We introduce an optimization model for reconstruction of sparse signals from 1-bit measurements. The model targets a solution that has the least l0-norm among all signals…
In this paper, {the goal is to design deterministic sampling patterns on the sphere and the rotation group} and, thereby, construct sensing matrices for sparse recovery of band-limited functions. It is first shown that random sensing…
The Restricted Isometry Property (RIP) is a fundamental property of a matrix enabling sparse recovery. Informally, an m x n matrix satisfies RIP of order k in the l_p norm if ||Ax||_p \approx ||x||_p for any vector x that is k-sparse, i.e.,…
Compressed Sensing decoding algorithms can efficiently recover an N dimensional real-valued vector x to within a factor of its best k-term approximation by taking m = 2klog(N/k) measurements y = Phi x. If the sparsity or approximate…
Since compressive sensing deals with a signal reconstruction using a reduced set of measurements, the existence of a unique solution is of crucial importance. The most important approach to this problem is based on the restricted isometry…
Compressed sensing has been a very successful high-dimensional signal acquisition and recovery technique that relies on linear operations. However, the actual measurements of signals have to be quantized before storing or processing.…
In deterministic compressive sensing, one constructs sampling matrices that recover sparse signals from highly incomplete measurements. However, the so-called square-root bottleneck limits the usefulness of such matrices, as they are only…
In this paper we generalize the 1-bit matrix completion problem to higher order tensors. We prove that when $r=O(1)$ a bounded rank-$r$, order-$d$ tensor $T$ in $\mathbb{R}^{N} \times \mathbb{R}^{N} \times \cdots \times \mathbb{R}^{N}$ can…
In the problem of one-bit compressed sensing, the goal is to find a $\delta$-close estimation of a $k$-sparse vector $x \in \mathbb{R}^n$ given the signs of the entries of $y = \Phi x$, where $\Phi$ is called the measurement matrix. For the…
What is the value of a single bit to a guesser? We study this problem in a setup where Alice wishes to guess an i.i.d. random vector, and can procure one bit of information from Bob, who observes this vector through a memoryless channel. We…
In the theory of compressed sensing, restricted isometry analysis has become a standard tool for studying how efficiently a measurement matrix acquires information about sparse and compressible signals. Many recovery algorithms are known to…
In [8] the author of this paper continued the research on the complex-valued discrete random variables $X_l(m,N)$ ($0\le l\le N-1$, $1\le M\le N)$ recently introduced and studied in [24]. Here we extend our results by considering $X_l(m,N)$…
The one-bit compressed sensing framework aims to reconstruct a sparse signal by only using the sign information of its linear measurements. To compensate for the loss of scale information, past studies in the area have proposed recovering…