Questions tagged [compressed-sensing]

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Basis pursuit algorithms for exponentially large matrices?

Are there any efficient algorithms/heuristics for basis pursuit for exponentially large matrices? That is $\arg\min_x\lVert x \rVert_0$ subject to $y = Ax$, where $A$ is an exponentially large matrix ...
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0answers
76 views

On sparse $0/1$ linear equations solvable with compressed sensing

If you have a system of $m$ linearly independent equations in $n$ variables with domain $0/1$ and we know there is at least one solution with at most $d$ variables to be $1$ then if $m$ at least a ...
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1answer
71 views

Additional structures for sparse recovery

The problem of sparse recovery using $l_1$ minimization is well known. Using random Gaussian matrices, we are able to achieve recovery with high probability in $O(k\log(d/k)$ measurements. It is ...
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1answer
208 views

Uniqueness of l1 minimization

Let $A \in \mathbb{R}^{m \times n}$. Is it true that $$\min \limits_{Q \in \mathbb{R}^{n \times m}}|I - QA|_{\infty} < \frac{1}{2}$$ is criteria for the uniqueness of the 1-sparse solution to $\...
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80 views

Does positivstellensatz and SOS proof system help here?

I have a system of $m$ homogeneous degree $2$ polynomial equations in $\mathbb Z[x_1,\dots,x_n]$ where $m=poly(n)$. Take $$f_1(x_1,\dots,x_n)=0$$ $$\dots$$ $$f_m(x_1,\dots,x_n)=0$$ to be the system. ...
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1answer
104 views

Two theorems about incoherence

These are two theorems I have heard being referred to in "folklore" but I cant find the proofs for these in any compressed sensing or high-dimensional probability reviews (like, https://www.math.uci....
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126 views

Restricted isometry property for Subgaussian matrices with non-diagonal covariance?

I was wondering if there were any results on whether the restricted isometry property held for Subgaussian or Gaussian random matrices that had covariance matrices that were not completely diagonal?
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0answers
73 views

Statistical properties from the projection on Hadamard matrix

I am using Hadamard matrix to generate the measurement matrix in compressive sensing for signal acquisition. Consider all the rows except the first row, which consists of all 1's, as a candidate for ...
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37 views

Question about a systematic row reduction algorithm for compressive sensing

Suppose a ''brute-force'' algorithm is designed to systematically select from the first $n$ columns of an $m \times (n+1),$ $m<n$ augmented matrix $G$ representing a consistent underdetermined ...
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1answer
59 views

Can there be underdetermined linear systems whose set of minimal support solutions is infinite?

Let $A \in \mathbb{F}^{m \times n},$ $m<n,$ and let $b \in \mathbb{F}^{m \times n}$ be such that the system $Ax=b$ is consistent. Does it follow that the set $X$ of minimal support solutions of ...
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0answers
67 views

Proving an Algorithm that generates minimal $\|x\|_0$ for the underdetermined system $Ax=b$

Let $A \in \mathbb {F}^{m \times n}$ with $m< n,$ $b \in \mathbb{F}^m$ and let $x$ be unknown in $\mathbb{F}^n.$ Assume $0<p<1.$ Then $$\arg \min\limits_{x: Ax=b} \|x\|_0 = \lim\limits_{p \to ...
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577 views

L0 norm compressed sensing vs L1 norm compressed sensing

Suppose we have an very efficient way to perform L0 norm compressed vs L1 norm compressed sensing. Specifically: L0 norm compressed sensing is: $$\eqalign{ & \min \quad {x^T}Qx + {b^T}x + \mu {\...
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1answer
74 views

NP-Hardness of finding minimal-support solutions of underdetermined systems over any field

Given a field $\mathbb{F}$ and a consistent underdetermined system $Ax=b$ over $\mathbb{F},$ $A\in \mathbb{F}^{m \times N}$ and $b \in \mathbb{F}^m,$ finding a vector $z \in \mathbb{F}^N$ such that $...
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108 views

NP-Hardness of Underdetermined Systems over $F_2$ implies the same for $F_{p^q}$

Assume that the yes-no problem of whether $x' \in F_{p^q}^{n}$ is a minimal support solution for a consistent underdetermined system $Ax=b,$ $A \in F_{p^q}^{m \times n}, b \in F_{p^q}^n$ is an NP-hard ...
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338 views

psd condition for matrix completion

The nuclear norm minimization for the matrix completion problem is given by \begin{align} \textrm{minimize } \quad &\|X\|_{*}\\ \textrm{subject to } \quad & X_{ij}=M_{ij} \quad \forall (i,j)...
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140 views

choosing regularization constant in compressive sensing

Given a compressive sensing formulation, $$\left\| {Ax - b} \right\|_2^2 + \mu {\left\| x \right\|_1}$$ And given curves (a) $\left\| {Ax - b} \right\|_2^2$ plotted against $\log \left( \mu \...
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2answers
450 views

Concentration of matrix norms under random projection.

Let X be a given matrix of dimension $p \times q$. Let $G$ be a $s \times p$ dimensional matrix of standard normal/Gaussian random variables. Are there cases where one can been able to quantify $P_G ...
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1answer
438 views

Restricted Isometry Property for Discrete Fourier Transform Matrix

I was wondering if the Restricted Isometry Property holds for Discrete Fourier Transform. In particular, I am interested in whether a subsampled DFT matrix has such property. Let$W \in \mathbb{C}^{d\...
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0answers
388 views

Can I use proximal algorithms on complex real-valued functions?

There is a plethora of literature in proximal operators and proximal optimization algorithms specially for Compressive sensing. A proximal operator is defined as \begin{equation} \operatorname{prox}_f(...
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1answer
79 views

Vanishing Restricted Isometric Constant

In compressed sensing, we are interested in the restricted isometry property. Suppose the design matrix is $n$ by $p$, consisting of $np$ iid $\mathcal{N}(0, 1/n)$ entries. Assume both $n$ and $p$ are ...
2
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1answer
210 views

Phase of the inner product between the elements of an ETF

I am doing research in compressive sampling for Cognitive Radio applications. While working on a project I came across with the following question: Is there any research about the phase of inner ...
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1answer
406 views

Finding sparsest solution of a linear system

I want to find the solution with most zero-components for the following problem: $Ax=b$ for $A\in \mathbb{R}^{k\times n}, b \in \mathbb{R}^{k},k<n$, where $x$ is real and has no additional ...
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1answer
1k views

Open problems in compressed sensing

What are the main open problems in compressed sensing? I am interested in theoretical as well as in numerical point of view.
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1answer
198 views

Can one always find sparse solutions to an $\ell^1$-minimization problem?

Consider $A\in\mathbb{R}^{m \times N}$ and $b \in \mathbb{R}^m$, with $m<N$. Is it true that the optimization problem $$\min \|x\|_1 \quad s.t. \;\; A x = b,$$ admits an $m$-sparse solution in ...
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123 views

RIP of the sensing matrix sampled from biased Bernoulli distribution

I have recently worked toward applying compressed sensing technique to the case of which sensing matrix is sampled from biased Bernoulli distribution. So far I have found literature demonstrating ...
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0answers
191 views

Question in Wainwright's paper about signed support recovery in lasso

Sharp thresholds for high dimensional and noisy sparsity recovery using $l_1$ constrained quadratic programming (Lasso) This paper is about support recovery guarantees of the Lasso. I have an issue ...
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1answer
577 views

Can sparse matrices satisfy the Null Space Property?

Definition A matrix $A \in \mathbb{C}^{m \times N}$ with $m < N$ satisfies the Null Space Property (NSP) of order $s$ if $$\|x_S\|_1 < \|x_{\bar{S}}\|_1, \quad \forall x \in \ker A \setminus \{...
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1answer
452 views

Restricted Isometry Property (Non Sparse Gaussian)

Let $x$ be a $N \times 1$ vector in $\mathbb{R}^{N}$ where $M$ components are zero and the remaining $N-M$ components are standard normal random variables. $x$ may not be sparse e.g. $M$ may be ...
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2answers
501 views

Approximating a subspace by sampling a base without replacement

Let $X$ be a $p \times n$ matrix, with $p > n$. Now, suppose I sample $m < n$ columns from $X$ at random, without replacement. I would like to characterize the distance between the subspace ...
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165 views

Do the Eigenvectors find by use PCA on a set of data point, a good replacement for Random Projection when I later on use L1Magic to reconstruct the sparse vector?

Concretely if I use the first k eigenvectors find by PCA with a point set A,to project another sparse vector b to k dimension subspace, then use L1-magic to recover b. Will this be better than a ...
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1answer
201 views

Is there any alternative characterization of sparsity of a signal in compressed sensing

The starting assumption for compressed sensing (CS) is that the underlying signal is sparse in some basis, e.g., there are a maximum of $s$ non-zero Fourier-coefficients for an $s$-sparse signal. And ...
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1answer
486 views

$\ell_o$ Minimization (Minimizing the support of a vector)

I have been looking into the problem $\min: \|x \|_0$ subject to$: Ax=b$. $\|x \|_0$ is not a linear function and can't be solved as a linear (or integer) program in its current form. Most of my time ...
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0answers
259 views

Multiobjective semidefinite programming

Let $C$ be size $n \times n^{2}$. Let $B$ be size $2^{g(n)} \times n^{2}$ where $g(n) > n$. There is only one $\mathcal{1}$ per row of $C$ and remaining entries of $C$ are $\mathcal{0}$. $B$ is ...
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1answer
184 views

Recovering a matrix instead of a vector

It is known that given corrupt measurements $y = Af+e \in \mathbb{R}^m$ with $f \in \mathbb{R}^n$ and $\|f\|_0 < m < n$, one can recover $f$ exactly by solving a convex optimization problem. ...
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3answers
380 views

On a randomized version of compressive sensing

The compressive sensing theory of Candes and Tao (See http://en.wikipedia.org/wiki/Compressed_sensing) relies highly on the fact that the underlying data (such as a signal or an image) is sparse or ...
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3answers
945 views

Decomposing a discrete signal into a sum of rectangle functions

Hello mathoverflow community ! I have a simple question that seems to have a non trivial answer. Given a discrete one dimensional signal $w(x)$ defined in a finite range, and the boxcar (rectangular)...
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764 views

Compressed Sensing with an Unusual Basis

I'm wondering if compressed sensing can be applied to a problem I have in the way I describe, and also whether it should be applied to this problem (or whether it's simply the wrong tool). I have a ...