Questions tagged [compressed-sensing]

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What does $\mathbb E_V \max_{x \in V,\,\|x\|=1} x^T Ax$ evaluate to when $V$ is random $k$-dim suspace of $\mathbb R^n$ and $A$ is fixed psd matrix?

Let $G_{k,n}$ be the grassmannian of $k$-dimensional vector spaces of $\mathbb R^n$. By the Courant–Fisher characterization, the $k$th largest eigenvalue of an $n \times n$ psd matrix $A$ is given by $...
dohmatob's user avatar
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5 votes
1 answer
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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 ...
dhasson's user avatar
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4 votes
1 answer
624 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 \{...
Paglia's user avatar
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4 votes
2 answers
268 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 $$\min \|x\|_1 \quad \text{s.t.} \quad A x = b,$$ admits an $m$-sparse solution ...
Paglia's user avatar
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4 votes
0 answers
140 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 ...
ElectricCat's user avatar
4 votes
0 answers
772 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 ...
DoubleJay's user avatar
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3 votes
1 answer
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.
Felix Goldberg's user avatar
3 votes
1 answer
85 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 ...
John Wong's user avatar
  • 753
3 votes
1 answer
269 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 $\...
love_backups's user avatar
3 votes
1 answer
299 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 ...
elexhobby's user avatar
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3 votes
0 answers
202 views

Compressed sensing for partitioning instead of recovery

Let $x_0 \in \mathbb{R}^{m}$ be a signal whose support $T_0 = \{ t \mid x_{0}(t) \neq 0\}$ is assumed to be of small cardinality. The recovery of $x_0$ from a small number of $n \ll m$ linear ...
J1996's user avatar
  • 131
2 votes
1 answer
74 views

Column subset selection with least angle optimization

Given a matrix $A$ I need to select a "representative" subset of its columns so that the each non-selected column is as close as possible to a selected one. Formally: Given $A \in \mathbb{R}...
Felix Goldberg's user avatar
2 votes
1 answer
105 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 $...
Thomas Rasberry's user avatar
2 votes
3 answers
401 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 ...
Daniel's user avatar
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2 votes
1 answer
247 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 ...
alira's user avatar
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2 votes
3 answers
1k 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)...
rodrigob's user avatar
  • 129
2 votes
0 answers
43 views

Combining SVD subspaces for low dimensional representations

Suppose we have matrix $A$ of size $N_t \times N_m$, containing $N_m$ measurements corrupted by some (e.g. Gaussian) noise. An SVD of this data $A = U_AS_A{V_A}^T$ can reveal the singular vectors $U_A$...
user2600239's user avatar
2 votes
0 answers
98 views

Sparse signal recovery (nonlinear case)

Let $K \subset \mathbb{R}^n$, it may be that $K$ is "very thin" (e.g. $K$ is a $k$-dimensional affine subset of $\mathbb{R}^n$, with $k \ll n$). I'm interested in the case where $K$ is ...
Sébastien Loisel's user avatar
2 votes
1 answer
225 views

Basis pursuit algorithms for exponentially large matrices?

Are there any efficient algorithms/heuristics for basis pursuit for exponentially large matrices? That is $$\begin{array}{ll} \underset{x \in \Bbb R^n}{\text{minimize}} & \lVert x \rVert_0\\ \text{...
user1576720's user avatar
2 votes
0 answers
80 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 ...
Amit's user avatar
  • 21
2 votes
0 answers
592 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(...
mermeladeK's user avatar
1 vote
2 answers
815 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 ...
Student's user avatar
  • 555
1 vote
1 answer
134 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....
gradstudent's user avatar
  • 2,136
1 vote
1 answer
734 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\...
Steve's user avatar
  • 1,117
1 vote
1 answer
205 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. ...
Richard's user avatar
  • 11
1 vote
1 answer
87 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 ...
Magi's user avatar
  • 281
1 vote
1 answer
515 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 ...
Mykie's user avatar
  • 189
1 vote
1 answer
218 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 ...
Arnab's user avatar
  • 615
1 vote
0 answers
55 views

Compressed sensing / compressive sensing: what is the lower bound on dimension of the measurement matrix?

I am looking for references discussing the formal requirements for the dimension $m$ of the "measurement" mixing matrix in compressive sensing (as in ${\bf y} = M {\bf x}$ where ${\bf y} \in ...
Zebra Fish's user avatar
1 vote
1 answer
134 views

Minimax estimation rate of sparse vector $w_\star$, w.r.t to mixed norm $\|\hat w_n-w_\star\| := \|\hat w_n - w_\star\|_2 + \|\hat w_n-w_\star\|_q$

Let $n,d,s$ be positive integers with $s \le d$, and let $B_0(d,s)$ be the set of all (real) $d$-dimensional vectors with at most $s$ nonzero components. Given an $n \times d$ matrix $X$ with rows $...
dohmatob's user avatar
  • 6,716
1 vote
0 answers
54 views

Good lower-bound for $\inf_{x \in \Delta_n} \|Gx\|$ where $G$ is an $N \times n$ random matrix with iid entries from $\mathcal N(0,1/\sqrt{N})$

Let $G$ be an $N \times n$ random matrix with independent entries distributed according to a centered Gaussian with variance $1/\sqrt{N}$ and let $n/N = \lambda \in (0, 1)$. Let $\Delta_n$ be the $(n-...
dohmatob's user avatar
  • 6,716
1 vote
0 answers
81 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 ...
VS.'s user avatar
  • 1,816
1 vote
0 answers
142 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?
bringingdownthegauss's user avatar
1 vote
0 answers
71 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 ...
Thomas Rasberry's user avatar
1 vote
0 answers
815 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 {\...
user40780's user avatar
  • 867
1 vote
0 answers
149 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 ...
Thomas Rasberry's user avatar
1 vote
0 answers
472 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)...
felasfaw's user avatar
  • 221
1 vote
0 answers
265 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 ...
user16007's user avatar
  • 780
0 votes
1 answer
68 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 ...
Thomas Rasberry's user avatar
0 votes
1 answer
73 views

Probability of accurate sparse recovery

Suppose $\mathbf{A}_{k\times n}$ ($k<n$) is a matrix whose entries are generated i.i.d. from Gaussian distribution and $\mathbf{s}_{n\times 1}$ is a sparse vector with $m$ sparsity (i.e., $\|\...
Math_Y's user avatar
  • 311
0 votes
1 answer
495 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 ...
Clark's user avatar
  • 11
0 votes
0 answers
65 views

Matrix Dantzig selector with many constraints

We observe data from the model $$ y_1=\mathcal{A}_1(X)+z_1\\ ...\\ y_k=\mathcal{A}_k(X)+z_k\\ $$ where $X$ is an unknown $n\times n$ matrix with rank at most $r$, each $\mathcal{A}_i:\mathbb{R}^{n\...
gondolf's user avatar
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0 votes
0 answers
122 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. ...
VS.'s user avatar
  • 1,816
0 votes
0 answers
40 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 ...
Thomas Rasberry's user avatar
0 votes
0 answers
145 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 \...
user40780's user avatar
  • 867
0 votes
0 answers
166 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 ...
gstar2002's user avatar
-1 votes
2 answers
572 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 ...
gappy3000's user avatar
  • 461