Sparse Reconstruction with Compressed Sensing: Difference between revisions

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Author: Ngoc Ly (SysEn 5800 Fall 2021)
Author: Ngoc Ly (SysEn 5800 Fall 2021)
[[File:The World is Sparse.png|center|thumb|600x600px]]
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==== Compressed Sensing (CS)====
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<sup>Compressed Sensing summary here
</sup>




Compression is synonymous with sparsity. So when we talk about compression we are actually referring to the sparsity. We introduce Compressed Sensing and then focus on reconstruction.
==  Introduction ==
 
=== Goal===
The goal of compressed sensing is to solve the underdetermined linear system in which the number of variables is much greater than the number of observations, resulting in an infinite number of signal coefficient vectors <math>x</math> for the same set of compressive measurements <math>y</math>. The objective is to reconstruct a vector <math>x</math> in a given of measurements <math>y</math> and a sensing matrix A. Instead of taking a large number of
high-resolution measurements and discarding the majority of them, consider taking far fewer random measurements and reconstructing the original <math>x</math> with high probability from its sparse representation.
 
=== Basic Idea and Notation ===
Begin with a linear equation <math>y = A x + e</math>, given a sensing matrix <math>A \in \mathbb{R}^{M \times N}</math> is random Gaussian or Bernouli will result in either exact or approximated optimum solution depending on how it is chosen, <math>x \in \mathbb{R}^{N}</math> is a signal vector with at most <math>k</math>-sparse entries, which means <math>x</math> has <math>k</math> non-zero entries, <math>[ N ] = \{ 1, \dots , N \} </math> be an index set, <math>y \in \mathbb{R}^{M}</math> is a compressed measurement vector, <math>[ M ] = \{ 1, \dots , M \} </math>, <math>e \in \mathbb{R}^{M}</math> is a noise vector and assumed to be bounded <math>\| e \|_2 \leq \eta</math> if it exists, and <math>M \ll N</math> read as <math>M</math> much less than <math>N</math>.
 
=== Sparsity  ===
 
A vector <math>x</math> is said to be <math>k</math>-sparse in <math>\mathbb{R}^N</math>
if it has at most <math>k</math> nonzero coefficients.
The support of <math>x</math> is <math>supp(x) = \{i \in [N] : x_i \neq 0 \}</math>, and <math>x</math> is a <math>k</math>-sparse signal when the cardinality <math>|supp(x)| \leq k</math>.
The set of <math>k</math>-sparse vectors is denoted by <math>\Sigma_k = \{x \in \mathbb{R}^N : \|x\|_0 \leq k \}</math>.
Consequently, there are <math>\binom{N}{k}</math> different subsets of <math>k</math>-sparse vectors. If a random <math>k</math>-sparse <math>x</math> is drawn uniformly from <math>\Sigma_k</math>,
its entropy, <math>\log \binom{N}{k}</math>, is approximately equivalent to <math>k \log \frac{N}{k}</math> bits are required for compression of <math>\Sigma_k</math> <ref name = "Laska"/><ref name = "Coluccia"/>.
 
The idea is to search for the sparsest <math>x \in \Sigma_k</math> from the measurement vector <math>y \in \mathbb{R}^M</math> and a sensing matrix <math>A \in \mathbb{R}^{M \times N}</math> with <math>M \ll N </math>. If the number of linear measurements is at least twice as its sparsity <math>x</math>, i.e., <math>M \geq 2k</math>, then there exists at most one signal <math>x \in \Sigma_k</math> that satisfies the constraint <math>y = A x</math> and produce the correct result for any <math>x \in \Sigma_k</math> <ref name = "Coluccia"/>. Hence, the reconstruction problem can be formulated as an <math>l_0</math>"norm" program.


==  Introduction ==
<math>\hat{x} = \underset{x \in \Sigma_k}{arg min} \|x\|_0 \quad s.t. \quad y = A x</math>
 
This optimization problem minimizes the number of nonzero entries of <math>x</math> subject to the constraint <math>y = Ax </math>, that is to find the sparsest element in the affine space  <math>\{ x \in \mathbb{R}^N : A x = y\}</math> <ref name = "Koep"/>. It turns out to be a combinatorial optimization problem, which is NP-Hard<ref name="Foucart"/> because it includes all possible sets of <math>k</math>-sparse out of <math>[N]</math>. Furthermore, if noise is present, the recovery is unstable [Buraniuk "compressed sensing"].
 
=== Restricted Isometry Property (RIP) ===
 
A matrix <math>A</math> is said to satisfy the RIP of order <math>k</math> if for all <math>x \in \Sigma_k</math> has a <math>\delta_k \in [0, 1)</math>. A restricted isometry constant (RIC) of <math>A</math> is the smallest <math>\delta_k</math> satisfying this condition <ref name = "CRT 2005"/><ref name = "Candes Tao"/><ref name = "Baraniuk 2008"/>.


=== sub module goal===
<math>(1 - \delta_k) \| x \|_2 ^2 \leq \| A x \|_2^2 \leq (1 + \delta_k) \| x \|_2 ^2</math>


The goal of compressed sensing is to interact with an underdetermined linear system in which the number of variables is much greater than the number of observations, resulting in an infinite number of signal coefficient vectors <math>x</math> for the same set of compressive measurements <math>y</math>. As a result, additional information is necessary for <math>x</math> to recover from <math>y</math>. The objective is to reconstruct a vector <math>x</math> in a given of measurements <math>y</math> and a sensing matrix A. Instead of taking a large number of
Under projections through matrix <math>A</math>, the restricted isometry property allows <math>k</math>-sparse vectors to have unique measurement vectors <math>y</math>. If <math>A</math> meets RIP, then <math>A</math> does not send two distinct <math>k</math>-sparse <math>x \in \Sigma_k</math> to the same measurement vector <math>y</math>, indicating that <math>x</math> is a unique solution under RIP.
high-resolution measurements and discarding the majority of them, consider taking way fewer random measurements and reconstructing the original <math>x</math> with high probability from its sparse representation.


=== sub modual ===


Begin with a linear equation <math>y = A x + e</math>, where <math>A \in \mathbb{R}^{M \times N}</math> is a sensing matrix that must be obtained and will result in either exact or approximated optimum solution depending on how it is chosen, <math>x \in \mathbb{R}^{N}</math> is a signal vector with at most <math>k</math>-sparse entries, which means <math>x</math> has <math>k</math> non-zero entries, <math>[ N ] = \{ 1, \dots , N \} </math>be an index set, <math>y \in \mathbb{R}^{M}</math> is a compressed measurement vector, <math>[ M ] = \{ 1, \dots , M \} </math>, <math>e \in \mathbb{R}^{M}</math> is a noise vector and assumed to be bounded <math>\| e \|_2 \leq \eta</math> if it exists, and <math>M \ll N</math>.
If the matrix <math>A \in \mathcal{R}^{M \times N}</math> satisfies the RIP condition of order <math>2k</math> and the constant <math>\delta_{2k} \in [0,1)</math>, there are two distinct <math>k</math>-sparse vectors in <math>\Sigma_{2k}</math>. When they are equal, the restricted isometry property holds. If <math>A</math> is a <math>2k</math>-order RIP matrix, it means that no two <math>k</math>-sparse vectors are mapped to the same measurement vector <math>y</math> by <math>A</math>.  In other words, when working with sparse vectors, the RIP ensures that the columns of <math>A</math> are nearly orthonormal. Furthermore, <math>A</math> is an approximately norm-preserving function, which means that it preserves its distance when mapping for <math>k</math>-sparse signals for all or more as <math>\delta_k</math> approaches zero. Candes, Romberg, and Tao  <ref name = "CRT 2005"/> proved that if <math>x</math> is <math>k</math>-sparse, and <math>A</math> satisfies the RIP of order <math>2k</math> with RIP-constant <math>\delta_{2k} < \sqrt(2) - 1</math>, then <math>\ell_1</math> gives a unique sparse solution. The <math>\ell_1</math> convex optimization problem is the same as the solution to the <math>\ell_0</math> program and can solve by the Linear Program <ref name = "Koep" /> <ref name = "Coluccia"/>. Hence, the <math>\ell_1</math> reconstruction problem is as followed which can be solved by basis pursuit <ref name = "CRT 2005"/> <ref name = "Donoho"/>.  


<math>\hat{x} = \underset{x \in \Sigma_k}{arg min} \|x\|_1 \quad s.t. \quad y = A x</math>


<!--
== Junk ==
The goal of compressed sensing is to being with the under determined linear system
The goal of compressed sensing is to being with the under determined linear system
<math>y = \Phi x + e</math>, Where <math>\Phi \in \mathbb{R}^{M \times N}</math>
<math>y = \Phi x + e</math>, Where <math>\Phi \in \mathbb{R}^{M \times N}</math>
Line 44: Line 67:
put defn of p norm here
put defn of p norm here


<math>x = \Psi \alpha</math> where <math>\Psi</math> is the sparsifying matrix and <math>\alpha</math> are coeficients
<math>x = \Psi \alpha</math> where <math>\Psi</math> is the sparsifying matrix and <math>\alpha</math> are coefficients
 
=== sub module sparsity  ===
 
A vector <math>x</math> is said to be <math>k</math>-sparse in <math>\mathbb{R}^N</math>
if it has at most <math>k</math> nonzero coefficients.
The support of <math>x</math> is <math>supp(x) = \{i \in [N] : x_i \neq 0 \}</math>, and <math>x</math> is a <math>k</math>-sparse signal when <math>|supp(x)| \leq k</math>.
The set of <math>k</math>-sparse vectors is denoted by <math>\Sigma_k = \{x \in \mathbb{R}^N : \|x\|_0 \leq k \}</math>.
Consequently, there are <math>\binom{N}{k}</math> different subsets of <math>k</math>-sparse vectors. If a random <math>k</math>-sparse <math>x</math> is drawn uniformly from <math>\Sigma_k</math>,
its entropy <math>\log \binom{N}{k}</math> is approximately equivalent to <math>k \log \frac{N}{k}</math> bits are required for compression of <math>\Sigma_k</math> ~cite(Measurements vs Bits).
 
 
The idea is to search for the sparsest <math>x \in \Sigma_k</math> from the measurement vector <math>y \in \mathbb{R}^M</math> and a sensing matrix <math>A \in \mathbb{R}^{M \times N}</math> with <math>M \ll N </math>. If the number of linear measurements is at least twice as its sparsity <math>x</math>, i.e., <math>M \geq 2k</math>, then there exists at most one signal <math>x \in \Sigma_k</math> that satisfies the constraint <math>y = A x</math> and produce the correct result for any <math>x \in \Sigma_k</math> [coluccia2015 book7]. Hence, the reconstruction problem can be formulated as an <math>l_0</math>"norm" program.
 
<math>\hat{x} = \underset{x \in \Sigma_k}{arg min} \|x\|_0 \quad s.t. \quad y = A x</math>
 
This optimization problem minimizes the number of nonzero entries of <math>x</math> subject to the constraint <math>y = Ax </math>, that is to find the sparsest element in the affine space  <math>\{ x \in \mathbb{R}^N : A x = y\}</math> [2019 book33].  It turns out to be a combinatorial optimization problem, which is NP-Hard because it includes all possible sets of <math>k</math>-sparse out of <math>[N]</math>. Furthermore, if noise is present, the recovery is unstable [Buraniuk "compressed sensing"].




Line 73: Line 80:
Another words we are interested in the smallest <math>supp(x)</math> , i.e. <math>min(supp(x))</math>
Another words we are interested in the smallest <math>supp(x)</math> , i.e. <math>min(supp(x))</math>


<!-- The problem formulation is to recover sparse data <math>\mathbf{x} \in \mathbb{R}^N</math>m -->
The problem formulation is to recover sparse data <math>\mathbf{x} \in \mathbb{R}^N</math>m  


=== sub module===
=== sub module===
Line 83: Line 90:


<math>\mathbf{\hat{x}} = \underset{\Sigma_k}{arg min} \| \mathbf{x}\|_0 \quad s.t. \quad \mathbf{y} = \Phi \mathbf{x}</math>, which is an combinatorial NP-Hard problem. Hence, if noise is presence the recovery is not stable. [Buraniuk "compressed sensing"]
<math>\mathbf{\hat{x}} = \underset{\Sigma_k}{arg min} \| \mathbf{x}\|_0 \quad s.t. \quad \mathbf{y} = \Phi \mathbf{x}</math>, which is an combinatorial NP-Hard problem. Hence, if noise is presence the recovery is not stable. [Buraniuk "compressed sensing"]
The goal of compressed sensing is to being with the under determined linear system
<math>y = \Phi x + e</math>, Where <math>\Phi \in \mathbb{R}^{M \times N}</math>
for
<math>M \ll N</math>
How can we reconstruct x from 
The goal is to reconstruct <math>x \in \mathbb{R}^N</math>given <math>y</math> and <math>\Phi</math>
Considerably fewer random measurements and reconstruct the original <math>x</math> with high probability from its sparse representation instead of taking a large number of high-resolution measurements and discarding the majority of them. being a random matrix


let <math>[ N ] = \{ 1, \dots , N \} </math>be an index set <math>[N]</math> enumerates the columns of <math>\Phi</math> and <math>x</math>. <math>\Phi</math> is an under determined systems with infinite solutions since <math>M \ll N</math>. Why <math>\ell_2</math> norm won't give sparse solutions, where asl <math>\ell_1</math> norm will return a sparse solution.


==== Notation =====
<math>x \in \mathbb{R}^N</math>often not really sparse but approximately sparse


=== Restricted Isometry Property (RIP) ===
<math>\mathbf{x} \in \mathbb{R}^{N}</math>
 
<math>\Phi \in \mathbb{R}^{M \times N}</math>for <math>M \ll N</math> Sensing matrix a Random Gaussian or Bernoulli matrix


A matrix <math>A</math> is said to satisfy the RIP of order <math>k</math> if for all <math>x \in \Sigma_k</math> has a <math>\delta_k \in [0, 1)</math>. A restricted isometry constant (RIC) of <math>A</math> is the smallest <math>\delta_k</math> satisfying this condition [2019 book38, coluccia2015 book7].
<math>y \in \mathbb{R}^M</math>are the observed y samples


<math>(1 - \delta_k) \| x \|_2 ^2 \leq \| A x \|_2^2 \leq (1 + \delta_k) \| x \|_2 ^2</math>
<math>e \in \mathbb{R}^M</math>noise vector <math>\| e \|_2 \leq \eta</math>


Under projections through matrix <math>A</math>, the restricted isometry property allows <math>k</math>-sparse vectors to have unique measurement vectors <math>y</math>. If <math>A</math> meets RIP, then <math>A</math> does not send two distinct <math>k</math>-sparse <math>x \in \Sigma_k</math> to the same measurement vector <math>y</math>, indicating that <math>x</math> is a unique solution under RIP.
put defn of p norm here


If the matrix <math>A \in \mathcal{R}^{M \times N}</math> satisfies the RIP condition of order <math>2k</math> and the constant <math>\delta_{2k} \in [0,1)</math>, there are two distinct <math>k</math>-sparse vectors in <math>\Sigma_{2k}</math>. When they are equal, the restricted isometry property holds. If <math>A</math> is a <math>2k</math>-order RIP matrix, it means that no two <math>k</math>-sparse vectors are mapped to the same measurement vector <math>y</math> by <math>A</math>.  In other words, when working with sparse vectors, the RIP ensures that the columns of <math>A</math> are nearly orthonormal. Furthermore, <math>A</math> is an approximately norm-preserving function, which means that it preserves its distance when mapping for <math>k</math>-sparse signals for all or more as <math>\delta_k</math> approaches zero. [Candes, Romberg, Tao[4]] demonstrate that if <math>x</math> is <math>k</math>-sparse, and <math>A</math> satisfies the RIP of order <math>2k</math> with RIP-constant <math>\delta_{2k} < \sqrt(2) - 1</math>, then <math>l_1</math> gives a unique sparse solution. The <math>l_1</mathL1 convex optimization problem is the same as the solution to the <math>l_0</math>L0 program and can be solved using the Linear Program.
<math>x = \Psi \alpha</math> where <math>\Psi</math> is the sparsifying matrix and <math>\alpha</math> are coefficients




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Restricted Isometry Constant (RIC) is the smallest <math>\delta_{|s}</math>  in  
Restricted Isometry Constant (RIC) is the smallest <math>\delta_{|s}</math>  in  
<math>\{\delta_k \in [0, 1): (1 - \delta_s) \| x \|_2 ^2 \leq \| \Phi x \|_2^2 \leq (1 + \delta_s) \| x \|_2 ^2\}</math>
<math>\{\delta_k \in [0, 1): (1 - \delta_s) \| x \|_2 ^2 \leq \| \Phi x \|_2^2 \leq (1 + \delta_s) \| x \|_2 ^2\}</math>


if <math>M \geq 2k</math> i.e. twise the sparsity, then there exists an unique <math>x</math> such that
if <math>M \geq 2k</math> i.e. twise the sparsity, then there exists an unique <math>x</math> such that
Line 133: Line 151:
From Results of Candes, Romberg, Tao, and Donoho
From Results of Candes, Romberg, Tao, and Donoho


<math>\mathbf{\hat{x}} = \underset{s}{arg min} \| \mathbf{x}\|_1 \quad s.t. \quad \mathbf{y} = \Phi \mathbf{x}</math>
Let <math>\Phi \in R^{M \times N}</math>, the mutual coherence <math>\mu_\Phi</math> is defined by:<nowiki></math></nowiki>


<math>\mathbf{\hat{x}} = \underset{s}{arg min} \| \mathbf{x}\|_1 \quad s.t. \quad \mathbf{y} = \Phi \mathbf{x}</math>
<math>\mu_{\Phi} = \underset{i \neq j} {\frac{| \langle a_i, a_j \rangle |}{ \| a_i \| \| a_j \|}}</math>
 
#TODO switch s to k
 
<math>(1 - \mu) \| x \|_2 ^2 \leq \| \Phi x \|_2^2 \leq (1 + \mu) \| x \|_2 ^2</math>
 
Welch bound <math>\mu_\Phi \geq \sqrt{\frac{n}{m(n-m)}}</math>> <math>\mu \geq \sqrt{\frac{N -M}{M(N-1)}}</math>>  is the coherence between <math>\Phi</math> and <math>\Psi</math>
We want a small <math>\mu_{\Phi}</math> because it will be close to the normal matrix, which satisfies RIP. Also, <math>\mu_{\Phi}</math> will be needed for the step size for the following IHT.
 
 
Need to make the connection of Coherence to RIP and RIC.
 
#TODO switch s to k
 
<math>(1 - \mu) \| x \|_2 ^2 \leq \| \Phi x \|_2^2 \leq (1 + \mu) \| x \|_2 ^2</math>
 
Gel’fand n-width
 
-->


== Theory ==
== Theory and Algorithmic Discussions ==


Two things need to be considered when recovering <math>x</math>
Two main things need to be considered when recovering <math>x</math>


* (1) '''The design of the sensing matrix'''
* (1) '''The design of the sensing matrix'''
Line 144: Line 184:
* (2) '''The recovery algorithm'''  
* (2) '''The recovery algorithm'''  


=== Sensing Matrix ===
==== Sparsity order <math>k</math> ====
Check if <math>\Phi</math> satisfies RIP
Checking <math>\Phi</math> satisfies RIP is combinatorial hard in general so it's unreasonable to ask a computer to verify a matrix satisfies RIP. In order to get around this problem, we need an understanding of what matrices satisfy RIP and recover <math>x</math> with high probability.


* '''Random Sensing matrices''': Gaussian, Bernoulli, Rademacher
<math>k \leq C_{M}\log\left(\frac{N}{M}\right)</math> <ref name="Cohen"/><ref name="Jia"/><ref name="ShuxingLiFinite"/>
* '''Deterministic Sensing Matrices''': binary, bipolar, ternary, Vandermond
TODO determining <math>k</math> for deterministic sensing matrices isn't the same as the random case <ref name="DeVore"/>.
* '''Structural Sensing Matrices''': Toeplitz, Circulant, Hadamard
* '''Optimized Sensing Matrices''': (Parkale, Nalbalwar, Sensing Matrices in Compressed Sensing)


Are some examples. Different sensing matrices are more suited for different problems, but in general, we want to use an alternative to Gaussian because it reduces the computational complexity.
==== Mutual Coherence ====


The recovery algorithm often refers to the measurement of quantities that are appropriate to the measurement matrix (sensing matrix), i.e., the coherence. A small coherence implies the RIP condition with high probability.  In general, the performance of the recovery algorithm gets better if the coherence is getting smaller. It means the columns of the matrices that have medium size are well-conditioned (governed).


==== Verification of the Sensing matrix ====
Let <math>A \in R^{M \times N}</math> and assume <math>\ell_2</math>-normalized columns of <math>A</math>, the mutual coherence <math>\mu = \mu(A)</math> is defined by


<!-- Gel’fand n-width -->
<math>\mu(A) = \underset{1 \leq i \neq j \leq N}{max} {\frac{| \langle a_i, a_j \rangle |}{ \| a_i \|_2 \| a_j \|_2}}</math>, where <math>a_i</math> is the <math>i</math>th column of <math>A</math><ref name = "Foucart"/>.


Definition Mutual Coherence
The feasibility of attaining the lower bounds for the coherence of a matrix <math>A \in R^{M \times N}</math>, <math>M < N</math> with <math>\ell_2</math>-normalized columns, and the columns of the matrix are equiangular tight frames defined as the Welch bound.


Let <math>\Phi \in R^{M \times N}</math>, the mutual coherence <math>\mu_\Phi</math> is defined by:<nowiki></math></nowiki>
<math>\mu(A) \geq \sqrt{\frac{N - M}{M(N-1)}}</math> <ref name = "Foucart"/> Sensing matrices using the Welch bound will imply a RIP of order <math>k = O (M^{1/2})</math>. <ref name="Jia"/><ref name="ShuxingLiFinite"/>  


<math>\mu_{\Phi} = \underset{i \neq j} {\frac{| \langle a_i, a_j \rangle |}{ \| a_i \| \| a_j \|}}</math><ref name=":1" />
However a sensing matrix <math>A</math> has two constraints a small coherence and <math>M \ll N</math> makes it impossible to satisfy the Welch bound<ref name = "Foucart"/>. <math>N</math> for an <math>\ell_2</math> must be <math>N \leq \frac{M(M+1)}{2}</math> for <math>A \in \mathbb{R}^{M \times N}</math>. This means that <math>N</math> can't be arbitrarily large and still satisfy the Welch bound. <ref name = "Foucart"/>


#TODO switch s to k
Let a matrix <math>A \in R^{M \times N}</math> with <math>\ell_2</math>-normalized columns and let <math>s \in [N]</math>. for all <math>k</math> sparse <math>x \in \Sigma_k</math>.


<math>(1 - \mu) \| x \|_2 ^2 \leq \| \Phi x \|_2^2 \leq (1 + \mu) \| x \|_2 ^2</math>
<math>(1 - \mu(s-1)) \| x \|_2 ^2 \leq \| A x \|_2^2 \leq (1 + \mu(s-1)) \| x \|_2 ^2</math> <ref name = "Foucart"/>


Welch bound <math>\mu_\Phi \geq \sqrt{\frac{n}{m(n-m)}}</math>> <ref name=":1" /> <math>\mu \geq \sqrt{\frac{N -M}{M(N-1)}}</math>>  is the coherence between <math>\Phi</math> and <math>\Psi</math>
=== Sensing Matrix ===
We want a small <math>\mu_{\Phi}</math> because it will be close to the normal matrix, which satisfies RIP. Also, <math>\mu_{\Phi}</math> will be needed for the step size for the following IHT.
Although the sensing matrix <math>A</math> satisfies RIP of order <math>k</math> in some situations, confirming a given matrix <math>A</math> meets RIP's criteria is NP-hard in general. As a result, designing an efficient sensing matrix is critical. These sensing matrices are responsible for signal compression at the encoder end and accurate or approximate reconstruction at the decoder end. Sensing matrices are classified as RIP or non-RIP. Non-RIP sensing matrices adhere to a weaker condition, statistical isometry property, while still ensuring high probability recover. <ref name="Jia"/> For signal compression, different sensing matrices are utilized in compressed sensing. There are random, deterministic, structural, and optimized sensing matrices are used in compressed sensing <ref name ="Parkale" />.


* '''Random Sensing Matrices'''
Some classes of random matrices satisfy RIP, specifically those matrices with independent and identically distributed (i.i.d.) entries drawn from a sub-Gaussian distribution. It requires the number of measurements, <math>M = O(k log(N/k))</math> <ref name = "CRT 2005"/> <ref name = "CRT 2006"/> <ref name = "Donoho"/>, to recover <math>x</math> with high probability. Other popular random sensing matrices are Gaussian, Bernoulli, or Rademacher distributions i.e. matrix with entries <math>\{0, \pm 1\}</math>. (give examples) However, the high memory storage costs of random dense matrices render them impractical for large-scale applications. Second, despite the fact that random matracies have a high probability of satisfying RIP, there is no efficient algorithm for verifying RIP for random matracies (Li, Gao, Ge, Zhang) <ref name = "ShuxingLiCurves"/>. Several researchers have turned to sparse measurement matrices such as binary or bi-adjacency matrices rather than random matrices to address this issue. Nonetheless, those sparse matrices are unstructured in the same way that acyclic networks or trees are. Fortunately, the new random Weibull matrices ~ cite (Xiaoya Zhang and Song Li) give time of failure example. are built with appropriate observations and provide exact sparse signal reconstruction with a greater probability <ref name ="Parkale" />.


Need to make the connection of Coherence to RIP and RIC.
===Weibull Example===


#TODO switch s to k
* '''Deterministic Sensing Matrices'''
Although deterministic sensing matrices are insufficient to satisfy the RIP condition, they can be used to provide instances for novel concepts. Unlike a random sensing matrix such Gaussian or Bernouli there is no such <math>k = M/\log{\left(\frac{N}{M}\right)}</math> for a deterministic construction<ref name="DeVore" />. The Vandermond <math>k \times N</math> matrices are one of the best deterministic matrices for recovering the <math>k</math>-sparse signal; however, the reconstruction technique gets unstable as <math>N</math> rises. Despite the lack of RIP support in the non-RIP deterministic sensing matrices, it successfully recovered the original sparse signal for the chirp function-based employing <math>M \times M^2</math> complex-valued deterministic matrices <ref name="Applebaum"/><ref name="Jia"/>. Proved binary, bipolar, and ternary matrices are deterministic constructions of sensing matrices that satisfy the RIP of order <math>k</math> <ref name = "Amini"/> Arash Amini and Farokh Marvasti. Because of the cyclic feature, the reconstruction process can be sped up using a fast Fourier transform (FFT) technique. Another type of deterministic CS sensing matrix is one based on mutual coherence. packing design in combinatorial design theory induces the connection between sensing matrices and finite geometry (Li, Ge) <ref name="ShuxingLiFinite"/>


<math>(1 - \mu) \| x \|_2 ^2 \leq \| \Phi x \|_2^2 \leq (1 + \mu) \| x \|_2 ^2</math>
(Deterministic Construction of Sparse Sensing Matrices via Finite Geometry Theorem II.1).
<math>\mu(A) \geq \frac{t - 1}{s}</math> <ref name = "ShuxingLiFinite"/>


=== Algorithms ===
The finite geometry-based sparse binary matrices were constructed with low coherence using a Steiner system (Shuxing Li and Gennian Ge) <ref name="ShuxingLiFinite"/> it's neat so give example. The sparseness property of matrices aids in reducing storage requirements and improving the reconstruction process <ref name ="Parkale" />.


Three big groups of algorithms are:<ref name=":0" />
===Finite Geometry Example===


* '''Optimization methods''': includes <math>\ell_1</math> minimization i.e. Basis Pursuit, and quadratically constraint <math>\ell_1</math>
* '''Structural Sensing Matrices'''
minimization i.e. basis pursuit denoising.
Sensing technologies require structured measurement matrices to perform a variety of tasks. Those matrices can be easily created with a small number of parameters. Furthermore, structured matrices can be used to speed the recovery performance of algorithms, making these matrices suitable for large matrices. The Toeplitz akin to a linear sparse ruler and the circulant matrix a square Toeplitz matrix akin to a circular sparse ruler. The Toeplitz structure is considered weakly stationary (second-order stationary) and a covariance matrix would be cinsidered Toeplitz more percicely circulant (Robert Gray) <ref name="RGray"/>. Applications or compressed senssing that follow a Toeplitz structure In terms of second-order statistics i.e. coveriance structures allows us to utilize statistical structures (Romero, Ariananda, Tian, Leus) <ref name="beyond sparsity"/>.
In terms of accuracy, signal reconstruction speed, and coherence, these matrices perform similar to i.i.d. Gaussian satisfies RIP with high probability. The new sparse block circulant matrix (SBCM) structure greatly reduces computational complexity. Other structural sensing matrices include the Hadamard matrix, which provides near-optimal assurances of recovery while requiring less complexity and so allowing for simple hardware implementation <ref name ="Parkale" />.


* '''Greedy methods''': include Orthogonal matching pursuit and Compressive Sampling Matching Pursuit (CoSaMP)
===Toeplitz matrix Example===


* '''thresholding-based methods''': such as Iterative Hard Thresholding(IHT) and Iterative Soft Thresholding, Approximate IHT or AM-IHT, and many more.
* '''Optimized Sensing Matrices'''
In order to accomplish high-quality signal reconstruction, the sensing matrices must satisfy RIP.
Regardless of what might be expected, the RIP is difficult to verify. Another method for validating
the RIP is to compute the mutual coherence between the sensing matrix and the sparse matrix or
figure the Gram matrix as <math>G = A^{T} A \in \mathbb{R}^{N \times N}</math>. The objective is to
improve the sensing matrix to lower coherence using numerous strategies such as the random-detecting
framework-based improvement strategy using the shrinkage technique and the irregular estimation of
sensing matrix improvement employing a symmetrical strategy. Another technique is to optimize the
sensing matrix using a block-sparsified dictionary approach, decreasing the total inter-block and
sub-block coherence of the dictionary matrix and thereby significantly increasing the reconstruction <ref name ="Parkale" />


More cutting-edge methods include dynamic programming.
=== Algorithms ===


We will cover one, i.e. IHT. WHY IHT THEN? Basis pursuit, matching pursuit type algorithms belong to a more general class of iterative thresholding algorithms. <ref name=":4" /> So IHT seems like the ideal place to start. If everything compliment with RIP, then IHT has fast convergence.
Several big groups of algorithms are:
* '''Optimization methods''': includes <math>\ell_1</math> minimization i.e. Basis Pursuit, and quadratically constraint <math>\ell_1</math> minimization i.e. basis pursuit denoising.


==== Algorithm IHT ====
* '''Greedy methods''': include orthogonal matching pursuit (OMP) and compressive sampling matching pursuit (CoSaMP).


The <math>\ell_1</math> convex program mentioned in introduction has an equivalent nonconstraint optimization program.
* '''Thresholding-based methods''': such as iterative hard thresholding (IHT) and iterative soft thresholding (IST), approximate IHT or AM-IHT.


<math>\underset{y}{min} \| \mathbf{y} - \Phi \mathbf{x} \|_2^2 + \lambda \| \mathbf{y} \|_0</math> (cite IT for sparse approximations)  ???
* '''Dynamic programming'''.  
<math>\hat{\mathbf{x}} = arg \underset{s}{min} \frac{1}{n} \| \mathbf{y} - \Phi \mathbf{x}\|_2^2 + \lambda \| \mathbf{x}\|_1</math> <ref name=":1" />. In statistics we call the <math>l_1</math> regularization LASSO with <math>\lambda</math> as the regularization parameter. This is the closest convex relaxation to <math>l_0</math> the first program menttioned in the introduction.[The Benefit of Group Sparsity]


<math>z_v^{(n)} = \nabla f_v(x^{(n)}) = - \Phi_v^T( \mathbf{y} - \Phi \mathbf{x})</math>
==== Algorithm IHT ====
Then
<math>x^{n+1} = \mathcal{H}\left( \mathbf{x}^{(n)} - \tau \sum_{j \in N}^{N} z_v^{(n)}\right)</math>


==== sub modual ====
The <math>\ell_1</math> convex program mentioned in introduction has an equivalent nonconstraint optimization program. <ref name = "Blumensath"/>


Define the threashholding operators as:
The threashholding operators is defined as:
<math> \mathcal{H}_s[\mathbf{x}] = \underset{z \in \sum_s}{argmin} \| x - \Phi \mathbf{x}\|_2</math>
<math> \mathcal{H}_k[x] = \underset{z \in \sum_k}{argmin} \| x - z\|_2</math>
selects the best-k term approximation for some k
selects the best-k term approximation for some k. The <math>\ell_2</math> was proved to be RIP of order
<math>3k</math>. <ref name = "Candes Tao"/> With the stopping criterion is <math>\| y - A x^{(n)}\|_2 \leq \epsilon</math>
<math> \ \iff \  \mbox{RIC}</math> <math>\delta_{3k} < \frac{1}{\sqrt{32}}</math> <ref name = "Blumensath"/>.


Stopping criterion is <math>\| y - \Phi \mathbf{x}^{(n)}\|_2 \leq \epsilon</math> iff RIC <math>\delta_{3s} < \frac{1}{\sqrt{32}}</math><ref name=":2" />
In addition the <math>\hat{f{x}} = arg \underset{x \in \Sigma_k}{min} \frac{1}{n} \| f{y} - A f{x}\|_2^2 + \lambda \| f{x}\|_1</math> in statistics the <math>\ell_1</math> regularization LASSO with <math>\lambda</math> as the regularization parameter. This is the closest convex relaxation to <math>\ell_0</math> the first program mentioned in the introduction.


* Input <math> \Phi, \mathbf{y}, \mathbf{e} \ \mbox{with} \ \mathbf{y} = \mathbf{\Phi} \mathbf{x} | \mathbf{e} and \mathfrak{M}</math>
Reducing the above loss function<math> \frac{1}{2}\| A x - y\|_2^2 </math> with the gradient
* output <math>IHT(\mathbf{y}, \mathbf{\Phi}, \mathcal{S}) </math>
<math>z_j^{(n)} = \nabla f_j(x^{(n)}) = - A_j^T( f{y} - A f{x})</math> for each iteration before pruning that is the hard threshold operator keeping the largest values while turning the values less than the threshold to zero.
* Set <math>x^{(0)} = \mathbf{0}</math>
* While Stopping criterion false do


**<math>x^{(n+1)} \leftarrow \mathcal{H}_{|s} \left[ x^{(n)} + \Phi^* (\mathbf{y} - \mathbf{\Phi x}^{(n)}) \right]</math>
Then
<math>x^{n+1} = \mathcal{H}\left( f{x}^{(n)} - \tau \sum_{j \in N}^{N} z_j^{(n)}\right)</math>


The IHT Algorithm reads as follow
* Input <math> A, y, e \ \mbox{with} \ y = A x + e</math> and <math>k</math>
* output <math>IHT(y, A, \mathcal{k}) </math>, an <math>k</math>-sparse solution to <math>x</math>
* Set <math>x^{(0)} = 0; n = 0</math>
* While stopping criterion false do
**<math>x^{(n+1)} \leftarrow \mathcal{H}_{k} \left[ x^{(n)} + A^{*} (y - A x^{(n)}) \right]</math>
**<math>n \leftarrow n + 1 </math>
**<math>n \leftarrow n + 1 </math>
**end while
**end while
*return: <math>IHT(\mathbf{y}, \mathbf{\Phi}, \mathfrak{M}) \leftarrow \mathbf{x}^{(n)}</math>
*return: <math>IHT(y, A, k) \leftarrow x^{(n)}</math>


<math>\Phi^*</math> is a Adjoint matrix i.e. the transpost of it's cofactor.
<math>A^*</math> is an Adjoint matrix i.e. the transpose of it's cofactor.


== Numerical Example ==
== Numerical Example ==
=== Deterministic Matrix Example ===
Question 1: Calculate the mutual coherence <math>\mu(A)</math>
Question 1: Does the sensing matrix satisfy the Welch?
<math>M = 5</math> <math>N = 10</math> <math>N \leq \frac{5(5+1)}{2}</math>
Question 2: What is the order of <math>k</math>?
=== A random matrix example ===
<math>  x = \left[
  0.1779,                                                                                                                                                     
  1.1649,                                                                                                                                                     
-0.8262 ,                                                                                                                                                     
-1.014 ,                                                                                                                                                     
-0.7975,                                                                                                                                                     
  0.1883  ,                                                                                                                                                      0.2648  ,                                                                                                                                                                         
-0.9036  ,                                                                                                                                                                       
  0.9409  ,                                                                                                                                                                            0.4295
  \right]
</math>
<math>
A =
\left[
\begin{matrix}
  1.24302 &    0.512088 &    0.727509 &  -0.621372 &  -0.167216 &  1.28429 &    1.80216 &  -0.860403 &  -0.190791 &  0.235666\\
-0.199659 &    0.197401 &  -0.496226 &    1.1034 &    -1.06153 &    0.0130745 &  -0.402581 &  0.788863 &  0.591947 &  -0.355693\\
  0.0202145 &  -0.0043102 &  -1.39167 &    -0.873447 &  0.253648 &  1.59332 &    0.988392 &  -1.32271 &  -0.951241 &  -0.192874\\
  0.238206 &    1.43934 &    -1.08785 &    -0.807931 &  0.285027 &  -2.64908 &    -0.140972 &  -0.215606 &  -0.760412 &  1.77824 \\
-0.17931 &    -1.31525 &    -0.0348558 &  -0.550808 &  0.460217 &  -1.5188 &    -1.45967 &    0.508893 &  -0.942668 &  1.1059\\
\end{matrix}
\right]
</math>
<math>\hat{x} = arg \underset{x \in \Sigma_x}{min} \frac{1}{a} \| y - A x\|_2^2 + \lambda \| x\|_1</math>
<math>a = 2</math>
<math>\lambda = 10^{-1}</math>
===Iteration 1===
calculate
<math>b = x + \frac{1}{a} * A^T (A x - y)</math>
<math>b = [-1.1194851676858286, -0.5425433278394107, 1.1296652382901562, 3.3691735377460557, -1.7975322402864389, 0.812781461654327, -1.5305641112291442, 2.4886377108681788, 2.7744632017898994, -2.450112036057253]</math>
====Thresholding ====
<math>
x^{(1)} =
\begin{cases}
b^{1} & if \ |b^{(1)}| > \lambda / 2a\\
        0 & if \ |b^{(1)}| \leq \lambda / 2a
\end{cases}
</math> <ref name = "Majumdar"/>
Also seen
<math>
x^{(1)} =
\begin{cases}
b^{1} & if \ |b^{(1)}| > \sqrt{\lambda}\\
        0 & if \ |b^{(1)}| \leq \sqrt{\lambda}
\end{cases}
</math> <ref name = "Rostami"/>
<math>x = [-1.1194851676858286, 0.0, 1.1296652382901562, 3.3691735377460557, -1.7975322402864389, 0.812781461654327, -1.5305641112291442, 2.4886377108681788, 2.7744632017898994, -2.450112036057253]</math>
Calculate the Error
<math> \|A * x - y \|_2 = 24.020025581623976 </math>
Calculate the new y
<math>y = A x = [-7.325091157735414, 10.392457780350403, -10.669635806747262, -13.670535847170495, -5.580526608842023]</math>
Check if error is less then <math>10 ^{-5} </math>
=== Iteration 2===
calculate
<math>b = x + \frac{1}{a} * A^T (A x - y)</math>
<math>b =  [-1.1194851676858286, 0.0, 1.1296652382901562, 3.3691735377460557, -1.7975322402864389, 0.812781461654327, -1.5305641112291442, 2.4886377108681788, 2.7744632017898994, -2.450112036057253]</math>
====Thresholding====
[[File:Reconstruction plot.png|thumb|right]]
<math>
x^{(2)} =
\begin{cases}
b^{2} & if \ |b^{(2)}| > \lambda / 2a\\
        0 & if \ |b^{(2)}| \leq \lambda / 2a
\end{cases}
</math>
<math>x = [-1.1194851676858286, 0.0, 1.1296652382901562, 3.3691735377460557, -1.7975322402864389, 0.812781461654327, -1.5305641112291442, 2.4886377108681788, 2.7744632017898994, -2.450112036057253]</math>
Calculate the Error
<math> \|A * x - y \|_2 = 0.0 </math>
Calculate the new y
<math>y = A x = [-7.325091157735414, 10.392457780350403, -10.669635806747262, -13.670535847170495, -5.580526608842023]</math>
Check if error is less then <math>10 ^{-5} </math>
Stopping condition meets.


== Applications and Motivations ==
== Applications and Motivations ==
Line 233: Line 394:
=== Low-Rank Matrices ===
=== Low-Rank Matrices ===


The Netflix Prize was accompanied by low-rank matrix recovery or the matrix completion problem.  The approach then fills in the missing values in the user's ratings for movies that the user hasn't seen. These estimates are based on ratings from other users, who have similar ratings if a matrix is created with all the users as rows and the movie titles as columns. Because some users' interests will be similar and therefore overlap, it is possible to reduce the degrees of freedom significantly. This low-rank structure is frequently assumed for the problem domain of collaborative filtering ~cited citations.
The Netflix Prize was accompanied by low-rank matrix recovery or the matrix completion problem.  The approach then fills in the missing values in the user's ratings for movies that the user hasn't seen. These estimates are based on ratings from other users, who have similar ratings if a matrix is created with all the users as rows and the movie titles as columns. Because some users' interests will be similar and therefore overlap, it is possible to reduce the degrees of freedom significantly. This low-rank structure is frequently assumed for the problem domain of collaborative filtering <ref name = "Bennett"/><ref name = "Koep"/>.


=== Dictionary Learning ===
=== Dictionary Learning ===


The goal in dictionary learning is to infer the original dictionary as possible. Instead of using a predefined dictionary, researchers have found that learning the dictionary by obtaining "dynamic features" from training data often yields representation. Biometric features can be taken from video clips of each subject in a dataset and used to populate the dictionary's columns. Using random projections and sparse representations for iris detection for noncontact biometrics-based authentication systems from video samples has been proposed ~cited citations.
The goal in dictionary learning is to infer the original dictionary as possible. Instead of using a predefined dictionary, researchers have found that learning the dictionary by obtaining "dynamic features" from training data often yields representation. Biometric features can be taken from video clips of each subject in a dataset and used to populate the dictionary's columns. Using random projections and sparse representations for iris detection for noncontact biometrics-based authentication systems from video samples has been proposed <ref name ="Pillai"/>.


=== Single-pixel cameras ===
=== Single-pixel cameras ===


Single-pixel cameras or single detector imaging are used in situations when detectors are either prohibitively expensive or difficult to miniaturize. A microarray is made up of a large number of miniature mirrors that can be individually turned on and off. The mechanism behind the random sampling, which results in low coherence between measurements, is the most important component of the single-pixel camera. This microarray reflects the light from the scene, and a lens combines all of the reflected beams into one sensor, which is the single detector of the camera used to capture the image ~cited citations.
Single-pixel cameras or single detector imaging are used in situations when detectors are either prohibitively expensive or difficult to miniaturize. A microarray is made up of a large number of miniature mirrors that can be individually turned on and off. The mechanism behind the random sampling, which results in low coherence between measurements, is the most important component of the single-pixel camera. This microarray reflects the light from the scene, and a lens combines all of the reflected beams into one sensor, which is the single detector of the camera used to capture the image <ref name="Baraniuk lecture notes" /> <ref name = "Burger"/>.


== Conclusion ==
== Conclusion ==
The theory was a buildup from what is an inverse problem and sparsity. It develops into the <math>\ell_0</math> norm and then concludes the <math>\ell_1</math> norm, which are sufficient conditions for basis pursuit. Candes, Romberg, Tao, and Donoho<ref name = "CRT 2005"/><ref name = "Donoho"/> were the first to overcome this problem.
Although the sensing matrix fulfills RIP of order k in some cases, establishing that a given matrix satisfies RIP's conditions is generally NP-hard. In many cases, verifying the sensing matrix isn't a reasonable task, so designing the sensing matrix is crucial. In addition to the computing costs, it must fulfill RIP and be well fitted to the problem domain. It demands some imagination as well as a grasp of the problem domain. It is possible to conclude that the sparse sensing matrix is the most significant components. To conclude, David Donoho saw sparsity everywhere and encouraged, mathematicians, engineers, and scientists to view problems through sparsity.
== References ==
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Latest revision as of 20:46, 22 October 2023

Author: Ngoc Ly (SysEn 5800 Fall 2021)


Introduction

Goal

The goal of compressed sensing is to solve the underdetermined linear system in which the number of variables is much greater than the number of observations, resulting in an infinite number of signal coefficient vectors for the same set of compressive measurements . The objective is to reconstruct a vector in a given of measurements and a sensing matrix A. Instead of taking a large number of high-resolution measurements and discarding the majority of them, consider taking far fewer random measurements and reconstructing the original with high probability from its sparse representation.

Basic Idea and Notation

Begin with a linear equation , given a sensing matrix is random Gaussian or Bernouli will result in either exact or approximated optimum solution depending on how it is chosen, is a signal vector with at most -sparse entries, which means has non-zero entries, be an index set, is a compressed measurement vector, , is a noise vector and assumed to be bounded if it exists, and read as much less than .

Sparsity

A vector is said to be -sparse in if it has at most nonzero coefficients. The support of is , and is a -sparse signal when the cardinality . The set of -sparse vectors is denoted by . Consequently, there are different subsets of -sparse vectors. If a random -sparse is drawn uniformly from , its entropy, , is approximately equivalent to bits are required for compression of [1][2].

The idea is to search for the sparsest from the measurement vector and a sensing matrix with . If the number of linear measurements is at least twice as its sparsity , i.e., , then there exists at most one signal that satisfies the constraint and produce the correct result for any [2]. Hence, the reconstruction problem can be formulated as an "norm" program.

This optimization problem minimizes the number of nonzero entries of subject to the constraint , that is to find the sparsest element in the affine space [3]. It turns out to be a combinatorial optimization problem, which is NP-Hard[4] because it includes all possible sets of -sparse out of . Furthermore, if noise is present, the recovery is unstable [Buraniuk "compressed sensing"].

Restricted Isometry Property (RIP)

A matrix is said to satisfy the RIP of order if for all has a . A restricted isometry constant (RIC) of is the smallest satisfying this condition [5][6][7].

Under projections through matrix , the restricted isometry property allows -sparse vectors to have unique measurement vectors . If meets RIP, then does not send two distinct -sparse to the same measurement vector , indicating that is a unique solution under RIP.


If the matrix satisfies the RIP condition of order and the constant , there are two distinct -sparse vectors in . When they are equal, the restricted isometry property holds. If is a -order RIP matrix, it means that no two -sparse vectors are mapped to the same measurement vector by . In other words, when working with sparse vectors, the RIP ensures that the columns of are nearly orthonormal. Furthermore, is an approximately norm-preserving function, which means that it preserves its distance when mapping for -sparse signals for all or more as approaches zero. Candes, Romberg, and Tao [5] proved that if is -sparse, and satisfies the RIP of order with RIP-constant , then gives a unique sparse solution. The convex optimization problem is the same as the solution to the program and can solve by the Linear Program [3] [2]. Hence, the reconstruction problem is as followed which can be solved by basis pursuit [5] [8].


Theory and Algorithmic Discussions

Two main things need to be considered when recovering

  • (1) The design of the sensing matrix
  • (2) The recovery algorithm

Sparsity order

[9][10][11] TODO determining for deterministic sensing matrices isn't the same as the random case [12].

Mutual Coherence

The recovery algorithm often refers to the measurement of quantities that are appropriate to the measurement matrix (sensing matrix), i.e., the coherence. A small coherence implies the RIP condition with high probability. In general, the performance of the recovery algorithm gets better if the coherence is getting smaller. It means the columns of the matrices that have medium size are well-conditioned (governed).

Let and assume -normalized columns of , the mutual coherence is defined by

, where is the th column of [4].

The feasibility of attaining the lower bounds for the coherence of a matrix , with -normalized columns, and the columns of the matrix are equiangular tight frames defined as the Welch bound.

[4] Sensing matrices using the Welch bound will imply a RIP of order . [10][11]

However a sensing matrix has two constraints a small coherence and makes it impossible to satisfy the Welch bound[4]. for an must be for . This means that can't be arbitrarily large and still satisfy the Welch bound. [4]

Let a matrix with -normalized columns and let . for all sparse .

[4]

Sensing Matrix

Although the sensing matrix satisfies RIP of order in some situations, confirming a given matrix meets RIP's criteria is NP-hard in general. As a result, designing an efficient sensing matrix is critical. These sensing matrices are responsible for signal compression at the encoder end and accurate or approximate reconstruction at the decoder end. Sensing matrices are classified as RIP or non-RIP. Non-RIP sensing matrices adhere to a weaker condition, statistical isometry property, while still ensuring high probability recover. [10] For signal compression, different sensing matrices are utilized in compressed sensing. There are random, deterministic, structural, and optimized sensing matrices are used in compressed sensing [13].

  • Random Sensing Matrices

Some classes of random matrices satisfy RIP, specifically those matrices with independent and identically distributed (i.i.d.) entries drawn from a sub-Gaussian distribution. It requires the number of measurements, [5] [14] [8], to recover with high probability. Other popular random sensing matrices are Gaussian, Bernoulli, or Rademacher distributions i.e. matrix with entries . (give examples) However, the high memory storage costs of random dense matrices render them impractical for large-scale applications. Second, despite the fact that random matracies have a high probability of satisfying RIP, there is no efficient algorithm for verifying RIP for random matracies (Li, Gao, Ge, Zhang) [15]. Several researchers have turned to sparse measurement matrices such as binary or bi-adjacency matrices rather than random matrices to address this issue. Nonetheless, those sparse matrices are unstructured in the same way that acyclic networks or trees are. Fortunately, the new random Weibull matrices ~ cite (Xiaoya Zhang and Song Li) give time of failure example. are built with appropriate observations and provide exact sparse signal reconstruction with a greater probability [13].

Weibull Example

  • Deterministic Sensing Matrices

Although deterministic sensing matrices are insufficient to satisfy the RIP condition, they can be used to provide instances for novel concepts. Unlike a random sensing matrix such Gaussian or Bernouli there is no such for a deterministic construction[12]. The Vandermond matrices are one of the best deterministic matrices for recovering the -sparse signal; however, the reconstruction technique gets unstable as rises. Despite the lack of RIP support in the non-RIP deterministic sensing matrices, it successfully recovered the original sparse signal for the chirp function-based employing complex-valued deterministic matrices [16][10]. Proved binary, bipolar, and ternary matrices are deterministic constructions of sensing matrices that satisfy the RIP of order [17] Arash Amini and Farokh Marvasti. Because of the cyclic feature, the reconstruction process can be sped up using a fast Fourier transform (FFT) technique. Another type of deterministic CS sensing matrix is one based on mutual coherence. packing design in combinatorial design theory induces the connection between sensing matrices and finite geometry (Li, Ge) [11]

(Deterministic Construction of Sparse Sensing Matrices via Finite Geometry Theorem II.1). [11]

The finite geometry-based sparse binary matrices were constructed with low coherence using a Steiner system (Shuxing Li and Gennian Ge) [11] it's neat so give example. The sparseness property of matrices aids in reducing storage requirements and improving the reconstruction process [13].

Finite Geometry Example

  • Structural Sensing Matrices

Sensing technologies require structured measurement matrices to perform a variety of tasks. Those matrices can be easily created with a small number of parameters. Furthermore, structured matrices can be used to speed the recovery performance of algorithms, making these matrices suitable for large matrices. The Toeplitz akin to a linear sparse ruler and the circulant matrix a square Toeplitz matrix akin to a circular sparse ruler. The Toeplitz structure is considered weakly stationary (second-order stationary) and a covariance matrix would be cinsidered Toeplitz more percicely circulant (Robert Gray) [18]. Applications or compressed senssing that follow a Toeplitz structure In terms of second-order statistics i.e. coveriance structures allows us to utilize statistical structures (Romero, Ariananda, Tian, Leus) [19]. In terms of accuracy, signal reconstruction speed, and coherence, these matrices perform similar to i.i.d. Gaussian satisfies RIP with high probability. The new sparse block circulant matrix (SBCM) structure greatly reduces computational complexity. Other structural sensing matrices include the Hadamard matrix, which provides near-optimal assurances of recovery while requiring less complexity and so allowing for simple hardware implementation [13].

Toeplitz matrix Example

  • Optimized Sensing Matrices

In order to accomplish high-quality signal reconstruction, the sensing matrices must satisfy RIP. Regardless of what might be expected, the RIP is difficult to verify. Another method for validating the RIP is to compute the mutual coherence between the sensing matrix and the sparse matrix or figure the Gram matrix as . The objective is to improve the sensing matrix to lower coherence using numerous strategies such as the random-detecting framework-based improvement strategy using the shrinkage technique and the irregular estimation of sensing matrix improvement employing a symmetrical strategy. Another technique is to optimize the sensing matrix using a block-sparsified dictionary approach, decreasing the total inter-block and sub-block coherence of the dictionary matrix and thereby significantly increasing the reconstruction [13]

Algorithms

Several big groups of algorithms are:

  • Optimization methods: includes minimization i.e. Basis Pursuit, and quadratically constraint minimization i.e. basis pursuit denoising.
  • Greedy methods: include orthogonal matching pursuit (OMP) and compressive sampling matching pursuit (CoSaMP).
  • Thresholding-based methods: such as iterative hard thresholding (IHT) and iterative soft thresholding (IST), approximate IHT or AM-IHT.
  • Dynamic programming.

Algorithm IHT

The convex program mentioned in introduction has an equivalent nonconstraint optimization program. [20]

The threashholding operators is defined as: selects the best-k term approximation for some k. The was proved to be RIP of order . [6] With the stopping criterion is [20].

In addition the in statistics the regularization LASSO with as the regularization parameter. This is the closest convex relaxation to the first program mentioned in the introduction.

Reducing the above loss function with the gradient for each iteration before pruning that is the hard threshold operator keeping the largest values while turning the values less than the threshold to zero.

Then

The IHT Algorithm reads as follow

  • Input and
  • output , an -sparse solution to
  • Set
  • While stopping criterion false do
    • end while
  • return:

is an Adjoint matrix i.e. the transpose of it's cofactor.

Numerical Example

Deterministic Matrix Example

Question 1: Calculate the mutual coherence

Question 1: Does the sensing matrix satisfy the Welch?

  

Question 2: What is the order of ?

A random matrix example

Iteration 1

calculate

Thresholding

[21]

Also seen

[22]

Calculate the Error

Calculate the new y

Check if error is less then

Iteration 2

calculate

Thresholding

Calculate the Error

Calculate the new y

Check if error is less then

Stopping condition meets.

Applications and Motivations

Low-Rank Matrices

The Netflix Prize was accompanied by low-rank matrix recovery or the matrix completion problem.  The approach then fills in the missing values in the user's ratings for movies that the user hasn't seen. These estimates are based on ratings from other users, who have similar ratings if a matrix is created with all the users as rows and the movie titles as columns. Because some users' interests will be similar and therefore overlap, it is possible to reduce the degrees of freedom significantly. This low-rank structure is frequently assumed for the problem domain of collaborative filtering [23][3].

Dictionary Learning

The goal in dictionary learning is to infer the original dictionary as possible. Instead of using a predefined dictionary, researchers have found that learning the dictionary by obtaining "dynamic features" from training data often yields representation. Biometric features can be taken from video clips of each subject in a dataset and used to populate the dictionary's columns. Using random projections and sparse representations for iris detection for noncontact biometrics-based authentication systems from video samples has been proposed [24].

Single-pixel cameras

Single-pixel cameras or single detector imaging are used in situations when detectors are either prohibitively expensive or difficult to miniaturize. A microarray is made up of a large number of miniature mirrors that can be individually turned on and off. The mechanism behind the random sampling, which results in low coherence between measurements, is the most important component of the single-pixel camera. This microarray reflects the light from the scene, and a lens combines all of the reflected beams into one sensor, which is the single detector of the camera used to capture the image [25] [26].

Conclusion

The theory was a buildup from what is an inverse problem and sparsity. It develops into the norm and then concludes the norm, which are sufficient conditions for basis pursuit. Candes, Romberg, Tao, and Donoho[5][8] were the first to overcome this problem.

Although the sensing matrix fulfills RIP of order k in some cases, establishing that a given matrix satisfies RIP's conditions is generally NP-hard. In many cases, verifying the sensing matrix isn't a reasonable task, so designing the sensing matrix is crucial. In addition to the computing costs, it must fulfill RIP and be well fitted to the problem domain. It demands some imagination as well as a grasp of the problem domain. It is possible to conclude that the sparse sensing matrix is the most significant components. To conclude, David Donoho saw sparsity everywhere and encouraged, mathematicians, engineers, and scientists to view problems through sparsity.


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