Sparse Reconstruction with Compressed Sensing

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Author: Ngoc Ly (SysEn 5800 Fall 2021)

Compressed Sensing (CS)

Compressed Sensing summary here


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.


Three big groups of algorithms are:[1]

Optimization methods: includes minimization i.e. Basis Pursuit, and quadratically constraint minimization i.e. basis pursuit denoising.

greedy include Orthogonal matching pursuit and Compressive Sampling Matching Pursuit (CoSaMP)

thresholding-based methods such as Iterative Hard Thresholding(IHT) and Iterative Soft Thresholding, Approximate IHT or AM-IHT, and many more.

More cutting-edge methods include dynamic programming.

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. [2] So IHT seems like the ideal place to start. If everything compliment with RIP, then IHT has fast convergence.

Introduction

Notation =

often not really sparse but approximately sparse

for Sensing matrix a Random Gaussian or Bernoulli matrix

are the observed y samples

noise vector

put defn of p norm here

where is the sparsifying matrix and are coeficients

sub module goal

s.t.

The goal of compressed sensing is to being with the under determined linear system , Where for How can we reconstruct x from The goal is to reconstruct given and Considerably fewer random measurements and reconstruct the original x 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 haphazard matrix

sub module sparsity

A vector is said to be sparse in , if it has at most nonzero coefficients.

The set of all k-sparse vectors is denoted by Failed to parse (unknown function "\Sum"): {\displaystyle \Sum_k = \{x | \|x\|_0 \leq k\}} Consequently there is different subsets of k-sparse vectors.

The problem formulation is to recover sparse data


The support of is we say is sparse when

We are interested in the smallest , i.e.

sub module RIP matracies

Sensing matrix must satisfy RIP i.e. Random Gaussian or Bernoulli matrixies satisfies A which is the number of measurements required for standard compressive sensing to recover with high probability.

sub modual

let be an index set enumerates the columns of and . is an under determined systems with infinite solutions since . Why norm won't give sparse solutions, where asl norm will return a sparse solution.


sub modual 3

Before we get into RIP lets talk about RIC

Restricted Isometry Constant (RIC) is the smallest that satisfies the RIP condition introduced by Candes, Tao

sub modual RIP

Random Gaussian and Bernoulli satisfies RIP

RIP defined as

sub module

Let satisfy RIP, Let be an index set For is a restriction on denoted by to k-sparse s.t. RIP is satisfied the i.e. and where the columns of is indexed by

In search for a unique solution we have the following optimization problem.

sub modual 4

, which is an combinatorial NP-Hard problem. Hence, if noise is presence the recovery is not stable. [Buraniuk "compressed sensing"]

From Results of Candes, Romberg, Tao, and Donoho

If satisfies RIP and is sparse the gives sparse solutions and is a unique. It is equivalent to convex optimization problem and can solve by Linear Program.

Theory

Verification of the Sensing matrix

Check if satisfies RIP Checking satisfies RIP is NP-complete in general so it's unreasonable to ask a computer to verify a matrix satisfies RIP. In order to get around this combinatorial hard problem, we need an understanding of what matrices satisfy RIP.

Random Sensing matrices: Gaussian, Bernoulli, Rademacher Deterministic Sensing Matrices: binary, bipolar, ternary, Vandermond 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.

Definition Mutual Coherence

Let , the mutual coherence is defined by:</math>

[3]

Welch bound > [3] > is the coherence between and We want a small because it will be close to the normal matrix, which satisfies RIP. Also, will be needed for the step size for the following IHT.


Need to make the connection of Coherence to RIP and RIC.

Algorithm IHT

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

(cite IT for sparse approximations)  ??? [3]. In statistics we call the regularization LASSO with as the regularization parameter. This is the closest convex relaxation to the first program menttioned in the introduction.[The Benefit of Group Sparsity]

Then

sub modual

Define the threashholding operators as: selects the best-k term approximation for some k

Stopping criterion is iff RIC [4]

  • Input
  • output
  • Set
  • While Stopping criterion false do
    • end while
  • return:

is a Adjoint matrix i.e. the transpost of it's cofactor.

Numerical Example

We will do some hacking to make use it works. If is a gaussian random matrix then we know that it satisfies RIP with high probability and IHT will reconstruct the the true signal find with minimization. (cite Ca, Ro, ta, Robust uncertainty exact signal) (cite Blu, Davies IHT for CS) Other words we don't really know in general if satisfies RIP in general unless we solve an NP-complete problem; however, we can cross our fingers that satisfies RIP with a high probability because it's Gaussian and not go through all the work for total verification of RIP. Donoho proves that nearly all matrices are sensing matrices.

Iterative Hard Thresholding IHT

Applications

Netflix problem

Conclusion

Referencse

[5] [6] [7] [8] [9] [10]

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  5. D. L. Donoho, “Compressed sensing,” vol. 52, pp. 1289–1306, 2006, doi: 10.1109/tit.2006.871582.
  6. E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory, vol. 51, no. 12, Art. no. 12, 2005, doi: 10.1109/TIT.2005.858979.
  7. D. L. Donoho, “Compressed sensing,” vol. 52, pp. 1289–1306, 2006, doi: 10.1109/tit.2006.871582.
  8. T. Blumensath and M. E. Davies, “Iterative Hard Thresholding for Compressed Sensing,” May 2008.
  9. S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing. New York [u.a.]: Birkhäuser, 2013.
  10. R. G. Baraniuk, “Compressive Sensing [Lecture Notes],” IEEE Signal Processing Magazine, vol. 24, no. 4, Art. no. 4, 2007, doi: 10.1109/MSP.2007.4286571.