Sparse Reconstruction with Compressed Sensing

Author: Ngoc Ly (SysEn 5800 Fall 2021)

Compressed Sensing (CS)

Compressed Sensing summary here

Introduction

$x\in \mathbb {R} ^{N}$often not really sparse but approximately sparse

$\Phi \in \mathbb {R} ^{M\times N}$for i$M\ll N$s a Random Gaussian or Bernoulli matrix

$y\in \mathbb {R} ^{M}$are the observed y samples

$e\in \mathbb {R} ^{M}$noise vector $\|e\|_{2}\leq \eta$k e k 2 ≤ η

s.t.

How can we reconstruct x from $y=\Phi x+e$

The goal is to reconstruct $x\in \mathbb {R} ^{N}$given $y$ and $\Phi$

Sensing matrix $\Phi$must satisfy RIP i.e. Random Gaussian or Bernoulli matrixies satisfies (cite)

let $[N]=\{1,\dots ,N\}$be an index set $[N]$ enumerates the columns of $\Phi$ and $x$ $\Phi$ is an under determined systems with infinite solutions. Why $l_{2}$ norm does not work

What is compression is synonymous with to the sparsity.

The problem formulation is to recover sparse data $\mathbf {x} \in \mathbb {R} ^{N}$

The support of $\mathbf {x}$ is $supp(\mathbf {x} )=\{i\in [N]:\mathbf {x} _{i}\neq 0\}$ we say $\mathbf {x}$ is $k$ sparse when $|supp(x)|\leq k$

We are interested in the smallest $supp(x)$ , i.e. $min(supp(x))$

Before we get into RIP lets talk about RIC

Restricted Isometry Constant (RIC) is the smallest $\delta _{|s}\ s.t.\ s\subseteq [N]$that satisfies the RIP condition introduced by Candes, Tao

Random Gaussian and Bernoulli satisfies RIP

Let $\Phi \in \mathbb {R} ^{M\times N}$ satisfy RIP, Let $[N]$ be an index set For $s$ is a restriction on $\mathbf {x}$ denoted by $x_{|s}$ $x\in \mathbb {R} ^{N}$ to $s$ k-sparse $\mathbf {x}$ s.t. RIP is satisfied the $s=|\Gamma |$ i.e. $s\subseteq [N]$and $\Phi _{|s}\subseteq \Phi$ where the columns of $\Phi _{|s}$ is indexed by $i\in S$

In search for a unique solution we have the following $l_{0}=|supp(x)|$ optimization problem. $\mathbf {\hat {s}} ={\underset {s}{argmin}}\|\mathbf {s} \|_{0}\quad s.t.\quad \mathbf {y} =\Phi \mathbf {s}$, which is an NP-Hard.

From Results of Candes, Romberg, Tao, and Donoho

If $\Phi$ satisfies RIP and $\mathbf {y}$ is sparse the $l_{0}$ has a unique solution. The equivalent $l_{1}$ convex program to the $l_{0}$ program. $\mathbf {\hat {s}} ={\underset {s}{argmin}}\|\mathbf {s} \|_{1}\quad s.t.\quad \mathbf {y} =\Phi \mathbf {s}$

�

x i , i f i ∈ S

( x | S ) i =

0 otherwise

RIP defined as

( 1 − δ s )k x k 22 ≤ k Φx k 22 ≤ ( 1 + δ s )k x k 22

3 Lemmas Page 267 Blumensath Davies IHT for CS

Lemma 1(Blumensath, Davis 2009 Iterative hard thresholding for compressed

sensing), For all index sets Γ and all Φ for which RIP holds with s = | Γ | that is

s = supp ( x )

1k Φ Γ T k 2 ≤

q

1 + δ | Γ | k y k 2

( 1 − δ | Γ | )k x Γ k 22 ≤ k Φ Γ T Φ Γ x Γ k 22 ≤ ( 1 + δ | Γ | )k x Γ k 22

and

k( I − Φ Γ T Φ Γ )k 2 ≤ δ | Γ | k x Γ k 2

SupposeΓ ∩ Λ = ∅

k Φ Γ T Φ Λ ) x Λ k 2 ≤ δ s k x Λ k 2

Lemma 2 (Needell Tropp, Prop 3.5 in CoSaMP: Iterative signal recovery

from incomplete and inaccurate √ samples)

If Φ satisfies RIP k Φx s k 2 ≤ 1 + δ s k x s k 2 , ∀ x s : k x s k 0 ≤ s, Then ∀ x

k Φx k 2 ≤

p

1 + δ s k x k 2 +

p

1 + δ s

k x k 1

sqrts

Lemma 3 (Needell Tropp, Prop 3.5 in CoSaMP: Iterative signal recovery

from incomplete and inaccurate samples)

Let x s be the best s-term approximation to x. Let x r = x − x s Let

y = Φx + e = Φx s + Φx r + e = Φx s + ẽ

If Φ satisfies RIP for sparsity s, then the norm of error ẽ is bounded by

k ẽ k 2 ≤

p

1 + δ s k x − x s k 2 +

p

1 + δ s

k x − x s k 1

√

+ k e k 2

s

∀ x

Theory

Gel’fend n-width

Errors E ( S, Φ, D )

Definition Mutual Coherence

Let $\Phi \in R^{M\times N}$, the mutual coherence $\mu _{\Phi }$ is defined by:</math>

$\mu _{\Phi }={\underset {i\neq j}{\frac {|\langle a_{i},a_{j}\rangle |}{\|a_{i}\|\|a_{j}\|}}}$

We want a small µ A because it will be close ot the normal matrix, which

$z_{v}(t)=\nabla f_{v}(x(t))=-\Phi _{v}^{T}(\mathbf {y} -\Phi \mathbf {x} )$ Then $x^{n+1}={\mathcal {H}}\left(\mathbf {x} ^{(n)}-\tau \sum _{j\in N}^{N}z_{v}^{(n)}\right)$

Define the threashholding operators as: ${\mathcal {H}}_{s}[\mathbf {x} ]={\underset {z\in \sum _{s}}{argmin}}\|x-\Phi \mathbf {x} \|_{2}$ selects the best-k term approximation for some k

will satisfies RIP

Stopping criterion is $\|y-\Phi \mathbf {x} ^{(n)}\|_{2}\leq \epsilon$

Algorithm IHT

• Initialize $\Phi ,\mathbf {y} ,\mathbf {e} \ {\mbox{with}}\ \mathbf {y} =\mathbf {\Phi } \mathbf {x} |\mathbf {e} and{\mathfrak {M}}$
• output $IHT(\mathbf {y} ,\mathbf {\Phi } ,{\mathfrak {M}})$
• Set $x^{(0)}=\mathbf {0}$
• While halting criterion false do
  • $x^{(n+1)}\leftarrow {\mathcal {H}}_{{\mathfrak {M}}\left[x^{(n)+\Phi ^{T}(\mathbf {y} -\mathbf {\Phi x} ^{(n)})}\right]}$
  • $n\leftarrow n+1$
  • end while
• return: $IHT(\mathbf {y} ,\mathbf {\Phi } ,{\mathfrak {M}})\leftarrow \mathbf {x} ^{(n)}$

Numerical Example

Iterative Hard Thresholding IHT

Check if $\Phi$ satisfies RIP with mutual coherence.