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}{arg min} \| \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}{arg min} \| \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

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

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.