# 4.7 Greedy algorithms

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In this module we provide an overview of some of the most common greedy algorithms and their application to the problem of sparse recovery.

## Setup

As opposed to solving a (possibly computationally expensive) convex optimization program, an alternate flavor to sparse recovery is to apply methods of sparse approximation . Recall that the goal of sparse recovery is to recover the sparsest vector $x$ which explains the linear measurements $y$ . In other words, we aim to solve the (nonconvex) problem:

$\underset{\mathcal{I}}{min}\left\{|\mathcal{I}|:y,=,\sum _{i\in \mathcal{I}},{\phi }_{i},{x}_{i}\right\},$

where $\mathcal{I}$ denotes a particular subset of the indices $i=1,...,N$ , and ${\phi }_{i}$ denotes the ${i}^{\mathrm{th}}$ column of $\Phi$ . It is well known that searching over the power set formed by the columns of $\Phi$ for the optimal subset ${\mathcal{I}}^{*}$ with smallest cardinality is NP-hard. Instead, classical sparse approximation methods tackle this problem by greedily selecting columns of $\Phi$ and forming successively better approximations to $y$ .

## Matching pursuit

Matching Pursuit (MP), named and introduced to the signal processing community by Mallat and Zhang  [link] , [link] , is an iterative greedy algorithm that decomposes a signal into a linear combination of elements from a dictionary. In sparse recovery, this dictionary is merely the sampling matrix $\Phi \in {\mathbb{R}}^{M×N}$ ; we seek a sparse representation ( $x$ ) of our “signal” $y$ .

MP is conceptually very simple. A key quantity in MP is the residual $r\in {\mathbb{R}}^{M}$ ; the residual represents the as-yet “unexplained” portion of the measurements. At each iteration of the algorithm, we select a vector from the dictionary that is maximally correlated with the residual $r$ :

${\lambda }_{k}=arg\underset{\lambda }{max}\frac{⟨{r}_{k},{\phi }_{\lambda }⟩{\phi }_{\lambda }}{\parallel {\phi }_{\lambda }{\parallel }^{2}}.$

Once this column is selected, we possess a “better” representation of the signal, since a new coefficient indexed by ${\lambda }_{k}$ has been added to our signal approximation. Thus, we update both the residual and the approximation as follows:

$\begin{array}{cc}\hfill {r}_{k}& ={r}_{k-1}-\frac{⟨{r}_{k-1},{\phi }_{{\lambda }_{k}}⟩{\phi }_{{\lambda }_{k}}}{\parallel {\phi }_{{\lambda }_{k}}{\parallel }^{2}},\hfill \\ \hfill {\stackrel{^}{x}}_{{\lambda }_{k}}& ={\stackrel{^}{x}}_{{\lambda }_{k}}+⟨{r}_{k-1},{\phi }_{{\lambda }_{k}}⟩.\hfill \end{array}$

and repeat the iteration. A suitable stopping criterion is when the norm of $r$ becomes smaller than some quantity. MP is described in pseudocode form below.

Inputs: Measurement matrix $\Phi$ , signal measurements $y$ Outputs: Sparse signal $\stackrel{^}{x}$ initialize: ${\stackrel{^}{x}}_{0}=0$ , $r=y$ , $i=0$ . while ħalting criterion false do 1. $i←i+1$ 2. $b←{\Phi }^{T}r$ {form residual signal estimate} 3. ${\stackrel{^}{x}}_{i}←{\stackrel{^}{x}}_{i-1}+\mathbf{T}\left(1\right)$ {update largest magnitude coefficient} 4. $r←r-\Phi {\stackrel{^}{x}}_{i}$ {update measurement residual} end while return $\stackrel{^}{x}←{\stackrel{^}{x}}_{i}$ 

Although MP is intuitive and can find an accurate approximation of the signal, it possesses two major drawbacks: (i) it offers no guarantees in terms of recovery error; indeed, it does not exploit the special structure present in the dictionary $\Phi$ ; (ii) the required number of iterations required can be quite large. The complexity of MP is $O\left(MNT\right)$   [link] , where $T$ is the number of MP iterations

## Orthogonal matching pursuit (omp)

Matching Pursuit (MP) can prove to be computationally infeasible for many problems, since the complexity of MP grows linearly in the number of iterations $T$ . By employing a simple modification of MP, the maximum number of MP iterations can be upper bounded as follows. At any iteration $k$ , Instead of subtracting the contribution of the dictionary element with which the residual $r$ is maximally correlated, we compute the projection of $r$ onto the orthogonal subspace to the linear span of the currently selected dictionary elements. This quantity thus better represents the “unexplained” portion of the residual, and is subtracted from $r$ to form a new residual, and the process is repeated. If ${\Phi }_{\Omega }$ is the submatrix formed by the columns of $\Phi$ selected at time step $t$ , the following operations are performed:

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