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This tour explores theoritical garantees for the performance of recovery using $\ell^1$ minimization.
options(warn=-1) # turns off warnings, to turn on: "options(warn=0)"
library(pracma)
library(imager)
# Importing the libraries
for (f in list.files(path="nt_toolbox/toolbox_general/", pattern="*.R")) {
source(paste("nt_toolbox/toolbox_general/", f, sep=""))
}
for (f in list.files(path="nt_toolbox/toolbox_signal/", pattern="*.R")) {
source(paste("nt_toolbox/toolbox_signal/", f, sep=""))
}
We consider the inverse problem of estimating an unknown signal $x_0 \in \RR^N$ from noisy measurements $y=\Phi x_0 + w \in \RR^P$ where $\Phi \in \RR^{P \times N}$ is a measurement matrix with $P \leq N$, and $w$ is some noise.
This tour is focused on recovery using $\ell^1$ minimization $$ x^\star \in \uargmin{x \in \RR^N} \frac{1}{2}\norm{y-\Phi x}^2 + \la \norm{x}_1. $$
Where there is no noise, we consider the problem $ \Pp(y) $ $$ x^\star \in \uargmin{\Phi x = y} \norm{x}_1. $$
We are not concerned here about the actual way to solve this convex problem (see the other numerical tours on sparse regularization) but rather on the theoritical analysis of wether $x^\star$ is close to $x_0$.
More precisely, we consider the following three key properties
\Pp(y) $ for $y=\Phi x_0$.
O(\norm{w})$ for $y=\Phi x_0+w$ if $\norm{w}$ is smaller than an arbitrary small constant that depends on $x_0$ if $\la$ is well chosen according to $\norm{w}$.
arbitrary.
Note that noise robustness implies identifiability, but the converse is not true in general.
The simplest criteria for identifiality are based on the coherence of the matrix $\Phi$ and depends only on the sparsity $\norm{x_0}_0$ of the original signal. This criteria is thus not very precise and gives very pessimistic bounds.
The coherence of the matrix $\Phi = ( \phi_i )_{i=1}^N \in \RR^{P \times N}$ with unit norm colum $\norm{\phi_i}=1$ is $$ \mu(\Phi) = \umax{i \neq j} \abs{\dotp{\phi_i}{\phi_j}}. $$
Compute the correlation matrix (remove the diagonal of 1's).
remove_diag = function(C){C - base::diag(base::diag(C))}
Correlation = function(Phi){remove_diag(abs(t(Phi) %*% Phi))}
Compute the coherence $\mu(\Phi)$.
mu = function(Phi){max(Correlation(Phi))}
The condition $$ \norm{x_0}_0 < \frac{1}{2}\pa{1 + \frac{1}{\mu(\Phi)}} $$ implies that $x_0$ is identifiable, and also implies to robustess to small and bounded noise.
Equivalently, this condition can be written as $\text{Coh}(\norm{x_0}_0)<1$ where $$ \text{Coh}(k) = \frac{k \mu(\Phi)}{ 1 - (k-1)\mu(\Phi) } $$
Coh = function(Phi, k){(k * mu(Phi)) / (1 - (k-1) * mu(Phi))}
Generate a matrix with random unit columns in $\RR^P$.
normalize = function(Phi){t(t(Phi) / sqrt(apply(Phi**2,2, sum)))}
PhiRand = function(P, N){normalize(matrix( rnorm(P*N,mean=0,sd=1), P, N))}
Phi = PhiRand(250, 1000)
Compute the coherence and the maximum possible sparsity to ensure recovery using the coherence bound.
c = mu(Phi)
print(paste("Coherence:", round(c, 2)))
print(paste("Sparsity max:", floor(1/2*(1 + 1/c))))
[1] "Coherence: 0.32" [1] "Sparsity max: 2"
Exercise 1
Display how the average coherence of a random matrix decays with the redundancy $\eta = P/N$ of the matrix $\Phi$. Can you derive an empirical law between $P$ and the maximal sparsity?
source("nt_solutions/sparsity_6_l1_recovery/exo1.R")
##Insert your code here.
In the following we will consider the support $$ \text{supp}(x_0) = \enscond{i}{x_0(i) \neq 0} $$ of the vector $x_0$. The co-support is its complementary $I^c$.
where = function(I)
{
n = length(I)
out = c()
for (i in 1:n)
{
if (I[i] == TRUE)
{
out = c(out, i)
}
}
return(out)
}
supp = function(s){where(abs(s) > 1e-5)}
cosupp = function(s){where(abs(s) < 1e-5)}
Given some support $ I \subset \{0,\ldots,N-1\} $, we will denote as $ \Phi = (\phi_m)_{m \in I} \in \RR^{N \times \abs{I}}$ the sub-matrix extracted from $\Phi$ using the columns indexed by $I$.
J.J. Fuchs introduces a criteria $F$ for identifiability that depends on the sign of $x_0$.
J.J. Fuchs. Recovery of exact sparse representations in the presence of bounded noise. IEEE Trans. Inform. Theory, 51(10), p. 3601-3608, 2005
Under the condition that $\Phi_I$ has full rank, the $F$ measure of a sign vector $s \in \{+1,0,-1\}^N$ with $\text{supp}(s)=I$ reads $$ \text{F}(s) = \norm{ \Psi_I s_I }_\infty \qwhereq \Psi_I = \Phi_{I^c}^* \Phi_I^{+,*} $$ where $ A^+ = (A^* A)^{-1} A^* $ is the pseudo inverse of a matrix $A$.
The condition $$ \text{F}(\text{sign}(x_0))<1 $$ implies that $x_0$ is identifiable, and also implies to robustess to small noise. It does not however imply robustess to a bounded noise.
Compute $\Psi_I$ matrix.
PsiI = function(Phi,I)
{
keep_indices = c()
for (i in 1:dim(Phi)[2])
{
if (!(i %in% I))
{
keep_indices = c(keep_indices, i)
}
}
return (t(Phi[,keep_indices]) %*% t(pinv(as.matrix(Phi[,I]))))
}
Compute $\text{F}(s)$.
F = function(Phi,s){base::norm(PsiI(Phi, supp(s)) %*% s[supp(s)], type="I")}
The Exact Recovery Criterion (ERC) of a support $I$, introduced by Tropp in
J. A. Tropp. Just relax: Convex programming methods for identifying sparse signals. IEEE Trans. Inform. Theory, vol. 52, num. 3, pp. 1030-1051, Mar. 2006.
Under the condition that $\Phi_I$ has full rank, this condition reads $$ \text{ERC}(I) = \norm{\Psi_{I}}_{\infty,\infty} = \umax{j \in I^c} \norm{ \Phi_I^+ \phi_j }_1. $$ where $\norm{A}_{\infty,\infty}$ is the $\ell^\infty-\ell^\infty$ operator norm of a matrix $A$.
erc = function(Phi, I){base::norm(PsiI(Phi, I), type="I")}
The condition $$ \text{ERC}(\text{supp}(x_0))<1 $$ implies that $x_0$ is identifiable, and also implies to robustess to small and bounded noise.
One can prove that the ERC is the maximum of the F criterion for all signs of the given support $$ \text{ERC}(I) = \umax{ s, \text{supp}(s) \subset I } \text{F}(s). $$
The weak-ERC is an approximation of the ERC using only the correlation matrix $$ \text{w-ERC}(I) = \frac{ \umax{j \in I^c} \sum_{i \in I} \abs{\dotp{\phi_i}{\phi_j}} }{ 1-\umax{j \in I} \sum_{i \neq j \in I} \abs{\dotp{\phi_i}{\phi_j}} }$$
setdiff = function(n, I)
{keep_indices = c()
for (i in 1:n)
{
if (!(i %in% I))
{
keep_indices = c(keep_indices, i)
}
}
return(keep_indices)
}
g = function(C,I){apply(as.matrix(C[,I]), 1, sum)}
werc_g = function(g,I,J){max(g[J])/(1 - max(g[I]))}
werc = function(Phi,I){werc_g(g(Correlation(Phi), I), I, setdiff(dim(Phi)[2], I))}
One has, if $\text{w-ERC}(I)>0$, for $ I = \text{supp}(s) $, $$ \text{F}(s) \leq \text{ERC}(I) \leq \text{w-ERC}(I) \leq \text{Coh}(\abs{I}). $$
This shows in particular that the condition $$ \text{w-ERC}(\text{supp}(x_0))<1 $$ implies identifiability and robustess to small and bounded noise.
Exercise 2
Show that this inequality holds on a given matrix. What can you conclude about the sharpness of these criteria ?
source("nt_solutions/sparsity_6_l1_recovery/exo2.R")
[1] "N = 2000 , P = 1990 , |I| = 6" [1] "F(s) = 0.19" [1] "ERC(I) = 0.23" [1] "w-ERC(s) = 0.27" [1] "Coh(|s|) = 1.65"
## Insert your code here.
Exercise 3
For a given matrix $\Phi$ generated using PhiRand, draw as a function of the sparsity $k$ the probability that a random sign vector $s$ of sparsity $\norm{s}_0=k$ satisfies the conditions $\text{F}(x_0)<1$, $\text{ERC}(x_0)<1$ and $\text{w-ERC}(x_0)<1$
source("nt_solutions/sparsity_6_l1_recovery/exo3.R")
#Insert your code here.
The restricted isometry constants $\de_k^1,\de_k^2$ of a matrix $\Phi$ are the smallest $\de^1,\de^2$ that satisfy $$ \forall x \in \RR^N, \quad \norm{x}_0 \leq k \qarrq (1-\de^1)\norm{x}^2 \leq \norm{\Phi x}^2 \leq (1+\de^2)\norm{x}^2. $$
E. Candes shows in
E. J. Cand s. The restricted isometry property and its implications for compressed sensing. Compte Rendus de l'Academie des Sciences, Paris, Serie I, 346 589-592
that if $$ \de_{2k} \leq \sqrt{2}-1 ,$$ then $\norm{x_0} \leq k$ implies identifiability as well as robustness to small and bounded noise.
The stability constant $\la^1(A), \la^2(A)$ of a matrix $A = \Phi_I$ extracted from $\Phi$ is the smallest $\tilde \la_1,\tilde \la_2$ such that $$ \forall \al \in \RR^{\abs{I}}, \quad (1-\tilde\la_1)\norm{\al}^2 \leq \norm{A \al}^2 \leq (1+\tilde \la_2)\norm{\al}^2. $$
These constants $\la^1(A), \la^2(A)$ are easily computed from the largest and smallest eigenvalues of $A^* A \in \RR^{\abs{I} \times \abs{I}}$
minmax = function(v){c(1 - min(v), max(v) - 1)}
ric = function(A){minmax(eigen(t(A) %*% A)$values)}
The restricted isometry constant of $\Phi$ are computed as the largest stability constants of extracted matrices $$ \de^\ell_k = \umax{ \abs{I}=k } \la^\ell( \Phi_I ). $$
The eigenvalues of $\Phi$ are essentially contained in the interval $ [a,b] $ where $a=(1-\sqrt{\be})^2$ and $b=(1+\sqrt{\be})^2$ with $\beta = k/P$ More precisely, as $k=\be P$ tends to infinity, the distribution of the eigenvalues tends to the Marcenko-Pastur law $ f_\be(\la) = \frac{1}{2\pi \be \la}\sqrt{ (\la-b)^+ (a-\la)^+ }. $
Exercise 4
Display, for an increasing value of $k$ the histogram of repartition of the eigenvalues $A^* A$ where $A$ is a Gaussian matrix of size $(P,k)$ and variance $1/P$. For this, accumulate the eigenvalues for many realizations of $A$.
source("nt_solutions/sparsity_6_l1_recovery/exo4.R")
## Insert your code here.
Exercise 5
Estimate numerically lower bound on $\de_k^1,\de_k^2$ by Monte-Carlo sampling of sub-matrices.
source("nt_solutions/sparsity_6_l1_recovery/exo5.R")
## Insert your code here.
We now consider a convolution dictionary $ \Phi $. Such a dictionary is used with sparse regulariz
Second derivative of Gaussian kernel $g$ with a given variance $\si^2$.
sigma = 6
g = function(x){(1 - (x**2 / sigma**2)) * exp(-x**2/(2*sigma**2))}
Create a matrix $\Phi$ so that $\Phi x = g \star x$ with periodic boundary conditions.
P = 1024
t = meshgrid(0:(P - 1),0:(P - 1))
Y = t$X
X = t$Y
Phi = normalize(g((X - Y + P/2.) %% P - P/2.))
To improve the conditionning of the dictionary, we sub-sample its atoms, so that $ P = \eta N > N $.
eta = 2.
N = P/eta
Phi = Phi[,seq(1, dim(Phi)[2], by=eta)]
Plot the correlation function associated to the filter. Can you determine the value of the coherence $\mu(\Phi)$?
c = t(Phi) %*% Phi
c = abs(c[,dim(c)[2]/2])
options(repr.plot.width=5, repr.plot.height=5)
plot(c[(length(c)/2 - 50):((length(c)/2) + 50)], type="l", xlab="", ylab="", col=4)
roll <- function( x , n ){
if( n == 0 )
return( x )
c( tail(x,n) , head(x,-n) )
}
twosparse = function(d){roll(c(1, rep(0, d), -1, rep(0, N - d- 2)), as.integer(N/2 - d/2))}
Display $x_0$ and $\Phi x_0$.
x0 = twosparse(50)
plot(x0, type="l", col=2, ylab="", xlab="")
plot(Phi %*% x0, type="l", col=4, ylab="", xlab="")
Exercise 6
Plot the evolution of the criteria F, ERC and Coh as a function of $d$. Do the same plot for other signs patterns for $x_0$. Do the same plot for a Dirac comb with a varying spacing $d$.
source("nt_solutions/sparsity_6_l1_recovery/exo6.R")
#Insert your code here.