Rank awarealgs small11

IDCOM, University of Edinburgh

Rank Aware Algorithms for Joint Sparse
Recovery

Mike Davies*
Joint work with Yonina Eldar‡ and Jeff Blanchard†

* Institute of Digital Communications, University of Edinburgh
‡ Technion, Israel † Grinnell College, USA


Outline of Talk

• Multiple Measurements vs Single Measurements
• Nec.+suff. conditions for Joint Sparse Recovery
• Reduced complexity combinatorial search
• Classical approaches to sparse MMV problem
– How good are SOMP and convex optimization?
• Rank Aware Pursuits
– Evolution of the rank of residual matrices
– A recovery guarantee
• Empirical simulations

Sparse Single Measurement Vector
Problem

m x1 mxn n x1

Measurements Measurement
Matrix

Sparse Signal
k nonzero elements

Given y ∈ Rm and Φ ∈ Rm×n with m < n ﬁnd:

x = argmin | supp(x)| s.t. Φx = y.
ˆ
x

Sparse Multiple Measurement Vector
Problem

m×l m×n n×l

Measurement
Measurements Matrix

row support

Sparse Signal
k nonzero rows

Given Y ∈ Rm×l and Φ ∈ Rm×n with m < n ﬁnd:
ˆ
X = argmin | supp(X)| s.t. ΦX = Y.
X


MMV uniqueness
Worst Case
• Uniqueness of solution for sparse MMV problem is equivalent to that for
SMV problem. Simply replicate SMV problem:
X = {x, x, . . . , x}
Hence nec. + suff. condition to uniquely determine each k-sparse vector x is
given by SMV condition:
spark(Φ)
| supp(X)| = k <
2
Rank 'r' Case
• If Rank(Y)=r then the necessary + sufficient conditions are less restrictive
[Chen & Huo 2006, D. & Eldar 2010]:
spark(Φ) − 1 + rank(Y)
| supp(X)| = k <
2
Equivalently we can replace rank(Y) with rank(X).

More measurements (higher rank) makes recovery easier!


MMV uniqueness
Generic scenario:
Typical matrices achieve maximal spark:

Φ ∈ Rm×n → spark(Φ) = m + 1

Typical matrices achieve maximal rank

X ∈ Rk×l → rank(X) = r = min{k, l}

Hence generically we have uniqueness if

m ≥ 2k − min{k, l} + 1 ≥ k + 1

When l ≥ k we typically only need k+1 measurements


Exhaustive search solution

How does the rank change the exhaustive search?
SMV exhaustive search:

ﬁnd , | | = k s.t. ΦX = Y

However since span(Y) ⊂ span(Φ ) and rank(Y) = r

∃γ⊂ , |γ| = k − r s.t. span([Φγ , Y ]) = span(Φ )

n
In fact we have a reduced k−r+1 combinatorial search.


Geometric Picture for MMV
φ1 φ2 Y = ΦΛ XΛ,:

φ3

2−sparse vector ∈ span(Y )

span(Y )

If X is k-sparse and rank(Y) = r there exists a (k-r+1)-sparse vector in span(Y)


Maximal Rank Exhaustive Search: MUSIC
When we have maximal rank(X) = k the exhaustive search is linear and
can be solved with a modified MUSIC algorithm.

Let U = orth(Y) This is an orthonormal basis for span(Φ )

Then under identifiablity conditions we have:

(I − UUT )φi 2 = 0, if and only if i ∈ .

(in practice select support by thresholding)

Theorem 1 (Feng 1996) Let Y = ΦX with | supp(X)| = k, rank(X) = k
ˆ
and k < spark(Φ) − 1. Then MUSIC is guaranteed to recover X (i.e. X = X).


Maximal rank problem is not NP-hard
Furthermore there is no constraint on
n!


Popular MMV solutions


Popular MMV sparse recovery solutions
Two classes of MMV sparse recovery algorithm:
greedy, e.g.
Algorithm 1 Simultaneous Orthogonal Matching Pursuit (SOMP)
1: initialization: R(0) = Y, X(0) = 0, 0 = ∅
2: for n = 1; n := n + 1 until stopping criterion do
3: in = argmaxi φT R(n−1) q
i
n n−1 n
4: = ∪i
(n) †
5: X n ,: = Φ n Y
6: R(n) = P ⊥(n) Y where P ⊥(n) := (I − Φ (n) Φ† (n) )
7: end for

and relaxed, e.g.

Algorithm 2 ℓ1 /ℓq Minimization
ˆ
X = argmin ||X||1,q s.t. ΦX = Y
X

Do such MMV solutions exploit the
rank?
Answer: NO. [D. & Eldar 2010]
Theorem 2 (SOMP is not rank aware) Let τ be given such that 1 ≤ τ ≤ k
and suppose that
max ||Φ† φj ||1 > 1
j∈

for some support , | | = k. Then there exists an X with supp(X) = and
rank(X) = τ that SOMP cannot recover.
SMV OMP Exact
Recovery condition

Proof - Rank r perturbation of rank 1 problem approaches rank 1 recovery
property due to continuity norm.

Do such MMV solutions exploit the
rank?
Answer: NO. [D. & Eldar 2010]
Theorem 3 (ℓ1 /ℓq minimization is not rank aware) Let τ be given such
that 1 ≤ τ ≤ k and suppose that there exists a z ∈ N (Φ) such that

||z ||1 > ||z c ||1

for some support , | | = k. Then there exists an X with supp(X) = ,
rank(X) l=Null
SMV 1 τ that the mixed norm solution cannot recover.
Space Property

Proof - Rank r perturbation of rank 1 problem approaches rank 1 recovery
property due to continuity of norm.


Rank Aware Pursuits


Rank Aware Selection
Aim: to select individual atoms in a similar manner to modified MUSIC

Rank Aware Selection [D. & Eldar 2010]
At the nth iteration make the following selection:

(n) (n−1)
= ∪ argmax ||φT U(n−1) ||2
i
i

where U(n−1) = orth(R(n−1) )

Properties:
1. Worst case behaviour does not approach SMV case.
2. When rank(R) = k it always selects a correct atom as with
MUSIC


Rank Aware OMP
Rank Aware OMP
Let's simply replace the selection step in SOMP with the rank aware
selection.

Does this provide guaranteed recovery in the full rank scenario?

Answer: NO.

Why?
We get rank degeneration of the residual matrix:

rank(R(i) ) ≤ min{rank(Y ), k − i}
As we take more steps the rank reduces to one while R(i) is typically still
k-sparse.
We lose the rank benefits as we iterate

Rank Aware Order Recursive Matching
Pursuit
The fix...
We can fix this problem by forcing the sparsity to also reduce as a
function of iteration. This is achieved by:
Algorithm 1 Rank Aware Order Recursive Matching Pursuit (RA-ORMP)
1: Initialize R(0) = Y, X(0) = 0, 0 = ∅, P ⊥ = I
(0)

2: for n = 1; n := n + 1 until stopping criterion do
3: Calculate orthonormal basis for residual: U(n−1) = Orth(R(n−1) )
4: in = argmaxi∈ (n−1) φT U(n−1) 2 / P ⊥
i (n−1) φi 2
n
5: = n−1 ∪ in
(n)
6: X n ,: = Φ† n Y
7: R(n) = P ⊥(n) Y where P ⊥ := (I − Φ
(n) (n) Φ† (n) )
8: end for

˜
R(n)is (k-n)-sparse in the modified dictionary φi = P ⊥(n) φi / P ⊥(n) φi 2


RA-OMP vs RA-ORMP
Comparison of how (typical) residual rank (–) and sparsity (–)
evolve as a function of iteration

RA-OMP RA-ORMP

k k

r r

k k
iteration # iteration #
- region where correct selection is not guaranteed

SOMP/RA-OMP/RA-ORMP
Comparison

SOMP RA−OMP RA=ORMP
1 1 1
Prob of Exact Recovery


0.8 0.8 0.8

0.6 0.6 0.6

0.4 0.4 0.4

0.2 0.2 0.2

0 0 0
0 10 20 30 0 10 20 30 0 10 20 30
Sparsity k Sparsity k Sparsity k

n = 256, m = 32, l = 1,2,4,8,16,32. Dictionary ~ i.i.d. Gaussian and X
coefficients ~ Gaussian i.i.d. (note that this is beneficial to SOMP!)


Rank Aware OMP
Alternative Solutions
Recently two independent solutions have been proposed that are variations on
a theme:
1. Compressive MUSIC [Kim et al 2010]
i. perform SOMP for k-r-1 steps but SOMP is rank blind
ii. apply modified MUSIC

2. Iterative MUSIC [Lee & Bresler 2010]
1. orthogonalize: U = orth(Y ) orthogonalization is not
2. apply SOMP to {Φ, U } for k-r-1 steps guaranteed beyond step 1
3. apply modified MUSIC

This motivates us to consider a minor modification of (2):

3 RA-OMP+MUSIC
i. perform RA-OMP for k-r-1 steps
ii. apply modified MUSIC


Recovery guarantee
Two nice rank aware solutions

a) Apply RA-OMP for k-r-1 steps then complete with modified MUSIC
b) Apply RA-ORMP for k steps (if first k-r steps make correct selection we
have guaranteed recovery)

we now have the following recovery guarantee [Blanchard & D.]:
Theorem 4 (MMV CS recovery) Assume XΛ ∈ Rn×r is in general position
for some support set Λ, |Λ| = k > r and let Φ is a random matrix independent
of X, Φi,j ∼ N (0, m−1 ). Then (a) and (b) can recover X from Y with high
probability if:
log N
m ≥ const.k +1
r

That is: as r increases the effect of the log N term diminishes

RA-OMP+MUSIC / RA-ORMP
Comparison
RA−OMP+MUSIC RA=ORMP
1 1


0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0
0 10 20 30 0 10 20 30
Sparsity k Sparsity k

n = 256, m = 32, l = 1,2,4,8,16,32. i.i.d. Gaussian Dictionary and X
coefficients ~ Gaussian i.i.d.


Empirical Phase Transitions
RA−OMP+MUSIC 16 16
RA−ORMPl l= 16=
RA−OMP = 16 l
SOMP l =
50

45

40

35

30
m

25

20

15

10

5

5 10 15 20 25 30 35 40 45 50
k

Gaussian dictionary "phase transitions" with Gaussian
significant coefficients

Correlated vs uncorrelated
coefficients
SOMP RA-ORMP
SOMP l = 16 RA−ORMP l = 16
50 50

45 45

40 40

35 35

30 30
m

m
25 25

20 20

15 15

10 10

5 5

5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50
k k

Gaussian dictionary "phase transitions" with uncorrelated
sparse coefficients

Correlated vs uncorrelated
coefficients
SOMP RA-ORMP
SOMP l = 16, highly correlated RA−ORMP l = 16 highly correlated
50 50

45 45

40 40

35 35

30 30
m

m
25 25

20 20

15 15

10 10

5 5

5 10 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50
k k

Gaussian dictionary "phase transitions" with highly correlated
sparse coefficients


Summary

• MMV problem is easier than SMV problem in general
• Don't dismiss using exhaustive search (not always NP-hard!)
• Good rank aware greedy algorithms exist

Questions
• Can we extend these ideas to IHT or CoSaMP?
• How can we incorporate rank awareness into convex optimization?


Workshop : Signal Processing with Adaptive Sparse
Structured Representations (SPARS '11)
June 27-30, 2011 - Edinburgh, (Scotland, UK)

Plenary speakers :

David L. Donoho, Stanford University, USA
Martin Vetterli, EPFL, Switzerland
Stephen J. Wright, University of Wisconsin, USA
David J. Brady, Duke University, Durham, USA
Yi Ma, University of Illinois at Urbana-Champaign, USA
Joel Tropp, California Institute of Technology, USA
Remi Gribonval, Centre de Recherche INRIA Rennes, France
Francis Bach, Laboratoire d'Informatique de l'E.N.S., France

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