Sparse recovery without
incoherence
Rachel Ward
University of Texas at Austin
February 2013
Research supported in part by the ONR and Alfred P. Sloan Foundation
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High dimensional data with low-dimensional structure is everywhere
Discrete images
Smooth functions
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Low-rank matrices
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The sparsity model
I
x ∈ RN is s-sparse if |{j : |xj | > 0}| ≤ s.
I
For general x ∈ RN , the best s-sparse approximation is
xs = arg min{kx − uk : u ∈ RN is s-sparse}
(from x to xs )
Compressed sensing
I
Instead of observing x = (xj )N
j=1 directly, acquire through
m N general linear measurements
I
Useful when it is expensive or impossible to acquire and store
the entire signal.
yk =
N
X
j=1
4
aj,k xj ,
k = 1, . . . , m
5
Sparsity in images
(xj1 ,j2 )
p
|xj1 +1,j2 − xj1 ,j2 |2 + |xj1 ,j2 +1 − xj1 ,j2 |2
Image as two-dimensional array of pixels (xj1 ,j2 ) ∈ Rn×n
Images are approximately sparse in spatial discrete differences /
wavelet bases / local cosine bases
MRI compressive imaging
In Magnetic Resonance Imaging (MRI), rotating magnetic field
measurements are Fourier transform measurements:
1X
yk1 ,k2 =
xj1 ,j2 e 2πi(k1 j1 +k2 j2 )/n , −n/2 + 1 ≤ k1 , k2 , ≤ n/2
n
j1 ,j2
Each measurement takes a certain amount of time — reduced
number of samples m n2 means reduced time of MRI scan.
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Polynomial interpolation
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0
−1
−0.8
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−0.4
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0
0.2
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Smooth functions can be characterized by decay rates in their
Fourier series / Legendre polynomial series expansions.
Multivariate polynomial interpolation: generalized Polynomial
Chaos expansions for Uncertainty Quantification
8
Incoherent sampling
Set-up
1. N-dimensional signal of interest x = (xj )N
j=1 with assumed
sparsity: x = Ψb and b is sparse
2. m × N measurement matrix A (m N)
3. Measurements y = Ax.
 
 

y  = 
A
 
 
 
 x 
 
 
 
Reconstructing sparse signals
Primer theorem
Assume A is an m × N matrix with the property that if x1 and x2
are s-sparse and not equal, then Ax1 6= Ax2 .
Let y = Ax for some s-sparse x. Then
x = arg min |{j : |uj | > 0}| such that
u∈RN
Au = y .
Sufficient condition for exact recovery: If Au = 0 then either u = 0
or u is not 2s-sparse.
Very sufficient condition is the null space property: If Au = 0, then
ku2s k1 ≤ 21 ku − u2s k1 .
Still, the support minimization problem is computationally
intractable in general.
10
11
Reconstructing sparse signals
Primer theorem
Assume A is an m × N matrix with the property that if x1 and x2
are s-sparse and not equal, then Ax1 6= Ax2 .
Let y = Ax for some s-sparse x. Then
x = arg min |{j : |uj | > 0}| such that
u∈RN
Au = y .
Sufficient condition for exact recovery: If Au = 0 then either u = 0
or u is not 2s-sparse.
Very sufficient condition is the null space property: If Au = 0, then
ku2s k1 ≤ 21 ku − u2s k1 .
Still, the support minimization problem is computationally
intractable in general.
12
Reconstructing sparse signals
Primer theorem
Assume A is an m × N matrix with the property that if x1 and x2
are s-sparse and not equal, then Ax1 6= Ax2 .
Let y = Ax for some s-sparse x. Then
x = arg min |{j : |uj | > 0}| such that
u∈RN
Au = y .
Sufficient condition for exact recovery: If Au = 0 then either u = 0
or u is not 2s-sparse.
Very sufficient condition is the null space property: If Au = 0, then
ku2s k1 ≤ 21 ku − u2s k1 .
Still, the support minimization problem is computationally
intractable in general.
13
Reconstructing sparse signals
Primer theorem
Assume A is an m × N matrix with the property that if x1 and x2
are s-sparse and not equal, then Ax1 6= Ax2 .
Let y = Ax for some s-sparse x. Then
x = arg min |{j : |uj | > 0}| such that
u∈RN
Au = y .
Sufficient condition for exact recovery: If Au = 0 then either u = 0
or u is not 2s-sparse.
Very sufficient condition is the null space property: If Au = 0, then
ku2s k1 ≤ 21 ku − u2s k1 .
Still, the support minimization problem is computationally
intractable in general.
14
Reconstructing sparse signals
Primer theorem
Assume A is an m × N matrix with the property that if x1 and x2
are s-sparse and not equal, then Ax1 6= Ax2 .
Let y = Ax for some s-sparse x. Then
x = arg min |{j : |uj | > 0}| such that
u∈RN
Au = y .
Sufficient condition for exact recovery: If Au = 0 then either u = 0
or u is not 2s-sparse.
Very sufficient condition is the null space property: If Au = 0, then
ku2s k1 ≤ 21 ku − u2s k1 .
Still, the support minimization problem is computationally
intractable in general.
Reconstructing sparse signals
Convex relaxation: `1 -minimization
Reconstruct sparse x from y = Ax by
x
#
= arg min
u∈RN
N
X
|uj | such that
Au = y .
j=1
If A satisfies the null space property1 , then
1. If x is s-sparse then x # = x:
2. If x is almost s-sparse, then
1
x#
Exact recovery
is close to x:
Stability
Candès, Romberg, Tao, Donoho, Cohen, Dahmen, DeVore, . . .
Reconstructing sparse signals
Convex relaxation: `1 -minimization
Reconstruct sparse x from y = Ax by
x
#
= arg min
u∈RN
N
X
|uj | such that
Au = y .
j=1
If A satisfies the null space property1 , then
1. If x is s-sparse then x # = x:
2. If x is almost s-sparse, then
1
16
x#
Exact recovery
is close to x:
Stability
Candès, Romberg, Tao, Donoho, Cohen, Dahmen, DeVore, . . .
Incoherent sampling
y = Ax
Let (Φ, Ψ) is a pair of orthonormal bases of RN .
1. Φ = (φj ) is used for sensing: A is a subset of rows of Φ∗
2. Ψ = (ψk ) is used to sparsely represent x: x = Ψb, and b is
assumed sparse
Definition
The coherence between Φ and Ψ is
√
µ(Φ, Ψ) = N max | hφj , ψk i |
1≤k,j≤N
If µ(Φ, Ψ) = C a constant, then Φ and Ψ are called incoherent.
Incoherent sampling
y = Ax
Let (Φ, Ψ) is a pair of orthonormal bases of RN .
1. Φ = (φj ) is used for sensing: A is a subset of rows of Φ∗
2. Ψ = (ψk ) is used to sparsely represent x: x = Ψb, and b is
assumed sparse
Definition
The coherence between Φ and Ψ is
√
µ(Φ, Ψ) = N max | hφj , ψk i |
1≤k,j≤N
If µ(Φ, Ψ) = C a constant, then Φ and Ψ are called incoherent.
Incoherent sampling
Example:
I
Ψ = Identity. Signal is sparse in canonical/Kroneker basis
I
Φ is discrete Fourier basis,
N−1
1
φj = √ e i2πjk/N
k=0
N
I
The Kronecker and Fourier bases are incoherent:
√
µ(Φ, Ψ) := N max | hφj , ψk i | = 1.
j,k
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Theorem (Sparse recovery via incoherent sampling2 )
Let (Φ, Ψ) be a pair of incoherent bases of RN , with µ(Φ, Ψ) ≤ K .
Let s ≥ 1,
Let m ≥ 100s K 2 log4 (N),
Select m (possibly not distinct) rows of Φ∗ i.i.d. from the uniform
distribution on {1, 2, . . . , N}. to form A : RN → Rm .
The following holds with exceedingly high probability: for all
x ∈ RN , given measurements y = Ax, the approximation
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
2
20
√1 kΨ∗ x
s
− (Ψ∗ x)s k1 .
Candès, Romberg, Tao ’06, Rudelson Vershynin ’08, ...
Theorem (Sparse recovery via incoherent sampling2 )
Let (Φ, Ψ) be a pair of incoherent bases of RN , with µ(Φ, Ψ) ≤ K .
Let s ≥ 1,
Let m ≥ 100s K 2 log4 (N),
Select m (possibly not distinct) rows of Φ∗ i.i.d. from the uniform
distribution on {1, 2, . . . , N}. to form A : RN → Rm .
The following holds with exceedingly high probability: for all
x ∈ RN , given measurements y = Ax, the approximation
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
2
√1 kΨ∗ x
s
− (Ψ∗ x)s k1 .
Candès, Romberg, Tao ’06, Rudelson Vershynin ’08, ...
Theorem (Sparse recovery via incoherent sampling2 )
Let (Φ, Ψ) be a pair of incoherent bases of RN , with µ(Φ, Ψ) ≤ K .
Let s ≥ 1,
Let m ≥ 100s K 2 log4 (N),
Select m (possibly not distinct) rows of Φ∗ i.i.d. from the uniform
distribution on {1, 2, . . . , N}. to form A : RN → Rm .
The following holds with exceedingly high probability: for all
x ∈ RN , given measurements y = Ax, the approximation
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
2
√1 kΨ∗ x
s
− (Ψ∗ x)s k1 .
Candès, Romberg, Tao ’06, Rudelson Vershynin ’08, ...
23
In practice, incoherent sampling is not always
possible. What then are optimal compressed
sampling strategies?
24
Compressed sensing MRI
Image as 2D array of pixels (xj1 ,j2 ) ∈ Rn×n
Natural images are sparsely represented in
2D wavelet bases
Sensing measurements in MRI are 2D
Fourier or K-space measurements,
n
φk1 ,k2 = n1 e i2π(j1 k1 +j2 k2 )/n j ,j =1 ,
1 2
−n/2 + 1 ≤ k1 , k2 ≤ n/2
Wavelet and Fourier bases are maximally coherent: µ(Ψ, Φ) =
√
N
Compressed sensing MRI
The coherence between frequency φk1 ,k2 and the entire bivariate
Haar wavelet basis Ψ = (ψI ) can be bounded by3
√
√
N
µ(φk1 ,k2 , Ψ) = N max | hφk1 ,k2 , ψI i | .
1/2
I
|k1 + 1|2 + |k2 + 1|2
When some elements of the sensing basis are more coherent with
the sparsity basis than others, is it best just to take the m most
coherent measurements?
3
25
Krahmer, W., 2012
Compressed sensing MRI
The coherence between frequency φk1 ,k2 and the entire bivariate
Haar wavelet basis Ψ = (ψI ) can be bounded by3
√
√
N
µ(φk1 ,k2 , Ψ) = N max | hφk1 ,k2 , ψI i | .
1/2
I
|k1 + 1|2 + |k2 + 1|2
When some elements of the sensing basis are more coherent with
the sparsity basis than others, is it best just to take the m most
coherent measurements?
3
26
Krahmer, W., 2012
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Compressed sensing MRI
Reconstructions of an 256 × 256 MRI image from m = .1 × (256)2
frequency measurements using total variation minimization
Pixel space / Frequency space
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Lowest frequencies
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Uniformly subsampled frequencies
Compressed sensing MRI
Several papers have proposed4 to sample K-space according to
densities scaling inversely to a power of the distance to the origin.
This is reminiscent of the coherence between frequency φk1 ,k2 and
bivariate Haar wavelet basis Ψ,
√
N
µ(φk1 ,k2 , Ψ) .
1/2
2
|k1 + 1| + |k2 + 1|2
Define the local coherence function µloc = (µj ) from an
orthonormal basis Φ to an orthonormal basis Ψ as
√
µk := µ(φk , Ψ) = N max | hφk , ψj i |
j
P
2
2
Note the inequality N1 N
j=1 µj ≤ (µ(Φ, Ψ)) .
For Fourier/Wavelets, average coherence is log(N), coherence is N.
4
28
Lustig, Donoho, Pauly 2007, Puy, Vandergheynst, Wiaux 2011
Compressed sensing MRI
Several papers have proposed4 to sample K-space according to
densities scaling inversely to a power of the distance to the origin.
This is reminiscent of the coherence between frequency φk1 ,k2 and
bivariate Haar wavelet basis Ψ,
√
N
µ(φk1 ,k2 , Ψ) .
1/2
2
|k1 + 1| + |k2 + 1|2
Define the local coherence function µloc = (µj ) from an
orthonormal basis Φ to an orthonormal basis Ψ as
√
µk := µ(φk , Ψ) = N max | hφk , ψj i |
j
P
2
2
Note the inequality N1 N
j=1 µj ≤ (µ(Φ, Ψ)) .
For Fourier/Wavelets, average coherence is log(N), coherence is N.
4
29
Lustig, Donoho, Pauly 2007, Puy, Vandergheynst, Wiaux 2011
Compressed sensing MRI
Several papers have proposed4 to sample K-space according to
densities scaling inversely to a power of the distance to the origin.
This is reminiscent of the coherence between frequency φk1 ,k2 and
bivariate Haar wavelet basis Ψ,
√
N
µ(φk1 ,k2 , Ψ) .
1/2
2
|k1 + 1| + |k2 + 1|2
Define the local coherence function µloc = (µj ) from an
orthonormal basis Φ to an orthonormal basis Ψ as
√
µk := µ(φk , Ψ) = N max | hφk , ψj i |
j
P
2
2
Note the inequality N1 N
j=1 µj ≤ (µ(Φ, Ψ)) .
For Fourier/Wavelets, average coherence is log(N), coherence is N.
4
30
Lustig, Donoho, Pauly 2007, Puy, Vandergheynst, Wiaux 2011
Theorem (Coherence-based sampling5 )
Consider a pair of orthonormal bases (Φ, Ψ) with local coherences
bounded by
µj = µ(φj , Ψ) ≤ κj
4
P
2
Let s ≥ 1, and suppose m & s N1 N
j=1 κj log (N).
Select m (possibly not distinct) rows of Φ∗ i.i.d. from the
multinomial distribution on {1, 2, . . . , N} with weights cκ2j to form
A : RN → Rm .
The following holds with exceedingly high probability for all
x ∈ RN . Given measurements y = Ax, the image
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
5
Krahmer, Rauhut, W ’12, ...
√1 kΨ∗ x
s
− (Ψ∗ x)s k1 .
Theorem (Coherence-based sampling5 )
Consider a pair of orthonormal bases (Φ, Ψ) with local coherences
bounded by
µj = µ(φj , Ψ) ≤ κj
4
P
2
Let s ≥ 1, and suppose m & s N1 N
j=1 κj log (N).
Select m (possibly not distinct) rows of Φ∗ i.i.d. from the
multinomial distribution on {1, 2, . . . , N} with weights cκ2j to form
A : RN → Rm .
The following holds with exceedingly high probability for all
x ∈ RN . Given measurements y = Ax, the image
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
5
Krahmer, Rauhut, W ’12, ...
√1 kΨ∗ x
s
− (Ψ∗ x)s k1 .
Theorem (Coherence-based sampling5 )
Consider a pair of orthonormal bases (Φ, Ψ) with local coherences
bounded by
µj = µ(φj , Ψ) ≤ κj
4
P
2
Let s ≥ 1, and suppose m & s N1 N
j=1 κj log (N).
Select m (possibly not distinct) rows of Φ∗ i.i.d. from the
multinomial distribution on {1, 2, . . . , N} with weights cκ2j to form
A : RN → Rm .
The following holds with exceedingly high probability for all
x ∈ RN . Given measurements y = Ax, the image
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
5
Krahmer, Rauhut, W ’12, ...
√1 kΨ∗ x
s
− (Ψ∗ x)s k1 .
34
Corollary for MRI compressive imaging
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Let n ∈ N. Let Ψ be the bivariate Haar wavelet basis. Form
sensing matrix A by selecting m & s · log5 (n) frequency
measurements (k1 , k2 ) i.i.d. from the multinomial distribution with
1
weights pk1 ,k2 ∝ (|k1 |+1)2 +(|k
2.
2 |+1)
The following holds with exceedingly high probability for all
x = (xj1 ,j2 ) ∈ Rn×n . Given measurements y = Ax, the image
x # = arg min kΨ∗ uk1 subject to y = Au
u∈RN
satisfies the error guarantee kx − x # k2 .
√1 kΨ∗ x
s
− (Ψ∗ x)s k1
Comparing different sampling schemes ...
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250
Low-frequencies only
Original MRI image
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Uniform
Equispaced radial lines
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(k1 , k2 ) ∼ (k12 + k22 )−1/2
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(k1 , k2 ) ∼ (k12 + k22 )−1
200
250
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Comparing different sampling schemes ...
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Original
150
MRI image
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150
150
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250
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150
200
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Equispaced radial lines
Low frequencies only
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(k1 , k2 ) ∼ (k12 + k22 )−1/2
250
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150
(k1 , k2 ) ∼ (k12 + k22 )−1
200
250
37
More examples of coherence-based
sampling
Polynomial interpolation
The Legendre polynomials, a smooth function, and its Legendre series coefficients.
I
I
I
The Legendre polynomials (Lj )j≥0 form an orthonormal basis
for L2 ([−1, 1]) with respect to the uniform measure,
R1
hLj , Lk i = −1 Lj (x)Lk (x)dx = δjk .
P
Smoothness assumption on f : f (x) ≈ N
j=0 cj Lj (x) and
|cj | ≤ j −α .
Approximate unknown f from sampling points
f (x1 ), f (x2 ), . . . , f (xm ). Sampling strategy?
39
Polynomial interpolation
The Legendre polynomials, a smooth function, and its Legendre series coefficients.
I
I
I
The Legendre polynomials (Lj )j≥0 form an orthonormal basis
for L2 ([−1, 1]) with respect to the uniform measure,
R1
hLj , Lk i = −1 Lj (x)Lk (x)dx = δjk .
P
Smoothness assumption on f : f (x) ≈ N
j=0 cj Lj (x) and
|cj | ≤ j −α .
Approximate unknown f from sampling points
f (x1 ), f (x2 ), . . . , f (xm ). Sampling strategy?
Polynomial interpolation
The Legendre polynomials, a smooth function, and its Legendre series coefficients.
I
I
I
I
6
40
Legendre √
polynomials are not uniformly bounded,
kLj k∞ = 2j + 1. However they satisfy
1/4
1
| Lj (x)| ≤ κ(x) = √2π 1−x
on the unit interval.
2
R1 2
−1 κ (x)dx ≤ 3 - Infinite-dimensional average local coherence
Coherence-based sampling implies a stable sampling strategy:
x1 , x2 , . . . , xm ∼ π(1−x1 2 )1/2 dx
Stability of Chebyshev sampling aligns with classical results on
Lagrange interpolation6
L. Brutman. Lebesgue functions for polynomial interpolation - a survey.
41
Polynomial interpolation
The Legendre polynomials, a smooth function, and its Legendre series coefficients.
I
I
I
I
6
Legendre √
polynomials are not uniformly bounded,
kLj k∞ = 2j + 1. However they satisfy
1/4
1
| Lj (x)| ≤ κ(x) = √2π 1−x
on the unit interval.
2
R1 2
−1 κ (x)dx ≤ 3 - Infinite-dimensional average local coherence
Coherence-based sampling implies a stable sampling strategy:
x1 , x2 , . . . , xm ∼ π(1−x1 2 )1/2 dx
Stability of Chebyshev sampling aligns with classical results on
Lagrange interpolation6
L. Brutman. Lebesgue functions for polynomial interpolation - a survey.
Polynomial interpolation
The Legendre polynomials, a smooth function, and its Legendre series coefficients.
I
I
I
I
6
Legendre √
polynomials are not uniformly bounded,
kLj k∞ = 2j + 1. However they satisfy
1/4
1
| Lj (x)| ≤ κ(x) = √2π 1−x
on the unit interval.
2
R1 2
−1 κ (x)dx ≤ 3 - Infinite-dimensional average local coherence
Coherence-based sampling implies a stable sampling strategy:
x1 , x2 , . . . , xm ∼ π(1−x1 2 )1/2 dx
Stability of Chebyshev sampling aligns with classical results on
Lagrange interpolation6
L. Brutman. Lebesgue functions for polynomial interpolation - a survey.
Polynomial interpolation
The Legendre polynomials, a smooth function, and its Legendre series coefficients.
I
I
I
I
6
Legendre √
polynomials are not uniformly bounded,
kLj k∞ = 2j + 1. However they satisfy
1/4
1
| Lj (x)| ≤ κ(x) = √2π 1−x
on the unit interval.
2
R1 2
−1 κ (x)dx ≤ 3 - Infinite-dimensional average local coherence
Coherence-based sampling implies a stable sampling strategy:
x1 , x2 , . . . , xm ∼ π(1−x1 2 )1/2 dx
Stability of Chebyshev sampling aligns with classical results on
Lagrange interpolation6
L. Brutman. Lebesgue functions for polynomial interpolation - a survey.
Polynomial Interpolation
uniformly distributed (left) and Chebyshev-distributed (right)
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Low-rank matrix approximation
Just as the convex relaxation of the sparsity of a vector is its `1
norm, the convex relaxation of the rank of a matrix is its nuclear
norm - the sum of its singular values.
Low-rank matrix completion guarantees via nuclear norm
minimization assume incoherence of underlying low-rank matrix7
Coherence-based sampling: incoherence condition can be removed
by row/column weighted sampling. Related to notion of weighted
matrix sampling8
7
8
Candes, Recht, Plan, Montanari, Keshavan, Oh . . .
Negahban, Wainwright 2012
Low-rank matrix approximation
Just as the convex relaxation of the sparsity of a vector is its `1
norm, the convex relaxation of the rank of a matrix is its nuclear
norm - the sum of its singular values.
Low-rank matrix completion guarantees via nuclear norm
minimization assume incoherence of underlying low-rank matrix7
Coherence-based sampling: incoherence condition can be removed
by row/column weighted sampling. Related to notion of weighted
matrix sampling8
7
8
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Candes, Recht, Plan, Montanari, Keshavan, Oh . . .
Negahban, Wainwright 2012
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Summary
Compressed sensing and related optimization problems often
assume incoherence between the sensing and sparsity bases to
derive sparse recovery guarantees.
Incoherence is restrictive and not achievable in many problems of
practical interest.
We introduced the concept of local coherence from one basis to
another, and showed that with a bound on the local coherence,
one may derive sampling strategies and sparse recovery results for
a wide range of new sensing problems.
More can be said: measurement error, instance-optimal results, . . .
Extensions
I
Compressed sensing video?
I
From orthonormal bases to redundant dictionaries
I
Implications for superresolution?
I
Incorporate structured sparsity constraints
I
...
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References
Rauhut, Ward, “Sparse Legendre expansions via
`1 -minimization.” Journal of approximation theory 164.5
(2012): 517-533.
Rauhut, Ward, “Sparse recovery for spherical harmonic
expansions.” arXiv preprint arXiv:1102.4097 (2011).
Burq, Dyatlov, Ward, Zworski, “Weighted eigenfunction
estimates with applications to compressed sensing.” SIAM
Journal on Mathematical Analysis 44.5 (2012), 3481-3501
Krahmer, Ward, “Beyond incoherence: stable and robust
sampling strategies for compressive imaging”. arXiv preprint
arXiv:1210.2380 (2012).
Chen, Bhojanapalli, Sanghavi, Ward, R. “Coherent Matrix
Completion.” In Proceedings of The 31st International
Conference on Machine Learning (2014) 674-682.