1. Sparse Representation and Compressed Sensing: Theory and Algorithms
Yi Ma1,2 Allen Yang3 John Wright1
1Microsoft Research Asia
2University of Illinois
at Urbana-Champaign
3University of California Berkeley
CVPR Tutorial, June 20, 2009
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2. MOTIVATION – Applications to a variety of vision problems
Face Recognition:
Wright et al PAMI ’09, Huang CVPR ’08, Wagner CVPR ’09 …
Image Enhancement and Superresolution:
Elad TIP ’06, Huang CVPR ‘08, …
Image Classification:
Mairal CVPR ‘08, Rodriguez ‘07, many others …
Multiple Motion Segmentation:
Rao CVPR ‘08, Elfamhir CVPR ’09 …
… and many others, including this conference
Today I will focus on one particular observation in automatic face recognition.

3. When and why can we expect such good performance?
MOTIVATION – Applications to a variety of vision problems
Face Recognition:
Wright et al PAMI ’09, Huang CVPR ’09, Wagner CVPR ’09 …
Image Enhancement and Superresolution:
Elad TIP ’06, Huang CVPR ‘08, …
Image Classification:
Mairal CVPR ‘08, Rodriguez ‘07, …
Multiple Motion Segmentation:
Rao CVPR ‘08, Elfamhir CVPR ’09 …
… and many others, including this conference
When and why can we expect such good performance?
A closer look at the theory …
Today I will focus on one particular observation in automatic face recognition.

4. SPARSE REPRESENTATION – Model problem
Underdetermined system of linear equations, ? …
Observation
Unknown
Today I will focus on one particular observation in automatic face recognition.
Two interpretations:
Compressed sensing: A as sensing matrix
Sparse representation: A as overcomplete dictionary

5. SPARSE REPRESENTATION – Model problem
Underdetermined system of linear equations, ? …
Observation
Unknown
Today I will focus on one particular observation in automatic face recognition.
Many more unknowns than observations → no unique solution.
Classical answer: minimum -norm solution
Emerging applications: instead desire sparse solutions

6. SPARSE SOLUTIONS – Uniqueness
Look for the sparsest solution:
- number of nonzero elements
Today I will focus on one particular observation in automatic face recognition.

7. SPARSE SOLUTIONS – Uniqueness
Look for the sparsest solution:
- number of nonzero elements
Is the sparsest solution unique?
- size of smallest set of linearly dependent columns of A.
Today I will focus on one particular observation in automatic face recognition.

8. SPARSE SOLUTIONS – Uniqueness
Look for the sparsest solution:
- number of nonzero elements
Is the sparsest solution unique?
- size of smallest set of linearly dependent columns of A.
Today I will focus on one particular observation in automatic face recognition.
Proposition [Gorodnitsky & Rao ‘97]:
If with ,
then is the unique solution to

9. SPARSE SOLUTIONS – So How Do We Compute It?
Looking for the sparsest solution:
Bad News: NP-hard in the worst case, hard to approximate within
certain constants [Amaldi & Kann ’95].
Today I will focus on one particular observation in automatic face recognition.

10. SPARSE SOLUTIONS – So How Do We Compute It?
Looking for the sparsest solution:
Bad News: NP-hard in the worst case, hard to approximate within
certain constants [Amaldi & Kann ’95].
Maybe we can still solve important cases?
Greedy algorithms:
Matching Pursuit, Orthogonal Matching Pursuit [Mallat & Zhang ‘93]
CoSAMP [Needell & Tropp ‘08]
Convex programming [Chen, Donoho & Saunders ‘94]
Today I will focus on one particular observation in automatic face recognition.

11. SPARSE SOLUTIONS – The Heuristic
Looking for the sparsest solution:
Intractable.
convex relaxation
Linear program,
solvable in
polynomial time.
Why ? Convex envelope of over the unit cube:
Today I will focus on one particular observation in automatic face recognition.
Rich applied history – geosciences, sparse coding in vision, statistics

12. EQUIVALENCE – A stronger motivation
In many cases, the solutions to (P0) and (P1) are exactly the same:
Theorem [Candes & Tao ’04, Donoho ‘04]:
For Gaussian , with overwhelming probability, whenever
Today I will focus on one particular observation in automatic face recognition.
“ -minimization recovers any sufficiently sparse solution”

13. GUARANTEES – “Well-Spread” A
Mutual coherence: largest inner product between distinct columns of A
Low mutual coherence: vectors are well-spread in the space
Today I will focus on one particular observation in automatic face recognition.

14. GUARANTEES – “Well-Spread” A
Mutual coherence:
Theorem [Elad & Donoho ’03, Gribvonel & Nielsen ‘03]:
minimization uniquely recovers any with
Strong point: checkable condition.
Weakness:
low coherence can only guarantee recovery up to nonzeros.
Today I will focus on one particular observation in automatic face recognition.

15. GUARANTEES – Beyond Coherence
Low coherence:
“any submatrix consisting of two columns of A is well-conditioned”
Stronger bounds by looking at larger submatrices?
Restricted Isometry Constants:
s.t. for all
-sparse ,
Today I will focus on one particular observation in automatic face recognition.
Low RIC:
“Column submatrices of A are uniformly well-conditioned”

16. GUARANTEES – Beyond Coherence
Restricted Isometry Constants:
s.t. for all
-sparse ,
Theorem [Candes & Tao ’04, Candes ‘07]:
If , then -minimization recovers any k-sparse .
Today I will focus on one particular observation in automatic face recognition.
For random A, this guarantees recovery up to linear sparsity:

17. GUARANTEES – Sharp Conditions?
Necessary and sufficient condition:
solves
iff
Today I will focus on one particular observation in automatic face recognition.
polytope spanned
by columns of A
and their negatives

18. GUARANTEES – Geometric Interpretation
Necessary and sufficient condition:
[Donoho ’06]
[Donoho + Tanner ’08]
uniquely recovers with support and signs iff
is a simplicial face of
Today I will focus on one particular observation in automatic face recognition.
Uniform guarantees for -sparse P centrally neighborly.

19. GUARANTEES – Geometric Interpretation
Geometric understanding gives sharp thresholds for sparse recovery with Gaussian A [Donoho & Tanner ‘08]:
Weak threshold
Failure almost always
Sparsity
Today I will focus on one particular observation in automatic face recognition.
Strong threshold
Success almost always
Success always
Aspect ratio of A

20. GUARANTEES – Geometric Interpretation
Explicit formulas in the wide-matrix limit [Donoho & Tanner ‘08]:
Today I will focus on one particular observation in automatic face recognition.
Weak threshold:
Strong threshold:

21. GUARANTEES – Noisy Measurements
What if there is noise in the observation?
Gaussian or bounded 2-norm
Natural approach: relax the constraint:
Studied in several literatures
Statistics – LASSO
Signal processing – BPDN.
Today I will focus on one particular observation in automatic face recognition.

22. GUARANTEES – Noisy Measurements
What if there is noise in the observation?
Natural approach:
Theorem [Donoho, Elad & Temlyakov ‘06]:
Recovery is stable:
Today I will focus on one particular observation in automatic face recognition.
See also [Candes-Romberg-Tao ‘06], [Wainwright ‘06],
[Meinshausen & Yu ’06], [Zhao & Yu ‘06], …

23. GUARANTEES – Noisy Measurements
What if there is noise in the observation?
Natural approach:
Theorem [Candes-Romberg-Tao ‘06]:
Recovery is stable – for A satisfying an appropriate condition,
Today I will focus on one particular observation in automatic face recognition.
– best S-term approximation
See also [Donoho ‘06], [Wainwright ‘06],
[Meinshausen & Yu ’06], [Zhao & Yu ‘06], …

24. CONNECTIONS – Sketching and Expanders
Similar sparse recovery problems explored in data streaming community:
Combinatorial algorithms → fast encoding/decoding at expense of
suboptimal # of measurements
Based on ideas from group testing, expander graphs 2 5 1
Sketch
Data stream …
Today I will focus on one particular observation in automatic face recognition.
[Gilbert et al ‘06],
[Indyk ‘08],
[Xu & Hassibi ‘08]

25. CONNECTIONS – High dimensional geometry
Sparse recovery guarantees can also be derived via probabilistic
constructions from high-dimensional geometry:
The Johnson-Lindenstrauss lemma
Dvoretsky’s almost-spherical section theorem:
There exist subspaces of dimension as high as
on which the and norms are comparable:
Given n points a random projection into
dimensions preserves pairwise distances:
Today I will focus on one particular observation in automatic face recognition.

26. THE STORY SO FAR – Sparse recovery guarantees
Sparse solutions can often be recovered by linear programming.
Performance guarantees for arbitrary matrices with
“uniformly well-spread columns”:
(in)-coherence
Restricted Isometry
Sharp conditions via polytope geometry
Very well-understood performance for random matrices
What about matrices arising in vision… ?
Today I will focus on one particular observation in automatic face recognition.

27. Combined training dictionary
PRIOR WORK - Face Recognition as Sparse Representation
Linear subspace model for images of same face under varying illumination:
Subject i Training
If test image is also of subject , then
for some
Can represent any test image wrt the entire training set as
Key point : e can be large, unbounded, *not* noise
Test image
Combined training dictionary
coefficients
corruption, occlusion

28. PRIOR WORK - Face Recognition as Sparse Representation
Underdetermined system of linear equations in unknowns :
Solution is not unique … but
should be sparse: ideally, only supported on images of the same subject
expected to be sparse: occlusion only affects a subset of the pixels
Seek the sparsest solution:
convex relaxation
Wright, Yang, Ganesh, Sastry, and Ma. Robust Face Recognition via Sparse Representation, PAMI 2008

29. GUARANTEES – What About Vision Problems?
Behavior under varying levels of random pixel corruption:
Recognition rate
99.3%
90.7%
37.5%
Today I will focus on one particular observation in automatic face recognition.
Can existing theory explain this phenomenon?

30. PRIOR WORK - Error Correction by minimization
Candes and Tao [IT ‘05]:
Apply parity check matrix s.t , yielding
Set
Recover from clean system
Underdetermined system in sparse e only
Succeeds whenever in the reduced system

31. PRIOR WORK - Error Correction by minimization
Candes and Tao [IT ‘05]:
Apply parity check matrix s.t , yielding
Set
Recover from clean system
Underdetermined system in sparse e only
Succeeds whenever in the reduced system
This work:
Instead solve
Can be applied when A is wide (no parity check).

32. PRIOR WORK - Error Correction by minimization
Candes and Tao [IT ‘05]:
Apply parity check matrix s.t , yielding
Set
Recover from clean system
Underdetermined system in sparse e only
Succeeds whenever in the reduced system
This work:
Instead solve
Succeeds whenever in the expanded system

33. GUARANTEES – What About Vision Problems?
Highly coherent
( volume )
very sparse: # images per subject,
often nonnegative (illumination cone models).
as dense as possible: robust to highest possible corruption.
Today I will focus on one particular observation in automatic face recognition.
Results so far: should not succeed.

34. SIMULATION - Dense Error Correction?
As dimension , an even more striking phenomenon emerges:

35. SIMULATION - Dense Error Correction?
As dimension , an even more striking phenomenon emerges:

36. SIMULATION - Dense Error Correction?
As dimension , an even more striking phenomenon emerges:

37. SIMULATION - Dense Error Correction?
As dimension , an even more striking phenomenon emerges:

38. SIMULATION - Dense Error Correction?
As dimension , an even more striking phenomenon emerges:

39. SIMULATION - Dense Error Correction?
As dimension , an even more striking phenomenon emerges:
Conjecture: If the matrices are sufficiently coherent, then for any error fraction , as , solving
corrects almost any error with .

40. DATA MODEL - Cross-and-Bouquet
Our model for should capture the fact that the columns are tightly clustered
around a common mean :
L^-norm of deviations well-controlled ( -> v )
Mean is mostly incoherent with standard (error) basis
We call this the “Cross-and-Bouquet’’ (CAB) model.

41. ASYMPTOTIC SETTING - Weak Proportional Growth
Observation dimension
Problem size grows proportionally:
Error support grows proportionally:
Support size sublinear in :

42. ASYMPTOTIC SETTING - Weak Proportional Growth
Observation dimension
Problem size grows proportionally:
Error support grows proportionally:
Support size sublinear in :
Sublinear growth of is necessary to correct arbitrary fractions of errors:
Need at least “clean” equations.
Empirical Observation:
If grows linearly in , sharp phase transition at

43. NOTATION - Correct Recovery of Solutions
Whether is recovered depends only on
Call recoverable if with these signs and support
and the minimizer is unique.

44. MAIN RESULT - Correction of Arbitrary Error Fractions
Recall notation:
“ recovers any sparse signal from almost any error with density less than 1”

45. SIMULATION - Arbitrary Errors in WPG
Fraction of correct successes for increasing m ( , )

46. SIMULATION - Phase Transition in Proportional Growth
What if grows linearly with m?
Asymptotically sharp phase transition, similar to that observed by Donoho and Tanner for homogeneous Gaussian matrices

47. SIMULATION - Comparison to Alternative Approaches
“L1 - [A I]”:
“L1 -  comp”:
“ROMP”: Regularized orthogonal matching pursuit
Candes + Tao ‘05
Needell + Vershynin ‘08

48. SIMULATION - Error Correction with Real Faces
For real face images, weak proportional growth corresponds to the setting where the total image resolution grows proportionally to the size of the database.
Fraction of correct recoveries
Above: corrupted images.
( % probability of correct recovery )
Below: reconstruction.

49. SUMMARY – Sparse Representation in Theory and Practice
So far:
Face recognition as a motivating example
Sparse recovery guarantees for generic systems
New theory and new phenomena from face data
After the break:
Algorithms for sparse recovery
Many more applications in vision and sensor networks
Matrix extensions: missing data imputation and robust PCA
Today I will focus on one particular observation in automatic face recognition.