Murad Megjhani MATH : 6397 1. Agenda Sparse Coding Dictionary Learning Problem Formulation (Kernel) Results and Discussions 2

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DESIGN OF NON-LINEAR KERNEL DICTIONARIES FOR

OBJECT RECOGNITION

Murad Megjhani

MATH : 6397

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Agenda

• Sparse Coding• Dictionary Learning• Problem Formulation (Kernel)• Results and Discussions

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MotivationGiven a 16x16(or nxn) image patch x, we can represent it using 256 real numbers(pixels).

Problem: Can we find or learn a better representation for this?

Given a set of images, learn a better way to represent image other than pixels.

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What is Sparse Linear Model

signal.abenRLet x

nxK

K1 R],...,d[dD Let

D is “adapted” to x if it can represent it with a few basis vectors—that is, there exists a sparse vector γ in Rk such that x ≈Dγ. We call γ the sparse code.

D

x

be a set of normalized “basis vectors”.Lets call it Dictionary.

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Sparse Coding Illustration Natural Images Bases [d1 , …, d64]

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» 0.8 * + 0.3 * + 0.5 *

x » 0.8 * d36 + 0.3 * d42

+ 0.5

* d63 [0, 0, …, 0, 0.8, 0, …, 0, 0.3, 0, …, 0, 0.5, …] = [γ1, …, γ64] (feature representation)

Test example

Compact & easilyinterpretable

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Notation

nxKD RKR

nRx nxNX R

KxNR

n dimensional Signal Vector

Matrix of N signal vectors of n dimension

Over-complete basis -Dictionary of size K (K>>n)

Sparse representation for input signal xMatrix of Sparse Vectors

nd R Atom : Elementary signal representing template

.

≈.

… …ΓX DN21 x..................x,x K21 ddd ..........,

N21 ..................,

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Sparse Coding Problem

)(γ ||D|| j

2

2jjγ j

xmin

ψ induces sparsity in γj

• The l0 “pseudo norm”. ||γ||0 ≡ #{i s.t. γ[i] ≠ 0} (NP-hard).

• The l1 norm. ||γ||1 ≡ ∑ | γ[i] | (convex).

data fitting Sparsity inducing regularization

This is a selection problem.

When ψ is the l1 norm, the problem is called LASSO [1] or Basis Pursuit [2].

When ψ is the l0 norm, the problem is called Matching Pursuit [3] or Orthogonal Matching Pursuit [4].

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Dictionary Learning Problem

)(γ ||DX||2

1j

2

FD

,min

data fitting Sparsity inducing regularization

Designed DictionaryD can be designed like in Haar [5], Curvelets [6], …

Learned Dictionary D can be created from data as in “Sparse coding with an over-complete basis set” [7] and K-SVD [8]

We will study two of these algorithms today and see how they can be Kernel-aized [9].

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• Next steps: given the previously found

atoms, find the next one to best fit the residual.

• The MP is one of the greedy algorithms that finds one atom at a time [4].

• Step 1: find the one atom that best matches the signal.

• The Orthogonal MP (OMP) is an improved version that re-evaluates the coefficients by Least-Squares after each round.

D

x

Sparse Coding Algorithms Matching Pursuit

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Orthogonal Matching PursuitT tosubject D

2

2

xmin

x r residual and 0; tscoefficien ; Set Active 0 S

for iter = 1…T doSelect the atom which most reduces the objective

rd 1-iiSi C

,maxargi

Update the active set: }{SS i

Update the residual Dri x

Update the coefficients

x

,ˆ

DDDD : OMP

rd]i[ : MP

T

S

1

S

T

SSS

1-ii

end for

1.

2.

3.

4.

5.

6.

7.

Input x, Dictionary D and Sparsity Threshold T

MP : Updates only one coefficient corresponding to the selected atom

OMP : Updates coefficient of all the coefficients in the active set

Initialize

.

≈.

… …ΓX D

Initialize D Sparse Code Dictionary Update

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Dictionary Learning Algorithms:

K-SVD

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The K-SVD algorithm:Train an explicit dictionary from examples

.

…

Set of Examples

X.

…ΓD

The examples are linear combinations of the atoms

Each representation uses at most T atoms

The target function to minimize:

T ||γ|| i s.t. ||DX|| 0i

2

FD,Γ Γmin

Input Output


K-SVD

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For the jth example

For sparse coding, use batch OMP, or any other sparse coding algorithm

.

≈.

… …ΓX D

The Sparse Coding Stage:

T ||γ|| i s.t. ||D-X 0i

2

F ΓΓ

||min

T ||γ|| i s.t. ||D- 0j

2

2jj

x||minj


K-SVD

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For the kth atom

(Residual)

.

≈.

… …ΓX D

T ||γ|| s.t. i ||γdγd X|| ||γdX|| 0i

2

F

T

kkkj

T

jj

2

F

K

1j

T

jjD D

minmin

T || s.t. i ||γdE|| 0i

2

F

T

kkkdk

||min kj

T

jjK γd X E

Dictionary Update Stage:

T ||γ|| i s.t. ||D-X 0i

2

F ΓD

||min


K-SVD

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We can do better!

.

≈.

… …Γ

sparsity?

X D


||γdE|| 2

F

T

kkkdk

min ||γdE|| 2

F

T

kkkd kk

,

min


K-SVD

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d ,γmin

Tk k

Only some of the examples use atom

dk

When updating γk, only re-compute

the coefficients for those examples

dk

γkT

Solve with SVD

We want to solve:

Ek

2

F

~ ~

~



K-SVD

N1i T tosubject DX0i

2

2D

...min,

Summary:

Input :X, Sparsity Threshold T

Initialization : Set the random normalized dictionary matrix D(0) ∈ Rn×K . Set J = 1Repeat until convergence ( or for some fixed number of iterations)

Sparse Coding Stage: Use any Pursuit algorithm to compute γi for i = 1…N

T ||γ|| i s.t. ||D|| 0i

2

2iii

xmin

Codebook Update Stage : for k = 1…K

• Define the group of examples that use dk, ωk = {i|1 ≤ i ≤ N, xi(k) ≠ 0}

• Compute

kj

T

jjK γd-XE

• Restrict Ek by choosing only the columns corresponding to those elements that initially used dk in their representation, and obtain R

KE

• Apply SVD decomposition TR

K VUE

Update : 1

R

k1k ).v11(ud ,, Set J = J+1

Output : D,Γ 19


K-SVD

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Non-Linear Dictionary Learning Problem Formulation

dictionarysought the is D

in XX nxN ΗSpaceFeature )(R

Goal is to learn the non-linear dictionary in the feature space H by solving

T ||γ|| i s.t. ||DX)(|| 0i

2

FD,Γ Γmin

Proposition 1**: There exists an optimal solution D* to (1) that has the following form

NxK Asomefor X)A(D R*

(1)

**For proof refer Appendix VI of H. Van Nguyen, V. M. Patel, N. M. Nasrabadi, and R. Chellappa, “Design of non-linear kernel dictionaries for object recognition.,” IEEE Trans. Image Process., vol. 22, no. 12, pp. 5123–35, Dec. 2013

T ||γ|| i s.t. ||X)A(X)(|| 0i

2

FD,Γ Γmin

Equation (1) now can be written as

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2

F

2

F ( ||(( ΓΓ- A-IX)X)AX) ||

tr

(( tr

ΓΓ

ΓΓ

A-IXXTA-I

A-IX)TX)TA-I

,K

Matrix. Kernel a is XX ,K

This gives the motivation to formulate the dictionary learning in functional spaceusing the kernel trick.

Non-Linear Dictionary Learning Problem Formulation

Kernel-Orthogonal Matching Pursuit

T tosubject A)X()(0

2

2

zmin

; r residual and 0; tscoefficien ; Set Active zS

for iter = 1…T do

iii

T

Si

aX d In |dr|C

)(Space Hilbertmaxargi

Update the active set: }{SS i

SSS A

Update the coefficients

T

S

1

S

T

SS

SS

T

S

1

S

T

SSS

X)A,(X)A(X,A

)( and (X)A D In DDDD

zKK

zzSpace Hilbertz

end for

1.

2.3.

4.

5.

6.

7.

Input: z, kernel ‘K’, Sparsity Threshold T, Coefficient Matrix A

Select the atom which most reduces the objective

i

T

SiiSi

i

T

S

SSSi

T

S

i

T

SS

i

T

i

T

a) X)(X,- Y)(z,( where )(

a) X)(X,- Y)(z,(

A where )aX()X(

)aX()AX(

)aX(r dr

C

KKmaxargi

KK

)()((z)

)()((z)

)(

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Initialize:

Output: γ

Kernel-Dictionary Learning Algorithms: K-SVD Summary:

Input: X, Sparsity Threshold T, Kernel function K

Initialization: Set the random normalized dictionary matrix A(0) ∈ Rn×K . Set J = 1

Repeat until convergence ( or for some fixed number of iterations)

Sparse Coding Stage: Use any KOMP to compute γi for i = 1…N given A(J-1)

Codebook Update Stage: for each column ak

(J-1) in A(J-1) k = 1…K• Define the group of examples that use ak,

ωk = {i|1 ≤ i ≤ N, xi(k) ≠ 0}• Compute

)γ(aM and )γa-(I where(X)M(X)

)γ(X)(a)γa-(X)(I

γ(X)aγ(X)a-(X)

γ(X)a-(X)

as written becan (X)A-(X)error The

kkkjjkkk

kkjj

kkjj

jj

kj

2

F

2

Fkj

2

Fkj

2

F

K

1j

2

F

EE ||||

||||

||||

||||

||||

• Restrict Ek by choosing only the columns corresponding to those elements that initially used ak in their representation, and obtain R

KE

• Apply SVD decomposition vEa updated choose

VV)X)(E(X,)(E

1

R

K

1(J)

k

TR

K

R

K

K

Set J = J+1

Output: A,Γ

N1i T tosubject A)X()X(0i

2

2D

...min,

kj

E )γa-(I jjk

Where v1 is the first vector of V corresponding to the largest singular value σ1

2 in Δ

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Results and Discussion• Synthetic data ( not discussed)• Kernel Sparse Representation• Digit Recognition• Caltech-101 and Caltech-256 Object Recognition

Kernel Sparse Representation

Compares the mean-squared-error (MSE) of an image from the USPS dataset when approximated using the first m dominant kernel PCA components and m = [1, . . . , 20] kernel dictionary atoms (i.e. T0 = m).

“It is clearly seen that the MSE decays much faster for kernel KSVD than kernel PCA with respect to the number of selected bases. This observation implies that the image is nonlinearly sparse and learning a dictionary in the high dimensional feature space can provide a better representation of data.”

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•Kernel Sparse Representation•Digit Recognition•Caltech-101 and Caltech-256 Object Recognition

Results and Discussion

Dataset • USPS handwritten digit database. • Image Size 16x16 making the dimension of the vector 256 and number

of classes = 10 ( number of digits).• Ntrain = 500 samples for training and Ntest = 200 samples for testing – for

each class.

Parameter Selection• The selection of parameters is done through a 5-fold cross-validation.• Parameters

o K (size of dictionary) = 300 atoms.o T0 (sparsity constraint)= 5.o maximum number of training iterations = 80.o Kernel Type : polynomial kernel of degree 4.

Digit Recognition

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Results and DiscussionDigit Recognition

Approach 1 : Distributive Approach

X1 = [xi1,…,xiN] ∈ R256×500 i=1...

X10 = [xi1,…,xiN] ∈ R256×500 i=10

compute

Training Examples A separate Dictionary for each class was learned using Kernel-KSVD

A1 R∈ 500×300 ...

A10 R∈ 500×300

Testing : Given a test image z R∈ 256 , perform KOMP

z R∈ 256

γ1 R∈ 300 using A1 ...

γ10 R∈ 300 using A10

r1

.

.

.r10

)r( i101i ...

minargclass

[1,...,10]i )A(A)A()( )( iiii ii

T

iii

2

2iii YYY2AXr ,Kz,Kzz,K(z)

Pre-images of learned atoms : Since the dictionary is learned in the kernel space the atoms in the dictionary need to be converted to back to the Euclidean space to view the atom**.

** J. T.-Y. Kwok and I. W.-H. Tsang, “The pre-image problem in kernel methods,” IEEE Trans. Neural Netw., vol. 15, no. 6, pp. 1517–1525, Nov. 2004.

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Testing : Given a test image z R∈ 256 , perform KOMP on joined Dictionary [A1,…,A10 ] Sparsity Constraint T = 10( in this case)

z R∈ 256 γ1 R∈ 3000 using [A1,…,A10]

r1

.

.

.r10

)r( i101i ...

minargclass

** J. T. Kwok and I. W. Tsang, “The pre-image problem in kernel methods.,” IEEE Trans. Neural Netw., vol. 15, no. 6, pp. 1517–25, Nov. 2004..

Digit Recognition

Approach 2 : Collective Approach

X1 = [xi1,…,xiN] ∈ R256×500 i=1...

X10 = [xi1,…,xiN] ∈ R256×500 i=10

compute

Training Examples A separate Dictionary for each class was learned using Kernel-KSVD

A1 R∈ 500×300 ...

A10 R∈ 500×300

Pre-images of learned atoms : Since the dictionary is learned in the kernel space the atoms in the dictionary need to be converted to back to the Euclidean space to view the atom**[10].

[1,...,10]i )A(A)A()( )( iiii ii

T

iii

2

2iii YYY2AXr ,Kz,Kzz,K(z)

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Comparison of digit recognition accuracies for different methods in the presence of Gaussian noise and missing-pixel effects. Red color and orange color represent the distributive and collective classification approaches for kernel KSVD, respectively. In order to avoid clutter, we only report distributive approach for kernel MOD which gives better performance for this dataset. (a) Missing pixels. (b) Gaussian noise.

Digit Recognition

Performance Comparison – KSVD, Kernel PCA, Kernel K-SVD and Kernel MOD under different scenarios – missing pixels, different noise.

Kernel KSVD classification accuracy versus the polynomial degree of the USPS dataset.

The second set of experiments examines the effects of parameters choices on the overall recognition performances. Figure shows the classification accuracy of kernel KSVD as we vary the degree of polynomial kernel. The best error rate of 1.6% is achieved with the polynomial degree of 4.

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Results and DiscussionCaltech-101 and Caltech-256 Object Recognition

Dataset • Caltech-101 database. • 101 object classes, and 1 background class – collected randomly

from Internet• Each category contains 31-800 images• Average size of the image = 300x300 pixels• Diverse and challenging dataset as it includes objects like

building, musical instruments, animals and natural scenes

Parameter Selection

• The selection of parameters is done through a 5-fold cross-validation• Parameters

o K (size of dictionary) = 300 atoms.o T0 (sparsity constraint)= 5.o Maximum number of training iterations =80.o Kernel Type : polynomial kernel of degree 4.

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Confusion matrix of Kernel KSVD recognition performances on Caltech 101 dataset The rows and columns correspond to true labels and predicted labels, respectively. The dictionary is learned from 3030 images where each class contributes 30 images. The sparsity is set at 30 for both training and testing. Although the confusion matrix contains all classes, only a subset of class labels is displayed for better legibility.

• Train on N images where N={5,10,15,20,25,30} and test on rest.

• Some categories are very small so end up with just one single image for testing.

• To compensate for the variations in class size they normalize the recognition results by the number of test images to get per-class accuracies.

• Final results is obtained by averaging per-class accuracies across 102 categories.

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#train samples 5 10 15 20 25 30

Griffin[11] 44.2 54.5 59.0 63.3 65.8 67.6

Germet[12] - - - - - 64.16

Yang[13] - - 67.0 - - 73.2

KSVD[8] 49.8 59.8 65.2 68.7 71.0 73.2

LC-KSVD[14] 54.0 63.1 67.7 70.5 72.3 73.6

D-Kernel (MOD) 53.8 64.0 70.2 73.8 76.4 78.1

C-Kernel (MOD) 56.2 67.7 72.4 75.6 77.5 80.0

D-Kernel (KSVD) 54.2 64.5 70.2 74.0 76.5 78.5

C-Kernel (KSVD) 56.5 67.2 72.5 75.8 77.6 80.1

Performance Comparison on Caltech – 101 Dataset

#train samples 15 30

Griffin[11] 28.3 34.1

Germet[12] - 27.2

Yang[13] 34.4 41.2

D-Kernel (MOD) 34.2 41.2

C-Kernel (MOD) 34.6 42.7

D-Kernel (KSVD) 34.5 41.4

C-Kernel (KSVD) 34.8 42.5

Performance Comparison on Caltech – 256 Dataset

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References1. R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” J. R. Stat. Soc., vol. Vol. 58, no. No. 1, pp. 267–288,

1996.

2. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic Decomposition by Basis Pursuit,” SIAM J. Sci. Comput., vol. 20, no. 1, pp. 33–61, Jan. 1998.

3. S. G. Mallat, “Matching pursuits with time-frequency dictionaries,” IEEE Trans. Signal Process., vol. 41, no. 12, pp. 3397–3415, 1993.

4. Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad, “Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers, 1993, pp. 40–44.

5. A. Haar, “Zur Theorie der orthogonalen Funktionensysteme,” Math. Ann., vol. 71, no. 1, pp. 38–53, Mar. 1911.

6. E. J. Candes and D. L. Donoho, “Curvelets: A surprisingly effective nonadaptive representation for objects with edges,” 2000.

7. B. A. Olshausen and D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?,” Vision Res., vol. 37, no. 23, pp. 3311–25, Dec. 1997.

8. M. Aharon, M. Elad, and A. Bruckstein, “K-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation,” IEEE Trans. Signal Process., vol. 54, no. 11, pp. 4311–4322, Nov. 2006.

9. H. Van Nguyen, V. M. Patel, N. M. Nasrabadi, and R. Chellappa, “Design of non-linear kernel dictionaries for object recognition.,” IEEE Trans. Image Process., vol. 22, no. 12, pp. 5123–35, Dec. 2013.

10. J. T. Kwok and I. W. Tsang, “The pre-image problem in kernel methods.,” IEEE Trans. Neural Netw., vol. 15, no. 6, pp. 1517–25, Nov. 2004.

11. G. Griffin, A. Holub, and P. Perona, “Caltech-256 Object Category Dataset.” California Institute of Technology, 10-Mar-2007.

12. J. C. van Gemert, J.-M. Geusebroek, C. J. Veenman, and A. W. M. Smeulders, “Kernel codebooks for scene categorization,” in Computer Vision--ECCV 2008, Springer, 2008, pp. 696–709.

13. T. Huang, “Linear spatial pyramid matching using sparse coding for image classification,” in 2009 IEEE Conference on Computer Vision and Pattern Recognition, 2009, pp. 1794–1801.

14. Z. Jiang, Z. Lin, and L. S. Davis, “Learning a discriminative dictionary for sparse coding via label consistent K-SVD,” in CVPR 2011, 2011, pp. 1697–1704.

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Thank You

Q&A(let Q be sparse )

Documents

Murad Megjhani MATH : 6397 1. Agenda Sparse Coding Dictionary Learning Problem Formulation (Kernel) Results and Discussions 2