LEARNING A LOW-RANK SHARED DICTIONARY FOR OBJECT CLASSIFICATIONTiep Vu, Vishal Monga

School of Electrical Engineering and Computer Science, The Pennsylvania State University, USA

Motivation

Typically, different objects usually share commonpatterns (brown bases in D).

⇒ the sparse code matrix is not block diagonal.

≈

[Y1 . . .Yc . . .YC ]

×

[D1 . . . Dc . . . DC ]

X11

Xcc

XCC

Main Contributions

I A new low-rank shared dictionary learning (LRSDL) framework forextracting discriminative and shared patterns.

I New accurate and efficient algorithms for selected existing andproposed dictionary learning methods.

I We derive the computational complexity of numerous dictionarylearning methods.

I Numerous sparse coding and dictionary learning algorithms in themanuscript are reproducible via a user-friendly toolbox.

LRSDL - Idea visualization

In real problems, different classes share somecommon features (represented by D0).

We model the shared dictionary D0 as alow-rank matrix.

≈

[Y1 . . .Yc . . .YC ] [D1 . . .Dc . . .DC D0]

×X

XX

X1 Xc

Xc

XC

X0

X1c

Xcc

XCc

X0cX0c

YcD0 X

0c

Yc

DcXcc

D1X

1c

DCXCc

No constraint

YcD0 X

0c

Yc

DcXcc

D1X

1c DCXCc

LRSDL

Yc = Yc −D0X0c

Goal:‖Yc −DcX

cc −D0X

0c‖2F small.

‖DiXic‖2F small (i 6= c).

m1

‖X1 −M1‖2F

mc

‖Xc −Mc‖2F

mC

‖XC −MC‖2F

m0

‖X0 −M0‖2F

m

Goal:

‖Xc−Mc‖2F (intra class) small.

‖Mc−M‖2F (inter class) lagre.

‖X0 −M0‖2F small.

LRSDL cost function and efficient algorithms

f (D,D0︸ ︷︷ ︸D̄

,X,X0

︸ ︷︷ ︸X̄

) =

f1(D̄,X̄)︷ ︸︸ ︷‖Y−D0X

0 −DX‖2F +∑

i=1,...,C

(‖Yi−D0X

0i −DiX

ii‖2F +

∑

j 6=i

‖D jXji ‖2F)+λ1‖X‖1

+λ2

( ∑

i=1,...,C

(‖Xi −Mi‖2F − ‖Mi −M‖2F

)+ ‖X‖2F+‖X0 −M0‖2F

)

︸ ︷︷ ︸f2(D̄,X̄)

+λ1‖X0‖1+η‖D0‖∗

Without red terms, LRSDL becomes FDDL (M. Yang, ICCV, 2011; IJCV, 2014).

LRSDL- cost function

Definition:

A11 . . . A1C

A21 . . . A2C

. . . . . . . . .AC1 . . . ACC

︸ ︷︷ ︸A

7→ A+

A11 . . . 00 . . . 0. . . . . . . . .0 . . . ACC

︸ ︷︷ ︸M(A)

⇒ function M(•) requires a low computational cost.

Lemma 1: Efficient FDDL solving D using ODL (J. Mairal, JML, 2010)

D = argminD−2trace(EDT ) + trace(FDTD)

where E = YM(X)T , F =M(XXT ).

Lemma 2: Efficient FDDL solving X using FISTA (A. Beck, JIS, 2009)

∂ 12 f1

∂X= M(DTD)X−M(DTY)

∂ 12 f2

∂X= 2X+M− 2

[M1 M2 . . . MC

].

Convergence rate comparison (cost and running time)

20 60 100

22

24

cost

FDDL (LRSDL)

10 30 50

50

150

250

DLSI (I. Ramirez, CVPR, 2010)

10 30 50

0

100

300

COPAR (S. Kong, ECCV, 2012)

20 60 100

02,000

7,000

12,000

02,000

7,000

12,000

iteration

runn

ing

tim

e(s

)

10 30 50

0500

2,000

4,000

iteration

10 30 50

0

2,000

4,000

iteration

Original Algorithms Proposed Efficient Algorithms

Figure: Original vs Proposed algorithm – convergence rate comparisons

Table: Complexity analysis for different dictionary learning methods

Method ComplexityPluggingnumbers

O-DLSI Ck(kd + dn + qkn) +Cqkd3 6.25× 1012

E-DLSI Ck(kd + dn + qkn) +Cd3 +Cqdk(qk + d) 3.75× 1010

O-FDDL C2dk(n +Ck +Cn) +Ck2q(d +C2n) 2.51× 1011

E-FDDL C2k((q + 1)k(d +Cn) + 2dn) 1.29× 1011

O-COPAR C3k2(2d +Ck + qn) +Cqkd3 6.55× 1012

E-COPAR C3k2(2d +Ck + qn) +Cd3 +Cqdk(qk + d) 3.38× 1011

LRSDL C2k((q + 1)k(d +Cn) + 2dn) +C2dkn + (q + q2)dk2 1.3× 1011

Simulated data

1

2

3

4 Shared

Basic elements Samples

1

2

3

4

DLSI bases

1

2

3

4

Accuracy: 95.15%.

LCKSVD11 bases

1

2

3

4

Accuracy: 45.15%.

LCKSVD21 bases

1

2

3

4

Accuracy: 48.15%.

FDDL bases

1

2

3

4

Accuracy: 97.25%.

COPAR bases

1

2

3

4

SharedAccuracy: 99.25%.

LRSDL bases

1

2

3

4

SharedAccuracy: 100%.

1 Z. Jiang, TPAMI, 2013

Datasets Effect of the shared dictionary

a) Extended YaleB b) AR face

c) AR gender

males females

bluebell fritillary sunflower daisy dandelion

d) Oxford Flower

laptop chair motorbike

e) Caltech 101

dragonfly air plane10 20 30 40 50 60 70 80

75

80

85

90

size of the shared dictionary (k0)

Overall accuray (%)

COPAR

LRSDL η = 0

LRSDL η = 0.01

LRSDL η = 0.1

Dependence of overall accuracy on theshared dictionary (AR gender dataset).

Overall accuracy (%) vs. # training samples per class

10 20 3070

80

90

100

YaleB

5 10 15 2040

60

80

100

AR face

50 150 250 350

90

95

AR gender

20 40 6060

70

80

90

Oxford Flower

10 20 30

50

60

70

Caltech 101 SRC2

LCKSVD1LCKSVD2DLSIFDDL

D2L2R2

COPARLRSDL

2J. Wright, TPAMI, 2009

Research was supported by an Office of Naval Research Grant no. N00014-15-1-2042. Email: tiepvu@psu.edu Website: http://signal.ee.psu.edu