Sparse Representation-based Object Recognition - Sparse Representations and Compressive Sensing for Imaging and Vision - page 69

N

N

where, φ : R → F ⊂ R is an implicit mapping projecting the vector x into a

higher dimensional space, F . Some commonly used kernels include polynomial

kernels

d

κ ( x , y )= ( x , y + c )

and Gaussian kernels

exp

2

−

x − y

κ ( x , y )= ,

c

where c and d are the parameters.
By substituting the mapped features to the formulation of sparse representation,
we arrive at kernel sparse representation

α α 1

) α , α = ˆ arg min subject to φ ( y )= φ ( B (5.14)

where with the abuse of notation we denote In other

words, kernel sparse representation seeks the sparse representation for a mapped
feature under the mapped dictionary in the high dimensional feature space. Problem
( 5.14 ) can be rewritten as

φ ( B )=[ φ ( b 1 ) ,···, φ ( b L )] .

) α

2

+ λ α 1 ,

ˆ

α = min

α φ (

y ) − φ ( B (5.15)

where corresponds to sparser solution. The objective

function in ( 5.15 ) can be simplified as

λ is a parameter and larger λ

) α

2

+ λ α min

α φ (

y ) − φ ( B

T T

= κ ( , )+ α K BB α − K B + λ α

y y 2 α

1

= g ( α )+ λ α ,

1

where

T T

( α )= κ ( , )+ α K

α −

K ,

g y y 2 α

BB B

L × L is a matrix with

K BB ∈ R K BB ( i , j )= κ ( b i , b j ) and

L × 1 T

K ∈ R =[ κ ( b i , y ) ,···, κ ( b L , y )] .

B

The objective is the same as that of sparse coding except for the definition of K

BB

and

B . Hence, the standard numerical tools for solving linear sparse representation
problem can be used to solve the above non-linear sparse coding problem [63].

K