Image category learning and classification via optimal linear combination of multiple partially matching kernels

Abstract

Multiple kernel learning (MKL) aims at simultaneously optimizing kernel weights while training the support vector machine (SVM) to get satisfactory classification or regression results. Recent publications and developments based on SVM have shown that by using MKL one can enhance interpretability of the decision function and improve classifier performance, which motivates researchers to explore the use of homogeneous model obtained as linear combination of various types of kernels. In this paper, we show that MKL problems can be solved efficiently by modified projection gradient method and applied for image categorization and object detection. The kernel is defined as a linear combination of feature histogram function that can measure the degree of similarity of partial correspondence between feature sets for discriminative classification, which allows recognition robust to within-class variation, pose changes, and articulation. We evaluate our proposed framework on the ETH-80 dataset for several multi-level image encodings for supervised and unsupervised object recognition and report competitive results.

Appendix

1.1 Derivation of the equivalence between Eqs. 4 and 5

Due to the relationship between $K_m(\cdot, \cdot)$ and $K^{\prime}_m,$ we have

$$ K_m(\cdot, \cdot)=\beta_mK^{\prime}_m $$

(24)

The inner production of ${\mathcal{H}}^{\prime}_m,$ the RKHS induced by $K^{\prime}_m,$ has the following form:

$$ \langle f,g \rangle_{{\mathcal{H}}^{\prime}_m}={\frac{1}{\beta_m}} \langle f,g \rangle_{{\mathcal{H}}_m} $$

(25)

where ${\mathcal{H}}_m$ is induced by $K_m,$ consider the following prime problem

$$ \begin{aligned} & \min_{{f},b,\xi}{\frac{1}{2}}\,\,||f||^2_{{\mathcal{H}}} + C\sum_i \xi_i\\ & {\rm s.t.} \; y_i\left(f(x_i)+b\right)\geq1-\xi_i, \; \forall i\in{1,\ldots,m}\\ & \xi_i\geq0. \; \forall i \in{1,\ldots,m}\\ \end{aligned} $$

(26)

then the following holds

$$ \begin{aligned} ||f||^2_{{\mathcal{H}}}&=\sum_m \langle f(\cdot),f(\cdot) \rangle_{{\mathcal{H}}^{\prime}_m}\\ & = \sum_m {\frac{1}{\beta_m}}||f_m||^2_{{\mathcal{H}}^{\prime}_m}\\ \end{aligned} $$

(27)

finally, the original optimization problem in the original RKHS ${\mathcal{H}}$ take the form of

$$ \begin{aligned} &\min_{{f_m},b,\xi,\beta}\ {\frac{1}{2}}\left(\sum_m {\frac{1}{\beta_m}}||f_m||^2_{{\mathcal{H}}^{\prime}_m}\right) + C\sum_i \xi_i\\ &s.t. \; y_i\left(\sum_mf_m(x_i)+b\right)\geq1-\xi_i, \; \forall i\in{1,\ldots,n}\\ & \sum_m \beta_m = 1, \beta_m\geq0. \; \forall m\in{1,\ldots,M}\\ \end{aligned} $$

(28)

in the RKHS ${\mathcal{H}}^{\prime}_m.$