Learning Discriminative Stein Kernel for SPD Matrices and Its Applications

Stein kernel has recently shown promising performance on classifying images represented by symmetric positive definite (SPD) matrices. It evaluates the similarity between two SPD matrices through their eigenvalues. In this paper, we argue that directly using the original eigenvalues may be problematic because: i) Eigenvalue estimation becomes biased when the number of samples is inadequate, which may lead to unreliable kernel evaluation; ii) More importantly, eigenvalues only reflect the property of an individual SPD matrix. They are not necessarily optimal for computing Stein kernel when the goal is to discriminate different sets of SPD matrices. To address the two issues in one shot, we propose a discriminative Stein kernel, in which an extra parameter vector is defined to adjust the eigenvalues of the input SPD matrices. The optimal parameter values are sought by optimizing a proxy of classification performance. To show the generality of the proposed method, three different kernel learning criteria that are commonly used in the literature are employed respectively as a proxy. A comprehensive experimental study is conducted on a variety of image classification tasks to compare our proposed discriminative Stein kernel with the original Stein kernel and other commonly used methods for evaluating the similarity between SPD matrices. The experimental results demonstrate that, the discriminative Stein kernel can attain greater discrimination and better align with classification tasks by altering the eigenvalues. This makes it produce higher classification performance than the original Stein kernel and other commonly used methods.Comment: 13 page

Feature and Region Selection for Visual Learning

Visual learning problems such as object classification and action recognition are typically approached using extensions of the popular bag-of-words (BoW) model. Despite its great success, it is unclear what visual features the BoW model is learning: Which regions in the image or video are used to discriminate among classes? Which are the most discriminative visual words? Answering these questions is fundamental for understanding existing BoW models and inspiring better models for visual recognition. To answer these questions, this paper presents a method for feature selection and region selection in the visual BoW model. This allows for an intermediate visualization of the features and regions that are important for visual learning. The main idea is to assign latent weights to the features or regions, and jointly optimize these latent variables with the parameters of a classifier (e.g., support vector machine). There are four main benefits of our approach: (1) Our approach accommodates non-linear additive kernels such as the popular $\chi^2$ and intersection kernel; (2) our approach is able to handle both regions in images and spatio-temporal regions in videos in a unified way; (3) the feature selection problem is convex, and both problems can be solved using a scalable reduced gradient method; (4) we point out strong connections with multiple kernel learning and multiple instance learning approaches. Experimental results in the PASCAL VOC 2007, MSR Action Dataset II and YouTube illustrate the benefits of our approach

Training Support Vector Machines Using Frank-Wolfe Optimization Methods

Training a Support Vector Machine (SVM) requires the solution of a quadratic programming problem (QP) whose computational complexity becomes prohibitively expensive for large scale datasets. Traditional optimization methods cannot be directly applied in these cases, mainly due to memory restrictions. By adopting a slightly different objective function and under mild conditions on the kernel used within the model, efficient algorithms to train SVMs have been devised under the name of Core Vector Machines (CVMs). This framework exploits the equivalence of the resulting learning problem with the task of building a Minimal Enclosing Ball (MEB) problem in a feature space, where data is implicitly embedded by a kernel function. In this paper, we improve on the CVM approach by proposing two novel methods to build SVMs based on the Frank-Wolfe algorithm, recently revisited as a fast method to approximate the solution of a MEB problem. In contrast to CVMs, our algorithms do not require to compute the solutions of a sequence of increasingly complex QPs and are defined by using only analytic optimization steps. Experiments on a large collection of datasets show that our methods scale better than CVMs in most cases, sometimes at the price of a slightly lower accuracy. As CVMs, the proposed methods can be easily extended to machine learning problems other than binary classification. However, effective classifiers are also obtained using kernels which do not satisfy the condition required by CVMs and can thus be used for a wider set of problems

Customizing kernel functions for SVM-based hyperspectral image classification

Previous research applying kernel methods such as support vector machines (SVMs) to hyperspectral image classification has achieved performance competitive with the best available algorithms. However, few efforts have been made to extend SVMs to cover the specific requirements of hyperspectral image classification, for example, by building tailor-made kernels. Observation of real-life spectral imagery from the AVIRIS hyperspectral sensor shows that the useful information for classification is not equally distributed across bands, which provides potential to enhance the SVM's performance through exploring different kernel functions. Spectrally weighted kernels are, therefore, proposed, and a set of particular weights is chosen by either optimizing an estimate of generalization error or evaluating each band's utility level. To assess the effectiveness of the proposed method, experiments are carried out on the publicly available 92AV3C dataset collected from the 220-dimensional AVIRIS hyperspectral sensor. Results indicate that the method is generally effective in improving performance: spectral weighting based on learning weights by gradient descent is found to be slightly better than an alternative method based on estimating ";relevance"; between band information and ground trut