Regularized kernel discriminant analysis (RKDA) performs linear discriminant analysis in the feature space via the kernel trick. Its performance depends on the selection of kernels. In this paper, we consider the problem of multiple kernel learning (MKL) for RKDA, in which the optimal kernel matrix is obtained as a linear combination of pre-specified kernel matrices. We show that the kernel learning problem in RKDA can be formulated as convex programs. First, we show that this problem can be formulated as a semidefinite program (SDP). Based on the equivalence relationship between RKDA and least square problems in the binary-class case, we propose a convex quadratically constrained quadratic programming (QCQP) formulation for kernel learning in RKDA. A semi-infinite linear programming (SILP) formulation is derived to further improve the efficiency. We extend these formulations to the multi-class case based on a key result established in this paper. That is, the multi-class RKDA kernel learning problem can be decomposed into a set of binary-class kernel learning problems which are constrained to share a common kernel. Based on this decomposition property, SDP formulations are proposed for the multi-class case. Furthermore, it leads naturally to QCQP and SILP formulations. As the performance of RKDA depends on the regularization parameter, we show that this parameter can also be optimized in a joint framework with the kernel. Extensive experiments have been conducted and analyzed, and connections to other algorithms are discussed.

We present a multi-label multiple kernel learning (MKL) formulation in which the data are embedded into a low-dimensional space directed by the instancelabel correlations encoded into a hypergraph. We formulate the problem in the kernel-induced feature space and propose to learn the kernel matrix as a linear combination of a given collection of kernel matrices in the MKL framework. The proposed learning formulation leads to a non-smooth min-max problem, which can be cast into a semi-infinite linear program (SILP). We further propose an approximate formulation with a guaranteed error bound which involves an unconstrained convex optimization problem. In addition, we show that the objective function of the approximate formulation is differentiable with Lipschitz continuous gradient, and hence existing methods can be employed to compute the optimal solution efficiently. We apply the proposed formulation to the automated annotation of Drosophila gene expression pattern images, and promising results have been reported in comparison with representative algorithms. 1

Maximum margin principle has been successfully applied to many supervised and semi-supervised problems in machine learning. Recently, this principle was extended for clustering, referred to as Maximum Margin Clustering (MMC) and achieved promising performance in recent studies. To avoid the problem of local minima, MMC can be solved globally via convex semi-definite programming (SDP) relaxation. Although many efficient approaches have been proposed to alleviate the computational burden of SDP, convex MMCs are still not scalable for medium data sets. In this paper, we propose a novel convex optimization method, LG-MMC, which maximizes the margin of opposite clusters via “Label Generation”. It can be shown that LG-MMC is much more scalable than existing convex approaches. Moreover, we show that our convex relaxation is tighter than state-of-art convex MMCs. Experiments on seventeen UCI datasets and MNIST dataset show significant improvement over existing MMC algorithms. 1

In this paper we describe our TRECVID 2008 video retrieval experiments. The MediaMill team participated in three tasks: concept detection, automatic search, and interactive search. Rather than continuing to increase the number of concept detectors available for retrieval, our TRECVID 2008 experiments focus on increasing the robustness of a small set of detectors. To that end, our concept detection experiments emphasize in particular the role of sampling, the value of color invariant features, the influence of codebook construction, and the effectiveness of kernel-based learning parameters. For retrieval, a robust but limited set of concept detectors necessitates the need to rely on as many auxiliary information channels as possible. Therefore, our automatic search experiments focus on predicting which information channel to trust given a certain topic, leading to a novel framework for predictive video retrieval. To improve the video retrieval results further, our interactive search experiments investigate the roles of visualizing preview results for a certain browse-dimension and active learning mechanisms that learn to solve complex search topics by analysis from user browsing behavior. The 2008 edition of the TRECVID benchmark has been the most successful MediaMill participation to date, resulting in the top ranking for both concept detection and interactive search, and a runner-up ranking for automatic retrieval. Again a lot has been learned during this year’s TRECVID campaign; we highlight the most important lessons at the end of this paper. 1

We consider the problem of high-dimensional non-linear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize non-linear interactions between the original variables. To select efficiently from these many kernels, we use the natural hierarchical structure of the problem to extend the multiple kernel learning framework to kernels that can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a graph-adapted sparsity-inducing norm, in polynomial time in the number of selected kernels. Moreover, we study the consistency of variable selection in high-dimensional settings, showing that under certain assumptions, our regularization framework allows a number of irrelevant variables which is exponential in the number of observations. Our simulations on synthetic datasets and datasets from the UCI repository show state-of-the-art predictive performance for non-linear regression problems. 1

Abstract not found

The Support Vector Machine (SVM) is an acknowledged powerful tool for building classifiers, but it lacks flexibility, in the sense that the kernel is chosen prior to learning. Multiple Kernel Learning (MKL) enables to learn the kernel, from an ensemble of basis kernels, whose combination is optimized in the learning process. Here, we propose Composite Kernel Learning to address the situation where distinct components give rise to a group structure among kernels. Our formulation of the learning problem encompasses several setups, putting more or less emphasis on the group structure. We characterize the convexity of the learning problem, and provide a general wrapper algorithm for computing solutions. Finally, we illustrate the behavior of our method on multi-channel data where groups correpond to channels. 1.

The choice of the kernel function which determines the mapping between the input space and the feature space is of crucial importance to kernel methods. The past few years have seen many efforts in learning either the kernel function or the kernel matrix. In this paper, we address this model selection issue by learning the hyperparameter of the kernel function for a support vector machine (SVM). We trace the solution path with respect to the kernel hyperparameter without having to train the model multiple times. Given a kernel hyperparameter value and the optimal solution obtained for that value, we find that the solutions of the neighborhood hyperparameters can be calculated exactly. However, the solution path does not exhibit piecewise linearity and extends nonlinearly. As a result, the breakpoints cannot be computed in advance. We propose a method to approximate the breakpoints. Our method is both efficient and general in the sense that it can be applied to many kernel functions in common use.

Abstract. Methods for inductive transfer take advantage of knowledge from previous learning tasks to solve a newly given task. In the context of supervised learning, the task is to find a suitable bias for a new dataset, given a set of known datasets. In this paper, we take a kernelbased approach to inductive transfer, that is, we aim at finding a suitable kernel for the new data. In our setup, the kernel is taken from the linear span of a set of predefined kernels. To find such a kernel, we apply convex optimization on two levels. On the base level, we propose an iterative procedure to generate kernels that generalize well on the known datasets. On the meta level, we combine those kernels in a minimization criterion to predict a suitable kernel for the new data. The criterion is based on a meta kernel capturing the similarity of two datasets. In experiments on small molecule and text data, kernel-based inductive transfer showed a statistically significant improvement over the best individual kernel in almost all cases.

For classification with multiple labels, a common approach is to learn a classifier for each label. With a kernel-based classifier, there are two options to set up kernels: select a specific kernel for each label or the same kernel for all labels. In this work, we present a unified framework for multi-label multiple kernel learning, in which the above two approaches can be considered as two extreme cases. Moreover, our framework allows the kernels shared partially among multiple labels, enabling flexible degrees of label commonality. We systematically study how the sharing of kernels among multiple labels affects the performance based on extensive experiments on various benchmark data including images and microarray data. Interesting findings concerning efficacy and efficiency are reported. 1