The role of kernels is central to machine learning. Motivated by the importance of power-law distributions in statistical modeling, in this paper, we propose the notion of power-law kernels to investigate power-laws in learning problem. We propose two power-law kernels by generalizing Gaussian and Laplacian kernels. This generalization is based on distributions, arising out of maximization of a generalized information measure known as nonextensive entropy that is very well studied in statistical mechanics. We prove that the proposed kernels are positive definite, and provide some insights regarding the corresponding Reproducing Kernel Hilbert Space (RKHS). We also study practical significance of both kernels in classification and regression, and present some simulation results.

One-against-all and one-against-one are two popular methodologies for reducing multiclass classification problems into a set of binary classifications. In this paper, we are interested in the performance of both one-against-all and one-against-one for classification algorithms, such as decision tree, naïve bayes, support vector machine, and logistic regression. Since both one-against-all and one-against-one work like creating a classification committee, they are expected to improve the performance of classification algorithms. However, our experimental results surprisingly show that one-against-all worsens the performance of the algorithms on most datasets. One-against-one helps, but performs worse than the same iterations of bagging these algorithms. Thus, we conclude that both one-against-all and one-against-one should not be used for the algorithms that can perform multiclass classifications directly. Bagging is better approach for improving their performance.

Classic sparse representation for classification (SRC) method fails to incorporate the label information of training images, and meanwhile has a poor scalability due to the expensive computation for l_1 norm. In this paper, we propose a novel subspace sparse coding method with utilizing label information to effectively classify the images in the subspace. Our new approach unifies the tasks of dimension reduction and supervised sparse vector learning, by simultaneously preserving the data sparse structure and meanwhile seeking the optimal projection direction in the training stage, therefore accelerates the classification process in the test stage. Our method achieves both flat and structured sparsity for the vector representations, therefore making our framework more discriminative during the subspace learning and subsequent classification. The empirical results on 4 benchmark data sets demonstrate the effectiveness of our method.

Resampling techniques such as bagging are often used in supervised learning to produce more accurate classifiers. In this work, we show that multiple-instance learning admits a different form of resampling, which we call "shuffling." In shuffling, we resample instances in such a way that the resulting bags are likely to be correctly labeled. We show that resampling results in both a reduction of bag label noise and a propagation of additional informative constraints to a multiple-instance classifier. We empirically evaluate shuffling in the context of multiple-instance classification and multiple-instance active learning and show that the approach leads to significant improvements in accuracy.

Road travel costs are important knowledge hidden in large-scale GPS trajectory data sets, the discovery of which can benefit many applications such as intelligent route planning and automatic driving navigation. While there are previous studies which tackled this task by modeling it as a regression problem with spatial smoothness taken into account, they unreasonably assumed that the latent cost of each road remains unchanged over time. Other works on route planning and recommendation that have considered temporal factors simply assumed that the temporal dynamics be known in advance as a parametric function over time, which is not faithful to reality. To overcome these limitations, in this paper, we propose an extension to a previous static trajectory regression framework by learning the temporal dynamics of road travel costs in an innovative non-parametric manner which can effectively overcome the temporal sparsity problem. In particular, we unify multiple different trajectory regression problems in a multi-task framework by introducing a novel cross-task regularization which encourages temporal smoothness on the change of road travel costs. We then propose an efficient block coordinate descent method to solve the resulting problem by exploiting its separable structures and prove its convergence to global optimum. Experiments conducted on both synthetic and real data sets demonstrate the effectiveness of our method and its improved accuracy on travel time prediction.

Non-negative tensor factorization (NTF) has attracted great attention in the machine learning community. In this paper, we extend traditional non-negative tensor factorization into a supervised discriminative decomposition, referred as Supervised Non-negative Tensor Factorization with Maximum-Margin Constraint(SNTFM2). SNTFM2 formulates the optimal discriminative factorization of non-negative tensorial data as a coupled least-squares optimization problem via a maximum-margin method. As a result, SNTFM2 not only faithfully approximates the tensorial data by additive combinations of the basis, but also obtains a strong generalization power to discriminative analysis (in particularfor classification in this paper). The experimental results show the superiority of our proposed model over state-of-the-art techniques on both toy and real world data sets.

This paper presents a new approach to representation discovery in reinforcement learning (RL) using basis adaptation. We introduce a general framework for basis adaptation as {\em nonlinear separable least-squares value function approximation} based on finding Frechet gradients of an error function using variable projection functionals. We then present a scalable proximal gradient-based approach for basis adaptation using the recently proposed mirror-descent framework for RL. Unlike traditional temporal-difference (TD) methods for RL, mirror descent based RL methods undertake proximal gradient updates of weights in a dual space, which is linked together with the primal space using a Legendre transform involving the gradient of a strongly convex function. Mirror descent RL can be viewed as a proximal TD algorithm using Bregman divergence as the distance generating function. We present a new class of regularized proximal-gradient based TD methods, which combine feature selection through sparse L1 regularization and basis adaptation. Experimental results are provided to illustrate and validate the approach.

Hierarchical classification (HC) plays an significant role in machine learning and data mining. However, most of the state-of-the-art HC algorithms suffer from high computational costs. To improve the performance of solving, we propose a stochastic perceptron (SP) algorithm in the large margin framework. In particular, a stochastic choice procedure is devised to decide the direction of next iteration. We prove that after finite iterations the SP algorithm yields a sub-optimal solution with high probability if the input instances are separable. For large-scale and high-dimensional data sets, we reform SP to the kernel version (KSP), which dramatically reduces the memory space needed. The KSP algorithm has the merit of low space complexity as well as low time complexity. The experimental results show that our KSP approach achieves almost the same accuracy as the contemporary algorithms on the real-world data sets, but with much less CPU running time.

Model selection based on classical information criteria, such as BIC, is generally computationally demanding, but its properties are well studied. On the other hand, model selection based on parameter shrinkage by $\ell_1$-type penalties is computationally efficient. In this paper we make an attempt to combine their strengths, and propose a simple approach that penalizes the likelihood with data-dependent $\ell_1$ penalties as in adaptive Lasso and exploits a fixed penalization parameter. Even for finite samples, its model selection results approximately coincide with those based on information criteria; in particular, we show that in some special cases, this approach and the corresponding information criterion produce exactly the same model. One can also consider this approach as a way to directly determine the penalization parameter in adaptive Lasso to achieve information criteria-like model selection. As extensions, we apply this idea to complex models including Gaussian mixture model and mixture of factor analyzers, whose model selection is traditionally difficult to do; by adopting suitable penalties, we provide continuous approximators to the corresponding information criteria, which are easy to optimize and enable efficient model selection.

We present ML4PG - a machine learning extension for Proof General. It allows users to gather proof statistics related to shapes of goals, sequences of applied tactics, and proof tree structures from the libraries of interactive higher-order proofs written in Coq and SSReflect. The gathered data is clustered using the state-of-the-art machine learning algorithms available in MATLAB and Weka. ML4PG provides automated interfacing between Proof General and MATLAB/Weka. The results of clustering are used by ML4PG to provide proof hints in the process of interactive proof development.