Laplacian Support Vector Machines Trained in the Primal

In the last few years, due to the growing ubiquity of unlabeled data, much effort has been spent by the machine learning community to develop better understanding and improve the quality of classifiers exploiting unlabeled data. Following the manifold regularization approach, Laplacian Support Vector Machines (LapSVMs) have shown the state of the art performance in semi--supervised classification. In this paper we present two strategies to solve the primal LapSVM problem, in order to overcome some issues of the original dual formulation. Whereas training a LapSVM in the dual requires two steps, using the primal form allows us to collapse training to a single step. Moreover, the computational complexity of the training algorithm is reduced from O(n^3) to O(n^2) using preconditioned conjugate gradient, where n is the combined number of labeled and unlabeled examples. We speed up training by using an early stopping strategy based on the prediction on unlabeled data or, if available, on labeled validation examples. This allows the algorithm to quickly compute approximate solutions with roughly the same classification accuracy as the optimal ones, considerably reducing the training time. Due to its simplicity, training LapSVM in the primal can be the starting point for additional enhancements of the original LapSVM formulation, such as those for dealing with large datasets. We present an extensive experimental evaluation on real world data showing the benefits of the proposed approach.Comment: 39 pages, 14 figure

Comparison of SVM Optimization Techniques in the Primal

This paper examines the efficacy of different optimization techniques in a primal formulation of a support vector machine (SVM). Three main techniques are compared. The dataset used to compare all three techniques was the Sentiment Analysis on Movie Reviews dataset, from kaggle.com

Representing data by sparse combination of contextual data points for classification

In this paper, we study the problem of using contextual da- ta points of a data point for its classification problem. We propose to represent a data point as the sparse linear reconstruction of its context, and learn the sparse context to gather with a linear classifier in a su- pervised way to increase its discriminative ability. We proposed a novel formulation for context learning, by modeling the learning of context reconstruction coefficients and classifier in a unified objective. In this objective, the reconstruction error is minimized and the coefficient spar- sity is encouraged. Moreover, the hinge loss of the classifier is minimized and the complexity of the classifier is reduced. This objective is opti- mized by an alternative strategy in an iterative algorithm. Experiments on three benchmark data set show its advantage over state-of-the-art context-based data representation and classification methods

Distributed Majorization-Minimization for Laplacian Regularized Problems

We consider the problem of minimizing a block separable convex function (possibly nondifferentiable, and including constraints) plus Laplacian regularization, a problem that arises in applications including model fitting, regularizing stratified models, and multi-period portfolio optimization. We develop a distributed majorization-minimization method for this general problem, and derive a complete, self-contained, general, and simple proof of convergence. Our method is able to scale to very large problems, and we illustrate our approach on two applications, demonstrating its scalability and accuracy.Comment: 18 pages, 3 figure

Supervised learning of sparse context reconstruction coefficients for data representation and classification

Context of data points, which is usually defined as the other data points in a data set, has been found to play important roles in data representation and classification. In this paper, we study the problem of using context of a data point for its classification problem. Our work is inspired by the observation that actually only very few data points are critical in the context of a data point for its representation and classification. We propose to represent a data point as the sparse linear combination of its context, and learn the sparse context in a supervised way to increase its discriminative ability. To this end, we proposed a novel formulation for context learning, by modeling the learning of context parameter and classifier in a unified objective, and optimizing it with an alternative strategy in an iterative algorithm. Experiments on three benchmark data set show its advantage over state-of-the-art context-based data representation and classification methods.Comment: arXiv admin note: substantial text overlap with arXiv:1507.0001

Semi-Supervised Representation Learning based on Probabilistic Labeling

In this paper, we present a new algorithm for semi-supervised representation learning. In this algorithm, we first find a vector representation for the labels of the data points based on their local positions in the space. Then, we map the data to lower-dimensional space using a linear transformation such that the dependency between the transformed data and the assigned labels is maximized. In fact, we try to find a mapping that is as discriminative as possible. The approach will use Hilber-Schmidt Independence Criterion (HSIC) as the dependence measure. We also present a kernelized version of the algorithm, which allows non-linear transformations and provides more flexibility in finding the appropriate mapping. Use of unlabeled data for learning new representation is not always beneficial and there is no algorithm that can deterministically guarantee the improvement of the performance by exploiting unlabeled data. Therefore, we also propose a bound on the performance of the algorithm, which can be used to determine the effectiveness of using the unlabeled data in the algorithm. We demonstrate the ability of the algorithm in finding the transformation using both toy examples and real-world datasets.Comment: 8 pages, 7 figure

Data-dependent kernels in nearly-linear time

We propose a method to efficiently construct data-dependent kernels which can make use of large quantities of (unlabeled) data. Our construction makes an approximation in the standard construction of semi-supervised kernels in Sindhwani et al. 2005. In typical cases these kernels can be computed in nearly-linear time (in the amount of data), improving on the cubic time of the standard construction, enabling large scale semi-supervised learning in a variety of contexts. The methods are validated on semi-supervised and unsupervised problems on data sets containing upto 64,000 sample points

Scaling Active Search using Linear Similarity Functions

Active Search has become an increasingly useful tool in information retrieval problems where the goal is to discover as many target elements as possible using only limited label queries. With the advent of big data, there is a growing emphasis on the scalability of such techniques to handle very large and very complex datasets. In this paper, we consider the problem of Active Search where we are given a similarity function between data points. We look at an algorithm introduced by Wang et al. [2013] for Active Search over graphs and propose crucial modifications which allow it to scale significantly. Their approach selects points by minimizing an energy function over the graph induced by the similarity function on the data. Our modifications require the similarity function to be a dot-product between feature vectors of data points, equivalent to having a linear kernel for the adjacency matrix. With this, we are able to scale tremendously: for $n$ data points, the original algorithm runs in $O(n^2)$ time per iteration while ours runs in only $O(nr + r^2)$ given $r$-dimensional features. We also describe a simple alternate approach using a weighted-neighbor predictor which also scales well. In our experiments, we show that our method is competitive with existing semi-supervised approaches. We also briefly discuss conditions under which our algorithm performs well.Comment: To be published as conference paper at IJCAI 2017, 11 pages, 2 figure

A Laplacian-Based Approach for Finding Near Globally Optimal Solutions to OPF Problems

A semidefinite programming (SDP) relaxation globally solves many optimal power flow (OPF) problems. For other OPF problems where the SDP relaxation only provides a lower bound on the objective value rather than the globally optimal decision variables, recent literature has proposed a penalization approach to find feasible points that are often nearly globally optimal. A disadvantage of this penalization approach is the need to specify penalty parameters. This paper presents an alternative approach that algorithmically determines a penalization appropriate for many OPF problems. The proposed approach constrains the generation cost to be close to the lower bound from the SDP relaxation. The objective function is specified using iteratively determined weights for a Laplacian matrix. This approach yields feasible points to the OPF problem that are guaranteed to have objective values near the global optimum due to the constraint on generation cost. The proposed approach is demonstrated on both small OPF problems and a variety of large test cases representing portions of European power systems.Comment: 11 pages, 4 figure

Multi-Label Learning with Global and Local Label Correlation

It is well-known that exploiting label correlations is important to multi-label learning. Existing approaches either assume that the label correlations are global and shared by all instances; or that the label correlations are local and shared only by a data subset. In fact, in the real-world applications, both cases may occur that some label correlations are globally applicable and some are shared only in a local group of instances. Moreover, it is also a usual case that only partial labels are observed, which makes the exploitation of the label correlations much more difficult. That is, it is hard to estimate the label correlations when many labels are absent. In this paper, we propose a new multi-label approach GLOCAL dealing with both the full-label and the missing-label cases, exploiting global and local label correlations simultaneously, through learning a latent label representation and optimizing label manifolds. The extensive experimental studies validate the effectiveness of our approach on both full-label and missing-label data