Chih-Jen Lin

Projected Gradient Methods for Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) can be formulated as a minimization problem with bound constraints. Although bound-constrained optimization has been studied extensively in both theory and practice, so far no study has formally applied its techniques to NMF. In this letter, we propose two projected gradient methods for NMF, both of which exhibit strong optimization properties. We discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach. A simple Matlab code is also provided.

Asymptotic behaviors of support vector machines with Gaussian kernel

Support vector machines (SVMs) with the gaussian (RBF) kernel have been popular for practical use. Model selection in this class of SVMs involves two hyper parameters: the penalty parameter C and the kernel width . This letter analyzes the behavior of the SVM classifier when these hyper parameters take very small or very large values. Our results help in understanding the hyperparameter space that leads to an efficient heuristic method of searching for hyperparameter values with small generalization errors. The analysis also indicates that if complete model selection using the gaussian kernel has been conducted, there is no need to consider linear SVM.

A note on Platt's probabilistic outputs for support vector machines

Abstract Platt’s probabilistic outputs for Support Vector Machines (Platt, J. in Smola, A., et al. (eds.) Advances in large margin classifiers. Cambridge, 2000) has been popular for applications that require posterior class probabilities. In this note, we propose an improved algorithm that theoretically converges and avoids numerical difficulties. A simple and ready-to-use pseudo code is included.

Combining SVMs with Various Feature Selection Strategies

This article investigates the performance of combining support vector machines (SVM) and various feature selection strategies. Some of them are filter-type approaches: general feature selection methods independent of SVM, and some are wrapper-type methods: modifications of SVM which can be used to select features. We apply these strategies while participating to the NIPS 2003 Feature Selection Challenge and rank third as a group.

A dual coordinate descent method for large-scale linear SVM

In many applications, data appear with a huge number of instances as well as features. Linear Support Vector Machines (SVM) is one of the most popular tools to deal with such large-scale sparse data. This paper presents a novel dual coordinate descent method for linear SVM with L1-and L2-loss functions. The proposed method is simple and reaches an ε-accurate solution in O(log(1/ε)) iterations. Experiments indicate that our method is much faster than state of the art solvers such as Pegasos, TRON, SVMperf, and a recent primal coordinate descent implementation.

Load forecasting using support vector Machines: a study on EUNITE competition 2001

Load forecasting is usually made by constructing models on relative information, such as climate and previous load demand data. In 2001, EUNITE network organized a competition aiming at mid-term load forecasting (predicting daily maximum load of the next 31 days). During the competition we proposed a support vector machine (SVM) model, which was the winning entry, to solve the problem. In this paper, we discuss in detail how SVM, a new learning technique, is successfully applied to load forecasting. In addition, motivated by the competition results and the approaches by other participants, more experiments and deeper analyses are conducted and presented here. Some important conclusions from the results are that temperature (or other types of climate information) might not be useful in such a mid-term load forecasting problem and that the introduction of time-series concept may improve the forecasting.

Predicting subcellular localization of proteins for Gram‐negative bacteria by support vector machines based on n‐peptide compositions

Gram‐negative bacteria have five major subcellular localization sites: the cytoplasm, the periplasm, the inner membrane, the outer membrane, and the extracellular space. The subcellular location of a protein can provide valuable information about its function. With the rapid increase of sequenced genomic data, the need for an automated and accurate tool to predict subcellular localization becomes increasingly important. We present an approach to predict subcellular localization for Gram‐negative bacteria. This method uses the support vector machines trained by multiple feature vectors based on n‐peptide compositions. For a standard data set comprising 1443 proteins, the overall prediction accuracy reaches 89%, which, to the best of our knowledge, is the highest prediction rate ever reported. Our prediction is 14% higher than that of the recently developed multimodular PSORT‐B. Because of its simplicity, this approach can be easily extended to other organisms and should be a useful tool for the high‐throughput and large‐scale analysis of proteomic and genomic data.

Training ν -Support Vector Classifiers: Theory and Algorithms

The -support vector machine (-SVM) for classification proposed by Schlkopf, Smola, Williamson, and Bartlett (2000) has the advantage of using a parameter on controlling the number of support vectors. In this article, we investigate the relation between -SVM and C-SVM in detail. We show that in general they are two different problems with the same optimal solution set. Hence, we may expect that many numerical aspects of solving them are similar. However, compared to regular C-SVM, the formulation of -SVM is more complicated, so up to now there have been no effective methods for solving large-scale -SVM. We propose a decomposition method for -SVM that is competitive with existing methods for C-SVM. We also discuss the behavior of -SVM by some numerical experiments.

Parallel Spectral Clustering in Distributed Systems

Spectral clustering algorithms have been shown to be more effective in finding clusters than some traditional algorithms, such as k-means. However, spectral clustering suffers from a scalability problem in both memory use and computational time when the size of a data set is large. To perform clustering on large data sets, we investigate two representative ways of approximating the dense similarity matrix. We compare one approach by sparsifying the matrix with another by the Nyström method. We then pick the strategy of sparsifying the matrix via retaining nearest neighbors and investigate its parallelization. We parallelize both memory use and computation on distributed computers. Through an empirical study on a document data set of 193,844 instances and a photo data set of 2,121,863, we show that our parallel algorithm can effectively handle large problems.

On the Convergence of Multiplicative Update Algorithms for Nonnegative Matrix Factorization

Nonnegative matrix factorization (NMF) is useful to find basis information of nonnegative data. Currently, multiplicative updates are a simple and popular way to find the factorization. However, for the common NMF approach of minimizing the Euclidean distance between approximate and true values, no proof has shown that multiplicative updates converge to a stationary point of the NMF optimization problem. Stationarity is important as it is a necessary condition of a local minimum. This paper discusses the difficulty of proving the convergence. We propose slight modifications of existing updates and prove their convergence. Techniques invented in this paper may be applied to prove the convergence for other bound-constrained optimization problems.