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Abstract. This paper gives an overview of the eigenvalue problems encountered in areas of data mining that are related to dimension reduction. Given some input high-dimensional data, the goal of dimension reduction is to map them to a lowdimensional space such that certain properties of the initial data are preserved. Optimizing the above properties among the reduced data can be typically posed as a trace optimization problem that leads to an eigenvalue problem. There is a rich variety of such problems and the goal of this paper is to unravel relations between them as well as to discuss effective solution techniques. First, we make a distinction between projective methods that determine an explicit linear projection from the high-dimensional space to the low-dimensional space, and nonlinear methods where the mapping between the two is nonlinear and implicit. Then, we show that all of the eigenvalue problems solved in the context of explicit projections can be viewed as the projected analogues of the so-called nonlinear or implicit projections. We also discuss kernels as a means of unifying both types of methods and revisit some of the equivalences between methods established in this way. Finally, we provide some illustrative examples to showcase the behavior and the particular characteristics of the various dimension reduction methods on real world data sets.

Is there a principled way to learn a probabilistic discriminative classifier from an unlabeled data set? We present a framework that simultaneously clusters the data and trains a discriminative classifier. We call it Regularized Information Maximization (RIM). RIM optimizes an intuitive information-theoretic objective function which balances class separation, class balance and classifier complexity. The approach can flexibly incorporate different likelihood functions, express prior assumptions about the relative size of different classes and incorporate partial labels for semi-supervised learning. In particular, we instantiate the framework to unsupervised, multi-class kernelized logistic regression. Our empirical evaluation indicates that RIM outperforms existing methods on several real data sets, and demonstrates that RIM is an effective model selection method. 1

Previous studies of Non-Parametric Kernel (NPK) learning usually reduce to solving some Semi-Definite Programming (SDP) problem by a standard SDP solver. However, time complexity of standard interior-point SDP solvers could be as high as O(n 6.5). Such intensive computation cost prohibits NPK learning applicable to real applications, even for data sets of moderate size. In this paper, we propose an efficient approach to NPK learning from side information, referred to as SimpleNPKL, which can efficiently learn non-parametric kernels from large sets of pairwise constraints. In particular, we show that the proposed SimpleNPKL with linear loss has a closed-form solution that can be simply computed by the Lanczos algorithm. Moreover, we show that the SimpleNPKL with square hinge loss can be re-formulated as a saddle-point optimization task, which can be further solved by a fast iterative algorithm. In contrast to the previous approaches, our empirical results show that our new technique achieves the same accuracy, but is significantly more efficient and scalable. 1.

The maximum margin clustering approach is a recently proposed extension of the concept of support vector machines to the clustering problem. Briefly stated, it aims at finding an optimal partition of the data into two classes such that the margin induced by a subsequent application of a support vector machine is maximal. We propose a method based on stochastic search to address this hard optimization problem. While a direct implementation would be infeasible for large data sets, we present an efficient computational shortcut for assessing the “quality ” of intermediate solutions. Experimental results show that our approach outperforms existing methods in terms of clustering accuracy. 1.

criterion (CLUHSIC) is a recent clustering algorithm that maximizes the dependence between cluster labels and data observations according to the Hilbert Schmidt independence criterion (HSIC). It is unique in that structure information on the cluster outputs can be easily utilized in the clustering process. However, while the choice of the loss function is known to be very important in supervised learning with structured outputs, we will show in this paper that CLUHSIC is implicitly using the often inappropriate zero-one loss. We propose an extension called CLUHSICAL (which stands for “Clustering using HSIC and loss”) which explicitly considers both the output dependency and loss function. Its optimization problem has the same form as CLUHSIC, except that its partition matrix is constructed in a different manner. Experimental results on a number of datasets with structured outputs show that CLUHSICAL often outperforms CLUHSIC in terms of both structured loss and clustering accuracy.

CLUHSIC is a recent clustering framework that unifies the geometric, spectral and statistical views of clustering. In this paper, we show that the recently proposed discriminative view of clustering, which includes the DIFFRAC and DisKmeans algorithms, can also be unified under the CLUH-SIC framework. Moreover, CLUHSIC involves integer programming and one has to resort to heuristics such as iterative local optimization. In this paper, we propose two relaxations that are much more disciplined. The first one uses spectral techniques while the second one is based on semidefinite programming (SDP). Experimental results on a number of structured clustering tasks show that the proposed method significantly outperforms existing optimization methods for CLUHSIC. Moreover, it can also be used in semi-supervised classification. Experiments on real-world protein subcellular localization data sets clearly demonstrate the ability of CLUHSIC in incorporating structural and evolutionary information. 1

Bag of features (BoF) representation has attracted an increasing amount of attention in large scale image processing systems. BoF representation treats images as loose collections of local invariant descriptors extracted from them. The visual codebook is generally constructed by using an unsupervised algorithm such as K-means to quantize the local descriptors into clusters. Images are then represented by the frequency histograms of the codewords contained in them. To build a compact and discriminative codebook, codeword selection has become an indispensable tool. However, most of the existing codeword selection algorithms are supervised and the human labeling may be very expensive. In this paper, we consider the problem of unsupervisedcodeword selection, and propose a novel algorithm called Discriminative Codeword Selection (DCS). Motivated from recent studies on discriminative clustering, the central idea of our proposed algorithm is to select those codewords so that the cluster structure of the image database can be best respected. Specifically, a multi-output linear function is fitted to model the relationship between the data matrix after codeword selection and the indicator matrix. The most discriminative codewords are thus defined as those leading to minimal fitting error. Experiments on image retrieval and clustering have demonstrated the effectiveness of the proposed method.

We present a new clustering algorithm by proposing a convex relaxation of hierarchical clustering, which results in a family of objective functions with a natural geometric interpretation. We give efficient algorithms for calculating the continuous regularization path of solutions, and discuss relative advantages of the parameters. Our method experimentally gives state-ofthe-art results similar to spectral clustering for non-convex clusters, and has the added benefit of learning a tree structure from the data. 1.

Previous studies of Non-Parametric Kernel Learning (NPKL) usually formulate the learning task as a Semi-Definite Programming (SDP) problem that is often solved by some general purpose SDP solvers. However, for N data examples, the time complexity of NPKL using a standard interiorpoint SDP solver could be as high as O(N 6.5), which prohibits NPKL methods applicable to real applications, even for data sets of moderate size. In this paper, we present a family of efficient NPKL algorithms, termed “SimpleNPKL”, which can learn non-parametric kernels from a large set of pairwise constraints efficiently. In particular, we propose two efficient SimpleNPKL algorithms. One is SimpleNPKL algorithm with linear loss, which enjoys a closed-form solution that can be efficiently computed by the Lanczos sparse eigen decomposition technique. Another one is SimpleNPKL algorithm with other loss functions (including square hinge loss, hinge loss, square loss) that can be re-formulated as a saddle-point optimization problem, which can be further resolved by a fast iterative algorithm. In contrast to the previous NPKL approaches, our empirical results show that the proposed new technique, maintaining the same accuracy, is significantly more efficient and scalable. Finally, we also demonstrate that the proposed new technique is also applicable to speed up many kernel learning tasks, including colored maximum variance unfolding, minimum volume embedding, and structure preserving embedding.