Recently, supervised dimensionality reduction has been gaining attention, owing to the realization that data labels are often available and strongly suggest important underlying structures in the data. In this paper, we present a novel convex supervised dimensionality reduction approach based on exponential family PCA and provide a simple but novel form to project new testing data into the embedded space. This convex approach successfully avoids the local optima of the EM learning. Moreover, by introducing a sample-based multinomial approximation to exponential family models, it avoids the limitation of the prevailing Gaussian assumptions of standard PCA, and produces a kernelized formulation for nonlinear supervised dimensionality reduction. A training algorithm is then devised based on a subgradient bundle method, whose scalability can be gained through a coordinate descent procedure. The advantage of our global optimization approach is demonstrated by empirical results over both synthetic and real data.

Principal component analysis (PCA) is an essential tool for data analysis and unsupervised dimensionality reduction, whose goal is to find, among linear combinations of the data variables, a sequence of orthogonal factors that most efficiently explain the variance of the observations. One of its main shortcomings is that, even if PCA finds a small number of important factors, the factor themselves typically involve all original variables. In the last decade, several alternatives to PCA which find sparse and potentially interpretable factors have been proposed, notably nonnegative matrix factorization (NMF) [2] and sparse PCA (SPCA) [3, 4, 5]. However, in many applications, only constraining the size of the factors does not seem appropriate because the considered factors are not only expected to be sparse but also to have a certain structure. In fact, the popularity of NMF for face image analysis owes essentially to the fact that the method happens to retrieve sets of variables that are localized on the face and capture some features or parts of the face which seem intuitively meaningful given our a priori.

Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To alleviate this drawback, various sparse PCA approaches were proposed in literature [15, 6, 17, 28, 8, 25, 18, 7, 16]. Despite success in achieving sparsity, some important properties enjoyed by the standard PCA are lost in these methods such as uncorrelation of PCs and orthogonality of loading vectors. Also, the total explained variance that they attempt to maximize can be too optimistic. In this paper we propose a new formulation for sparse PCA, aiming at finding sparse and nearly uncorrelated PCs with orthogonal loading vectors while explaining as much of the total variance as possible. We also develop a novel augmented Lagrangian method for solving a class of nonsmooth constrained optimization problems, which is well suited for our formulation of sparse PCA. We show that it converges to a feasible point, and moreover under some regularity assumptions, it converges to a stationary point. Additionally, we propose two nonmonotone gradient methods for solving the augmented Lagrangian subproblems, and establish their global and local convergence. Finally, we compare our sparse PCA approach with several existing methods on synthetic, random, and real data, respectively. The computational results demonstrate that the sparse PCs produced by our approach substantially outperform those by other methods in terms of total explained variance, correlation of PCs, and orthogonality of loading vectors.

This paper focuses on applications of sparse principal component analysis to clustering and feature selection problems, with a particular focus on gene expression data analysis. Sparse methods have had a significant impact in many areas of statistics, in particular regression and classification (see [CT05], [DT05] and [Vap95] among others). As in these areas, our motivation for developing sparse multivariate visualization tools is the potential of these methods for yielding statistical results that are both more interpretable and more robust than classical analyses, while giving up little statistical efficiency. Principal component analysis (PCA) is a classic tool for analyzing large scale multivariate data. It seeks linear combinations of the data variables (often called factors or principal components) that capture a maximum amount of variance.

Given a sample covariance matrix, we examine the problem of maximizing the variance explained by a linear combination of the input variables while constraining the number of nonzero coefficients in this combination. This is known as sparse principal component analysis and has a wide array of applications in machine learning and engineering. We formulate a new semidefinite relaxation to this problem and derive a greedy algorithm that computes a full set of good solutions for all target numbers of non zero coefficients, with total complexity O(n^3), where n is the number of variables. We then use the same relaxation to derive sufficient conditions for global optimality of a solution, which can be tested in O(n^3) per pattern. We discuss applications in subset selection and sparse recovery and show on artificial examples and biological data that our algorithm does provide globally optimal solutions in many cases.

This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic. Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. We prove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence this allows to infer stability results for these estimated spaces.

This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic. Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. We prove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence this allows to infer stability results for these estimated spaces.

This paper presents a non-asymptotic statistical analysis of Kernel-PCA with a focus different from the one proposed in previous work on this topic. Here instead of considering the reconstruction error of KPCA we are interested in approximation error bounds for the eigenspaces themselves. Weprove an upper bound depending on the spacing between eigenvalues but not on the dimensionality of the eigenspace. As a consequence thisallows to infer stability results for these estimated spaces.

Eigenvoice speaker adaptation has been shown to be effective when only a small amount of adaptation data is available. At the heart of the method is principal component analysis (PCA) employed to find the most important eigenvoices. In this paper, we postulate that nonlinear PCA, in particular kernel PCA, may be even more effective. One major challenge is to map the feature-space eigenvoices back to the observation space so that the state observation likelihoods can be computed during the estimation of eigenvoice weights and subsequent decoding. Our solution is to compute kernel PCA using composite kernels, and we will call our new method kernel eigenvoice speaker adaptation. On the TIDIGITS corpus, we found that compared with a speaker-independent model, our kernel eigenvoice adaptation method can reduce the word error rate by 28-33% while the standard eigenvoice approach can only match the performance of the speaker-independent model.

Eigenvoice speaker adaptation has been shown to be effective when only a small amount of adaptation data is available. At the heart of the method is principal component analysis (PCA) employed to find the most important eigenvoices. In this paper, we postulate that nonlinear PCA, in particular kernel PCA, may be even more effective. One major challenge is to map the feature-space eigenvoices back to the observation space so that the state observation likelihoods can be computed during the estimation of eigenvoice weights and subsequent decoding. Our solution is to compute kernel PCA using composite kernels, and we will call our new method kernel eigenvoice speaker adaptation. On the TIDIGITS corpus, we found that compared with a speaker-independent model, our kernel eigenvoice adaptation method can reduce the word error rate by 28-33% while the standard eigenvoice approach can only match the performance of the speaker-independent model.