Rank Approximation Of Matrices Research Articles

A QUINTET SINGULAR VALUE DECOMPOSITION THROUGH EMPIRICAL MODE DECOMPOSITIONS

A particular quintet singular valued decomposition (Quintet-SVD) is introduced in this paper via empirical mode decompositions (EMDs). The Quintet-SVD results in four specific orthogonal matrices with a diagonal matrix of singular values. Furthermore, this paper shows relationships between the Quintet-SVD and traditional SVD, generalized low rank approximations of matrices (GLRAM) of one single matrix, and EMDs. One application of the Quintet-SVD for speech enhancement is shown and compared with an application of traditional SVD.

Adaptation of Hidden Markov Models Using Model-as-Matrix Representation

In this paper, we describe basis-based speaker adaptation techniques using the matrix representation of training models. Bases are obtained from training models by decomposition techniques for matrix-variate objects: two-dimensional principal component analysis (2DPCA) and generalized low rank approximations of matrices (GLRAM). The motivation for using matrix representation is that the sample covariance matrix of training models can be more accurately computed and the speaker weight becomes a matrix. Speaker adaptation equations are derived in the maximum-likelihood (ML) framework, and the adaptation equations can be solved using the maximum-likelihood linear regression technique. Additionally, novel applications of probabilistic 2DPCA and GLRAM to speaker adaptation are presented. From the probabilistic 2DPCA/GLRAM of training models, speaker adaptation equations are formulated in the maximum a posteriori (MAP) framework. The adaptation equations can be solved using the MAP linear regression technique. In the isolated-word experiments, the matrix representation-based methods in the ML and MAP frameworks outperformed maximum-likelihood linear regression adaptation, MAP adaptation, eigenvoice, and probabilistic PCA-based model for adaptation data longer than 20 s. Furthermore, the adaptation methods using probabilistic 2DPCA/GLRAM showed additional performance improvement over the adaptation methods using 2DPCA/GLRAM for small amounts of adaptation data.

Bilinear Lanczos components for fast dimensionality reduction and feature extraction

Matrix-based methods such as generalized low rank approximations of matrices (GLRAM) have gained wide attention from researchers in pattern recognition and machine learning communities. In this paper, a novel concept of bilinear Lanczos components (BLC) is introduced to approximate the projection vectors obtained from eigen-based methods without explicit computing eigenvectors of the matrix. This new method sequentially reduces the reconstruction error for a Frobenius-norm based optimization criterion, and the resulting approximation performance is thus improved during successive iterations. In addition, a theoretical clue for selecting suitable dimensionality parameters without losing classification information is presented in this paper. The BLC approach realizes dimensionality reduction and feature extraction by using a small number of Lanczos components. Extensive experiments on face recognition and image classification are conducted to evaluate the efficiency and effectiveness of the proposed algorithm. Results show that the new approach is competitive with the state-of-the-art methods, while it has a much lower training cost.

Speaker adaptation using generalised low rank approximations of training matrices

A speaker adaptation method based on the low rank approximation of matrices (GLRAM) of training models is described. In the method, each model is represented as a matrix, and a set of such training matrices is decomposed into a set of speaker weights and two basis matrices for row and column spaces by reducing both row and column ranks of the training models. As a result, the speaker weight becomes a matrix, the row and column dimensions of which can be adjusted. In the isolated-word experiment, the proposed method showed better performance than both eigenvoice and MLLR for the adaptation data of about 20 s or longer.

Comments on “An analytical algorithm for generalized low-rank approximations of matrices”

Low rank approximations of matrices have been widely used in pattern recognition and machine learning. Based on a sequence of matrices, a generalized low rank approximation problem was presented and an iterative scheme was given by Liang and Shi recently proposed an analytical scheme for this approximation problem. In this paper, we identify the weakness in their scheme and prove that their algorithm is incorrect.

Non-iterative generalized low rank approximation of matrices

As an extension to 2DPCA, generalized low rank approximation of matrices (GLRAM) applies two-sided (i.e., the left and right) rather than single-sided (i.e., the left or the right alone) linear projecting transform(s) to each 2D image for compression and feature extraction. Its advantages over 2DPCA include higher compression ratio, superior classification performance, etc. However, GLRAM can only adopt an iterative rather than analytical approach to get the left and right projecting transforms and lacks a criterion to automatically determine the dimensionality of the projected matrix. In this paper, a novel non-iterative GLRAM (NIGLRAM) is proposed to overcome the above shortcomings. Experimental results on ORL and AR face datasets and COIL-20 object dataset show that NIGLRAM can get not only so-needed closed-form transforms but also comparable performance to GLRAM.

Resistant lower rank approximation of matrices by iterative majorization

It is commonly known that many techniques for data analysis based on the least squares criterion are very sensitive to outliers in the data. Gabriel and Odoroff (1984) suggested a resistant approach for lower rank approximation of matrices. In this approach, weights are used to diminish the influence of outliers on the low-dimensional representation. The present paper uses iterative majorization to provide for a general algorithm for such resistant lower rank approximations which guarantees convergence. It is shown that the weights can be chosen in different ways corresponding with different objective functions. Some possible extensions of the algorithm are discussed.

LORANK: A Package for Lower Rank Approximation of Matrices

LORANK: A Package for Lower Rank Approximation of Matrices