Generalized low-rank approximation of matrices based on multiple transformation pairs (original) (raw)

A novel extension of Generalized Low-Rank Approximation of Matrices based on multiple-pairs of transformations

ArXiv, 2018

Dimensionality reduction is a main step in the learning process which plays an essential role in many applications. The most popular methods in this field like SVD, PCA, and LDA, only can be applied to data with vector format. This means that for higher order data like matrices or more generally tensors, data should be fold to the vector format. So, in this approach, the spatial relations of features are not considered and also the probability of over-fitting is increased. Due to these issues, in recent years some methods like Generalized low-rank approximation of matrices (GLRAM) and Multilinear PCA (MPCA) are proposed which deal with the data in their own format. So, in these methods, the spatial relationships of features are preserved and the probability of overfitting could be fallen. Also, their time and space complexities are less than vector-based ones. However, because of the fewer parameters, the search space in a multilinear approach is much smaller than the search space o...

Low-Rank Matrix Approximation Using Point-Wise Operators

IEEE Transactions on Information Theory, 2000

The problem of extracting low dimensional structure from high dimensional data arises in many applications such as machine learning, statistical pattern recognition, wireless sensor networks, and data compression. If the data is restricted to a lower dimensional subspace, then simple algorithms using linear projections can find the subspace and consequently estimate its dimensionality. However, if the data lies on a low dimensional but nonlinear space (e.g., manifolds), then its structure may be highly nonlinear and hence linear methods are doomed to fail.

RES-PCA: A Scalable Approach to Recovering Low-Rank Matrices

2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR), 2019

Robust principal component analysis (RPCA) has drawn significant attentions due to its powerful capability in recovering low-rank matrices as well as successful appplications in various real world problems. The current state-ofthe-art algorithms usually need to solve singular value decomposition of large matrices, which generally has at least a quadratic or even cubic complexity. This drawback has limited the application of RPCA in solving real world problems. To combat this drawback, in this paper we propose a new type of RPCA method, RES-PCA, which is linearly efficient and scalable in both data size and dimension. For comparison purpose, AltProj, an existing scalable approach to RPCA requires the precise knowlwdge of the true rank; otherwise, it may fail to recover low-rank matrices. By contrast, our method works with or without knowing the true rank; even when both methods work, our method is faster. Extensive experiments have been performed and testified to the effectiveness of proposed method quantitatively and in visual quality, which suggests that our method is suitable to be employed as a lightweight , scalable component for RPCA in any application pipelines.

Accurate and fast matrix factorization for low-rank learning

ArXiv, 2021

In this paper, we tackle two important problems in low-rank learning, which are partial singular value decomposition and numerical rank estimation of huge matrices. By using the concepts of Krylov subspaces such as Golub-Kahan bidiagonalization (GK-bidiagonalization) as well as Ritz vectors, we propose two methods for solving these problems in a fast and accurate way. Our experiments show the advantages of the proposed methods compared to the traditional and randomized singular value decomposition methods. The proposed methods are appropriate for applications involving huge matrices where the accuracy of the desired singular values and also all of their corresponding singular vectors are essential. As a real application, we evaluate the performance of our methods on the problem of Riemannian similarity learning between two different image datasets of MNIST and USPS.

Tensor Robust Principal Component Analysis via Non-Convex Low Rank Approximation

Applied Sciences, 2019

Tensor Robust Principal Component Analysis (TRPCA) plays a critical role in handling high multi-dimensional data sets, aiming to recover the low-rank and sparse components both accurately and efficiently. In this paper, different from current approach, we developed a new t-Gamma tensor quasi-norm as a non-convex regularization to approximate the low-rank component. Compared to various convex regularization, this new configuration not only can better capture the tensor rank but also provides a simplified approach. An optimization process is conducted via tensor singular decomposition and an efficient augmented Lagrange multiplier algorithm is established. Extensive experimental results demonstrate that our new approach outperforms current state-of-the-art algorithms in terms of accuracy and efficiency.

Multilinear Subspace Regression: An Orthogonal Tensor Decomposition Approach

A multilinear subspace regression model based on so called latent variable decomposition is introduced. Unlike standard regression methods which typically employ matrix (2D) data representations followed by vector subspace transformations, the proposed approach uses tensor subspace transformations to model common latent variables across both the independent and dependent data. The proposed approach aims to maximize the correlation between the so derived latent variables and is shown to be suitable for the prediction of multidimensional dependent data from multidimensional independent data, where for the estimation of the latent variables we introduce an algorithm based on Multilinear Singular Value Decomposition (MSVD) on a specially defined cross-covariance tensor. It is next shown that in this way we are also able to unify the existing Partial Least Squares (PLS) and N-way PLS regression algorithms within the same framework. Simulations on benchmark synthetic data confirm the advantages of the proposed approach, in terms of its predictive ability and robustness, especially for small sample sizes. The potential of the proposed technique is further illustrated on a real world task of the decoding of human intracranial electrocorticogram (ECoG) from a simultaneously recorded scalp electroencephalograph (EEG).

LLORMA: Local Low-Rank Matrix Approximation

J. Mach. Learn. Res., 2016

Matrix approximation is a common tool in recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the partially observed matrix is low-rank. In this paper, we propose, analyze, and experiment with two procedures, one parallel and the other global, for constructing local matrix approximations. The two approaches approximate the observed matrix as a weighted sum of low-rank matrices. These matrices are limited to a local region of the observed matrix. We analyze the accuracy of the proposed local low-rank modeling. Our experiments show improvements in prediction accuracy over classical approaches for recommendation tasks.

Multiview Subspace Clustering Using Low-Rank Representation

IEEE Transactions on Cybernetics, 2021

Multiview subspace clustering is one of the most widely used methods for exploiting the internal structures of multiview data. Most previous studies have performed the task of learning multiview representations by individually constructing an affinity matrix for each view without simultaneously exploiting the intrinsic characteristics of multiview data. In this paper, we propose a multiview low-rank representation (MLRR) method to comprehensively discover the correlation of multiview data for multiview subspace clustering. MLRR considers symmetric low-rank representations (LRRs) to be an approximately linear spatial transformation under the new base, i.e., the multiview data themselves, to fully exploit the angular information of the principal directions of LRRs, which is adopted to construct an affinity matrix for multiview subspace clustering, under a symmetric condition. MLRR takes full advantage of LRR techniques and a diversity regularization term to exploit the diversity and consistency of multiple views, respectively, and this method simultaneously imposes a symmetry constraint on LRRs. Hence, the angular information of the principal directions of rows is consistent with that of columns in symmetric LRRs. The MLRR model can be efficiently calculated by solving a convex optimization problem. Moreover, we present an intuitive fusion strategy for symmetric LRRs from the perspective of spectral clustering to obtain a compact representation, which can be shared by multiple views and comprehensively represents the intrinsic features of multiview data. Finally, the experimental results based on benchmark datasets demonstrate the effectiveness and robustness of MLRR compared with several state-of-the-art multiview subspace clustering algorithms.

A fast approach for dimensionality reduction with image data

Pattern Recognition, 2005

An important objective in image analysis is dimensionality reduction. The most often used data-exploratory technique with this objective is principal component analysis, which performs a singular value decomposition on a data matrix of vectorized images. When considering an array data or tensor instead of a matrix, the high-order generalization of PCA for computing principal components offers multiple ways to decompose tensors orthogonally. As an alternative, we propose a new method based on the projection of the images as matrices and show that it leads to a better reconstruction of images than previous approaches.

N-Dimensional Principal Component Analysis

2010

In this paper, we first briefly introduce the multidimensional Principal Component Analysis (PCA) techniques, and then amend our previous N-dimensional PCA (ND-PCA) scheme by introducing multidirectional decomposition into ND-PCA implementation. For the case of high dimensionality, PCA technique is usually extended to an arbitrary n-dimensional space by the Higher-Order Singular Value Decomposition (HO-SVD) technique. Due to the size of tensor, HO-SVD implementation usually leads to a huge matrix along some direction of tensor, which is always beyond the capacity of an ordinary PC. The novelty of this paper is to amend our previous ND-PCA scheme to deal with this challenge and further prove that the revised ND-PCA scheme can provide a near optimal linear solution under the given error bound. To evaluate the numerical property of the revised ND-PCA scheme, experiments are performed on a set of 3D volume datasets.