Low-Rank Preserving Projections

Abstract

As one of the most popular dimensionality reduction techniques, locality preserving projections (LPP) has been widely used in computer vision and pattern recognition. However, in practical applications, data is always corrupted by noises. For the corrupted data, samples from the same class may not be distributed in the nearest area, thus LPP may lose its effectiveness. In this paper, it is assumed that data is grossly corrupted and the noise matrix is sparse. Based on these assumptions, we propose a novel dimensionality reduction method, named low-rank preserving projections (LRPP) for image classification. LRPP learns a low-rank weight matrix by projecting the data on a low-dimensional subspace. We use the L21 norm as a sparse constraint on the noise matrix and the nuclear norm as a low-rank constraint on the weight matrix. LRPP keeps the global structure of the data during the dimensionality reduction procedure and the learned low rank weight matrix can reduce the disturbance of noises in the data. LRPP can learn a robust subspace from the corrupted data. To verify the performance of LRPP in image dimensionality reduction and classification, we compare LRPP with the state-of-the-art dimensionality reduction methods. The experimental results show the effectiveness and the feasibility of the proposed method with encouraging results.

Flexible unsupervised feature extraction for image classification

“…In this section, we evaluate our algorithm on several well-known databases (AR, Extended YaleB, LFWcrop, Fifteen Scene Categories and Caltech101) whose details are presented in Table 1. We compare our method with PCA [28], LPP [11], NPE [14] and LRPP [21]. For LPP, NPE, LRPP and our method, we first use PCA to reduce dimensionality of all original datasets (except for Fifteen Scene Category database) to be 200, and then extract feature by these four methods, respectively.…”

“…For example, Xu et al [20] tried to preserve the global and local structures of data by imposing joint low-rank and sparse constraints on the reconstruction coefficient matrix. Lu et al [21] proposed a method named low-rank preserving projections (LRPP) which learns a low-rank weight matrix by projecting the data on a low-dimensional subspace. In addition, LRPP advocates the uses of the L21 norm as a sparse constraint on the noise matrix and the nuclear norm as a low-rank constraint on the weight matrix, which preserve the global structure of the data during the dimensionality reduction procedure.…”

Reconstruction of Undersampled Big Dynamic MRI Data Using Non-Convex Low-Rank and Sparsity Constraints

“…To achieve an efficient solution, the iterative singular value thresholding scheme [60,61], originally proposed for unclear norm minimization, can be extended to solve (13) which has a non-convex low-rank constraint. Let ${\tilde{P}}^k=L^k+{\mathit{Z}}_1^k/{\alpha }_1$, the solution $P^{k+1}$ is given by: $P^{k+1}={{\displaystyle \sum }}_{i=1}^{\min \left\{N_x{N}_y,N_t\right\}}\max \left({\sigma }_i-\frac{{\mu }_1{\sigma }_i^{p-1}}{{\alpha }_1},0\right)u_i{v}_i^{\mathrm{T}}$ where the superscript $\mathrm{T}$ denotes the transpose (conjugate transpose) operator for real (complex) matrices or vectors.…”

Gaussian Process Graph-Based Discriminant Analysis for Hyperspectral Images Classification

“…Recently, the representation-based algorithms have also been introduced to the GE framework to construct various similarity graphs [31]. For example, Sparse Representation (SR), Collaborative Representation (CR) and Low Rank Representation (LRR) [32] are utilized to constitute the sparse graph ( 1 graph), collaborative graph ( 2 graph) and low-rank graph, leading to Sparsity Preserving Projection (SPP) [33], Collaborative Representation based Projection (CRP) [34] and Low Rank Preserving Projections (LRPP) [35], respectively.…”