Sparse Low-Rank Matrix Approximation for Data Compression

• Published in: IEEE transactions on circuits and systems for video technology 2017-05, Vol.27 (5), p.1043-1054
• Main Authors: Hou, Junhui, Chau, Lap-Pui, Magnenat-Thalmann, Nadia, He, Ying
• Format: Article
• Language: English
• Subjects: Approximation; Approximation error; Coherence; Data compression; Error detection; low-rank matrix; Mathematical analysis; Matrix decomposition; Matrix methods; optimization; orthogonal transform; Orthogonality; Pattern analysis; Signal processing; Software; Sparse matrices; sparsity; State of the art; Test procedures; Transforms; Wavelet transforms
• ISSN: 1051-8215; 1558-2205
• DOI: 10.1109/TCSVT.2015.2513698

Low-rank matrix approximation (LRMA) is a powerful technique for signal processing and pattern analysis. However, its potential for data compression has not yet been fully investigated. In this paper, we propose sparse LRMA (SLRMA), an effective computational tool for data compression. SLRMA extends conventional LRMA by exploring both the intra and inter coherence of data samples simultaneously. With the aid of prescribed orthogonal transforms (e.g., discrete cosine/wavelet transform and graph transform), SLRMA decomposes a matrix into a product of two smaller matrices, where one matrix is made up of extremely sparse and orthogonal column vectors and the other consists of the transform coefficients. Technically, we formulate SLRMA as a constrained optimization problem, i.e., minimizing the approximation error in the least-squares sense regularized by the \ell _{0} -norm and orthogonality, and solve it using the inexact augmented Lagrangian multiplier method. Through extensive tests on real-world data, such as 2D image sets and 3D dynamic meshes, we observe that: 1) SLRMA empirically converges well; 2) SLRMA can produce approximation error comparable to LRMA but in a much sparse form; and 3) SLRMA-based compression schemes significantly outperform the state of the art in terms of rate-distortion performance.