Lp Norm Relaxation Approach for Large Scale Data Analysis: A Review

In the last few years there has been a lot of attention to the techniques of sparse representation and low-rank approximation. These techniques have shown important applications in many areas in image and video processing, signal analysis, computer vision and machine learning. These techniques can also be used in unsupervised and dictionary learning to uncover high-order relations in the data and to train deep neural networks. This paper is a review to discuss the latest developments and show the effectiveness of these techniques in large scale data processing problems.

[1]  Rayan Saab,et al.  Sparse Recovery by Non-convex Optimization -- Instance Optimality , 2008, ArXiv.

[2]  R. Gribonval,et al.  Highly sparse representations from dictionaries are unique and independent of the sparseness measure , 2007 .

[3]  Zhihua Zhang,et al.  A Feasible Nonconvex Relaxation Approach to Feature Selection , 2011, AAAI.

[4]  Tong Zhang,et al.  Analysis of Multi-stage Convex Relaxation for Sparse Regularization , 2010, J. Mach. Learn. Res..

[5]  Cun-Hui Zhang Nearly unbiased variable selection under minimax concave penalty , 2010, 1002.4734.

[6]  Tien D. Bui,et al.  Non-convex online robust PCA: Enhance sparsity via ℓp-norm minimization , 2017, Comput. Vis. Image Underst..

[7]  Armando Manduca,et al.  Highly Undersampled Magnetic Resonance Image Reconstruction via Homotopic $\ell_{0}$ -Minimization , 2009, IEEE Transactions on Medical Imaging.

[8]  Rick Chartrand,et al.  Exact Reconstruction of Sparse Signals via Nonconvex Minimization , 2007, IEEE Signal Processing Letters.

[9]  R. Chartrand,et al.  Restricted isometry properties and nonconvex compressive sensing , 2007 .

[10]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[11]  J. Horowitz,et al.  Asymptotic properties of bridge estimators in sparse high-dimensional regression models , 2008, 0804.0693.

[12]  Taner Ince,et al.  Nonconvex compressed sensing with partially known signal support , 2013, Signal Process..

[13]  Wotao Yin,et al.  Iteratively reweighted algorithms for compressive sensing , 2008, 2008 IEEE International Conference on Acoustics, Speech and Signal Processing.

[14]  Guangming Shi,et al.  Compressive Sensing via Nonlocal Low-Rank Regularization , 2014, IEEE Transactions on Image Processing.

[15]  Christian Jutten,et al.  A Fast Approach for Overcomplete Sparse Decomposition Based on Smoothed $\ell ^{0}$ Norm , 2008, IEEE Transactions on Signal Processing.

[16]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[17]  Jianqing Fan,et al.  Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties , 2001 .

[18]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[19]  Maryam Fazel,et al.  Iterative reweighted algorithms for matrix rank minimization , 2012, J. Mach. Learn. Res..

[20]  E. Candès The restricted isometry property and its implications for compressed sensing , 2008 .

[21]  Donald Geman,et al.  Nonlinear image recovery with half-quadratic regularization , 1995, IEEE Trans. Image Process..

[22]  Xuelong Li,et al.  Fast and Accurate Matrix Completion via Truncated Nuclear Norm Regularization , 2013, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[23]  Rick Chartrand,et al.  Nonconvex Compressed Sensing and Error Correction , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[24]  Sungyoung Lee,et al.  Compressive sensing: From theory to applications, a survey , 2013, Journal of Communications and Networks.

[25]  Bernhard Schölkopf,et al.  Use of the Zero-Norm with Linear Models and Kernel Methods , 2003, J. Mach. Learn. Res..

[26]  Stephen P. Boyd,et al.  Log-det heuristic for matrix rank minimization with applications to Hankel and Euclidean distance matrices , 2003, Proceedings of the 2003 American Control Conference, 2003..

[27]  S. Foucart,et al.  Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 , 2009 .