Fast greedy algorithms for dictionary selection with generalized sparsity constraints

In dictionary selection, several atoms are selected from finite candidates that successfully approximate given data points in the sparse representation. We propose a novel efficient greedy algorithm for dictionary selection. Not only does our algorithm work much faster than the known methods, but it can also handle more complex sparsity constraints, such as average sparsity. Using numerical experiments, we show that our algorithm outperforms the known methods for dictionary selection, achieving competitive performances with dictionary learning algorithms in a smaller running time.

[1]  Bogdan Dumitrescu,et al.  Dictionary Learning Algorithms and Applications , 2018 .

[2]  Alexandros G. Dimakis,et al.  Streaming Weak Submodularity: Interpreting Neural Networks on the Fly , 2017, NIPS.

[3]  Jiebo Luo,et al.  Adaptive Greedy Dictionary Selection for Web Media Summarization , 2017, IEEE Transactions on Image Processing.

[4]  Morteza Zadimoghaddam,et al.  Probabilistic Submodular Maximization in Sub-Linear Time , 2017, ICML.

[5]  Andreas Krause,et al.  Submodular Dictionary Selection for Sparse Representation , 2010, ICML.

[6]  Andreas Krause,et al.  Learning Sparse Combinatorial Representations via Two-stage Submodular Maximization , 2016, ICML.

[7]  Alexandros G. Dimakis,et al.  On Approximation Guarantees for Greedy Low Rank Optimization , 2017, ICML.

[8]  Kjersti Engan,et al.  Method of optimal directions for frame design , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[9]  Michael Elad,et al.  Double Sparsity: Learning Sparse Dictionaries for Sparse Signal Approximation , 2010, IEEE Transactions on Signal Processing.

[10]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[11]  M. Elad,et al.  $rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation , 2006, IEEE Transactions on Signal Processing.

[12]  Balas K. Natarajan,et al.  Sparse Approximate Solutions to Linear Systems , 1995, SIAM J. Comput..

[13]  Christos Boutsidis,et al.  Greedy Minimization of Weakly Supermodular Set Functions , 2015, APPROX-RANDOM.

[14]  Tengyuan Liang,et al.  Adaptive Feature Selection: Computationally Efficient Online Sparse Linear Regression under RIP , 2017, ICML.

[15]  Jianguo Zhang,et al.  The PASCAL Visual Object Classes Challenge , 2006 .

[16]  Guillermo Sapiro,et al.  Non-Parametric Bayesian Dictionary Learning for Sparse Image Representations , 2009, NIPS.

[17]  A. Bruckstein,et al.  K-SVD : An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation , 2005 .

[18]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[19]  藤重 悟 Submodular functions and optimization , 1991 .

[20]  Alexandros G. Dimakis,et al.  Restricted Strong Convexity Implies Weak Submodularity , 2016, The Annals of Statistics.

[21]  Junzhou Huang,et al.  Learning with structured sparsity , 2009, ICML '09.

[22]  Guillermo Sapiro,et al.  Online Learning for Matrix Factorization and Sparse Coding , 2009, J. Mach. Learn. Res..

[23]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[24]  Jiebo Luo,et al.  Towards Scalable Summarization of Consumer Videos Via Sparse Dictionary Selection , 2012, IEEE Transactions on Multimedia.

[25]  Sanjeev Arora,et al.  New Algorithms for Learning Incoherent and Overcomplete Dictionaries , 2013, COLT.

[26]  Prateek Jain,et al.  Learning Sparsely Used Overcomplete Dictionaries via Alternating Minimization , 2013, SIAM J. Optim..

[27]  Amin Karbasi,et al.  Online Continuous Submodular Maximization , 2018, AISTATS.

[28]  Matthew J. Streeter,et al.  An Online Algorithm for Maximizing Submodular Functions , 2008, NIPS.

[29]  Abhimanyu Das,et al.  Submodular meets Spectral: Greedy Algorithms for Subset Selection, Sparse Approximation and Dictionary Selection , 2011, ICML.

[30]  Sotirios A. Tsaftaris,et al.  Explicit Shift-Invariant Dictionary Learning , 2014, IEEE Signal Processing Letters.

[31]  Andreas Krause,et al.  Greedy Dictionary Selection for Sparse Representation , 2011, IEEE Journal of Selected Topics in Signal Processing.

[32]  Martin J. Wainwright,et al.  A unified framework for high-dimensional analysis of $M$-estimators with decomposable regularizers , 2009, NIPS.