Sparse coding with learnable dictionaries

November 18, 2014 — March 2, 2023

convolution
high d
Hilbert space
linear algebra
nonparametric
optimization
regression
signal processing
sparser than thou
Figure 1

Adaptive dictionaries for sparse coding. How is this different from matrix factorisation, you ask? It is not. AFAICT these are emphases of the same thing.

(Bruno A. Olshausen and Field 1996) kicked this area off by arguing sparse coding tricks are revealing of what the brain does.

For a walk through of one version of this, see Theano example of dictionary learning by Daniel LaCombe, who bases his version on (Ngiam et al. 2011; Hyvärinen, Hurri, and Hoyer 2009; Hahn et al. 2015).

See (Mairal, Bach, and Ponce 2014) for some a summary of methods to 2009 in basis learning.

Question: how do you do adaptive sparse coding in a big data / offline setting?

TRANSFORM LEARNING: Sparse Representations at Scale.

We have proposed several methods for batch learning of square or overcomplete sparsifying transforms from data. We have also investigated specific structures for these transforms such as double sparsity, union-of-transforms, and filter bank structures, which enable their efficient learning or usage. Apart from batch transform learning, our group has investigated methods for online learning of sparsifying transforms, which are particularly useful for big data or real-time applications.

Huh.

0.1 Codings with desired invariances

I would like to find bases robust against certain transformations, especially phase/shift-robust codings, although doing this naively can be computationally expensive outside of certain convenient bases. (Sorry, that’s not very clear; I need to return to this section to polish it up. 🏗)

One method is “Shift Invariant Sparse coding”, (Blumensath and Davies 2004) and there are various extensions and approximations out there. (Grosse et al. (2007) etc) One way is to include multiple shifted copies of your atoms, another is to actually shift them in a separate optimisation stage. Both these get annoying in the time domain for various reasons. (Lattner, Dorfler, and Arzt 2019) presents an adaptive sparse coding method preserving desired invariants.

spams does a huge variety of off-the-shelf sparse codings, although none of them are flexible. Nonetheless it does some neat things fast.

SPAMS (SPArse Modeling Software) is an optimization toolbox for solving various sparse estimation problems.

  • Dictionary learning and matrix factorization (NMF, sparse PCA,…)
  • Solving sparse decomposition problems with LARS, coordinate descent, OMP, SOMP, proximal methods
  • Solving structured sparse decomposition problems (l1/l2, l1/linf, sparse group lasso, tree-structured regularization, structured sparsity with overlapping groups,…).

1 References

Blumensath, and Davies. 2004. “On Shift-Invariant Sparse Coding.” In Independent Component Analysis and Blind Signal Separation.
Charles, Balavoine, and Rozell. 2016. “Dynamic Filtering of Time-Varying Sparse Signals via L1 Minimization.” IEEE Transactions on Signal Processing.
Garg, Rish, Cecchi, et al. 2017. “Neurogenesis-Inspired Dictionary Learning: Online Model Adaption in a Changing World.” In arXiv:1701.06106 [Cs, Stat].
Gehler, and Nowozin. n.d. “Let the Kernel Figure It Out; Principled Learning of Pre-Processing for Kernel Classifiers.”
Gregor, and LeCun. 2010. “Learning fast approximations of sparse coding.” In Proceedings of the 27th International Conference on Machine Learning (ICML-10).
———. 2011. “Efficient Learning of Sparse Invariant Representations.” arXiv:1105.5307 [Cs].
Grosse, Raina, Kwong, et al. 2007. “Shift-Invariant Sparse Coding for Audio Classification.” In The Twenty-Third Conference on Uncertainty in Artificial Intelligence (UAI2007).
Hahn, Lewkowitz, Lacombe, et al. 2015. “Deep Learning Human Actions from Video via Sparse Filtering and Locally Competitive Algorithms.” Multimedia Tools and Applications.
Henaff, Jarrett, Kavukcuoglu, et al. 2011. “Unsupervised Learning of Sparse Features for Scalable Audio Classification.” In ISMIR.
Hyvärinen, and Hoyer. 2000. “Emergence of Phase- and Shift-Invariant Features by Decomposition of Natural Images into Independent Feature Subspaces.” Neural Computation.
Hyvärinen, Hurri, and Hoyer. 2009. Natural Image Statistics: A Probabilistic Approach to Early Computational Vision.
Knudson, Yates, Huk, et al. 2014. “Inferring Sparse Representations of Continuous Signals with Continuous Orthogonal Matching Pursuit.” In Advances in Neural Information Processing Systems 27.
Kreutz-Delgado, Murray, Rao, et al. 2003. “Dictionary Learning Algorithms for Sparse Representation.” Neural Computation.
Lattner, Dorfler, and Arzt. 2019. “Learning Complex Basis Functions for Invariant Representations of Audio.” In Proceedings of the 20th Conference of the International Society for Music Information Retrieval.
Lewicki, M S, and Sejnowski. 1999. “Coding Time-Varying Signals Using Sparse, Shift-Invariant Representations.” In NIPS.
Lewicki, Michael S., and Sejnowski. 2000. “Learning Overcomplete Representations.” Neural Computation.
Mairal, Bach, Ponce, et al. 2009. “Online Dictionary Learning for Sparse Coding.” In Proceedings of the 26th Annual International Conference on Machine Learning. ICML ’09.
———, et al. 2010. “Online Learning for Matrix Factorization and Sparse Coding.” The Journal of Machine Learning Research.
Mairal, Bach, and Ponce. 2014. Sparse Modeling for Image and Vision Processing.
Meinshausen, and Yu. 2009. “Lasso-Type Recovery of Sparse Representations for High-Dimensional Data.” The Annals of Statistics.
Ngiam, Chen, Bhaskar, et al. 2011. “Sparse Filtering.” In Advances in Neural Information Processing Systems 24.
Olshausen, B. A., and Field. 1996. “Natural image statistics and efficient coding.” Network (Bristol, England).
Olshausen, Bruno A., and Field. 1996. “Emergence of Simple-Cell Receptive Field Properties by Learning a Sparse Code for Natural Images.” Nature.
Olshausen, Bruno A, and Field. 2004. “Sparse Coding of Sensory Inputs.” Current Opinion in Neurobiology.
Qian, Hong, Cai, et al. 2016. “Non-Negative Matrix Factorization with Sinkhorn Distance.” In Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence. IJCAI’16.
Rubinstein, Bruckstein, and Elad. 2010. “Dictionaries for Sparse Representation Modeling.” Proceedings of the IEEE.
Scetbon, Cuturi, and Peyré. 2021. “Low-Rank Sinkhorn Factorization.” In Proceedings of the 38th International Conference on Machine Learning.
Schmitz, Heitz, Bonneel, et al. 2018. “Wasserstein Dictionary Learning: Optimal Transport-Based Unsupervised Nonlinear Dictionary Learning.” SIAM Journal on Imaging Sciences.
Shen, and Li. 2010. “On the Dual Formulation of Boosting Algorithms.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Simoncelli, and Olshausen. 2001. “Natural Image Statistics and Neural Representation.” Annual Review of Neuroscience.
Soh, and Chandrasekaran. 2017. “A Matrix Factorization Approach for Learning Semidefinite-Representable Regularizers.” arXiv:1701.01207 [Cs, Math, Stat].
Yaghoobi, Nam, Gribonval, et al. 2013. “Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling.” IEEE Transactions on Signal Processing.
Zhang. 2021. “A Unified Framework for Non-Negative Matrix and Tensor Factorisations with a Smoothed Wasserstein Loss.”