Linear expansion with dictionaries of basis functions, with respect to which you wish your representation to be sparse; i.e. in the statistical case, basissparse regression. But even outside statistics, you wish simply to approximate some data compactly. My focus here is on the noisyobservation case, although the same results are recycled enough throughout the field.
Note that there are two ways you can get your representation to be sparse;
 you know that your signal happens to be compressible, in the sense that under some transform its coefficient vector is mostly zeros, even in a plain old orthogonal basis expansion.
 you are using a redundant dictionary such that you won’t need most of it to represent even a dense signal.
I should break these two notions apart here. For now, I’m especially interested in adaptive bases.
This is merely a bunch of links to important articles at the moment; I should do a little exposition one day.
Decomposition of stuff by matching pursuit, wavelets, curvelets, chirplets, framelets, shearlets, camelhairbrushlets, contentspecific basis dictionaries, designed or learned. Mammals visual cortexes seem to use something like this, if you squint right at the evidence.
To discuss:
 connection to mixture models.
 Sampling complexity versus approximation complexity
 am especially interested in approaches where we learn the transform or the basis dictionary unsupervised
Learnable codings
Adaptive dictionaries!
I want to generalise or extend this idea, ideally in some shiftinvariant way (see below.)
Oldhausen and Field (OlFi96a) 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 NCBK11, HyHH09 and HLLB15.
See MBPS09 for some a summary of methods to 2009 in basis learning.
Question: how do you do this in a big data / offline setting?
TRANSFORM LEARNING: Sparse Representations at Scale:
Analytical sparsifying transforms such as Wavelets and DCT have been widely used in compression standards. Recently, the datadriven learning of sparse models such as the synthesis dictionary model have become popular especially in applications such as denoising, inpainting, compressed sensing, etc. Our group’s research at the University of Illinois focuses on the datadriven adaptation of the alternative sparsifying transform model, which offers numerous advantages over the synthesis dictionary model.
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, unionoftransforms, 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 realtime applications.
Huh.
Shiftinvariant codings
I would like to find a general way of doing this in a phase/shiftrobust fashion, 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.)
It can be better if you know you have have constraints, say, a convex combination (e.g. mixtures) or positive definite bases. (e.g. in kernel methods)
One method is “Shift Invariant Sparse coding”, ( BlDa04) and there are various versions out there. (GRKN07 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.
Affine tight framelets (DHRS03) and their presumably lesscomputationallytractable, more flexible cousins, shearlets also sound interesting here. For reasons I do not yet understand I am told they can naturally be used on sundry graphs and manifolds, not just lattices, is traditional in DSP. I saw Xiaosheng Zhuang present these (see, e.g. HaZZ16 and WaZh16, where WaZh16 demonstrates a Fast Framelet Transform which is supposedly as computationally as cheap as the FFT.)
I have some ideas I call learning gamelan which relate to this.
Optimisation procedures
TBD.
Implementations
This boils down to clever optimisation to make the calculations tractable.
all the wavelet toolkits.
 scipy’s wavelet transform has no frills and little coherent explanation, but it goes
 pywavelets does various fancy wavelets and seems to be a standard for python.
 Matlab’s Wavelet toolbox seems to be the reference.
 scikitlearn dictionary learning version here
 also pydbm
 Fancy easy GPU wavelet implementation, PyTorchWavelets.

SParse Optimization Research COde (SPORCO) is an opensource Python package for solving optimization problems with sparsityinducing regularization, consisting primarily of sparse coding and dictionary learning, for both standard and convolutional forms of sparse representation. In the current version, all optimization problems are solved within the Alternating Direction Method of Multipliers (ADMM) framework. SPORCO was developed for applications in signal and image processing, but is also expected to be useful for problems in computer vision, statistics, and machine learning.
Sparsefiltering: Unsupervised feature learning based on sparsefiltering
This implements the method described Jiquan Ngiam, Pang Wei Koh, Zhenghao Chen, Sonia Bhaskar, Andrew Y. Ng: Sparse Filtering. NIPS 2011: 11251133 and is based on the Matlab code provided in the supplementary material
spams does a variety of sparse codings, although non of them accepting pluggable models. (see optimisation)
To read
 AGMM15
 Arora, S., Ge, R., Ma, T., & Moitra, A. (2015) Simple, Efficient, and Neural Algorithms for Sparse Coding. ArXiv:1503.00778.
 BaJo06
 Bach, F. R., & Jordan, M. I.(2006) Learning spectral clustering, with application to speech separation. Journal of Machine Learning Research, 7(Oct), 1963–2001.
 BCDH10
 Baraniuk, R. G., Cevher, V., Duarte, M. F., & Hegde, C. (2010) Modelbased compressive sensing. IEEE Transactions on Information Theory, 56(4), 1982–2001. DOI.
 BLMM12
 Barthélemy, Q., Larue, A., Mayoue, A., Mercier, D., & Mars, J. I.(2012) Shift & 2D rotation invariant sparse coding for multivariate signals. IEEE Transactions on Signal Processing, 60(4), 1597–1611.
 BePR11
 Bertin, K., Pennec, E. L., & Rivoirard, V. (2011) Adaptive Dantzig density estimation. Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 47(1), 43–74. DOI.
 BlDa04
 Blumensath, T., & Davies, M. (2004) On ShiftInvariant Sparse Coding. In C. G. Puntonet & A. Prieto (Eds.), Independent Component Analysis and Blind Signal Separation (Vol. 3195, pp. 1205–1212). Berlin, Heidelberg: Springer Berlin Heidelberg
 BlDa06
 Blumensath, T., & Davies, M. (2006) Sparse and shiftInvariant representations of music. IEEE Transactions on Audio, Speech and Language Processing, 14(1), 50–57. DOI.
 BJPD17
 Bora, A., Jalal, A., Price, E., & Dimakis, A. G.(2017) Compressed Sensing using Generative Models. ArXiv:1703.03208 [Cs, Math, Stat].
 Boye11
 Boyes, G. (2011) DictionaryBased Analysis/Synthesis and Structured Representations of Musical Audio. . McGill University
 CaCS08
 Cai, J.F., Chan, R. H., & Shen, Z. (2008) A frameletbased image inpainting algorithm. Applied and Computational Harmonic Analysis, 24(2), 131–149. DOI.
 CVVR11
 CarabiasOrti, J. J., Virtanen, T., VeraCandeas, P., RuizReyes, N., & CanadasQuesada, F. J.(2011) Musical Instrument Sound MultiExcitation Model for NonNegative Spectrogram Factorization. IEEE Journal of Selected Topics in Signal Processing, 5(6), 1144–1158. DOI.
 ChDo94
 Chen, S., & Donoho, D. L.(1994) Basis pursuit. In 1994 Conference Record of the TwentyEighth Asilomar Conference on Signals, Systems and Computers, 1994 (Vol. 1, pp. 41–44 vol.1). DOI.
 Daub88
 Daubechies, I. (1988) Orthonormal bases of compactly supported wavelets. Communications on Pure and Applied Mathematics, 41(7), 909–996. DOI.
 DaDD04
 Daubechies, I., Defrise, M., & De Mol, C. (2004) An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Communications on Pure and Applied Mathematics, 57(11), 1413–1457. DOI.
 DHRS03
 Daubechies, I., Han, B., Ron, A., & Shen, Z. (2003) Framelets: MRAbased constructions of wavelet frames. Applied and Computational Harmonic Analysis, 14(1), 1–46. DOI.
 Davi98
 Davis, G. M.(1998) A waveletbased analysis of fractal image compression. IEEE Transactions on Image Processing, 7(2), 141–154. DOI.
 DaMZ94
 Davis, G. M., Mallat, S. G., & Zhang, Z. (1994) Adaptive timefrequency decompositions. Optical Engineering, 33(7), 2183–2191. DOI.
 Devo98
 DeVore, R. A.(1998) Nonlinear approximation. Acta Numerica, 7, 51–150. DOI.
 Dong15
 Dong, B. (2015) Sparse representation on graphs by tight wavelet frames and applications. Applied and Computational Harmonic Analysis. DOI.
 DoJo95
 Donoho, D. L., & Johnstone, I. M.(1995) Adapting to Unknown Smoothness via Wavelet Shrinkage. Journal of the American Statistical Association, 90(432), 1200–1224. DOI.
 DJKP95
 Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., & Picard, D. (1995) Wavelet Shrinkage: Asymptopia?. Journal of the Royal Statistical Society. Series B (Methodological), 57(2), 301–369.
 DuKL06
 Du, P., Kibbe, W. A., & Lin, S. M.(2006) Improved peak detection in mass spectrum by incorporating continuous wavelet transformbased pattern matching. Bioinformatics, 22(17), 2059–2065. DOI.
 EkTS11
 Ekanadham, C., Tranchina, D., & Simoncelli, E. P.(2011) Recovery of Sparse TranslationInvariant Signals With Continuous Basis Pursuit. IEEE Transactions on Signal Processing, 59(10), 4735–4744. DOI.
 FaLi01
 Fan, J., & Li, R. (2001) Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties. Journal of the American Statistical Association, 96(456), 1348–1360. DOI.
 GRCL17
 Garg, S., Rish, I., Cecchi, G., & Lozano, A. (2017) NeurogenesisInspired Dictionary Learning: Online Model Adaption in a Changing World. In arXiv:1701.06106 [cs, stat].
 GiNi09
 Giné, E., & Nickl, R. (2009) Uniform limit theorems for wavelet density estimators. The Annals of Probability, 37(4), 1605–1646. DOI.
 GiSB16
 Giryes, R., Sapiro, G., & Bronstein, A. M.(2016) Deep Neural Networks with Random Gaussian Weights: A Universal Classification Strategy?. IEEE Transactions on Signal Processing, 64(13), 3444–3457. DOI.
 Good97
 Goodwin, M. M.(1997) Adaptive Signal Models: Theory, Algorithms, and Audio Applications (phdthesis). . University of California, Berkeley, Berkeley, CA
 Good01
 Goodwin, M. M.(2001) Multiscale overlapadd sinusoidal modeling using matching pursuit and refinements. In IEEE Workshop on Applications of Signal Processing to Audio and Acoustics.
 GoVe99
 Goodwin, M. M., & Vetterli, M. (1999) Matching pursuit and atomic signal models based on recursive filter banks. IEEE Trans. Signal Process., 47(7), 1890–1902. DOI.
 GoVe97
 Goodwin, M., & Vetterli, M. (1997) Atomic decompositions of audio signals. In 1997 IEEE ASSP Workshop on Applications of Signal Processing to Audio and Acoustics, 1997 (p. 4 pp.). DOI.
 GrLe10
 Gregor, K., & LeCun, Y. (2010) Learning fast approximations of sparse coding. In Proceedings of the 27th International Conference on Machine Learning (ICML10) (pp. 399–406).
 GrLe11
 Gregor, K., & LeCun, Y. (2011) Efficient Learning of Sparse Invariant Representations. ArXiv:1105.5307 [Cs].
 GRKN07
 Grosse, R., Raina, R., Kwong, H., & Ng, A. Y.(2007) ShiftInvariant Sparse Coding for Audio Classification. In The TwentyThird Conference on Uncertainty in Artificial Intelligence (UAI2007) (Vol. 9, p. 8).
 GuPe16
 Gupta, P., & Pensky, M. (2016) Solution of linear illposed problems using random dictionaries. ArXiv:1605.07913 [Math, Stat].
 HLLB15
 Hahn, W. E., Lewkowitz, S., Lacombe Jr, D. C., & Barenholtz, E. (2015) Deep learning human actions from video via sparse filtering and locally competitive algorithms.
 HaZZ16
 Han, B., Zhao, Z., & Zhuang, X. (2016) Directional tensor product complex tight framelets with low redundancy. Applied and Computational Harmonic Analysis, 41(2), 603–637. DOI.
 HaZh15
 Han, B., & Zhuang, X. (2015) Smooth affine shear tight frames with MRA structure. Applied and Computational Harmonic Analysis, 39(2), 300–338. DOI.
 HaSh00
 Han, K., & Shin, H. (n.d.) Functional Linear Regression for Functional Response via Sparse Basis Selection.
 HaSG06
 Harte, C., Sandler, M., & Gasser, M. (2006) Detecting Harmonic Change in Musical Audio. In Proceedings of the 1st ACM Workshop on Audio and Music Computing Multimedia (pp. 21–26). New York, NY, USA: ACM DOI.
 HJKL11
 Henaff, M., Jarrett, K., Kavukcuoglu, K., & LeCun, Y. (2011) Unsupervised learning of sparse features for scalable audio classification. In ISMIR.
 HuCB08
 Huang, C., Cheang, G. L. H., & Barron, A. R.(2008) Risk of penalized least squares, greedy selection and l1 penalization for flexible function libraries.
 HyHo00
 Hyvärinen, A., & Hoyer, P. (2000) Emergence of Phase and ShiftInvariant Features by Decomposition of Natural Images into Independent Feature Subspaces. Neural Computation, 12(7), 1705–1720. DOI.
 HyHH09
 Hyvärinen, A., Hurri, J., & Hoyer, P. O.(2009) Natural Image Statistics: A Probabilistic Approach to Early Computational Vision. (Vol. 39). Springer Science & Business Media
 JaPl11
 Jafari, M. G., & Plumbley, M. D.(2011) Fast Dictionary Learning for Sparse Representations of Speech Signals. IEEE Journal of Selected Topics in Signal Processing, 5(5), 1025–1031. DOI.
 JGPZ10
 Jaillet, F., Gribonval, R., Plumbley, M. D., & Zayyani, H. (2010) An L1 criterion for dictionary learning by subspace identification. In 2010 IEEE International Conference on Acoustics, Speech and Signal Processing (pp. 5482–5485). DOI.
 Jung13
 Jung, A. (2013) An RKHS Approach to Estimation with Sparsity Constraints. In Advances in Neural Information Processing Systems 29.
 KoCo16
 Koch, P., & Corso, J. J.(2016) Sparse Factorization Layers for Neural Networks with Limited Supervision. ArXiv:1612.04468 [Cs, Stat].
 KWSR16
 Koppel, A., Warnell, G., Stump, E., & Ribeiro, A. (2016) Parsimonious Online Learning with Kernels via Sparse Projections in Function Space. ArXiv:1612.04111 [Cs, Stat].
 LBRN07
 Lee, H., Battle, A., Raina, R., & Ng, A. Y.(2007) Efficient sparse coding algorithms. Advances in Neural Information Processing Systems, 19, 801.
 LeSe99
 Lewicki, M. S., & Sejnowski, T. J.(1999) Coding timevarying signals using sparse, shiftinvariant representations. In NIPS (Vol. 11, pp. 730–736). Denver, CO: MIT Press
 LeSe00
 Lewicki, M. S., & Sejnowski, T. J.(2000) Learning Overcomplete Representations. Neural Computation, 12(2), 337–365. DOI.
 LiTX16
 Liu, T., Tao, D., & Xu, D. (2016) DimensionalityDependent Generalization Bounds for $k$Dimensional Coding Schemes. ArXiv:1601.00238 [Cs, Stat].
 MGVB11
 Mailhé, B., Gribonval, R., Vandergheynst, P., & Bimbot, F. (2011) Fast orthogonal sparse approximation algorithms over local dictionaries. Signal Processing, 91(12), 2822–2835. DOI.
 MaBP14
 Mairal, J., Bach, F., & Ponce, J. (2014) Sparse Modeling for Image and Vision Processing. Foundations and Trends® in Comput Graph. Vis., 8(2–3), 85–283. DOI.
 MBPS09
 Mairal, J., Bach, F., Ponce, J., & Sapiro, G. (2009) Online Dictionary Learning for Sparse Coding. In Proceedings of the 26th Annual International Conference on Machine Learning (pp. 689–696). New York, NY, USA: ACM DOI.
 MBPS10
 Mairal, J., Bach, F., Ponce, J., & Sapiro, G. (2010) Online learning for matrix factorization and sparse coding. The Journal of Machine Learning Research, 11, 19–60.
 Mall89
 Mallat, S. G.(1989) Multiresolution approximations and wavelet orthonormal bases of L²(R). Transactions of the American Mathematical Society, 315(1), 69–87. DOI.
 MaZh93
 Mallat, S. G., & Zhang, Z. (1993) Matching pursuits with timefrequency dictionaries. IEEE Transactions on Signal Processing, 41(12), 3397–3415.
 MaZh92
 Mallat, S., & Zhang, Z. (1992) Adaptive timefrequency decomposition with matching pursuits. In TimeFrequency and TimeScale Analysis, 1992., Proceedings of the IEEESP International Symposium (pp. 7–10). DOI.
 MaMD14
 Marcus, G., Marblestone, A., & Dean, T. (2014) The atoms of neural computation. Science (New York, N.Y.), 346(6209), 551–552. DOI.
 Mlyn13
 Mlynarski, W. (2013) Sparse, complexvalued representations of natural sounds learned with phase and amplitude continuity priors. ArXiv Preprint ArXiv:1312.4695.
 MoPe10
 Mondal, D., & Percival, D. B.(2010) Mestimation of wavelet variance. Annals of the Institute of Statistical Mathematics, 64(1), 27–53. DOI.
 MøSH07
 Mørup, M., Schmidt, M. N., & Hansen, L. K.(2007) Shift invariant sparse coding of image and music data. Journal of Machine Learning Research.
 NCBK11
 Ngiam, J., Chen, Z., Bhaskar, S. A., Koh, P. W., & Ng, A. Y.(2011) Sparse Filtering. In J. ShaweTaylor, R. S. Zemel, P. L. Bartlett, F. Pereira, & K. Q. Weinberger (Eds.), Advances in Neural Information Processing Systems 24 (pp. 1125–1133). Curran Associates, Inc.
 OlFi96a
 Olshausen, B. A., & Field, D. J.(1996a) Emergence of simplecell receptive field properties by learning a sparse code for natural images. Nature, 381(6583), 607–609. DOI.
 OlFi96b
 Olshausen, B. A., & Field, D. J.(1996b) Natural image statistics and efficient coding. Network (Bristol, England), 7(2), 333–339. DOI.
 OlFi04
 Olshausen, B. A., & Field, D. J.(2004) Sparse coding of sensory inputs. Current Opinion in Neurobiology, 14(4), 481–487. DOI.
 OpWY01
 Opsomer, J., Wang, Y., & Yang, Y. (2001) Nonparametric Regression with Correlated Errors. Statistical Science, 16(2), 134153.
 OyBZ17
 Oyallon, E., Belilovsky, E., & Zagoruyko, S. (2017) Scaling the Scattering Transform: Deep Hybrid Networks. ArXiv Preprint ArXiv:1703.08961.
 PfBr17
 Pfister, L., & Bresler, Y. (2017) Automatic parameter tuning for image denoising with learned sparsifying transforms. . Presented at the ICASSP
 PABD06
 Plumbley, M. D., Abdallah, S. A., Blumensath, T., & Davies, M. E.(2006) Sparse representations of polyphonic music. Signal Processing, 86(3), 417–431. DOI.
 QiCh94
 Qian, S., & Chen, D. (1994) Signal representation using adaptive normalized Gaussian functions. Signal Processing, 36(1), 1–11. DOI.
 RaBr15
 Ravishankar, S., & Bresler, Y. (2015) Efficient Blind Compressed Sensing Using Sparsifying Transforms with Convergence Guarantees and Application to MRI. ArXiv:1501.02923 [Cs, Stat].
 RuBE10
 Rubinstein, R., Bruckstein, A. M., & Elad, M. (2010) Dictionaries for Sparse Representation Modeling. Proceedings of the IEEE, 98(6), 1045–1057. DOI.
 RuZE08
 Rubinstein, R., Zibulevsky, M., & Elad, M. (2008) Efficient implementation of the KSVD algorithm using batch orthogonal matching pursuit. CS Technion, 40.
 Shen10
 Shen, Z. (2010) Wavelet frames and image restorations. In Scopus (pp. 2834–2863). World Scientific
 SiOl01
 Simoncelli, E. P., & Olshausen, B. A.(2001) Natural Image Statistics and Neural Representation. Annual Review of Neuroscience, 24(1), 1193–1216. DOI.
 SmLe06
 Smith, E. C., & Lewicki, M. S.(2006) Efficient auditory coding. Nature, 439(7079), 978–982. DOI.
 SoCh17
 Soh, Y. S., & Chandrasekaran, V. (2017) A Matrix Factorization Approach for Learning SemidefiniteRepresentable Regularizers. ArXiv:1701.01207 [Cs, Math, Stat].
 ToCo98
 Torrence, C., & Compo, G. P.(1998) A Practical Guide to Wavelet Analysis. Bulletin of the American Meteorological Society, 79(1), 61–78.
 ToFr11
 Tošić, I., & Frossard, P. (2011) Dictionary learning: What is the right representation for my signal?. IEEE Signal Processing Magazine, 28(2), 27–38. DOI.
 TsDo06a
 Tsaig, Y., & Donoho, D. L.(2006a) Breakdown of equivalence between the minimal norm solution and the sparsest solution. Signal Processing, 86(3), 533–548. DOI.
 TsDo06b
 Tsaig, Y., & Donoho, D. L.(2006b) Extensions of compressed sensing. Signal Processing, 86(3), 549–571. DOI.
 VaMB11
 Vainsencher, D., Mannor, S., & Bruckstein, A. M.(2011) The Sample Complexity of Dictionary Learning. Journal of Machine Learning Research, 12(Nov), 3259–3281.
 Vett99
 Vetterli, M. (1999) Wavelets: approximation and compression–a review. In AeroSense’99 (Vol. 3723, pp. 28–31). International Society for Optics and Photonics DOI.
 WaZh17
 Wang, Y. G., & Zhu, H. (2017) Localized Tight Frames and Fast Framelet Transforms on the Simplex. ArXiv:1701.01595 [Cs, Math].
 WaZh16
 Wang, Y. G., & Zhuang, X. (2016) Tight framelets and fast framelet transforms on manifolds. ArXiv:1608.04026 [Math].
 WaST14
 Wang, Y.X., Smola, A., & Tibshirani, R. J.(2014) The Falling Factorial Basis and Its Statistical Applications. ArXiv:1405.0558 [Stat].
 WeVe12
 Weidmann, C., & Vetterli, M. (2012) Rate Distortion Behavior of Sparse Sources. IEEE Transactions on Information Theory, 58(8), 4969–4992. DOI.
 YNGD13
 Yaghoobi, M., Nam, S., Gribonval, R., & Davies, M. E.(2013) Constrained Overcomplete Analysis Operator Learning for Cosparse Signal Modelling. IEEE Transactions on Signal Processing, 61(9), 2341–2355. DOI.
 Zhua15
 Zhuang, X. (2015) Smooth affine shear tight frames: digitization and applications. In SPIE Optical Engineering+ Applications (Vol. 9597, pp. 959709–959709). International Society for Optics and Photonics
 Zhua16
 Zhuang, X. (2016) Digital Affine Shear Transforms: Fast Realization and Applications in Image/Video Processing. SIAM Journal on Imaging Sciences, 9(3), 1437–1466. DOI.