Sparse stochastic processes identification and sampling

Discrete sample representation of sparse continuous stochastic processes

Sampling and estimation theory for SDEs driven by LΓ©vy noise. which produces a nice inference theory and gives us a machinery for producing prior for Bayesian sensing problems where the signal is known to be non-Gaussian. I have not got much to say about this yet. In particular I should say what β€œsparse” implies in this context. πŸ—

Related maybe, signatures of rough paths.


Amini, Arash, Michael Unser, and Farokh Marvasti. 2011. β€œCompressibility of Deterministic and Random Infinite Sequences.” IEEE Transactions on Signal Processing 59 (11): 5193–5201.
Bolin, David. 2014. β€œSpatial MatΓ©rn Fields Driven by Non-Gaussian Noise.” Scandinavian Journal of Statistics 41 (3): 557–79.
Bostan, E., U. S. Kamilov, M. Nilchian, and M. Unser. 2013. β€œSparse Stochastic Processes and Discretization of Linear Inverse Problems.” IEEE Transactions on Image Processing 22 (7): 2699–2710.
Bruna, Joan, and Stephane Mallat. 2013. β€œInvariant Scattering Convolution Networks.” IEEE Transactions on Pattern Analysis and Machine Intelligence 35 (8): 1872–86.
β€”β€”β€”. 2019. β€œMultiscale Sparse Microcanonical Models.” arXiv:1801.02013 [Math-Ph, Stat], May.
Bruna, Joan, StΓ©phane Mallat, Emmanuel Bacry, and Jean-FranΓ§ois Muzy. 2015. β€œIntermittent Process Analysis with Scattering Moments.” The Annals of Statistics 43 (1): 323–51.
Bruti-Liberati, Nicola, and Eckhard Platen. 2007. β€œStrong Approximations of Stochastic Differential Equations with Jumps.” Journal of Computational and Applied Mathematics, Special issue on evolutionary problems, 205 (2): 982–1001.
Cheng, Sihao, and Brice MΓ©nard. 2021. β€œHow to Quantify Fields or Textures? A Guide to the Scattering Transform.” arXiv.
Hanson, Floyd B. 2007. β€œStochastic Processes and Control for Jump-Diffusions.” SSRN Scholarly Paper ID 1023497. Rochester, NY: Social Science Research Network.
Mallat, StΓ©phane. 2012. β€œGroup Invariant Scattering.” Communications on Pure and Applied Mathematics 65 (10): 1331–98.
Maravic, I., and M. Vetterli. 2005. β€œSampling and Reconstruction of Signals with Finite Rate of Innovation in the Presence of Noise.” IEEE Transactions on Signal Processing 53 (8): 2788–2805.
Marziliano, P., M. Vetterli, and T. Blu. 2006. β€œSampling and Exact Reconstruction of Bandlimited Signals with Additive Shot Noise.” IEEE Transactions on Information Theory 52 (5): 2230–33.
Matheron, G. 1973. β€œThe Intrinsic Random Functions and Their Applications.” Advances in Applied Probability 5 (3): 439–68.
Meidan, R. 1980. β€œOn the Connection Between Ordinary and Generalized Stochastic Processes.” Journal of Mathematical Analysis and Applications 76 (1): 124–33.
Oyallon, Edouard, Eugene Belilovsky, and Sergey Zagoruyko. 2017. β€œScaling the Scattering Transform: Deep Hybrid Networks.” arXiv Preprint arXiv:1703.08961.
Sprechmann, Pablo, Joan Bruna, and Yann LeCun. 2014. β€œAudio Source Separation with Discriminative Scattering Networks.” arXiv:1412.7022 [Cs], December.
Sun, Qiyu, and Michael Unser. 2012. β€œLeft-Inverses of Fractional Laplacian and Sparse Stochastic Processes.” Advances in Computational Mathematics 36 (3): 399–441.
Tan, V. Y. F., and V. K. Goyal. 2008. β€œEstimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic Algorithm.” IEEE Transactions on Signal Processing 56 (10): 5135–46.
Unser, M. 2015. β€œSampling and (Sparse) Stochastic Processes: A Tale of Splines and Innovation.” In 2015 International Conference on Sampling Theory and Applications (SampTA), 221–25.
Unser, Michael A., and Pouya Tafti. 2014. An Introduction to Sparse Stochastic Processes. New York: Cambridge University Press.
Unser, M., P. D. Tafti, A. Amini, and H. Kirshner. 2014. β€œA Unified Formulation of Gaussian Vs Sparse Stochastic Processes - Part II: Discrete-Domain Theory.” IEEE Transactions on Information Theory 60 (5): 3036–51.
Unser, M., P. D. Tafti, and Q. Sun. 2014. β€œA Unified Formulation of Gaussian Vs Sparse Stochastic Processesβ€”Part I: Continuous-Domain Theory.” IEEE Transactions on Information Theory 60 (3): 1945–62.
Yadrenko, Mikhail Iosifovich. 1983. Spectral theory of random fields. Translation series in mathematics and engineering. New York, NY: Optimization Software.

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