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Asymptotic Analysis of higher-order scattering transform of Gaussian processes

Publication ,  Journal Article
Liu, GR; Sheu, YC; Wu, HT
Published in: Electronic Journal of Probability
January 1, 2022

We analyze the scattering transform with the quadratic nonlinearity (STQN) of Gaussian processes without depth limitation. STQN is a nonlinear transform that involves a sequential interlacing convolution and nonlinear operators, which is motivated to model the deep convolutional neural network. We prove that with a proper normalization, the output of STQN converges to a chi-square process with one degree of freedom in the finite dimensional distribution sense, and we provide a total variation distance control of this convergence at each time that converges to zero at an exponential rate. To show these, we derive a recursive formula to represent the intricate nonlinearity of STQN by a linear combination of Wiener chaos, and then apply the Malliavin calculus and Stein’s method to achieve the goal.

Duke Scholars

Published In

Electronic Journal of Probability

DOI

EISSN

1083-6489

Publication Date

January 1, 2022

Volume

27

Related Subject Headings

  • Statistics & Probability
  • 0105 Mathematical Physics
  • 0104 Statistics
 

Citation

APA
Chicago
ICMJE
MLA
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Liu, G. R., Sheu, Y. C., & Wu, H. T. (2022). Asymptotic Analysis of higher-order scattering transform of Gaussian processes. Electronic Journal of Probability, 27. https://doi.org/10.1214/22-EJP766
Liu, G. R., Y. C. Sheu, and H. T. Wu. “Asymptotic Analysis of higher-order scattering transform of Gaussian processes.” Electronic Journal of Probability 27 (January 1, 2022). https://doi.org/10.1214/22-EJP766.
Liu GR, Sheu YC, Wu HT. Asymptotic Analysis of higher-order scattering transform of Gaussian processes. Electronic Journal of Probability. 2022 Jan 1;27.
Liu, G. R., et al. “Asymptotic Analysis of higher-order scattering transform of Gaussian processes.” Electronic Journal of Probability, vol. 27, Jan. 2022. Scopus, doi:10.1214/22-EJP766.
Liu GR, Sheu YC, Wu HT. Asymptotic Analysis of higher-order scattering transform of Gaussian processes. Electronic Journal of Probability. 2022 Jan 1;27.

Published In

Electronic Journal of Probability

DOI

EISSN

1083-6489

Publication Date

January 1, 2022

Volume

27

Related Subject Headings

  • Statistics & Probability
  • 0105 Mathematical Physics
  • 0104 Statistics