Spectral deconvolution of unitarily invariant matrix models

Tarrago, Pierre

doi:10.48550/arxiv.2006.09356

“…The subordination function w f p in (2.13) is estimated using the second equality of Lemma 2.7. In parallel with our work, Tarrago [26] has used the formula (2.13) to perform spectral deconvolution in a more general setting (including the multiplicative free convolution), but neither the approximation of w f p by its estimator w n f p defined Theorem-Definition 2.8 below nor the (classical) deconvolution of the Cauchy distribution are treated, which are key difficulties encountered in our paper. Tarrago uses a different approach based on concentration inequalities when we use fluctuations in view of the work of Février and Dallaporta [16].…”

Section: Construction Of the Estimator Of Pmentioning

confidence: 99%

Statistical deconvolution of the free Fokker-Planck equation at fixed time

Maïda

¹

,

Nguyen

²

,

Ngoc

³

et al. 2022

Bernoulli

1

0

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We are interested in reconstructing the initial condition of a non-linear partial differential equation (PDE), namely the Fokker-Planck equation, from the observation of a Dyson Brownian motion at a given time t > 0. The Fokker-Planck equation describes the evolution of electrostatic repulsive particle systems, and can be seen as the large particle limit of correctly renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation can be written as the free convolution of the initial condition and the semi-circular distribution. We propose a nonparametric estimator for the initial condition obtained by performing the free deconvolution via the subordination functions method. This statistical estimator is original as it involves the resolution of a fixed point equation, and a classical deconvolution by a Cauchy distribution. This is due to the fact that, in free probability, the analogue of the Fourier transform is the R-transform, related to the Cauchy transform. In past literature, there has been a focus on the estimation of the initial conditions of linear PDEs such as the heat equation, but to the best of our knowledge, this is the first time that the problem is tackled for a non-linear PDE. The convergence of the estimator is proved and the integrated mean square error is computed, providing rates of convergence similar to the ones known for non-parametric deconvolution methods. Finally, a simulation study illustrates the good performances of our estimator.

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“…This calculus was implemented symbolically as RTNI [FKN19] with Mathematica and Python, for tensor networks with some of nodes being Haardistributed unitary matrices of symbolic dimensions. So far, RTNI has been used for quantum neural networks and quantum circuits [CSV + 21, PCW + 21, SCCC22, LNS + 23, WLL + 22, LLH + 22], more broadly quantum information and quantum physics [FK19, Liu20, KNP + 21, RGPA21, IEH + 23, HBS23, CKF23] and mathematics [Tar20]. Now, we make further steps to develop Python-based PyRTNI2.…”

Section: Introductionmentioning

confidence: 99%

RTNI—A symbolic integrator for Haar-random tensor networks

Fukuda¹,

König²,

Nechita³

2019

J. Phys. A: Math. Theor.

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We provide a computer algebra package called Random Tensor Network Integrator (RTNI). It allows to compute averages of tensor networks containing multiple Haar-distributed random unitary matrices and deterministic symbolic tensors. Such tensor networks are represented as multigraphs, with vertices corresponding to tensors or random unitaries and edges corresponding to tensor contractions. Input and output spaces of random unitaries may be subdivided into arbitrary tensor factors, with dimensions treated symbolically. The algorithm implements the graphical Weingarten calculus and produces a weighted sum of tensor networks representing the average over the unitary group. We illustrate the use of this algorithmic tool on some examples from quantum information theory, including entropy calculations for random tensor network states as considered in toy models for holographic duality. Mathematica and Python implementations are supplied.

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Statistical deconvolution of the free Fokker-Planck equation at fixed time

Maïda¹,

Nguyen²,

Ngoc³

et al. 2020

Preprint

0

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We are interested in reconstructing the initial condition of a non-linear partial differential equation (PDE), namely the Fokker-Planck equation, from the observation of a Dyson Brownian motion at a given time t > 0. The Fokker-Planck equation describes the evolution of electrostatic repulsive particle systems, and can be seen as the large particle limit of correctly renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation can be written as the free convolution of the initial condition and the semi-circular distribution. We propose a nonparametric estimator for the initial condition obtained by performing the free deconvolution via the subordination functions method. This statistical estimator is original as it involves the resolution of a fixed point equation, and a classical deconvolution by a Cauchy distribution. This is due to the fact that, in free probability, the analogue of the Fourier transform is the R-transform, related to the Cauchy transform. In past literature, there has been a focus on the estimation of the initial conditions of linear PDEs such as the heat equation, but to the best of our knowledge, this is the first time that the problem is tackled for a non-linear PDE. The convergence of the estimator is proved and the integrated mean square error is computed, providing rates of convergence similar to the ones known for non-parametric deconvolution methods. Finally, a simulation study illustrates the good performances of our estimator.

show abstract

Spectral deconvolution of unitarily invariant matrix models

Cited by 3 publications

References 32 publications

Statistical deconvolution of the free Fokker-Planck equation at fixed time

Statistical deconvolution of the free Fokker-Planck equation at fixed time

RTNI—A symbolic integrator for Haar-random tensor networks

Statistical deconvolution of the free Fokker-Planck equation at fixed time

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