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High-Resolution Tensor-Network Fourier Methods for Exponentially Compressed Non-Gaussian Aggregate Distributions

Juan José Rodríguez-Aldavero, Juan José García-Ripoll

Abstract

Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components $D$ grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small-$D$ regime, the characteristic function undergoes a sharp bond-dimension collapse for $D \gtrsim 300$ components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of $N = 2^{30}$ frequency modes on standard hardware, far beyond the $N = 2^{24}$ ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.

High-Resolution Tensor-Network Fourier Methods for Exponentially Compressed Non-Gaussian Aggregate Distributions

Abstract

Characteristic functions of weighted sums of independent random variables exhibit low-rank structure in the quantized tensor train (QTT) representation, also known as matrix product states (MPS), enabling up to exponential compression of their fully non-Gaussian probability distributions. Under variable independence, the global characteristic function factorizes into local terms. Its low-rank QTT structure arises from intrinsic spectral smoothness in continuous models, or from spectral energy concentration as the number of components grows in discrete models. We demonstrate this on weighted sums of Bernoulli and lognormal random variables. In the former, despite an adversarial, incompressible small- regime, the characteristic function undergoes a sharp bond-dimension collapse for components, enabling polylogarithmic time and memory scaling. In the latter, the approach reaches high-resolution discretizations of frequency modes on standard hardware, far beyond the ceiling of dense implementations. These compressed representations enable efficient computation of Value at Risk (VaR) and Expected Shortfall (ES), supporting applications in quantitative finance and beyond.
Paper Structure (41 sections, 4 theorems, 71 equations, 16 figures, 3 algorithms)

This paper contains 41 sections, 4 theorems, 71 equations, 16 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Fourier spectral method
  4. Characteristic function
  5. Cumulative distribution function
  6. Spectral filtering
  7. Spectral reconstruction algorithm
  8. QTT-based Fourier spectral method
  9. Quantized tensor trains
  10. Finite-precision algebra
  11. Fourier spectral approximation using QTT
  12. Construction of the filtered characteristic function
  13. Dirichlet convolution and Fourier inversion
  14. Computational complexity
  15. Complexity for dense-vector Fourier approximations
  16. ...and 26 more sections

Key Result

Theorem 1

Berry--Esseen theorem Let $X_1, \ldots, X_D$ be independent random variables with $\mathbb{E}\left[ X_d \right]=0$, variances $\sigma_d^2$, and finite third absolute moments $\mathbb{E}\left[ |X_d|^3 \right]$. Denote $\sigma^2 = \sum_{d=1}^{D} \sigma_d^2$ and $\rho = \sum_{d=1}^{D} \mathbb{E}\left[ where $F_G$ denotes the CDF of the standard Gaussian distribution and $C$ is a universal constant.

Figures (16)

  • Figure 1: QTT diagram with $n$ rank-3 cores, carrying free indices $s_k$ of size $\ell_k$ and mutually contracted along virtual bonds $\alpha_k$ of dimension $\chi_k$.
  • Figure 2: Experimental setting used for the discrete experiments. (a) Beta $\mathcal{B}(2, 10)$ distribution used to sample the weights. (b) Spectral filters considered, including the raised cosine filter $\sigma_{\mathrm{rc}}$ (blue, $q=2$), sharpened raised cosine filter $\sigma_{\mathrm{src}}$ (orange, $q=8$), and exponential (Gaussian) filter $\sigma_{\mathrm{exp}}$ (green).
  • Figure 3: Spectral reconstruction of the binomial distribution with $D=3$ Bernoulli variables and 256 frequency modes, using the filters shown in Fig. \ref{['fig:filters']}b. (a) Exact PMF and spectral reconstructions. (b) Reconstructed CDF. (c) Pointwise error of the CDF, showing Gibbs oscillations surrounding the jump discontinuities.
  • Figure 4: Error analysis for a binomial model with $D=3$ components. (a) Decay of the median error (50% quantile of pointwise errors), reflecting the filter-dependent spectral convergence rates in smooth regions. (b) Decay of the global $\rm{L}^1$ error and the widths of the Gibbs bands.
  • Figure 5: Error analysis for WPB models of increasing size $D$. (a) Error in the global $\rm{L}^1$ norm, largely independent of $D$. (b) Error in the pointwise $\rm{L}^\infty$ norm, showing an exponential decay of Gibbs oscillation amplitudes due to diminishing jump heights. (c) Median error, deteriorating with increasing $D$ as the measure of the smooth region shrinks.
  • ...and 11 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof