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

TL;DR

This paper introduces a tensor network Fourier method that achieves exponential compression of non-Gaussian distributions' characteristic functions, enabling efficient risk measure computations for large-scale problems.

Contribution

It presents a novel tensor network approach to represent and compute characteristic functions of sums of independent variables with exponential compression.

Findings

Achieves exponential compression of characteristic functions in non-Gaussian distributions.

Enables high-resolution Fourier analysis with billions of frequency modes on standard hardware.

Supports efficient computation of financial risk measures like VaR and ES.

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.