High-Resolution Tensor-Network Fourier Methods for Exponentially Compressed Non-Gaussian Aggregate Distributions
This enables efficient risk calculations like VaR and ES in quantitative finance, though it is incremental as it builds on existing tensor methods.
The paper tackled the problem of compressing non-Gaussian aggregate distributions by exploiting low-rank tensor structures in characteristic functions, achieving exponential compression with polylogarithmic scaling for large component counts and handling up to 2^30 frequency modes on standard hardware.
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.