Stochastic trace estimation with tensor train random vectors

2026-06-14

AI Summary

This work addresses the high computational and storage costs of traditional stochastic trace estimation in tensor-structured settings, where unstructured test vectors lead to prohibitive overhead and low-rank tensor-product vectors suffer from exponentially growing sample complexity with tensor order. To overcome these limitations, the paper proposes using Gaussian random Tensor Train (TT) vectors as structured probes, which significantly reduce resource requirements while preserving accuracy. The key contributions include the first proof of dimension-independent error bounds under appropriate TT ranks, and the recovery—within a median-of-means framework—of the classical Gaussian estimator’s dependence on accuracy and failure probability. Furthermore, the authors establish an unbiased subspace embedding theory: when the TT rank satisfies \( r \geq d-1 \), only \( O(\varepsilon^{-2}(k + \log(1/\delta))) \) samples suffice to reliably embed a \( k \)-dimensional subspace, and under spectral tail conditions, the Nyström++ estimator achieves an \( O(\varepsilon^{-1}) \) sample complexity.

Abstract

Stochastic trace estimation is a standard tool for approximating the trace of a large-scale matrix available only through matrix-vector products. However, in tensor-structured settings, unstructured Gaussian or Rademacher test vectors may be prohibitively expensive to store and compute with, while cheaper rank-one tensor-product vectors can require sample complexities that grow exponentially with the tensor order. This work studies Gaussian random tensor train vectors as a structured alternative for stochastic trace estimation. We show that, with a suitable choice of the tensor train rank, random tensor train vectors recover dimension-independent guarantees for the Girard--Hutchinson estimator. In particular, a median-of-means variant with tensor train rank $r \geq d-1$ achieves the same dependence on the accuracy $\varepsilon$ and failure probability $δ$ as the classical estimator based on unstructured Gaussian vectors. We further prove an oblivious subspace injection result for sketches formed from independent Gaussian random tensor train vectors: tensor train rank $r\geq d-1$ and $\mathcal{O}(\varepsilon^{-2}(k+\log(1/δ)))$ samples suffice for a $k$-dimensional target subspace. Finally, we investigate the use of such sketches within the Nyström++ framework. We show that the resulting estimator can achieve the desired $\mathcal{O}(\varepsilon^{-1})$ sample complexity under an additional spectral-tail condition. These results provide clarififcation on both the potential and the limitations of random tensor train vectors in stochastic trace estimation.

Problem

• stochastic trace estimation
• tensor train
• sample complexity
• structured random vectors
• matrix trace approximation

Innovation

• tensor train
• stochastic trace estimation
• structured random vectors
• subspace embedding
• Nyström++