Tensor train vectors cut trace estimation costs without losing accuracy

Stochastic trace estimation is crucial for large-scale matrices accessible only through matrix-vector products, but tensor-structured settings pose a storage dilemma: unstructured Gaussian or Rademacher test vectors become prohibitively expensive, while cheap rank-one tensor-product vectors require sample complexities that grow exponentially with the tensor order. Bujanović, Kressner, and Olić (arXiv:2606.15679) propose Gaussian random tensor train vectors as a structured alternative. They prove that with a suitable tensor train rank (r ≥ d-1), these vectors recover dimension-independent guarantees for the Girard–Hutchinson estimator. Their median-of-means variant achieves the same dependence on accuracy ε and failure probability δ as the classical unstructured estimator, making it both memory-efficient and statistically sound.

The authors further demonstrate an oblivious subspace injection result: with tensor train rank r ≥ d-1 and O(ε⁻²(k + log(1/δ))) samples, the sketch captures a k-dimensional target subspace. In the Nyström++ framework, this approach yields an estimator with O(ε⁻¹) sample complexity under an additional spectral-tail condition, paving the way for scalable kernel matrix approximation. These theoretical clarifications highlight both the potential—dimension-independent guarantees with structured storage—and the limitations of random tensor train vectors, offering a practical bridge between efficiency and accuracy for high-order tensor data.