Factors Governing TCI Convergence Rate

Determine the factors that govern the empirical error convergence rate O(1/n_s^a) (with exponent a > 1/2) observed for Tensor Cross Interpolation when approximating multidimensional functions, such as integrands arising in quantum impurity models and related Feynman-diagram calculations, in order to explain what controls the observed convergence behavior.

Background

Tensor Cross Interpolation (TCI) is a recent tensor-network-based approach that can learn and integrate high-dimensional functions by exploiting low-rank structure. In applications to quantum impurity models, TCI has been reported to converge faster than Monte Carlo sampling with an error scaling of O(1/n_sa) for some a > 1/2.

Despite these empirical observations, the paper explicitly notes that the determinants of this convergence rate are not currently understood. Clarifying which structural properties of the target functions (e.g., smoothness, discontinuities, variable couplings, symmetries) drive the rate would guide algorithm design and parameter choices for TCI in more complex multiorbital electron-phonon settings.

References

It offers faster convergence than QMC, with an error rate of $O(1/n_{\mathrm{s}a)$, where $a > 1/2$. However, the factors affecting this rate are still unclear.

— Low-rank quantics tensor train representations of Feynman diagrams for multiorbital electron-phonon models  (2405.06440 - Ishida et al., 2024) in Main text, Introduction (TCI paragraph), first page

This further validates the approach as fairly controlled and strengthens the conjectured connection between the TCI decomposition error and the DMFT convergence.

— Accelerating a Strong-Coupling Non-Equilibrium Steady-State Impurity Solver using (Quantics) Tensor Trains  (2608.13146 - Schindler et al., 13 Aug 2026) in Appendix, Section “Convergence of the DMFT self-consistency in the photodoped case” (Section \ref{app:tfla_convergence})