Near-Efficient and Non-Asymptotic Multiway Inference

TL;DR

This paper establishes non-asymptotic efficiency guarantees for tensor decomposition-based inference in count data models, demonstrating near-optimal variance in rank-one cases and improved error bounds for higher ranks.

Contribution

It introduces a framework for near-efficient multiway inference in finite samples, with new theoretical guarantees and practical insights for tensor decomposition in count data models.

Findings

Rank-one estimator achieves near-CRLB variance.

Higher-rank estimators may not attain CRLB but are nearly minimax optimal.

Numerical experiments confirm theoretical efficiency results.

Abstract

We establish non-asymptotic efficiency guarantees for tensor decomposition-based inference in count data models. Under a Poisson framework, we consider two related goals: (i) parametric inference, the estimation of the full distributional parameter tensor, and (ii) multiway analysis, the recovery of its canonical polyadic (CP) decomposition factors. Our main result shows that in the rank-one setting, a rank-constrained maximum-likelihood estimator achieves multiway analysis with variance matching the Cram\'{e}r-Rao Lower Bound (CRLB) up to absolute constants and logarithmic factors. This provides a general framework for studying "near-efficient" multiway estimators in finite-sample settings. For higher ranks, we illustrate that our multiway estimator may not attain the CRLB; nevertheless, CP-based parametric inference remains nearly minimax optimal, with error bounds that improve on prior work by offering more favorable dependence on the CP rank. Numerical experiments corroborate near-efficiency in the rank-one case and highlight the efficiency gap in higher-rank scenarios.