Near-Efficient and Non-Asymptotic Multiway Inference

2025-11-07

AI Summary

This paper addresses multiway statistical inference for low-rank tensor models with count-valued (Poisson-distributed) data, tackling the lack of non-asymptotic efficiency guarantees in existing tensor decomposition methods. We propose a CP decomposition inference framework based on rank-constrained maximum likelihood estimation. For the rank-one case, we construct, for the first time, a “nearly efficient” estimator whose variance asymptotically attains the Cramér–Rao lower bound (CRLB). For higher CP ranks, we derive a sharper error upper bound—improving upon prior work both in its dependence on the CP rank and in achieving near minimax optimality. Our theoretical analysis establishes non-asymptotic efficiency guarantees under finite samples, and extensive numerical experiments confirm that the proposed estimator significantly outperforms state-of-the-art baselines in estimation accuracy. The core contribution is the first multiway inference theory for Poisson tensor models that simultaneously ensures statistical efficiency and computational tractability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Personalized content ranking and presentation

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.

Problem

• Establish non-asymptotic efficiency guarantees for tensor decomposition inference
• Achieve near-efficient multiway analysis matching Cramér-Rao Lower Bound
• Provide CP-based parametric inference with improved rank dependence

Innovation

• Rank-constrained maximum-likelihood estimator for tensor decomposition
• Achieves near-efficient variance matching Cramér-Rao bound
• Provides non-asymptotic guarantees for multiway count data inference