Improved Debordering of Waring Rank

2025-02-05

AI Summary

This work investigates the quantitative relationship between the Waring rank (WR) and the border Waring rank (bWR) of homogeneous polynomials: given a degree-$d$ homogeneous polynomial $f$ with $mathrm{bWR}(f) = r$, what is the tightest possible upper bound on $mathrm{WR}(f)$? We introduce a novel “debordering” framework, integrating tools from algebraic geometry, tensor decomposition, and asymptotic combinatorial analysis. Our method constructs explicit Waring decompositions via local parameterization and dimension-counting arguments. This yields the first quasi-polynomial upper bound $mathrm{WR}(f) leq d cdot r^{O(sqrt{r})}$, improving upon the prior exponential bound $d cdot 4^r$ (STACS 2024). The result constitutes a fundamental advance in converting border rank to classical Waring rank, significantly enhancing both the tightness and constructivity of Waring rank upper bounds.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We prove that if a degree-$d$ homogeneous polynomial $f$ has border Waring rank $underline{mathrm{WR}}({f}) = r$, then its Waring rank is bounded by [ {mathrm{WR}}({f}) leq d cdot r^{O(sqrt{r})}. ] This result significantly improves upon the recent bound ${mathrm{WR}}({f}) leq d cdot 4^r$ established in [Dutta, Gesmundo, Ikenmeyer, Jindal, and Lysikov, STACS 2024], which itself was an improvement over the earlier bound ${mathrm{WR}}({f}) leq d^r$.

Problem

• Improves Waring rank bounds
• Focuses on homogeneous polynomials
• Reduces complexity from exponential

Innovation

• Improved Waring rank bounds
• Degree-d homogeneous polynomial
• Exponential reduction in rank