Affine Rank Minimization is ER Complete

2026-02-15

AI Summary

This work investigates the affine rank minimization problem ARM(k)—determining whether there exists a real matrix of rank at most a fixed constant \(k\) satisfying given affine constraints. By constructing a polynomial-time reduction from the Existential Theory of the Reals (ETR), it establishes for the first time that ARM(3) is ETR-complete, and further proves that ARM(k) remains ETR-complete for every fixed integer \(k \geq 1\). The core technical contributions include the development of a non-interfering embedding gadget for rank enforcement and a canonical decomposition encoding scheme. These innovations precisely characterize the computational complexity boundary of low-rank affine feasibility problems, revealing that even systems consisting solely of affine equalities together with a fixed-rank constraint already possess the full expressive power of real algebraic computation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Learning Preferences or Rankings

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: Fairness-aware retrieval and ranking

Abstract

We study the decision problem Affine Rank Minimization, denoted ARM(k). The input consists of rational matrices A_1,...,A_q in Q^{m x n} and rational scalars b_1,...,b_q in Q. The question is whether there exists a real matrix X in R^{m x n} such that trace(A_l^T X) = b_l for all l in {1,...,q} and rank(X)<= k. We first prove membership: for every fixed k>= 1, ARM(k) lies in the existential theory of the reals by giving an explicit existential encoding of the rank constraint using a constant-size factorization witness. We then prove existential-theory-of-reals hardness via a polynomial-time many-one reduction from ETR to ARM(k), where the target instance uses only affine equalities together with a single global constraint rank(X)<= k. The reduction compiles an ETR formula into an arithmetic circuit in gate-equality normal form and assigns each circuit quantity to a designated entry of X. Affine semantics (constants, copies, addition, and negation) are enforced by linear constraints, while multiplicative semantics are enforced by constant-size rank-forcing gadgets. Soundness is certified by a fixed-rank gauge submatrix that removes factorization ambiguity. We prove a composition lemma showing that gadgets can be embedded without unintended interactions, yielding global soundness and completeness while preserving polynomial bounds on dimension and bit-length. Consequently, ARM(k) is complete for the existential theory of the reals; in particular, ARM(3) is complete. This shows that feasibility of purely affine constraints under a fixed constant rank bound captures the full expressive power of real algebraic feasibility.

Problem

• Affine Rank Minimization
• existential theory of the reals
• rank constraint
• matrix feasibility
• computational complexity

Innovation

• Affine Rank Minimization
• Existential Theory of the Reals
• Rank-Forcing Gadgets
• Factorization Witness
• ETR-Completeness