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Affine Rank Minimization is ER Complete

Angshul Majumdar

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.

Affine Rank Minimization is ER Complete

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.
Paper Structure (90 sections, 40 theorems, 91 equations, 1 figure)

This paper contains 90 sections, 40 theorems, 91 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation
  3. Complexity background
  4. Contributions
  5. Technical overview
  6. Related work
  7. Rank minimization and relaxations.
  8. Hardness of rank problems in discrete settings.
  9. $\exists\mathbb{R}$-completeness as the right lens.
  10. Preliminaries
  11. Existential theory of the reals (ETR)
  12. Syntax and encoding.
  13. The class $\exists\mathbb{R}$.
  14. Standard normal forms.
  15. Circuit normal form.
  16. ...and 75 more sections

Key Result

Lemma 2.2

Given any ETR instance $\exists x:\varphi(x)$, one can construct in polynomial time an equisatisfiable instance in equality form (Definition def:etr_equality_form) with polynomial blow-up in bit-size.

Figures (1)

  • Figure 1: Determinant-forcing multiplication gadget. The $2\times2$ block is combined with the fixed $3\times3$ gauge block; any violation of $z=xy$ induces a nonzero $4\times4$ determinant.

Theorems & Definitions (94)

  • Definition 2.1: ETR equality form
  • Lemma 2.2: Eliminating inequalities and Boolean structure
  • proof
  • Definition 2.3: Arithmetic circuit
  • Definition 2.4: Circuit size and bit-size
  • Lemma 2.5: Circuit constraints are exact
  • proof
  • Definition 2.6: $\mathrm{ARM}(k)$
  • Lemma 2.7: Rank-factorization equivalence
  • proof
  • ...and 84 more