Limits of Rank Recovery in Bilinear Observation Problems

TL;DR

This paper investigates the fundamental limits of recovering the effective rank in bilinear observation problems, showing that certain dimensional deficits are stable and cannot be resolved by numerical refinement alone, but require structural problem modifications.

Contribution

It provides a detailed analysis of the nullspace structure and demonstrates that rank recovery necessitates changing the problem formulation rather than just numerical refinement.

Findings

Extended rank plateaus indicate persistent dimensional deficits.

Nullspace exhibits organized structure with sector concentration.

Rank recovery requires structural modifications to the problem.

Abstract

Bilinear observation problems arise in many physical and information-theoretic settings, where observables and states enter multiplicatively. Rank-based diagnostics are commonly used in such problems to assess the effective dimensionality accessible to observation, often under the implicit assumption that rank deficiency can be resolved through numerical refinement. Here we examine this assumption by analyzing the rank and nullity of a bilinear observation operator under systematic tolerance variation. Rather than focusing on a specific reconstruction algorithm, we study the operator directly and identify extended rank plateaus that persist across broad tolerance ranges. These plateaus indicate stable dimensional deficits that are not removed by refinement procedures applied within a fixed problem definition. To investigate the origin of this behavior, we resolve the nullspace into algebraic sectors defined by the block structure of the variables. The nullspace exhibits a pronounced but nonexclusive concentration in specific sectors, revealing an organized internal structure rather than uniform dimensional loss. Comparing refinement with explicit modification of the problem formulation further shows that rank recovery in the reported setting requires a change in the structure of the observation problem itself. Here, "problem modification" refers to changes that alter the bilinear observation structure (e.g., admissible operator/state families or coupling constraints), in contrast to refinements that preserve the original formulation such as tolerance adjustment and numerical reparameterizations. Together, these results delineate limits of rank recovery in bilinear observation problems and clarify the distinction between numerical refinement and problem modification in accessing effective dimensional structure.