Limits of Rank Recovery in Bilinear Observation Problems

TL;DR

The paper addresses whether rank deficiency in bilinear observation problems can be overcome by numerical refinement or is a structural feature of the fixed formulation. It analyzes the rank and nullity of a bilinear observation operator under systematic tolerance variation, revealing extended rank plateaus that persist across refinements. By decomposing the nullspace into algebraic sectors, it shows organized localization, concentrated in the block-off-diagonal sector but not exclusively. The key finding is that rank recovery requires explicit modification of the observation structure rather than further numerical refinement, providing a concrete operational distinction for interpreting rank-based diagnostics in bilinear inverse problems.

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.

Figures (5)

• Figure 1: Conceptual structure of the bilinear observation setting. (a) Observation as a bilinear map from measurement operators and states. (b) Accessibility as a structural property, with an accessible subspace and a structurally inaccessible nullspace. (c) Distinction between refinement within a fixed formulation and explicit problem modification.
• Figure 2: Rank and nullity under tolerance variation. The rank of the bilinear observation operator is shown as a function of numerical tolerance. Extended plateau regions indicate stable rank values that persist across refinement within a fixed problem formulation. Nullity is shown as a derived quantity, defined as the difference between the ambient dimension and the measured rank.
• Figure 3: Robustness of rank plateaus across configurations. Rank-versus-tolerance profiles are shown for multiple experimental configurations. The persistence of plateau structure demonstrates that the observed rank behavior is reproducible within the reported settings.
• Figure 4: Sector localization of the nullspace. The nullspace is resolved into algebraic sectors defined by the block structure of the variables. A pronounced but non-exclusive concentration is observed in the block-off-diagonal sector, indicating organized nullspace structure.
• Figure 5: Refinement versus problem modification. Refinement procedures preserve the observed rank plateaus within a fixed problem formulation. In contrast, explicit modification of the observation structure enables recovery of the maximal rank in the reported setting.