Extended Rank Plateaus in Geometry & Numerics

• Extended rank plateaus are stable intervals characterized by constant effective rank, reflecting intrinsic structural properties rather than numerical artifacts.
• In metric geometry, they emerge as unique tangent cones at infinity of quasi-minimizing cycles, providing insight into large-scale Euclidean structures in CAT(0) spaces.
• In bilinear observation problems, these plateaus offer robust rank diagnostics that remain invariant under numerical refinement, guiding necessary modifications to measurement models.

Extended rank plateaus arise as distinct, stable intervals in the effective rank of operators in both geometric group theory (specifically, in the study of spaces with higher asymptotic rank) and in numerical analysis of bilinear observation problems. In both contexts, the phenomenon denotes the persistence of certain rank values—across wide parameter or tolerance ranges—reflecting intrinsic structural features rather than artifacts of parameter choice or numerics. In geometric settings, extended rank plateaus are formalized as unique tangent cones at infinity associated to quasi-minimizing cycles in spaces of higher asymptotic rank, capturing their large-scale asymptotic geometry. In bilinear observation problems, they manifest as maximal intervals of constant numerical rank under threshold variation, signifying limits of rank recoverability under numeric refinement.

1. Asymptotic Rank and Platonic Structure in Metric Spaces

In metric geometry, the asymptotic rank $\mathrm{asrk}(X)$ of a proper space $(X,d)$ is the largest $k$ such that $X$ admits quasi-isometric embeddings of arbitrarily large balls in $\mathbb{R}^k$. The study of higher rank hyperbolicity, as initiated by Kleiner and Lang (Kleiner et al., 2018), replaces geodesic lines—central to the geometry of hyperbolic (rank one) spaces—with $n$-dimensional, locally finite, quasi-minimizing cycles with polynomial ($r^n$) volume growth. In this framework, the large-scale geometry of these cycles mimics that of Euclidean $n$-flats. The tangent cones at infinity associated to such cycles, constructed as Gromov–Hausdorff limits of rescalings ("blow-ups"), are designated as extended rank plateaus. They embody the limiting geometric structure of these cycles and provide an essential invariant for the category of proper CAT(0) or convex-bicombing spaces with $\mathrm{asrk}(X)=n\ge2$ (Kleiner et al., 2018).

2. Formal Definition and Construction in Metric Geometry

Given a quasi-minimizing $n$-cycle $S\in Z_{n,\mathrm{loc}}(X)$ with controlled volume growth and asymptotic density $0_\infty(S)<\infty$, there exists a unique (up to $F$-equivalence) conical representative $S_{p,0}$ from which, by projection and metric rescaling, one extracts an integral current $E$ in the metric cone $C_T X$ over the Tits boundary $\partial_T X$. The pair $(C_T X, E)$ is called the extended rank plateau associated to $S$ [(Kleiner et al., 2018), §4.2]. The support of $E$ is the unique flat cone in $C_T X$ capturing the asymptotic shape of $S$. For every asymptotic boundary datum $R\in Z_{n-1,c}(\partial_T X)$, there exists a unique area-minimizing cycle $S$ with limit set determined by $R$, whose tangent cone becomes the plateau.

3. Extended Rank Plateaus in Bilinear Observation Operators

In the setting of bilinear observation problems (Choi, 13 Jan 2026), extended rank plateaus refer to intervals of the tolerance parameter $\varepsilon$ over which the numerical rank of the design matrix $A$ remains strictly constant. Explicitly, for $A\in \mathbb{C}^{N \times D_{\mathrm{amb}}}$ constructed by stacking features $$\varphi(E_i,\rho_j)=\mathrm{vec}(E_i\otimes \rho_j^{\mathsf T})$$, the $\varepsilon$-numerical rank is defined as $$\mathrm{rank}_\varepsilon(A)=|\{k : \sigma_k(A) > \varepsilon \sigma_1(A)\}|$$, where $\{\sigma_k\}$ are the singular values of $A$.

Plateaus are maximal intervals $[\varepsilon_k, \varepsilon_{k+1}]$ such that $\mathrm{rank}_\varepsilon(A)=r_k$ is constant; jumps occur only at the boundaries. For example, in a $256\times 256$ configuration, plateaus at numerical ranks 200, 220, 228, 240 are observed over decades of tolerance variation, and similar robustness is found for other configurations (Choi, 13 Jan 2026).

Configuration	Plateau Intervals ($\varepsilon$)	Plateau Ranks
A ($256\times256$)	$[10^{-12},10^{-9}]$, $(10^{-9},10^{-6}]$, $(10^{-6},10^{-3}]$, $(10^{-3},10^{-2}]$	200, 220, 228, 240
B,C ($400\times256$)	$[10^{-12},10^{-8}]$, $(10^{-8},10^{-5}]$, $(10^{-5},10^{-2}]$	230, 245, 250

4. Origin and Structure of Plateau Nullspaces

The nullspace corresponding to an extended rank plateau demonstrates pronounced algebraic organization. Given the block structure $$\mathbb{C}^{d^2\times d^2}\simeq\mathbb{C}^{d\times d}\otimes\mathbb{C}^{d\times d}$$, projections $P_{DD}$, $P_{DO}$, $P_{OD}$, $P_{OO}$ are defined onto diagonal and off-diagonal sectors (Diagonal$\otimes$Diagonal, etc.). Empirically, the "off-off" sector ($OO$) dominates, with $w_{OO}\approx 0.65$–$0.75$ of the total nullity in all configurations, while diagonal contributions are minimal ($w_{DD}\approx0.02$–$0.05$) (Choi, 13 Jan 2026). This reveals that the missing rank does not arise from numerical instability but reflects a persistent, algebraically concentrated deficit rooted in the structural symmetries or constraints of the original problem.

5. Plateau Robustness: Numerical Refinement versus Problem Modification

Extended rank plateaus are robust to all forms of numerical refinement that preserve the underlying operator or problem definition. Adjustments such as finer $\varepsilon$ grids, higher-precision arithmetic, or reparameterization (e.g., realification) do not alter the plateau ranks; $\mathrm{rank}_\varepsilon(A)$ remains strictly constant inside each plateau. Only explicit modification of the problem—such as changing the admissible sets $\{\rho_j\}$, altering coupling constraints, or introducing new measurement operators—restores full rank at previous plateau-locked tolerances (Choi, 13 Jan 2026). This distinction sharply delineates structural (formulation-based) limitations from numerical or threshold artifacts.

6. Applications and Implications in Geometry and Data Analysis

In metric geometry, extended rank plateaus of cycles underpin the extension of quasi-isometries from spaces to their Tits boundaries, with direct consequences for quasi-isometric rigidity, large-scale invariants, and structural stability in higher rank spaces such as symmetric spaces and Euclidean buildings (Kleiner et al., 2018). In bilinear observation problems, plateau detection clarifies the limits of dimensionality accessible to observation under the current setup and provides diagnostic power by separating deficiencies due to numerical implementation from those due to intrinsic algebraic structure. A plausible implication is that rank-based diagnostics in data-driven settings must be interpreted in light of possible plateau phenomena and that only substantive change in the model or measurement protocol can break intrinsic dimensional degeneracies.

7. Summary: Foundational Status and Interpretation Guidelines

Extended rank plateaus constitute a robust invariant in both geometric and algebraic contexts. They arise from deep structural properties—be it the asymptotic cones of metric spaces or the algebraic structure of bilinear maps—and serve as a diagnostic for irreducible dimensional deficits inaccessible through infinitesimal or numerical adjustment. Their detection indicates intrinsic constraints of the formulation and serves as a guide for problem modification when full effective rank is required. Failure to recover rank under numerical refinement is therefore best interpreted as reflecting foundational limitations of the current setup rather than a deficiency in computational protocol (Kleiner et al., 2018, Choi, 13 Jan 2026).