Stable Rank and Distance Preservation

• Stable Rank is a spectral invariant that quantifies the effective dimension or spread in data, ensuring the stability of topological features.
• Distance preservation guarantees that metric and topological summaries remain stable under perturbations and embeddings, facilitating reliable dimension reduction.
• Applications span persistent homology, JL embeddings, and graph clustering, supporting scalable, noise-resistant geometric and topological data analysis.

Stable rank is a categorical and spectral invariant underlying robust distance preservation in persistent homology, metric embeddings, and geometric data analysis. It measures the effective complexity (in terms of spread or dimension) of objects such as persistence modules or point clouds and provides a foundation for both the stability of topological invariants and dimension reduction schemes. Distance preservation addresses how metrics or topological summaries are stably retained under perturbations, embedding mappings, or categorical generalizations.

1. Stable Rank: Definitions and General Principles

Stable rank arises in distinct but related contexts: vector spaces, persistence modules, difference matrices, and regular categories.

• In spectral embedding and metric geometry, the stable rank of a real matrix $M$ is defined as $\mathrm{sr}(M) = \|M\|_F^2 / \|M\|_2^2$—the ratio of Frobenius norm squared to operator norm squared—which quantifies the number of significant singular directions beyond simple rank (Deshpande et al., 2015, Casey, 2023).
• In persistence, the stable rank generalizes the classical Betti number: for a diagram $F: (\mathbb{R}, \le) \to \mathbf{R}$, $p_F(u,v) = r(\mathrm{Im}[F(u) \to F(v)])$ records the integer-valued invariant (e.g., dimension) induced by a rank function $r$ satisfying monotonicity and subadditivity on a regular category $(\mathbf{R}, r)$ (Bergomi et al., 2019).
• In one-parameter persistence, given a pseudometric $d_C$ between modules (from a contour $C$), the stabilized rank $\widehat{\mathrm{rank}}_C(V)(t)$ is the minimum rank among modules $W$ within $d_C$ distance $t$ of $V$ (Chachólski et al., 2019, Agerberg et al., 2023).

2. Stable Rank and Distance Preservation in Metric Embeddings

Stable rank directly governs the quality of average distance preservation in spectral embeddings, particularly in $\ell_2^2$ to $\ell_1$ dimension reduction:

• For $n$ points $x_1, ..., x_n \in \mathbb{R}^d$ satisfying $\ell_2^2$ triangle inequalities (i.e., squared distances forming a metric), the stable rank of the difference matrix $M$ controls the distortion of embeddings $f:\mathbb{R}^d \to \ell_1$. Specifically:
  • There exists an explicit linear embedding with contraction and average distortion $O(\mathrm{sr}(M))$. That is, for all $i, j$:

  $\|f(x_i) - f(x_j)\|_2 \leq \|x_i - x_j\|_2^2$

  while

  $$\sum_{i<j} \|f(x_i) - f(x_j)\|_2 \geq \frac{1}{\mathrm{sr}(M)} \sum_{i<j} \|x_i - x_j\|_2^2$$

  (Deshpande et al., 2015). - High stable rank implies strong aggregate distance preservation for most pairs, and is essential for efficient approximation algorithms (e.g., for Sparsest Cut) (Deshpande et al., 2015).

• Bulk Johnson–Lindenstrauss Lemmas: If one tolerates an $\eta$ fraction of distances being distorted, only the minimal stable rank $R$ of batches of difference vectors limits the necessary target dimension $m$ for random projections. The formula

$m = O(\epsilon^{-2} \log(4e/\eta) \log N / R)$

quantifies how high stable rank enables drastic dimensionality reduction with controlled distance distortion in almost all pairs (Casey, 2023).

3. Rank-Based Persistence and Categorical Stability

The categorical axiomatization of stable rank enables broad generalization of persistence and its stability:

• In a ranked category $(\mathbf{R}, r)$, the rank function is required to satisfy:
  1. Monotonicity under monomorphisms: $A \hookrightarrow B \implies r(A) \leq r(B)$,
  2. Monotonicity under regular epimorphisms: $B \twoheadrightarrow D \implies r(B) \geq r(D)$,
  3. Subadditivity in pullback squares: $r(B) - r(A) \geq r(D) - r(C)$. (Bergomi et al., 2019).

• Persistence functions $p_F(u,v)$ induced by such categories generalize Betti numbers and recover classical stable ranks in vector space settings. Moreover, these persistence invariants are stable under function perturbations:

$$\|f-g\|_{\infty} \geq d_I(F,G) \geq d_B(\mathcal{D}F, \mathcal{D}G)$$

where $d_I$ is interleaving distance, $d_B$ is colored bottleneck distance, and $\mathcal{D}F$ denotes the multicolored persistence diagram (Bergomi et al., 2019).

• In semisimple Abelian categories, equality holds:

$$d_I(F,G) = d_B(\mathcal{D}_\mathcal{C}F, \mathcal{D}_\mathcal{C}G)$$

confirming that stable rank—as encoded in colored diagrams—fully captures categorical distance preservation (Bergomi et al., 2019).

4. Stability of Rank Invariants in Persistent Homology

Rank invariants underpin the stability of multidimensional and one-parameter persistence modules:

• For $X$ a triangulable space and $f, g: X \to \mathbb{R}^n$ continuous, the multidimensional rank invariant $\rho_{f,k}(u,v)$ tracks the dimension of persistent homology classes across lower-level sets. The matching (bottleneck) distance $D_\mathrm{match}(\rho_f, \rho_g)$ between their rank invariants is stably bounded:

$$D_\mathrm{match}(\rho_{f,k}, \rho_{g,k}) \leq \|f - g\|_\infty$$

(0908.0064, Frosini et al., 2010). This uniform bound generalizes the classical bottleneck stability.

• For two tame persistence modules $V, W$, the interleaving distance $d_I$ lower-bounds the uniform difference of rank invariants:

$$d_I(V,W) \geq \sup_{u \preceq v} |\rho_V(u,v) - \rho_W(u,v)|$$

(Landi, 2014).

• In domain perturbation, encoding sets via distance functions or densities (Hausdorff, symmetric difference, sup-norm) yields stability bounds for the matching distance between rank invariants in terms of the underlying set metric (Frosini et al., 2010).

5. Stable Rank Invariants and 1-Lipschitz Robustness

Hierarchical stabilization converts discrete rank invariants into robust, 1-Lipschitz stable rank functions:

• For a chosen contour $C$, the pseudometric $d_C$ between persistence modules induces the stabilized rank

$$\widehat{\mathrm{rank}}_C(V)(t) = \min\{\mathrm{rank}(W) \mid d_C(V, W) \leq t\}$$

which is nonincreasing, additive, and monotonic (Chachólski et al., 2019, Agerberg et al., 2023).

• The interleaving distance between stable ranks is bounded by the pseudometric:

$$d_C(V, W) \geq d_\bowtie(\widehat{\mathrm{rank}}_C(V), \widehat{\mathrm{rank}}_C(W))$$

and

$$|\widehat{\mathrm{rank}}_C(V)(t) - \widehat{\mathrm{rank}}_C(W)(t)| \leq d_C(V, W)$$

(Chachólski et al., 2019, Agerberg et al., 2023).

• In the context of algebraic Wasserstein distances, stable rank functions are efficiently computable and can be tuned via interpretable parameters (norm order $p$, contour $C$) to reflect task-specific, robust geometry. These invariants are 1-Lipschitz and support metric learning pipelines (Agerberg et al., 2023).

6. Applications and Algorithmic Implications

Stable rank and distance preservation are foundational in several algorithmic and geometric contexts:

• Sparsest Cut and spectral clustering: Stable rank enables polynomial-time cut rounding in graphs with low threshold-rank SDP relaxations, with quality directly controlled by stable rank rather than ambient dimension (Deshpande et al., 2015).
• Dimension reduction: Bulk JL embeddings and SR-based dimension formulas allow substantial reduction in target dimensions when intrinsic stable rank is high, crucial for scalable geometric data analysis and privacy-preserving random projections (Casey, 2023).
• Persistent homology: Stable rank invariants facilitate reliable, noise-resistant classification (e.g., in point-processes, time-series, and artery tree data), with accuracy sensitively tunable via contour choices (Chachólski et al., 2019, Agerberg et al., 2023).

Context	Stable Rank Role	Distance Preservation Mechanism
Metric embeddings	Limits average distortion	Spectral embeddings: $\ell_2^2 \to \ell_1$
JL Lemmas	Enables reduced target dimension	Bulk distance preservation in random projections
Persistent homology	Provides robust invariant feature map	Bottleneck/interleaving stability (1-Lipschitz)
Wasserstein invariants	Allows task-adaptive feature learning	Parameterized pseudometrics, interpretable tuning

In summary, stable rank quantifies the effective dimension, spread, or complexity governing the stability of geometric and topological invariants under perturbation and embedding. Its presence ensures that metric distortion or topological loss is uniformly controlled in algorithms, categorical frameworks, and data analysis settings. The robust, 1-Lipschitz behavior of stabilized rank invariants underlies their suitability for practical, discriminative, and reliable applications across mathematics, computer science, and applied topology.