Characteristic Lattice Algorithm

• Characteristic Lattice Algorithm (CLA) is a grid-based data reduction framework that partitions data into lattice-aligned cells for efficient analysis in TDA and lattice theory.
• The method guarantees stability with explicit error bounds in persistent homology, making large-scale or high-dimensional dataset analysis practical.
• CLA extends to computing lattice invariants and incorporating denoising via statistical thresholds, supporting robust classification and efficient invariant enumeration.

The Characteristic Lattice Algorithm (CLA) is a grid-based data reduction and structural analysis framework with central applications in topological data analysis (TDA) and the theory of automorphism groups of lattices. The CLA formalism was independently developed in multiple contexts: as a scalable preprocessing tool for persistent homology to render TDA feasible on large or high-dimensional datasets (Choi et al., 2023), as a denoising and data-reduction method with statistical guarantees (Choi et al., 31 Mar 2026), and as a computational method for enumerating characteristic masses of lattices in the arithmetic theory of automorphic forms (2002.03707). Across these domains, CLA organizes data into lattice-aligned cells, allowing succinct, representative sampling and efficient structure-preserving computation.

1. Precise Formulation and Objectives

For TDA and geometric data analysis, CLA takes as input a finite point cloud $X \subset \mathbb{R}^m$ and a user-selected scale parameter $\delta > 0$. The algorithm outputs a reduced point cloud $X^*_\delta$ with $|X^*_\delta| \leq |X|$ whose persistent homology (computed via the Vietoris–Rips complex) is provably close, under the bottleneck distance, to that of $X$. The fundamental objective is tunable data reduction: the user can decrease $|X|$ to any desired degree by increasing $\delta$, with an explicit upper bound on the maximal error in persistent barcodes. This makes large-scale persistent homology computation practical on high-dimensional or dense data (Choi et al., 2023).

In its original group-theoretic context, CLA computes the distribution (“masses”) of conjugacy classes in the isometry group $O(L)$ of an integral lattice $L$, organized by their characteristic polynomials. This involves organizing group elements by their action on a root lattice and its irreducible components, facilitating enumeration of invariants and explicit dimension formulas for orthogonal modular forms (2002.03707).

2. Algorithmic Workflow

Characteristic Lattice Subsampling for Point Clouds

Given $X = \{x^1, \dots, x^n\} \subset \mathbb{R}^m$ and scale $\delta > 0$0:

1. Partition $\delta > 0$1 into $\delta > 0$2-dimensional cubes (cells) of side length $\delta > 0$3, aligned with the integer lattice $\delta > 0$4.
2. For each cell $\delta > 0$5, if $\delta > 0$6, select a representative $\delta > 0$7 (either any $\delta > 0$8 or the cell center).
3. The reduced set is $\delta > 0$9.

Pseudocode is as follows: $O(L)$0 Here, choose_representative may select the point itself, or the cell center $X^*_\delta$0.

Characteristic Masses in Lattice Theory

For a lattice $X^*_\delta$1 specified by a Gram matrix $X^*_\delta$2:

1. Enumerate root vectors $X^*_\delta$3 via shortest-vector searches.
2. Compute the Weyl group $X^*_\delta$4 and “umbral” subgroup $X^*_\delta$5 using the stabilizer of a Weyl vector and permutation representation.
3. Decompose $X^*_\delta$6 into irreducible components and compute local masses for each cycle (irreducible ADE block).
4. Combine local masses via factorization identities (cf. Proposition 3.12 in (2002.03707)) to enumerate the number of elements of $X^*_\delta$7 with a given characteristic polynomial.

3. Stability and Error Analysis

For point cloud subsampling, CLA is accompanied by a stability theorem (Choi et al., 2023). Denoting $X^*_\delta$8 as the $X^*_\delta$9-th Vietoris–Rips barcode and $|X^*_\delta| \leq |X|$0 the bottleneck distance,

$|X^*_\delta| \leq |X|$1

for all $|X^*_\delta| \leq |X|$2. The proof relies on the fact that the Hausdorff distance between $|X^*_\delta| \leq |X|$3 and $|X^*_\delta| \leq |X|$4 is at most the diagonal of a cube, $|X^*_\delta| \leq |X|$5, and the classical stability of persistent homology under Gromov–Hausdorff perturbation.

When the “center-of-cube” version is used, an analogous bound holds between the two types of output:

$|X^*_\delta| \leq |X|$6

Error bounds are thus fully controlled by the lattice cell diameter, and the user can adjust $|X^*_\delta| \leq |X|$7 so that the topological summary remains faithful at the desired scale.

4. Complexity, Data Reduction, and Parameter Guidance

The reduced set cardinality after CLA satisfies:

$|X^*_\delta| \leq |X|$8

where $|X^*_\delta| \leq |X|$9 denotes projection onto the $X$0-th coordinate. When this product is strictly smaller than $X$1, the reduction is guaranteed. Empirically, up to 95% reduction in Vietoris–Rips computation time for 3D data is observed when halving the point count (Choi et al., 2023).

For the group-theoretic version, the root enumeration is exponential in $X$2 but practical up to $X$3, the permutation-group routines for conjugacy class enumeration are tractable for the moderate-sized stabilizers appearing in root lattices, and precomputed cyclotomic table lookups allow mass computation for all $X$4.

5. Extensions, Denoising, and Statistical Guarantees

The Refined Characteristic Lattice Algorithm (RCLA) introduces a count threshold $X$5 per cell, integrating denoising and reduction (Choi et al., 31 Mar 2026). For each lattice cell, only those with $X$6 (where $X$7) are retained, and the rest are discarded as likely noise. Stability is established under a homogeneous Poisson noise model: with high probability,

$X$8

for all $X$9. Parameter selection for $|X|$0 can be performed automatically via nearest-neighbor distance quantiles and Poisson model quantile estimates, subject to a user-specified false-positive budget. This approach outperforms both pure CLA and several dedicated denoisers in classification and bottleneck metrics, especially in high-noise regimes.

For general metric spaces, a plausible implication is to replace the Euclidean grid by coverings with balls of radius $|X|$1, picking one representative per (non-empty) ball, conceptually aligning with $|X|$2-nets.

6. Limitations, Applications, and Extensions

The principal limitation is the trade-off between reduction and topological fidelity: increasing $|X|$3 aggressively decreases $|X|$4 and computational time, but may merge or lose features close together at the scale $|X|$5. Uniform grids may under-sample low-density or small-cluster regions; possible extensions include adaptive grids, kd-tree decompositions, and hybrid approaches with witness complexes or coarsening.

Applications span high-dimensional genomics, computer vision (large-scale point sets), and time-series embedding (Takens' embeddings with $|X|$6). In arithmetic geometry, the algorithmic structure is essential for explicit mass formulae and for computational approaches to modular forms and automorphic invariants.

7. Case Studies and Empirical Validation

Experiments on synthetic 2D spheres and random point clouds with $|X|$7 reduce to $|X|$8–$|X|$9 points at $\delta$0, with persistence diagrams matching up to bottleneck $\delta$1. For $\delta$2 in 2D, TDA runtime drops from $\delta$3 seconds to $\delta$4, an approximately $\delta$5 speed-up; in 3D with $\delta$6, a $\delta$7 reduction is obtained at $\delta$8 size reduction. For binary classification using persistent barcodes processed with CLA, 100% accuracy is retained post-reduction.

RCLA achieves robust classification (mean accuracy $\delta$9 in multiclass 3D shape tasks under injected Poisson noise) and exhibits greater bottleneck stability and variance reduction compared to Adaptive DBSCAN, LDOF, and LUNAR denoisers in high-noise synthetic settings (Choi et al., 31 Mar 2026).

• "Effective data reduction algorithm for topological data analysis" (Choi et al., 2023)
• "Denoising data reduction algorithm for Topological Data Analysis" (Choi et al., 31 Mar 2026)
• "The Characteristic Masses of Niemeier Lattices" (2002.03707)