arXiv subjects

Uzu Lim

Publications and source records attributed to Uzu Lim.

8 recordsLinked to original sources

arXiv2026

Universal topological statistics on triangulated singular spaces

We prove a universality theorem for random persistent homology over a class of triangulable spaces. More precisely, let $M \subset \mathbb{R}^D$ be a compact $C^2$-triangulable space satisfying a geometric quality condition and let $f: M \to \mathbb{R}$ be a probability density. Then the expected persistence ratio measure computed from the \v{C}ech or Vietoris-Rips complex of a Poisson point process with intensity $nf$ has a universal limit independent of $(M, f)$. Since smooth manifolds, algebraic varieties, semialgebraic sets and Whitney stratified spaces are all triangulable spaces, our theorem applies to a large class of non-Euclidean spaces. Beyond persistent homology, our proof covers a general class of scale-invariant functionals. It relies on a geometric transfer method that adapts constructions in Euclidean space to triangulable spaces through successive approximations by Freudenthal-Kuhn triangulations, and control of interference across singular strata.

math.PR↗

arXiv2025

Cover Learning for Large-Scale Topology Representation

Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation or infer local geometry by constructing a graph on the input data. More recently, topological data analysis popularized the use of simplicial complexes to represent data topology with two main methodologies: topological inference with geometric complexes and large-scale topology visualization with Mapper graphs -- central to these is the nerve construction from topology, which builds a simplicial complex given a cover of a space by subsets. While successful, these have limitations: geometric complexes scale poorly with data size, and Mapper graphs can be hard to tune and only contain low dimensional information. In this paper, we propose to study the problem of learning covers in its own right, and from the perspective of optimization. We describe a method for learning topologically-faithful covers of geometric datasets, and show that the simplicial complexes thus obtained can outperform standard topological inference approaches in terms of size, and Mapper-type algorithms in terms of representation of large-scale topology.

cs.LG↗

arXiv2024

Fibers of point cloud persistence

Persistent homology (PH) studies the topology of data across multiple scales by building nested collections of topological spaces called filtrations, computing homology and returning an algebraic object that can be vizualised as a barcode--a multiset of intervals. The barcode is stable and interpretable, leading to applications within mathematics and data science. We study the spaces of point clouds with the same barcode by connecting persistence with real algebraic geometry and rigidity theory. Utilizing a semi-algebraic setup of point cloud persistence, we give lower and upper bounds on its dimension and provide combinatorial conditions in terms of the local and global rigidity properties of graphs associated with point clouds and filtrations. We prove that for generic point clouds in $\mathbb{R}^d$ ($d \geq 2$), a point cloud is identifiable up to isometry from its VR persistence if the associated graph is globally rigid, and locally identifiable up to isometry from its \v{C}ech persistence if the associated hypergraph is rigid.

math.AT↗

arXiv2024

Cohomology of p-adic fields and Local class field theory

In this expository article, we outline a basic theory of group (co)homology and prove a cohomological formulation of the Local Reciprocity Law: $${\rm Gal}(L/K)^{\rm ab} \cong H_T^{-2}({\rm Gal}(L/K),\mathbb{Z}) \cong H_T^{0}({\rm Gal}(L/K),L^\times) \cong \frac{K^\times}{{\rm Nm}_{L/K}(L^\times)}$$ We first recall basic facts about local fields and homological algebra. Then we define group (co)homology, Tate cohomology, and furnish a toolbox. The Local Reciprocity Law is proven in an abstract cohomological setting, then applied to the case of local fields.

math.NT↗

arXiv2023

HADES: Fast Singularity Detection with Local Measure Comparison

We introduce Hades, an unsupervised algorithm to detect singularities in data. This algorithm employs a kernel goodness-of-fit test, and as a consequence it is much faster and far more scaleable than the existing topology-based alternatives. Using tools from differential geometry and optimal transport theory, we prove that Hades correctly detects singularities with high probability when the data sample lives on a transverse intersection of equidimensional manifolds. In computational experiments, Hades recovers singularities in synthetically generated data, branching points in road network data, intersection rings in molecular conformation space, and anomalies in image data.

cs.LG↗

arXiv2023

Strange Random Topology of the Circle

We characterise high-dimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers growing arbitrarily large over shrinking intervals of filtration radii. Using the fact that the homotopy type of the random Cech complex is either an odd-dimensional sphere or a bouquet of even-dimensional spheres, we give probabilistic bounds of the homotopy types. By departing from the conventional practice of scaling down filtration radii as the sample size grows large, our findings indicate that the full breadth of filtration radii leads to interesting systematic behaviour that cannot be regarded as "topological noise".

math.PR↗

arXiv2022

Harmonic Forms, Hodge Theory and the Kodaira Embedding Theorem

In this expository article, we outline the theory of harmonic differential forms and its consequences. We provide self-contained proofs of the following important results in differential geometry: (1) Hodge theorem, which states that for a compact complex manifold, the de Rham cohomology group is isomorphic to the group of harmonic forms, (2) Hodge decomposition theorem, which states that for a K\"ahler manifold, the de Rham cohomology group decomposes into the Dolbeault cohomology groups, and (3) The Kodaira Embedding theorem, which gives a criterion of when a compact complex manifold is in fact a smooth complex projective variety. The basic theory of vector bundles is also contained for completeness.

math.HO↗

arXiv2021

Tangent Space and Dimension Estimation with the Wasserstein Distance

Consider a set of points sampled independently near a smooth compact submanifold of Euclidean space. We provide mathematically rigorous bounds on the number of sample points required to estimate both the dimension and the tangent spaces of that manifold with high confidence. The algorithm for this estimation is Local PCA, a local version of principal component analysis. Our results accommodate for noisy non-uniform data distribution with the noise that may vary across the manifold, and allow simultaneous estimation at multiple points. Crucially, all of the constants appearing in our bound are explicitly described. The proof uses a matrix concentration inequality to estimate covariance matrices and a Wasserstein distance bound for quantifying nonlinearity of the underlying manifold and non-uniformity of the probability measure.

math.ST↗