Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2202.14021

A probabilistic result on impulsive noise reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators

Patrizio Frosini·Ivan Gridelli·Andrea Pascucci
View original source ↗Read source document ↗Export metadata ↓

Abstract

In recent years, group equivariant non-expansive operators (GENEOs) have attracted attention in the fields of Topological Data Analysis and Machine Learning. In this paper we show how these operators can be of use also for the removal of impulsive noise and to increase the stability of TDA in the presence of noisy data. In particular, we prove that GENEOs can control the expected value of the perturbation of persistence diagrams caused by uniformly distributed impulsive noise, when data are represented by $L$-Lipschitz functions from $\mathbb{R}$ to $\mathbb{R}$.

Explore related subjects

cs.CGmath.ATmath.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Patrizio Frosini, Ivan Gridelli, Andrea Pascucci. 2022-04-24. A probabilistic result on impulsive noise reduction in Topological Data Analysis through Group Equivariant Non-Expansive Operators. https://doi.org/10.3390/e25081150

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tight Fréchet bounds for $λ$-low density curves

The Fréchet distance is a well-studied similarity measure between curves. We computing the Fréchet distance between $λ$-low-density curves, the most general of realistic curve assumptions, where every ball of radius $r$ intersects at most $λ$ edges of length at least $r$. Previous algorithms either assumed constant $λ$ or had no tight dependence on $λ$. For two $n$-vertex $λ$-low-density curves in $\mathbb{R}^d$, we give a $(1+\varepsilon)$-approximation algorithm for the continuous and discrete Fréchet distance running in $ \tilde{O}\!\left(\frac{λ^{2/d}n^{2-2/d}}{\varepsilon^2}\right) $ time. Our key insight is a tight property of simplifying $λ$-low density curves: the simplification of any $n$-vertex $λ$-low-density curve is $O(λ^{1/d}n^{1-1/d})$-low-density. We show this is tight, and this provides the structural property under simplification that was previously known for $c$-packed curves. We provide matching lower bounds for $n$ and $λ$: assuming the Orthogonal Vectors Hypothesis, for every $δ>0$, we rule out algorithms with running time $O\!\left( \left( \frac{λ^{2/d}n^{2-2/d}} {\varepsilon^{2-4/d}} \right)^{1-δ} \right). $ We extend our techniques to the map matching problem, where we also give tight bounds.

cs.CG↗
arXiv2026

Perfect Rectangular Tilings with Two Colors

We study a finite tiling problem, where tiles are unit squares whose four edges are colored with one of two colors. We ask whether a given rectangle admits a perfect rectangular tiling: every cell of the rectangle is occupied by one tile, neighboring edge colors match, and exactly $n_i$ tiles of type $i$ are used, where rotations of the tiles are allowed. Our problem is related to classical Wang tilings, more general finite tile-placement problems, and edge placement puzzles. But in our problem, the multiplicities of the tile types are part of the input and the tile alphabet is fixed and extremely small; thus the complexity of the problem arises from the interaction between the rectangle dimensions and the prescribed tile multiplicities. We provide a comprehensive study of the perfect rectangular tiling problem. For this we consider all classes of subsets of the six possible tile types for two-colored edges, and we characterize for each class whether multiplicities either always allow a perfect rectangular tiling or whether their existence can be decided efficiently.

cs.CG↗
arXiv2026

A constant-factor approximation of the Gromov-Hausdorff distance in the plane

We give the first polynomial-time constant-factor approximation of the Gromov-Hausdorff distance d_GH between finite point sets in the Euclidean plane; in fixed Euclidean dimension such an approximation was previously known only on the line (Majhi, Vitter and Wenk, 2024). Global alignment cannot succeed: the classical dimension drop defeats alignment by isometries, a multiplicity gap defeats alignment by bijections, and a reflection barrier defeats sorting under any single global reflection pattern. The algorithm is therefore local. Guessing the images of one diameter pair pins every point's longitudinal coordinate to within O(d_GH). Heights are read in windows whose height spread is at most a fixed multiple of their length, where a chain argument makes every compatible match local in the plane. One reflection sign per window is then chosen by 2-SAT; at the right frame and guess, any solution yields a correspondence of distortion O(d_GH). For the bijective relative of d_GH, half the least additive distortion over bijections, the same scheme reduces the planar problem to a matching question that we leave open.

cs.CG↗