Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometric Localization of Homology Cycles

Amritendu Dhar, Vijay Natarajan, Abhishek Rathod

TL;DR

This work tackles the NP-hard problem of localizing homology classes by introducing the geometry-aware $\ell_2$-radius objective, which measures the radius of the smallest sphere enclosing a cycle. The authors develop polynomial-time approximation algorithms for localizing a cycle within a given homology class, computing minimum homology bases, and constructing minimum persistent homology bases, along with a notion of approximate stability under persistence. A central result is that optimal persistent bases can be achieved by selecting minimal radiants per bar, with practical, scalable algorithms that run in $O(|P|N^3\log N)$ time. Experimental results on diverse datasets show the proposed cycles are tight and of high quality, often outperforming state-of-the-art methods such as PersLoop, and demonstrate the practical viability of geometry-aware homology localization in TDA.

Abstract

Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes geometrically and admits a stability property under the setting of persistent homology. We present a geometric optimization of the cycles that is computable in polynomial time and is stable in an approximate sense. Tailoring our search criterion to different settings, we obtain various optimization problems like optimal homologous cycle, minimum homology basis, and minimum persistent homology basis. In practice, the (trivial) exact algorithm is computationally expensive despite having a worst case polynomial runtime. Therefore, we design approximation algorithms for the above problems and study their performance experimentally. These algorithms have reasonable runtimes for moderate sized datasets and the cycles computed by these algorithms are consistently of high quality as demonstrated via experiments on multiple datasets.

Geometric Localization of Homology Cycles

TL;DR

This work tackles the NP-hard problem of localizing homology classes by introducing the geometry-aware -radius objective, which measures the radius of the smallest sphere enclosing a cycle. The authors develop polynomial-time approximation algorithms for localizing a cycle within a given homology class, computing minimum homology bases, and constructing minimum persistent homology bases, along with a notion of approximate stability under persistence. A central result is that optimal persistent bases can be achieved by selecting minimal radiants per bar, with practical, scalable algorithms that run in time. Experimental results on diverse datasets show the proposed cycles are tight and of high quality, often outperforming state-of-the-art methods such as PersLoop, and demonstrate the practical viability of geometry-aware homology localization in TDA.

Abstract

Computing an optimal cycle in a given homology class, also referred to as the homology localization problem, is known to be an NP-hard problem in general. Furthermore, there is currently no known optimality criterion that localizes classes geometrically and admits a stability property under the setting of persistent homology. We present a geometric optimization of the cycles that is computable in polynomial time and is stable in an approximate sense. Tailoring our search criterion to different settings, we obtain various optimization problems like optimal homologous cycle, minimum homology basis, and minimum persistent homology basis. In practice, the (trivial) exact algorithm is computationally expensive despite having a worst case polynomial runtime. Therefore, we design approximation algorithms for the above problems and study their performance experimentally. These algorithms have reasonable runtimes for moderate sized datasets and the cycles computed by these algorithms are consistently of high quality as demonstrated via experiments on multiple datasets.
Paper Structure (25 sections, 19 theorems, 51 equations, 7 figures, 1 table, 5 algorithms)

This paper contains 25 sections, 19 theorems, 51 equations, 7 figures, 1 table, 5 algorithms.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. The $\ell_2\text{--radius}$ metric
  4. Computing optimal homologous cycle
  5. Notations and Conventions.
  6. Optimal homology basis
  7. Optimal persistent homology basis
  8. Experiments
  9. Homology localization.
  10. Optimal homology basis.
  11. Optimal persistent homology cycle.
  12. Execution time.
  13. Persistent homology
  14. Interleavings and matchings for persistence modules
  15. Approximate stability for --radius
  16. ...and 10 more sections

Key Result

Theorem 1

Let $J$ be the indexing set for the intervals in the barcode $\mathcal{B}_p(\mathcal{F})$ of filtration $\mathcal{F}$. Then, an indexed set of $p$-cycles $\{ \zeta_j \mid j \in J\}$ is a persistent $p$-basis for a filtration $\mathcal{F}$ if and only if $\zeta_j \in \mathcal{R}([b_j,d_j))$ for every

Figures (7)

  • Figure 1: (top) The localization algorithm computes optimal (blue) 1-cycles that are homologous to input 1-cycles (red). (bottom) Minimum 1-homology basis.
  • Figure 2: The green persistent 1-homology cycles computed by our algorithm are tighter than the red (or yellow) cycles computed by PersLoop deyperstwo.
  • Figure 3: Two $1$-cycles $\xi_1$ and $\xi_2$ appear at time (index) $1$ and $2$ respectively. In the upper filtration, the green triangles appear at time $3$ filling up the annulus and the golden triangle appears at time $4$ giving rise to the bars $[2,3)$ and $[1,4)$ with the minimal (lengthwise) persistent cycles $\xi_1+\xi_2$ and $\xi_1$ resp. In the lower filtration the golden triangle comes first and green triangles next giving rise to slightly perturbed bars $[1,3)$ and $[2,4)$. Their minimal persistent cycles change to $\xi_1$ and $\xi_2$, resp. suggesting that the cycles may change considerably with respect to the length function.
  • Figure 4: Consider the $\mathop{\mathrm{\check{C}{ech}}}\nolimits$ filtrations on the point set shown in the top figure, denoted by $P$, which is $\delta$-perturbed to obtain another point set $Q$ shown in the bottom figure. In $\mathop{\mathrm{\check{\mathsf{C}}}}\nolimits(P)$, the cycle on the left supported by the inner rim of black grid points, and the cycle on the right supported by black points are born at $b$ and die at $d$. Owing to a $\delta$-perturbation, where $\delta_1, \delta_2 < \delta$, in $\mathop{\mathrm{\check{\mathsf{C}}}}\nolimits(Q)$, the first cycle is born at ${b+\delta_1}$ and dies at ${d+\delta_1}$, whereas the second cycle is born at ${b+\delta_2}$ and dies at ${d+\delta_2}$. If one uses Bjerkevik's approach to obtain a $\delta$-matching, the bar $[b,d)$ (top left) may be matched to either the bar $[b+\delta_1,d+\delta_1)$ (bottom left) or $[b+\delta_2,d+\delta_2)$ (bottom right), whereas the bar $[b,d)$ (top right) may also be matched to either the bar $[b+\delta_1,d+\delta_1)$ (bottom left) or $[b+\delta_2,d+\delta_2)$ (bottom right). Thus, a naive approach to $\delta$-matching is not approximately stable as the radius values of the enclosing spheres of matched representatives can be arbitrarily far apart. Our version of matching ensures that for this example, $[b,d)$ (top left) is matched to $[b+\delta_1,d+\delta_1)$ (bottom left) and $[b,d)$ (top right) is matched to the bar $[b+\delta_2,d+\delta_2)$ (bottom right).
  • Figure 5: The localization algorithm computes optimal (blue) 1-cycles that are homologous to input 1-cycles (red).
  • ...and 2 more figures

Theorems & Definitions (60)

  • Definition 1
  • Definition 2: Persistent cycles
  • Definition 3: Persistent basis
  • Theorem 1: deypersone
  • Remark 4.1
  • Remark 4.2
  • Proposition 4
  • Remark 4.3
  • Definition 5
  • Definition 6: $\epsilon$-shifts of persistence modules
  • ...and 50 more