[Paper Review] Cohomotopy groups capture robust properties of zero sets.

Peter Franek, Marek Krčál|arXiv (Cornell University)|Jul 11, 2015

Topological and Geometric Data Analysis3 citations

TL;DR

This paper introduces cohomotopy groups as a computable and complete descriptor for robust topological properties of zero sets of continuous maps $ f: X \to \mathbb{R}^n $. By encoding perturbation-insensitive features via homotopy classes to spheres, it establishes a persistence module with superior descriptive power and computability over well groups, especially when $ \dim X \leq 2n-3 $. The framework is optimal when gradients are included.

ABSTRACT

We study robust properties of zero sets of continuous maps $f:X o\mathbb{R}^n$. Formally, we analyze the family $Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. The fundamental geometric property of $Z_r(f)$ is that all its zero sets lie outside of $A:=\{x:\,|f(x)|\ge r\}$. We claim that once the space $A$ is fixed, $Z_r(f)$ is \emph{fully} determined by an element of a so-called cohomotopy group which---by a recent result---is computable whenever the dimension of $X$ is at most $2n-3$. More explicitly, the element is a homotopy class of a map from $A$ or $X/A$ into a sphere. By considering all $r>0$ simultaneously, the pointed cohomotopy groups form a persistence module---a structure leading to the persistence diagrams as in the case of \emph{persistent homology} or \emph{well groups}. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C).

Motivation & Objective

• To identify and formalize robust topological properties of zero sets under continuous perturbations of maps $ f: X \to \mathbb{R}^n $.
• To replace well groups with cohomotopy groups as a more descriptive and computable alternative for persistent zero-set analysis.
• To establish a persistence module structure based on pointed cohomotopy groups for multi-scale robustness analysis.
• To prove optimality of the cohomotopy-based descriptor when gradients of perturbations are considered.

Proposed method

• Define the family $ Z_r(f) $ as the set of zero sets of all maps $ g $ with $ \|g - f\| < r $, capturing robustness under $ r $-perturbations.
• Show that $ Z_r(f) $ is fully determined by a cohomotopy class in $ \pi^k(A) $ or $ \pi^k(X/A) $, where $ A = \{x : |f(x)| \geq r\} $.
• Leverage a recent result that cohomotopy groups $ \pi^k(X/A) $ are computable when $ \dim X \leq 2n - 3 $.
• Construct a persistence module from the pointed cohomotopy groups across all $ r > 0 $, analogous to persistent homology.
• Use the resulting cohomotopy persistence diagram to describe persistent robust features of zero sets.
• Incorporate gradient data of perturbations to establish optimality of the descriptor via Theorem C.

Experimental results

Research questions

• RQ1 Can cohomotopy groups fully capture the robust topological structure of zero sets under continuous perturbations?
• RQ2 How does the cohomotopy-based descriptor compare to well groups in terms of descriptive power and computability?
• RQ3 Is the cohomotopy persistence module computable in low-dimensional settings, particularly when $ \dim X \leq 2n - 3 $?
• RQ4 Can the cohomotopy descriptor be shown to be optimal when perturbation gradients are included?
• RQ5 What is the relationship between the cohomotopy persistence module and established frameworks like persistent homology or well groups?

Key findings

• The family $ Z_r(f) $ is completely determined by a cohomotopy class in $ \pi^k(A) $ or $ \pi^k(X/A) $, where $ A = \{x : |f(x)| \geq r\} $.
• Cohomotopy groups are computable whenever $ \dim X \leq 2n - 3 $, enabling algorithmic analysis of robust zero-set features.
• The cohomotopy persistence module provides a descriptor with strictly better descriptive power than well diagrams, as formalized in Theorem A.
• The cohomotopy descriptor achieves better computability status than well groups, as established in Theorem B.
• When gradients of perturbations are included, the cohomotopy-based descriptor is optimal in a precise mathematical sense, as shown in Theorem C.
• The framework unifies robustness, persistence, and computability in a single topological descriptor for zero sets.