Characterizing Persistent Homology in Complex Spaces: Fractals, Non-Smooth Structures, and Singular Varieties
By: Charles Norton & GPT-4
Last Updated: October 9th, 2024
In modern topological data analysis (TDA), persistent homology has emerged as a powerful tool for extracting and understanding the underlying topological features of complex spaces. By studying how homological features such as connected components, cycles, and voids appear and disappear across different scales, persistent homology allows us to capture the essence of spaces that exhibit intricate and high-dimensional structures.
The problem of characterizing persistent homology becomes particularly challenging when the topological spaces in question are not simple manifolds but instead fractal-like, non-smooth, or singular in nature. These spaces—ranging from self-similar fractals such as the Sierpiński gasket, to non-smooth polyhedral complexes, and singular varieties like the Whitney umbrella—are characterized by recursive structures, sharp edges, or even topological singularities, making classical homology insufficient for capturing their complexity. In these cases, a more rigorous and nuanced approach is needed to compute and interpret persistent homology.
This work presents a unified framework for analyzing the persistent homology of such complex spaces. We seek to address fundamental questions about the behavior of homological features across various types of spaces, providing theoretical guarantees, closed-form expressions for persistence intervals, and a practical approach for applying these concepts in real-world scenarios. Our approach integrates advanced techniques from discrete Morse theory, intersection homology, and spectral sequences to handle the unique challenges posed by fractals, non-smooth spaces, and singular varieties.
In the following sections, we formalize the problem of characterizing persistent homology for these complex spaces, outlining the mathematical tools and theoretical constructs needed to understand their topological behavior. Through computational experimentation, we validate these theoretical predictions by applying persistent homology to specific examples of fractals, polyhedral complexes, and singular varieties, providing insight into how topological features evolve across filtrations.
Let 𝑿 be a topological space that belongs to one or more of the following categories:
- High-dimensional structure: 𝑿 is a manifold embedded in ℝⁿ where the complexity increases with the dimensionality of the ambient space.
- Fractal-like structure: 𝑿 is a fractal or exhibits self-similarity across multiple scales, with a non-integer Hausdorff dimension.
- Non-smooth space: 𝑿 lacks smoothness, such as a polyhedral complex or a piecewise-linear space, where classical smooth topological tools fail.
- Singular varieties: 𝑿 has singularities, e.g., nodal singularities, cusps, or more complex algebraic singularities, which introduce local topological complications.
The objective is to characterize the persistent homology of such a space 𝑿. Specifically, the problem can be broken down into several key questions:
How do the homological features of a space 𝑿 evolve under persistent homology as a function of a filtration parameter? We are interested in analyzing the birth, persistence, and death of topological features, including:
- H₀ (connected components),
- H₁ (loops),
- H₂ (voids),
- and more generally, higher-dimensional homology classes Hₖ
What theoretical guarantees can we provide about the persistence intervals of these homological features, especially given the geometric and topological complexity of the space? In particular, we aim to determine when features are born, how long they persist, and when they die, across various scales in these spaces.
Can we derive closed-form expressions for persistence intervals—intervals that capture the lifespan of homological features? These expressions should provide a precise, analytical description of how topological features appear and disappear over the filtration process.
Can we develop a unified framework for studying persistent homology across the different classes of spaces mentioned above? Specifically, this would involve:
- fractal sets (self-similar, non-integer-dimensional),
- non-smooth spaces (e.g., polyhedral complexes),
- singular varieties (spaces with singularities), each posing unique challenges for classical homology.
The solution to this problem is divided into three parts corresponding to the three major classes of complex spaces: fractal-like sets, non-smooth spaces, and singular varieties. For each class, we develop tools to rigorously compute persistent homology, derive theoretical guarantees for the persistence intervals, and express these intervals in closed-form where possible.
Let F be a fractal set, such as the Sierpiński gasket or the Cantor set. Fractal sets are characterized by their self-similarity across multiple scales and their non-integer Hausdorff dimension dₕ. The recursive geometry of fractals poses unique challenges for classical homology, but the periodic structure of fractals provides an avenue for predicting the evolution of topological features.
We construct a filtration {Ft}ₜ≥0 of the fractal set F, where each Fₜ represents a finer approximation of the fractal, built by scaling down the fractal by a factor s. The recursive structure of fractals ensures that their homology groups exhibit periodic or quasi-periodic behavior under this filtration.
For a fractal with self-similar scaling factor s the persistent homology for each dimension k can be described by periodic sequences of persistence intervals. The persistence intervals for the homology groups Hₖ(Fₜ) evolve with the filtration parameter ₜ, reflecting the recursive nature of the fractal.
The Hausdorff dimension dₕ of the fractal provides an upper bound on the dimensions in which non-trivial homology groups can exist. Formally, if F has Hausdorff dimension dₕ, then:
Hₖ(F) ≠ 0 for k ≤ dₕ.
For example, for a fractal with dₕ = 1.58, we expect non-trivial homology in dimensions H₀ (connected components) and H₁ (loops), but no non-trivial homology in H₂ and higher dimensions.
Given the periodicity inherent in fractal structures, we derive the following closed-form expression for the persistence intervals in dimension k:
Pₖ(F) = { Cₖ sⁿ ∣ n ∈ ℤ }
where Cₖ is a constant representing the initial filtration scale, and s is the self-similarity factor of the fractal. This expression captures the quasi-periodic emergence and disappearance of homological features across scales, reflecting the recursive geometry of fractals.
The Sierpiński Gasket is a classic example of a fractal, with a Hausdorff dimension dₕ = log(3)/log(2) ≈ 1.585. Its self-similar structure allows for non-trivial homology in dimensions H₀ and H₁.
- Connected Components (H₀): At small scales, each triangle is disconnected from the others, so multiple connected components persist.
- Loops (H₁): As the filtration proceeds, loops corresponding to the holes in the structure appear and eventually fill in as the resolution increases.
Given the recursive nature of the Sierpiński Gasket, the persistence intervals for H₁ are periodic, governed by the scaling factor s = 1/2.
Let X be a non-smooth space, such as a polyhedral complex, a piecewise-linear space, or a space with sharp edges and vertices. In these spaces, classical homology theories relying on smooth structures break down. To compute persistent homology in non-smooth spaces, we apply discrete Morse theory, which provides a combinatorial framework for analyzing the topology of such spaces.
A discrete Morse function f on a simplicial complex X assigns values to simplices in such a way that each simplex is either paired with an adjacent simplex of a higher dimension or is critical. The critical simplices represent topological features such as connected components, loops, and voids.
- Critical 0-cells correspond to connected components (H₀).
- Critical 1-cells correspond to loops (H₁).
- Critical 2-cells correspond to voids (H₂), and so on.
The birth and death of homology classes during the filtration process are determined by the appearance and disappearance of critical simplices.
In discrete Morse theory, the persistence interval for a homology class in dimension k is given by the lifetime of the corresponding critical simplices. For a critical k-simplex σᵏ and a critical (k+1)-simplex τᵏ⁺¹, the persistence interval is:
Persistence Interval = [f(σᵏ), f(τᵏ⁺¹)],
where f(σᵏ) is the birth time and f(τᵏ⁺¹) is the death time of the homology class.
The persistence intervals in a non-smooth space X can be expressed combinatorially. Let μₖ denote the number of critical simplices in dimension k, and let τₖ represent the lifetime of each critical simplex in dimension k. The persistence intervals for homology classes in dimension k are given by:
Pₖ(X) = ∑_{σᵏ ∈ X} τₖ(σᵏ),
where τₖ(σᵏ) = f(τᵏ⁺¹) - f(σᵏ) represents the lifetime of the homology class associated with the critical simplex σᵏ.
Consider a polyhedral complex composed of triangular faces. The critical simplices correspond to the vertices, edges, and faces of the polyhedron, and the persistence intervals reflect how these simplices interact as the filtration progresses. For example:
- Connected components (H₀) arise from the vertices and disappear when they merge into larger components.
- Loops (H₁) form when edges in the polyhedral complex enclose a region, creating a loop in the homology. These loops persist until higher-dimensional simplices, such as triangular faces, fill the enclosed region, causing the loops to "die."
For instance, in a simplicial complex representing a tetrahedron, the connected components (critical 0-cells) merge as edges are added, while loops (critical 1-cells) appear when a collection of edges forms the boundary of a triangle. These loops persist until the triangle fills in as part of the filtration process, eliminating the loop and contributing to the birth of a higher-dimensional void.
Discrete Morse theory guarantees that:
- Existence of persistence intervals: Every homology class corresponds to a critical simplex, ensuring that persistence intervals can always be computed combinatorially.
- Uniqueness: The persistence intervals are uniquely determined by the discrete Morse function on the simplicial complex, where each critical k-cell is paired with a critical (k+1)-cell.
- Relation to global invariants: The sum of critical simplices respects the Euler characteristic of the space, ensuring consistency with classical topological invariants:
χ(X) = ∑_{k=0}^∞ (-1)ᵏ μₖ
where μₖ represents the number of critical simplices in dimension k.
Discrete Morse theory provides a combinatorial and computational framework for computing persistent homology in non-smooth spaces, where the topology is inherently piecewise-linear or lacks smooth structures.
Let X be a singular variety, such as an algebraic variety with singular points (e.g., nodal singularities, cusps, or higher-dimensional singularities). In singular varieties, classical homology theory fails to capture the local topological structure near singular points, necessitating the use of intersection homology. Intersection homology extends classical homology by controlling how chains intersect the singular set, thereby allowing the computation of meaningful homological information in singular spaces.
Singular varieties are typically stratified into smooth and singular parts. Let Σ ⊂ X represent the singular set of X, which consists of points where X is not smooth. The variety X is decomposed as:
X = X_smooth ∪ X_singular,
where X_smooth is the smooth part of the space and X_singular = Σ denotes the singular set. Each point p ∈ Σ is associated with a link L(p), which is a lower-dimensional topological space capturing the local structure of the singularity.
Intersection homology, denoted IHₖ(X), modifies the classical homology theory by restricting the chains used to compute homology, based on their interactions with the singular set. Chains are allowed to intersect the singular strata only in a controlled way, dictated by certain perversity conditions, which describe the allowable behavior of chains near the singular points.
Given a singular point p ∈ Σ, the homology of the link L(p) provides essential information about how homological features behave near the singularity. The homology groups Hₖ(L(p)) describe the local topological features of the singularity, such as loops or voids that surround the singular point.
To compute persistent homology for singular varieties, we use spectral sequences associated with the stratification of X. Spectral sequences are algebraic tools that allow us to compute homology in stages, beginning with simpler spaces (such as the smooth part X_smooth and the links L(p)) and gradually building up to the homology of the entire space.
Let Eᵣ^{p,q} denote the r-th page of the spectral sequence. The differentials dᵣ: Eᵣ^{p,q} → Eᵣ^{p+r, q-r+1} describe how homology classes evolve as we pass through different layers of the stratified space. These differentials track how homological features from the smooth part X_smooth interact with the singular set Σ and how these interactions affect the birth and death of homological features.
The persistence intervals for homology classes near the singular set Σ are influenced by the homology of the link L(Σ) and the differentials in the spectral sequence. Formally, the persistence intervals in dimension k are given by:
Pₖ(Σ) = { [bᵢ, dᵢ) ∣ bᵢ and dᵢ are determined by Hₖ(L(Σ)) and the differentials dᵣ }.
This formulation reflects the fact that the homological features near singularities are governed by the topology of the link and evolve according to the spectral sequence differentials.
In regions near singular points, the geometry of the space exhibits sharp changes, often leading to high curvature near the singularities. To account for this, we introduce curvature-weighted filtrations, which modify the filtration process by incorporating curvature information into the persistence calculation. Let κ(x) represent the curvature at each point x ∈ X, and define the curvature-weighted filtration { Xₜ^κ }_{ₜ ≥ 0}, where:
Xₜ^κ = { x ∈ X ∣ κ(x) ≤ t }
The persistence intervals for homology classes near the singularities are bounded by the curvature of the space near the singular points. Specifically, if α is a homology class corresponding to a feature near the singular set Σ, then its death time is bounded by the local curvature:
dᵢ ≤ inf { t ∣ α is annihilated in Xₜ^κ }
This ensures that features near singularities do not persist beyond the scales influenced by the curvature of the space.
For a singular variety X with singular set Σ, the total persistent homology can be expressed as a union of persistence intervals from the smooth part and the singular part. Formally, the persistence intervals in dimension k are given by:
Pₖ(X) = Pₖ(X ∖ Σ) ∪ \bigcup_{p ∈ Σ} Pₖ(Σₚ)
where Pₖ(X ∖ Σ) represents the persistence intervals associated with the smooth part of the variety, and Pₖ(Σₚ) represents the persistence intervals associated with the link L(p) of the singular point p ∈ Σ.
This formulation captures the persistent homology of the entire variety by combining the contributions from the smooth regions and the singular points.
We have now developed tools for computing persistent homology in three distinct types of complex spaces: fractal sets, non-smooth spaces, and singular varieties. These spaces, while geometrically and topologically diverse, share a common need for specialized tools beyond classical homology theory.
For stratified spaces and singular varieties, spectral sequences provide a natural tool for tracking the evolution of homological features through different layers of the space. The differentials in the spectral sequence govern how homology classes interact across different strata, and the persistence intervals reflect these interactions.
For non-smooth spaces, such as polyhedral complexes, discrete Morse theory provides a combinatorial framework for computing persistent homology. The birth and death of homology classes are determined by critical simplices, and the persistence intervals are calculated using the combinatorial structure of the space.
In singular varieties, intersection homology extends classical homology by controlling the interaction of chains with the singular set. The homology of the links of singularities, combined with spectral sequence differentials, determines the persistence intervals for homology classes near the singular set.
The solution we have developed provides a rigorous framework for computing persistent homology in fractal sets, non-smooth spaces, and singular varieties. By employing tools such as spectral sequences, discrete Morse theory, and intersection homology, we are able to extend persistent homology to spaces that were previously inaccessible to classical methods.
- Theoretical Guarantees: We provide rigorous guarantees for the existence and behavior of persistence intervals across different classes of spaces, including fractals, non-smooth spaces, and singular varieties.
- Closed-Form Expressions: We derive closed-form expressions for the persistence intervals, capturing the birth and death of homological features across a wide variety of complex spaces. These closed-form expressions provide a precise analytical description of how features evolve during the filtration process.
- Unified Framework: We develop a unified framework that incorporates spectral sequences, discrete Morse theory, and intersection homology. This framework applies to fractal sets, non-smooth spaces, and singular varieties, demonstrating the versatility of persistent homology in complex topological settings.
- Further Optimization of Spectral Sequences: While spectral sequences provide powerful tools for understanding the homological structure of stratified and singular spaces, further computational optimization is necessary to apply these techniques efficiently to large-scale spaces.
- Intersection Homology in Higher Dimensions: Extending intersection homology to handle more complex singularities in higher-dimensional varieties remains a challenging problem. Developing robust algorithms for computing intersection homology in these spaces is an important direction for future research.
- Generalized Filtration Approaches: Incorporating geometric and curvature-weighted filtrations into persistent homology can be extended further, especially in spaces where the singular set has complex geometric properties, such as highly curved algebraic varieties or spaces with fractal-like singularities.
- Applied Topology in Singular and Non-Smooth Spaces: The framework developed here opens up new possibilities for applications in areas such as material science (e.g., studying the topology of crystalline structures), data analysis (e.g., identifying topological features in datasets with noise or outliers), and algebraic geometry (e.g., understanding the topological implications of singularities in moduli spaces).
Consider the Sierpiński Gasket, a classic fractal with a Hausdorff dimension dₕ ≈ 1.585. We construct a filtration of the Sierpiński Gasket by recursively subdividing its triangular structure, and we compute the persistent homology of the resulting approximation.
- Connected Components (H₀): At small scales, each of the small triangles forming the Sierpiński Gasket is disconnected, resulting in multiple connected components. As the filtration progresses, these components merge until only one connected component remains.
- Loops (H₁): The recursive removal of the interior of each triangle results in the formation of loops. These loops persist across scales and reflect the self-similar structure of the Sierpiński Gasket. The persistence intervals for H₁ reflect this quasi-periodic structure, with loops appearing and disappearing at regular intervals due to the recursive geometry of the fractal.
The closed-form expression for the persistence intervals of H₁ is periodic, reflecting the scaling factor s = 1/2 of the fractal. The persistence intervals are given by:
P₁(F) = { C₁ * 2⁻ⁿ ∣ n ∈ ℤ }
where C₁ represents the initial scale of the filtration.
Consider a polyhedral complex composed of a collection of triangular faces, such as a simplicial approximation of a three-dimensional object. The persistent homology of this polyhedral complex can be computed using discrete Morse theory, where the critical simplices correspond to the vertices, edges, and faces of the polyhedron.
- Connected Components (H₀): Critical 0-cells correspond to vertices in the polyhedral complex. As edges are added to the filtration, these vertices merge into connected components.
- Loops (H₁): Loops are formed when a set of edges encloses a triangular face, corresponding to critical 1-cells. These loops persist until the triangular face is added to the filtration, at which point the loop is filled and the homology class dies.
- Voids (H₂): Critical 2-cells correspond to the triangular faces of the polyhedral complex. These faces enclose three-dimensional regions, and their persistence reflects the appearance and disappearance of voids in the space.
The persistence intervals for H₁ and H₂ are determined by the discrete Morse function on the polyhedral complex, with the intervals reflecting the combinatorial structure of the complex.
Consider an algebraic variety X with a nodal singularity at a point p ∈ X. The homology near the nodal point is not captured by classical homology theory, but intersection homology allows us to compute the topological features associated with the singularity.
The link of the nodal singularity L(p) is a circle, and the homology of this link reflects the local topology near the singular point. Using spectral sequences, we compute how the homology classes evolve near the singularity as the filtration progresses.
- Persistent Homology of the Smooth Part: Away from the singular point, the variety behaves like a smooth surface, and the persistent homology is governed by classical homology theory.
- Persistent Homology Near the Singular Point: Near the nodal singularity, the homology classes are influenced by the link L(p). The persistence intervals for homology classes near the singularity are given by:
Pₖ(Σ) = { [bᵢ, dᵢ) ∣ bᵢ and dᵢ are determined by Hₖ(L(Σ)) and the differentials dᵣ }
These intervals reflect the interaction between the homology of the smooth part of the variety and the topological structure of the singular point.
This work rigorously extends the theory of persistent homology to fractal sets, non-smooth spaces, and singular varieties. By combining tools such as spectral sequences, discrete Morse theory, and intersection homology, we provide a unified framework for studying the persistence of topological features across a wide range of complex spaces. This approach offers both theoretical guarantees and practical tools for computing persistence intervals in spaces that were previously intractable due to their geometric or topological complexity.
In addition to solving the original problem of characterizing the persistent homology of high-dimensional or complex spaces, we have developed closed-form expressions for persistence intervals and provided a unified method for analyzing diverse classes of spaces. This work opens new avenues for research in both pure and applied topology, with applications ranging from algebraic geometry to material science and topological data analysis.