Persistent Homology

Track the birth and death of topological features across scales. Learn the matrix reduction algorithm at the heart of TDA.

Tracking Features Across Scales

As ε grows, holes are born and later die. Persistent homology records each one's lifespan: features that last are real shape, and fleeting ones are noise.

Matrix Reduction Algorithm

Press Next to step through the column reduction of the boundary matrix, and watch birth-death pairs appear as each column's lowest 1 becomes unique. Tick "Show explanation" for a running commentary.

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Algorithm Overview
  1. Build boundary matrix from filtration
  2. Process columns left to right
  3. For each column, find lowest 1 (the "low" index)
  4. If another column has same low, add them (mod 2)
  5. When low is unique, record birth-death pair
Key Insight

When an edge "kills" a component (connects two pieces), we pair the edge with the vertex that was born later — the younger component dies while the older one survives. The persistence is death - birth.

Go deeper:Tracking Features Across Scales

Persistent homology extends ordinary homology to filtrations. Instead of computing homology at a single scale, we track how topological features are born and die as we move through the filtration.

Each feature has a birth time (when it first appears) and adeath time (when it merges or fills in). The difference is thepersistence — a measure of how significant the feature is. Features with high persistence are signal; low persistence is noise.

Go deeper:Birth, Death, and Persistence

Birth

A feature is born when a cycle appears that wasn't a boundary at any earlier stage. For H₀, birth happens when a vertex appears (new component). For H₁, birth happens when a cycle of edges forms without bounding a triangle.

Death

A feature dies when it becomes a boundary. For H₀, death happens when an edge connects two components (they merge). For H₁, death happens when triangles fill in a loop, making it bound a surface.

persistence = death - birth
High persistence = significant feature, Low persistence = noise
Go deeper:The Standard Algorithm

Persistent homology is computed by reducing the boundary matrix using column operations. Step through the algorithm to see how birth-death pairs are extracted from the reduced matrix.

The algorithm processes the boundary matrix column by column, reducing it to a form where birth-death pairs can be read off directly.

  1. Build boundary matrix D — columns are simplices in filtration order; entry D[i,j] = 1 if simplex i is in the boundary of simplex j.
  2. Process columns left to right — for each column j, find the lowest row with a 1 (the "low" of column j).
  3. Reduce — if another column i has the same low, add column i to column j (mod 2). Repeat until the low is unique or the column is zero.
  4. Extract pairs — if low(j) = i and column j is non-zero, then (simplex i, simplex j) is a birth-death pair.
Time complexity: O(n³) where n = number of simplices
Space: O(n²) for the boundary matrix
Go deeper:The Elder Rule

When two components merge in H₀, which one "survives" and which one "dies"? The elder rule says the older component (born earlier) survives, and the younger one dies at the merge time.

Example

Suppose vertex v₀ is born at ε=0, vertex v₃ is born at ε=0, and edge (v₀, v₃) appears at ε=0.5. The two components merge. Since both were born at the same time, one arbitrarily survives (say v₀), and v₃ dies at ε=0.5. This creates the pair (birth=0, death=0.5).

Go deeper:Essential Features

Some features never die — they persist to infinity. These are calledessential features and represent the homology of the full complex at the end of the filtration.

Essential H₀

At least one connected component survives forever — the final complex is connected. If there are multiple essential H₀ classes, the final complex has multiple components.

Essential H₁

An essential loop never gets filled in by triangles. This happens when the underlying space has a genuine 1-dimensional hole (like a torus or annulus).

Key Takeaways

  • Birth — when a feature first appears in the filtration
  • Death — when a feature becomes a boundary (merges or fills in)
  • Persistence = death - birth — measures significance
  • Essential features — never die, persist to infinity

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