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The Poset of Cancellations in a Filtered Complex

Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros

Abstract

Motivated by questions about simplification and topology optimization, we take a discrete approach toward the dependency of topology simplifying operations and the reachability of perfect Morse functions. Representing the function by a filter on a Lefschetz complex, and its (non-essential) topological features by the pairing of its cells via persistence, we simplify using combinatorially defined cancellations. The main new concept is the depth poset on these pairs, whose linear extensions are schedules of cancellations that trim the Lefschetz complex to its essential homology. One such linear extensions is the cancellation of the pairs in the order of their persistence. An algorithm that constructs the depth poset in two passes of standard matrix reduction is given and proven correct.

The Poset of Cancellations in a Filtered Complex

Abstract

Motivated by questions about simplification and topology optimization, we take a discrete approach toward the dependency of topology simplifying operations and the reachability of perfect Morse functions. Representing the function by a filter on a Lefschetz complex, and its (non-essential) topological features by the pairing of its cells via persistence, we simplify using combinatorially defined cancellations. The main new concept is the depth poset on these pairs, whose linear extensions are schedules of cancellations that trim the Lefschetz complex to its essential homology. One such linear extensions is the cancellation of the pairs in the order of their persistence. An algorithm that constructs the depth poset in two passes of standard matrix reduction is given and proven correct.
Paper Structure (14 sections, 13 theorems, 6 equations, 8 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 13 theorems, 6 equations, 8 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Cancellations in Lefschetz Complexes
  3. Lefschetz Complexes
  4. Cancellations
  5. Shallow and Other Birth-death Pairs
  6. Persistent Homology
  7. Shallow Pairs
  8. Shallow Orders
  9. The Depth Poset
  10. Partial Order on Birth-death Pairs
  11. Transposing and Canceling Shallow Pairs
  12. Lazy Reduction with Clearing
  13. Relations from Book-keeping
  14. Discussion

Key Result

Theorem 2.1

If $X$ is a regular complex, then the homology of the free chain complex defined by its Lefschetz complex is isomorphic to the singular homology of $X$.

Figures (8)

  • Figure 1: Upper left: a generic smooth function with $8$ minima and $8$ maxima on a circle. Upper right: simplified version of the function after canceling all $5$ shallow min-max pairs, which are indicated by red arrows. The $5$ cancellations turn a former non-shallow min-max pair shallow, whose cancellation leads to the further simplified version of the function at the lower left. The cancellation of the last birth-death pair, which is now shallow, produces a function with a single minimum and a single maximum (not shown). Lower right: the depth poset, whose relations express the dependencies between the cancellations: its linear extensions are sequences such that each pair is shallow at the time it is canceled.
  • Figure 2: The effect of canceling $s < t$ on the Lefschetz complex on the left and the boundary matrix on the right. If in addition $x$ were also incident to $y$, then the cancellation would removed this incidence, leaving $y$ without child and $x$ without parent (not shown).
  • Figure 3: Far left: a cylinder cut along the edge $\tt AB$, which connects the points $\tt A$ and $\tt B$ on its two boundary circles (represented by the edges $\tt AA$ and $\tt BB$), and (an artistic sketch of) a Dunce hat attached to $\tt AA$ three times. Far right: after canceling the Dunce hat and $\tt AA$, we get an upside-down urn cut along the edge connecting $\tt A$ (to which the Dunce hat contracted) to $\tt B$. In the middle: the Lefschetz complexes before and after the cancellation of the Dunce hat.
  • Figure 4: Left: the $s<t$ is a birth-death pair if the alternating sum of ranks of the four lower left minors is positive. Right: the pair is shallow if furthermore row $s$ and column $t$ to the left and below the common entry are zero.
  • Figure 5: From left to right: a filtered graph, its merge tree (dendogram), its depth poset above the nodes and arcs ordered according to the filter, and the depth poset after swapping nodes $\tt a$ and $\tt d$ in the filter. The swap only causes nodes $\tt a$ and $\tt d$ to trade places in the dendogram, which does not affect the structure of the merge tree. In contrast, it shrinks the depth poset from four to two relations.
  • ...and 3 more figures

Theorems & Definitions (25)

  • Definition 1: Lefschetz Complex
  • Theorem 2.1: McCord McC66
  • Definition 2: Cancellation
  • Proposition 2.2
  • Definition 3: Birth-death Pairs
  • Definition 4: Shallow Pairs
  • Lemma 3.1: Cohen-Steiner et al. 2006
  • Theorem 3.2: Canceling a Shallow Pair
  • proof
  • Definition 5: Shallow Orders
  • ...and 15 more