Topological optimization with birth and death cochains

Abstract

We introduce the notion of birth and death cochains as generalized versions of birth and death simplices in persistent cohomology. We show that birth and death cochains (unlike birth and death simplices) are always unique for a given persistent cohomology class. We use birth and death cochains to define birth and death content as generalizations of birth and death times. We then demonstrate the advantages of using that birth and death content as loss functions on a variety of topological optimization tasks with point clouds, time series and scalar fields. We close with a novel application of topological optimization to a dataset of arctic ice images.

Figures (27)

• Figure 1: The classical notion of birth simplex (in green) and death simplex (purple) for degree-$1$ persistent homology.
• Figure 1: The birth cochain for the cohomology class $\alpha$ born between $K$ and $L$, where the edge values show the coefficients in each $1$-chain.
• Figure 1: Tracking $b$, $b \pm \varepsilon$, $d$ and $d \pm \varepsilon$ for the run in \ref{['fig:smallcochains']}, where $\varepsilon$ is defined as a relative value via $0.05(d-b)$ for this run.
• Figure 1: Maximizing an degree-$1$ feature using either birth and death cochains, or birth and death simplices. We show how the normalized persistence changes over gradient ascent iterations (left), and the initial point cloud (in black) vs the final point cloud for each method (right).
• Figure 1: MNIST images (top row) are corrupted by a horizontal band of dark gray pixels and slight noising (second row). We attempt to repair them by reducing the death time (third row) or death content (bottom row) of degree-$0$ features.

Theorems & Definitions (29)

• Definition 3.1
• Definition 3.2: Birth cochain
• Definition 3.3: Death cochain
• Remark 3.4: Uniqueness of birth and death cochains
• Remark 3.5: Homogeneity of birth and death cochains
• Definition 3.6: $\varepsilon$-birth and death cochains
• Remark 3.7
• Proposition 4.1
• Proof 1
• Proposition 4.2