Inverse problems on persistence diagrams
Ippei Obayashi
Jan. 7, 2021
Abstract
In this talk, I will discuss a kind of inverse problem from a persistence diagram to a pointcloud.
Since the solution is not unique, we need an additional constraint to remove the ambiguity. In
our research we consider the problem to find a pointcloud whose k-th persistence diagram is ourdesired diagram and nearest to the given pointcloud. The key mathematical idea is piecewise
differentiability of the map from pointclouds to persistence diagrams. This research is a joint
work with M. Gamerio (Universidade de São Paulo, Brazil) and H. Hiraoka (Kyoto University,Japan) [1].

1Introduction

Persistent homology (PH) is homology theory on filtrations, and it can summarize the shape of data
quantitatively. PH is one of the most important tool of topological data analysis, and it is applied tovarious data analysis studies such as molecular phylogenetics [2], materials science [3], and structural
biology [4].
Figure 1 shows the outline of PH. To characterize the shape of the input pointcloud from theviewpoint of topology, we put discs (for 2d) or balls (for 3d) with the same radius on all points(Fig. 1(a)). To capture the topological information of various scales, we consider the process ofincreasing balls as the arrow in Fig. 1(a) indicates. We analyze the appearances and disappearances ofhomology generators such as holes and cavities in the process. The theory of PH enables us to makea set of pairs (strictly saying, a multiset of pairs) of appearance and disappearance. The theory also
ensures the uniqueness of the set of pairs and gives the algorithm to compute the pairs.
The radii of the appearance of homology generators are called birth times, and the radii of disappearance is called death times, and the pair of a birth time and a death time is called birth-deathpairs. The multiset of all birth-death pairs is called a persistence diagram (PD) (Fig. 1(b)). For eachbirth-death pair, (death time) − (birth time) is called a lifetime.

(a)radiusA holeappearbirthDivided
One hole
intodisappers
two holes
birth
death

Another holedisappears

deathFigure 1: Outline of persistent homology1(b)2Inverse problem on PD

In some applications of PH to materials science [3], we applied PH to the molecular dynamical simulation data and we found that PH can capture some typical geometric structures of materials in atomic
configurations and the important change the structures by the simulation conditions. The facts saysthat PDs correlate to the physical properties of the materials and it suggest the possibility to designmaterials from the viewpoint of PH. The first step to realize the PH-based materials design is to findthe way to construct a pointcloud whose k-th persistence diagram is the desired diagram.
In fact the problem is ill-posed, it means that the solution is not unique. Here we consider the theadditional constraint that the pointcloud should be as close as possible to the given pointcloud. The
constraint comes from the original research question of materials science. We cannot realize any atomic
configuration in nature, therefore it is natural to consider how to realize better atomic configuration
with smallest displacement. Of course it is difficult to make such atomic configuration in the realworld, but our PH-designed structure suggests an insight for new material design.
Finally we formalize the inverse problem from a PD to a pointcloud as follows.
Problem 1. For a given PD, Dk , and a pointcloud, P0 , find a pointcloud P satisfying the following
conditions:minimize d(P, P0 ), subject to Dk (P ) = Dk ,where d is a distance between pointcloud and Dk (P ) is the k-th persistence diagram of P .
Here we consider `2 norm as the distance by fixing the number of points N and the order of points.
Finally the problem can be formalized as follows:
Problem 2. For a given PD Dk and a pointcloud P0 , find a pointcloud P satisfying the following
conditions:(1)minimize kP − P0 k2 , subject to P ∈ R3N and Dk (P ) = Dk .
Numerically solve the problem, we use the Newton-Raphson method by pseudo-inverse. The
method solves the problem if Dk is a differentiable map from R3N to RM . The first difficulty toapply the Newton-Raphson method is the fact that the number of pairs in Dk (P ) is not constant. We
can avoid the difficulty by the stability theorem of PH. We can show the following fact.
Fact 1. Let Dk (P ; L) be {(b, d) ∈ Dk (P ) | d − b > L}, and P̃ be a pointcloud. We assume that any
birth-death pair (b, d) ∈ DK (P ; L) does not satisfy L −  < b − d < L + . Then the number of pairsin Dk (P ; L) is the same as Dk (P̃ ; L) if kP − P̃ k∞ < , where k · k∞ is the maximum norm on R3N .
Moreover the map P 7→ Dk (P ; L) is continuous in this neighborhood.
This fact means that we can define the map with good properties by ignoring birth-death pairs
with small lifetime from a persistence diagram.
The second difficulty is Dk (P ; L) is not differentiable. However we can show the following proposition.
Proposition 1. If P satisfies the general position condition, Dk (P ; L) is differentiable in its smallneighborhood. Moreover we can explicitly write the map by using the coordinates of points P . Eachelement of the map is a rational function.
Finally we write the iteration step of the Newton-Raphson method as follows:
P (j+1) = P (j) − (dDk (P (j) ; L))† Dk (P (j) ; L),(2)

where dDk (P ; L) is the total derivative of D(P ; L) given by Proposition 1 and A† means the Moore–Penrose
pseudo-inverse of matrix A.

2

3Discussions and Conclusions

The key mathematical fact is the piecewise differentiability of the map P 7→ Dk (P ; L). The differentiability is useful for not only Newton-Raphson method, but other optimization methods. We canapply gradient decent to persistence diagram. One example is the following minimization problem.
Problem 3. For a given PD Dk and a pointcloud P0 , find a pointcloud P satisfying the following
conditions:minimize kP − P0 |22 + λkDk (P ; L) − Dk k22 ,(3)where λ is a weight parameter.
This minimization problem gives the similar solution of Problem 1. In some sense we can regardkDk (P ; L) − Dk k22 as “topological cost function”.
Some studies after our study utilize the piecewise differentiability of persistence diagrams and theconcept of topological cost function. Hu et al. [5] applied the idea to image processing. We candefine persistence diagram on any dimensional bitmap data and the map from bitmaps to persistence
diagrams is also piecewise differentiable if we ignore pairs with small lifetime. In the paper the authors
construct a neural network for image segmentation with this topological cost function. The costfunction penalize the topologically bad result. Integrating Dk (P ; L) to the neural network is not so
difficult since piecewise differentiability is sufficient for backward propagation, fundamental neuralnetwork learning mechanism.
We also have an idea that we can utilize the topological cost function to material design. For thedesign, we need various physical constraints. To integrate such constraints with PH, we consider thefollowing “total cost function” and minimize it.
(energetic cost function comes from physical property)+(material manufacturing cost function)+(topological cost function)(4)+··· .
This is just an idea, but I believe that integrating PH to various optimization problems has interestingapplications.

References[1] M. Gameiro, Y. Hiraoka, and I. Obayashi. Continuation of point clouds via persistence diagrams.
Physica D: Nonlinear Phenomena, 334(1), 118–132 (2016)[2] J. M. Chan, G. Carlsson, and R. Rabadan. Topology of viral evolution. PNAS 110 (46), 18566–18571 (2013)[3] A. Hirata, T. Wada, I. Obayashi and Y. Hiraoka. Structural changes during glass formation
extracted by computational homology with machine learning. Communications Materials 1, 98(2020)[4] T. Ichinomiya, I. Obayashi, and Y. Hiraoka. Protein folding analysis using features obtained bypersistent homology. Biophysical Journal 118, 2926–2937 (2020)[5] X. Hu, F. Li, D. Samaras, and C. Chen. Topology-Preserving Deep Image Segmentation. Advancesin Neural Information Processing Systems 32, 5657–5668 (2019)3