# Registered Data

## [00533] Recovery and robustness of geometric fingerprints for point clouds and data

**Session Date & Time**:- 00533 (1/3) : 1C (Aug.21, 13:20-15:00)
- 00533 (2/3) : 1D (Aug.21, 15:30-17:10)
- 00533 (3/3) : 1E (Aug.21, 17:40-19:20)

**Type**: Proposal of Minisymposium**Abstract**: The aim of our mini-symposium is to connect communities interested in the problem of condensing information from a dataset to a less complex geometric/statistical "summary", sometimes called a fingerprint. We will concentrate especially on, Distance histograms, Persistence diagrams, as well as spectral fingerprints and other geometric fingerprints. Questions relevant to applications, including topics such as resistence or stability to noise/error of a given fingerprint ("robustness" problems), or injectivity of the fingerprint (relevant for "recovery" problems) will be our focus during the minisymposium.**Organizer(s)**: Mircea Petrache, Rodolfo Viera**Classification**:__53C23__,__55N31__,__68T09__,__52C35__,__68R12__,__Distance histograms, Persistence diagrams, Fourier fingerprints__**Speakers Info**:- Facundo Memoli (Ohio State University)
- Jose Perea (Northeastern University)
- Tom Needham (Florida State University)
- Matti Lassas (University of Helsinki)
- Theresa Heiss (IST Austria)
- Shuai Huang (Emory University)
- Vitaly Kurlin (University of Liverpool)
- Yohai Rheani (Technion)
- Rodolfo Viera (Pontificia Catolica Universidad de Chile)
**Mircea Petrache**(Pontificia Catolica Universidad de Chile)

**Talks in Minisymposium**:**[03736] Distance Distributions and Inverse Problems for Metric Measure Spaces****Author(s)**:**Tom Needham**(Florida State University)- Facundo Mémoli (The Ohio State University)

**Abstract**: Distance distributions, or distance histograms, are widely used invariants of metric measure spaces. We will discuss theoretical properties of these (and other, related) invariants. In particular, we are interested in their injectivity properties: given a class of metric measure spaces, can one characterize those spaces which are not distinguished by these invariants? We will explain results for several families of metric measure spaces, such as Riemannian manifolds and metric graphs.

**[04178] Recovering discrete Fourier spectra from random perturbations****Author(s)**:- Mircea Petrache (Pontificia Catolica Universidad de Chile)
**Rodolfo Viera**(Pontificia Universidad Católica de Chile)

**Abstract**: In this talk I will discuss the behaviour of the Fourier Transform of (quasi-)periodic sets under random perturbations. We will see that for i.i.d random perturbations of a quasi-periodic set X in the Euclidean space, the effect of the perturbations is almost surely that of multiplying the Fourier Transform of X by a weight which depends on the law of the perturbation. Also we will see quantitative versions of the previous discussion in finite groups which we will use to obtain, after passing to the limit, the almost sure recovery of the Fourier Transform of lattices in some non-abelian instances, such as the Heisenberg group.

**[04313] Curvature sets and curvature measures over persistence diagrams****Author(s)**:**Facundo Memoli**(Ohio State University)

**Abstract**: We study an invariant (i.e. a feature) of compact metric spaces which combines the notion of curvature sets introduced by Gromov in the 1980s together with the notion of Vietoris-Rips persistent homology. For given integers k≥0 and n≥1 these invariants arise by considering the degree k Vietoris-Rips (VR) persistence diagrams of all finite point clouds with cardinality at most n sampled from a given metric space. We call these invariants \emph{persistence sets}. This family of invariants contains the usual VR persistence diagram of the original space (when n is large enough). We argue that for a certain range of values of parameters n and k, (1) the family of these invariants 'sees' information not detected by the VR persistence diagrams of the whole space and (2) computing these invariants is significantly easier than computing the usual VR persistence diagrams. We establish stability results for our persistence sets and also precisely characterize some of them in the case of spheres with geodesic and Euclidean distances. We identify a rich family of metric graphs for which the invariant determined by n=4 and k=1 fully recovers their homotopy type. Along the way we prove some novel properties of VR persistence diagrams.

**[04676] Persistent cycle registration and topological bootstrap****Author(s)**:**Yohai Reani**(Viterbi Faculty of Electrical Engineering, Technion - Israel Institute of Technology)- Omer Bobrowski (Viterbi Faculty of Electrical Engineering, Technion - Israel Institute of Technology)

**Abstract**: In this talk we present a novel approach for comparing the persistent homology representations of two spaces (filtrations) directly in the data space. We do so by defining a correspondence relation between such representations and devising a method, based on persistent homology variants, for its efficient computation. We demonstrate our new framework in the context of topological inference, where we use statistical bootstrap-like methods to differentiate between real phenomena and "noise" in point cloud data.

**[04866] Reconstruction of manifolds from point clouds and inverse problems****Author(s)**:**Matti Lassas**(University of Helsinki)- Charles Fefferman (Princeton University)
- Sergei Ivanov (Steklov Institute of Mathematics)
- Hariharan Narayanan (Tata Institute for Fundamental Research)
- Jinpeng Lu (University of Helsinki)

**Abstract**: We consider a geometric problem on how a Riemannian manifold can be constructed to approximate a given discrete metric space. This problem is closely related to invariant manifold learning, where a Riemannian manifold $(M,g)$ needs to be approximately constructed from the noisy distances $d(X_j,X_k)+\eta_{jk}$ of points $X_1,X_2,\dots,X_N$, sampled from the manifold $M$. Here, $d(X_j,X_k)$ are the distance of the points $X_j,X_k\in M$ and $\eta_{jk}$ are random measurement errors. The values $d(X_j,X_k)$ can be considered as distance fingerprints of the manifold $M$. We also consider applications of the results in inverse problems encountered in medical and seismic imaging. In these problems, an unknown wave speed in a domain needs to be determined from indirect measurements. Moreover, we discuss a problem analogous to the above one, where distances are measured from points in a small subset $U\subset M$ to points in a discrete subset of $M$ and the errors are deterministic.

**[04956] An information-theoretic perspective on the turnpike and beltway problems****Author(s)**:**Shuai Huang**(Emory University)

**Abstract**: Reconstructing a set of points on a line or a loop from their unlabelled pairwise distances is known as the turnpike or beltway problem. Some point configurations are easy to reconstruct, while others are more difficult. We show that the difficulty of problem can be characterized by the mutual information $I(X;Y)$ between the point variable $X$ and distance variable $Y$. Experiments show that $I(X;Y)$ decreases when there are more repeated distances.

**[05121] Learning with persistence diagrams****Author(s)**:**Jose Perea**(Northeastern University)- Iryna Hartsock (University of Florida)
- Alex Elchesen (Colorado State University)
- Tatum Rask (Colorado State University)

**Abstract**: Persistence diagrams are common descriptors of the topological structure of data appearing in various classification and regression tasks. They can be generalized to Radon measures supported on the birth-death plane and endowed with an optimal transport distance. Examples of such measures are expectations of probability distributions on the space of persistence diagrams. In this talk, I will present methods for approximating continuous functions on the space of Radon measures supported on the birth-death plane, as well as their utilization in supervised learning tasks.

**[05127] The Density Fingerprint of a Periodic Set and Persistent Homology****Author(s)**:- Herbert Edelsbrunner (Institute of Science and Technology Austria)
**Teresa Heiss**(Institute of Science and Technology Austria)- Vitaliy Kurlin (University of Liverpool)
- Philip Smith (University of Liverpool)
- Mathijs Wintraecken (Institute of Science and Technology Austria)

**Abstract**: Modeling a crystal as a periodic point set, we present a fingerprint consisting of density functions. The density fingerprint is invariant under isometries, continuous, and complete in the generic case, which are necessary features for reliable comparison of crystals. The fingerprint has a fast algorithm based on Brillouin zones and related inclusion-exclusion formulae, which we have implemented. I will discuss the connection with persistent homology, suggesting a possible extension of the fingerprint.