Poster Session

Single cell transcriptomics and topological data analysis

Djordje Baralic
Mathematical Institute SANU, Serbia

A simple permutoassociahedron

Nicolas Berkouk
INRIA Saclay, France

Stable Resolutions of Multi-Parameter persistence modules

Multi-graded betti numbers, one of the only invariant we have to study multi-parameter persistence modules, are highly not stable with respect to the interleaving distance. However, they arise from the existence of free minimal resolutions and we propose to show how we can equip the derived category of persistent modules (the one in which resolutions « naturally » live) with a derived distance to make those resolutions stable. As a byproduct, we get the non-existence of a right (or left) exact functor from the category of modules to the one that decomposes in thin summands.

Nello Blaser
University of Bergen, Norway

Topological data analysis to identify blood cells

Mass cytometry can be used to measure up to forty different parameters for millions of cells. We used topological methods to analyse and visualize these large high-dimensional point clouds and compare the results of traditional methods used in cytometry to the results of topological methods. We found that topological methods could be used to quickly identify large cell populations and that these cell populations agree with the ones found by traditional methods.

Morten Brun
University of Bergen, Norway

Sparse Covers

We construct sparse approximations of filtered Cech, Rips and Voronoi complexes. The sparse Cech and Rips complexes appeared first in papers of Sheehy and Cavanna-Jahnseir-Sheehy. The sparse Voronoi complex is closely related to witness complexes. These sparse complexes are constructed from an arbitrary finite metric space, and they have a version for every choice of interleaving constant greater than one. The sparse Cech, Rips and Voronoi complexes are subcomplexes of nerves of sparse covers of the underlying finite metric space, and most of out reasoning takes place on the level of such covers.

Morten Brun
University of Bergen, Norway

Sparse Dowker Nerves

Ben Cassidy
Columbia University, USA

Topological Change Points for Time-Varying Brain Activity Networks

A common problem in the statistical analysis of multivariate time series is to identify and characterise change points where the probability distribution underlying the system of interest shifts from one state to another throughout time. But most existing change point statistics are either univariate measures or a simple function of the empirical covariance matrix, and are not suitable for detecting changes in time-varying networks.
We present new approaches for identifying and characterising change points in time-varying brain activity networks constructed from functional MRI data. The methods are based on persistent homology and build on our recent work for non-stationary network system identification. We demonstrate examples on real and simulated data.

Andrew Chen
Columbia University, USA

Functional Data Analysis using a Topological Summary Statistic: the Smooth Euler Characteristic Transform

We introduce a novel statistic, the smooth Euler characteristic transform (SECT), which is designed to integrate shape information into regression models by representing shapes and surfaces as a collection of curves with little to no loss in information. As a result, for the first time detailed geometric and topological information can be utilized as informative predictor variables in an a large catalogue of functional data analytic methodologies. We illustrate the utility of the SECT in a radiogenomics context by showing that topological summaries of tumors, assayed by magnetic resonance imaging (MRI), are better predictors of clinical outcomes in patients with glioblastoma multiforme (GBM). We show that SECT features outperform previous topological summary statistics, and alone explain more of the variance in patient survival than gene expression, volumetric features, and morphometric features.

Jérémy Cochoy
INRIA / Paris-Sud, France

Multidimensional persistence modules

Persistence is a tool which allow to recover topological property of a sampled surface from a point cloud.
In the construction, an algebraic object appear : the persistence module. The generalisation of this object with
index set of higher dimension still present a challenge
for their understanding.
This poster will present persistence in the multidimensional setting and some of it's properties.

Ioan Filip
Columbia University, USA

Towards an algebraic-geometric formulation of evolution

TAGS_apps_in_bio.pdf

Oliver Gäfvert
KTH Royal Institute of Technology, Sweden

Topology-based metric learning

Metric learning (or similarity learning) is the task of learning a metric that, in some sense, maximally separates two sets of objects. Using a parametrized class of pseudometrics on persistence modules, derived from what we call persistence contours. We define a scheme for learning a pseudometric on persistence modules that maximally (in some sense) separates between two given sets of persistence modules. In the one-dimensional case this means that we're optimizing the parameters of a pseudometric on the space of barcodes in such a way that it ''maximally'' separates two given sets of barcodes. The resulting metric can then be used for classifying unseen barcodes into the two sets.

Barbara Giunti
Università di Pavia, Italy

Persistent chain complexes

Persistent homology of the Vietoris-Rips complex has proven to be an effective tool to extract geometric information from data sets. Homology, however, is a drastic simplification and in certain situations might remove too much information. This project aims to investigate invariants, extracted not from the homology but rather from the persistent chain complex. For this approach to be useful, a decomposition structure theorem for persistence chain complexes is necessary. That is the crucial fundament theorem of this project. This presentation aims to describe the indecomposable and present an algorithm of how to enumerate indecomposable summands for a persistent chain complex.

Gillian Grindstaff
The University of Texas at Austin, USA

Geometric comparison of phylogenetic trees on varying taxa sets

As discovered by Billera, Holmes, and Vogtmann in 2001, the set of phylogenetic trees on n taxa form a CAT(0) contractible geodesic metric space, referred to as BHV space, which allows for the adaptation of statistical techniques to sets of phylogenies. However, the problem of comparing or combining trees on different sets of taxa has not been possible using the same methods. Davidson et al. recently created a combinatorial algorithm to extend tree topologies to higher dimensional sectors of BHV space. We adapt their algorithm for metric trees to compute the subspace of extensions of a tree on fewer than n taxa to n-dimensional BHV space. We also define several metrics which can be used to measure the compatibility of multiple metric tree fragments, and the degree to which a collection of fragments can be realized as subtrees of the same tree structure.

Marc Harkonen
Georgia Institute of Technology, USA

Holonomic extended least angle regression

Model selection algorithms are used to automatically choose the statistical model or family of models that best fit our data. For example in generalized linear models each parameter corresponds to one covariate, and then one might ask which of these are actually useful, or the most “impactful” in our fitting procedure, and how do we actually decide which covariates to include in the model. There have been numerous attempts at automatizing this process. Most notably, the Least Angle Regression algorithm, or LARS, by Efron et al. is a computationally efficient algorithm that ranks the covariates of a linear model without being too greedy. The LARS algorithm was extended in 2010 by Hirose and Komaki for a class of distributions in the generalized linear model by using properties of the manifold of exponential families as a dually flat manifold. However this extension assumes that the normalizing constant of the joint distribution of observations is “easy” to compute. This is often not the case, for example the normalizing constant may contain an integral that has no analytic solution.
However, if the normalizing constant satisfies a holonomic system, we can circumvent this issue by using a method inspired by the Holonomic Gradient Method from Nakayama et al. In this poster we present a modification of the holonomic gradient method and add it to the extended LARS algorithm. We call this the holonomic extended least angle regression algorithm, or HELARS. The algorithm was implemented using the statistical software R, and was tested with real and simulated datasets.

Gregory Henselman-Petrusek
Princeton University, USA

Matroid Structure in Computational Topology

The proposed poster would highlight recent developments in matroid theory, homological algebra, and computation, with a view to statistical topology. Several of the premier applications in this field rest on the decomposition of homological persistence modules into 1-dimensional factors that realize, collectively, the structure of a finitary matroid. The combinatorial properties of this object have profound consequences in statistical topology and homological algebra, with numerous connections to Lie, Hodge, and Gromov-Witten theory. As an elementary application, we present the Eirene library for homological persistence, an open-source platform for persistence module decomposition that outperforms, in several cases of interest, the current state-of-the-art by orders of magnitude in both time and memory.

Shaoxiong Hu
Queen Mary University of London, United Kingdom

Model Selection by Algebraic Criteria

Théo Lacombe
Inria Saclay, France

Statistical tools for persistence diagrams via smoothed Optimal Transport

Standard metrics on persistence diagrams (Bottleneck or Wasserstein) provide a powerful theoretical framework to compare topological descriptors. However, their combinatorial nature makes them hard to handle from a statistical perspective. By leveraging recent progress in optimal transport theory, we provide an approximation of these metrics which presents interesting properties for statistical and numerical purposes. We show applications to the so-called Wasserstein barycenter problem for persistence diagrams and other numerical problems.

Zhangsheng Lai
Singapore University of Design and Technology, Republic of Singapore

Activity-based Graphical Models for Time Series

Given a directed graph, possibly with cycles and self-loops, we define a continuous-time Markov chain where the nodes are binary random variables and where the off-diagonal entries of the transition rate matrix are defined by monomials in the model parameters. We call this Markov chain a McCulloch-Pitts process, and its defining monomials are known as activities. Such processes play an important role in biologically plausible deep learning, because they explain an experimentally-observed phenomenon in mammalian brains known as Spike Timing Dependent Plasticity. The learning rules for the model are simple, local and distributed, so we may finally be able to train massive neural networks. The model is a toric variety in the space of stochastic rate matrices, and learning corresponds to movement along this toric variety to a point that is closest to the empirical distribution by means of multiplicative weight updates. This is joint work with Chris Hillar and Shaowei Lin.

Jānis Lazovskis
University of Illinois at Chicago, USA

Sheaves on configuration spaces

Combining all the configuration spaces of a manifold gives the Ran space, which has a natural topology on it. There exists a constructible sheaf on this space, which can be described explicitly or by using a duality between such sheaves and functors of exit paths. Further, taking the product of the Ran space with non-zero real numbers gives a sheaf that classifies the Cech (or Vietoris-Rips) complexes that come from point samples of the base manifold. In other words, this sheaf contains the same information as the persistent homology of all point clouds on the manifold.

Kang-Ju Lee
Seoul National University, South Korea

Simplicial networks and effective resistance

We introduce the notion of effective resistance for a \emph{simplicial network} $(X,R)$ where $X$ is a simplicial complex and $R$ is a set of resistances for the top simplices, and prove two formulas generalizing previous results concerning effective resistance for resistor networks. Our approach, based on combinatorial Hodge theory, is to assign a unique harmonic class to a \emph{current generator} $\sigma$, an extra top-dimensional simplex to be attached to $X$. We will show that the harmonic class gives rise to the \emph{current} $I_{\sigma}$ and the \emph{voltage} $V_{\sigma}$ for $X\cup\sigma$, satisfying Thompson's energy-minimizing principle and Ohm's law for simplicial networks.

The effective resistance $R_{\sigma}$ of a current generator $\sigma$ shall be defined as a ratio of the $\sigma$-components of $V_{\sigma}$ and $I_{\sigma}$. By introducing \emph{potential} for voltage vectors, we present a formula for $R_\sigma$ via the inverse of the weighted combinatorial Laplacian of $X$ in codimension one. We also derive a formula for $R_{\sigma}$ via weighted high-dimensional tree-numbers for $X$, providing a combinatorial interpretation for $R_{\sigma}$. As an application, we generalize Foster's Theorem, and discuss various high-dimensional examples.

This is a joint work with Woong Kook.

Kang-Ju Lee
Seoul National University, South Korea

Simplicial networks and effective resistance

Poster.pdf

Barbara Mahler
University of Oxford, United Kingdom

Analysis of contagion maps on a class of networks that are spatially embedded in a torus

The spreading of real-world contagions is often guided by the geometry of the underlying domain. One example are contagious diseases that historically spread gradually along part of the earth's surface.
In this case contagions propagate as a wave front, passing between geometrically close entities. However, in today's world with modern transport and communication technology there are many scenarios where, even in the presence of a well-defined underlying geometry, such as the spherical surface of the earth, a contagion spreads via connections that are not intrinsically geometric.
Examples of such scenarios include the spreading of an infectious disease via passengers on a long-distance flight, or the dissemination of information via internet communication. Here the contagion jumps across space to distant locations rather that strictly following the geometry of the underlying domain.
Following the methodology introduced in [1], we study such phenomena by considering a threshold contagion model on a family of networks that can be construed as being embedded in a torus and have both `geometric edges' that respect the geometry of the underlying torus, and `non-geometric' edges that are not constrained by the underlying geometry and can connect nodes across a long range. We investigate what propagation pattern a contagion follows and how much this pattern is influenced by the structure of the underlying space.

[1] D. Taylor et al, “Topological data analysis of contagion maps for examining spreading processes on networks,” Nature communications, vol. 6, 2015.

Siddharth Pritam
Inria, Sophia Antipolis France, France

Simplifying a simplicial tower for efficient computation of Persistent Homology

We present a novel approach to reduce the size of a simplicial tower ( simplicial complexes connected through general simplicial maps) whille preserving their persistent homology. This technique is easily extendable in the case of Zigzag persistence as well.

Wanli Qiao
George Mason University, USA

Optimal Bandwidth for Nonparametric Estimation of Density Level Sets

Bandwidth selection is crucial in the kernel estimation of density level sets. Risk based on the symmetric difference between the estimated and true level sets is usually used to measure their proximity. In this paper we provide an asymptotic Lp approximation to this risk, where p is characterized by the weight function in the risk. In particular the excess risk corresponds to an L2 type of risk, and is adopted in an optimal bandwidth selection rule for nonparametric level set estimation of d-dimensional density functions (d >= 1).

Henri Riihimäki
Tampere University of Technology + KTH Stockholm, Finland

Persistence contours for quantifying topological difference in 1D persistent homology

Different studies applying persistent homology in data analysis are giving evidence that topological features in different scales are important for the analysis task at hand. This is in contrast to the traditional view in persistence putting focus on features with high persistence. We develop tools called persistence contours to quantify topology at different scales with different emphasis. This leads to a simple functional invariant for persistence modules. Persistence contours also give general metrics between persistence objects.

Yitzchak Solomon
Brown University, USA

Barcode Embeddings, Persistence Distortion, and Inverse Problems for Metric Graphs

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we demonstrate the generic injectivity of a rich homology-based invariant first defined by Dey, Wang, and Shi, which we think of as embedding a metric graph in the barcode space.

Gard Spreemann
EPFL, Switzerland

A notion of homological clustering

Spectral clustering of graphs has long been a standard tool. We propose a similar method of clustering simplicial complexes based on the simplicial Laplacian.

Stefan Stein
University of Warwick, United Kingdom

Asymptotic Equivalence between nonparametric density estimation and Gaussian white noise

Following a paper of Nussbaum (1996) I present the key steps in proving asymptotic equivalence in the Le Cam sense between nonparametric density estimation and Gaussian white noise. The poster will also include a short introduction to Le Cam theory from its classical perspective in terms of risk functions as well as from a category theoretical point of view.

Elizabeth Vidaurre
University of Rochester, USA

Cohomology of real moment-angle complexes

: Certain subspaces of a product of spaces whose factors are labeled by the vertices of a simplicial complex are referred to as "polyhedral product spaces". Polyhedral products are given by taking the union of subproducts depending on the face category of a fixed simplicial complex on $m$ vertices and a labelled family of $m$ topological pairs. Such polyhedral products are realized by objects studied in combinatorics, commutative algebra and algebraic geometry. There are applications of polyhedral products to topological robotics.

Madeleine Weinstein
UC Berkeley, USA

Offset hypersurfaces and persistent homology of algebraic varieties

We present connections between the true persistent homology of algebraic varieties and their corresponding offset hypersurfaces. By Hardt's theorem, true persistent homology events occur at algebraic numbers. By studying the offset hypersurface, we characterize when true persistent homology events occur. The degree of the offset hypersurface is bounded by the Euclidean Distance Degree of the starting variety, so this way we show a bound on the degree of the algebraic numbers at which the homological events occur.