Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space is complicated. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important topic currently researched in TDA. In this paper, our main contribution consists of three components. First, we develop a general and unifying framework of vectorizing diagrams that we call the Persistence Curves (PCs), and show that several well-known summaries, such as Persistence Landscapes, fall under the PC framework. Second, we propose several new summaries based on PC framework and provide a theoretical foundation for their stability analysis. Finally, we apply proposed PCs to two applications---texture classification and determining the parameters of a discrete dynamical system; their performances are competitive with other TDA methods.

Persistence Curves: A canonical framework for summarizing persistence diagrams.

In this survey, we review the literature on inverse problems in topological persistence theory. The first half of the survey is concerned with the question of surjectivity, i.e. the existence of right inverses, and the second half focuses on injectivity, i.e. left inverses. Throughout, we highlight the tools and theorems that underlie these advances, and direct the reader’s attention to open problems, both theoretical and applied.

Inverse Problems in Topological Persistence.

Future grid management systems will coordinate distributed production and storage resources to manage, in a cost effective fashion, the increased load and variability brought by the electrification of transportation and by a higher share of weather dependent production. Electricity demand forecasts at a low level of aggregation will be key inputs for such systems. We focus on forecasting demand at the individual household level, which is more challenging than forecasting aggregate demand, due to the lower signal-to-noise ratio and to the heterogeneity of consumption patterns across households. We propose a new ensemble method for probabilistic forecasting, which borrows strength across the households while accommodating their individual idiosyncrasies. In particular, we develop a set of models or 'experts' which capture different demand dynamics and we fit each of them to the data from each household. Then we construct an aggregation of experts where the ensemble weights are estimated on the whole data set, the main innovation being that we let the weights vary with the covariates by adopting an additive model structure. In particular, the proposed aggregation method is an extension of regression stacking (Breiman, 1996) where the mixture weights are modelled using linear combinations of parametric, smooth or random effects. The methods for building and fitting additive stacking models are implemented by the gamFactory R package, available at this https URL.

Additive stacking for disaggregate electricity demand forecasting

We define notions of differentiability for maps from and to the space of persistence barcodes. Inspired by the theory of diffeological spaces, the proposed framework uses lifts to the space of ordered barcodes, from which derivatives can be computed. The two derived notions of differentiability (respectively from and to the space of barcodes) combine together naturally to produce a chain rule that enables the use of gradient descent for objective functions factoring through the space of barcodes. We illustrate the versatility of this framework by showing how it can be used to analyze the smoothness of various parametrized families of filtrations arising in topological data analysis.

A Framework for Differential Calculus on Persistence Barcodes

Solving optimization tasks based on functions and losses with a topological flavor is a very active, growing field of research in data science and Topological Data Analysis, with applications in non-convex optimization, statistics and machine learning. However, the approaches proposed in the literature are usually anchored to a specific application and/or topological construction, and do not come with theoretical guarantees. To address this issue, we study the differentiability of a general map associated with the most common topological construction, that is, the persistence map. Building on real analytic geometry arguments, we propose a general framework that allows us to define and compute gradients for persistence-based functions in a very simple way. We also provide a simple, explicit and sufficient condition for convergence of stochastic subgradient methods for such functions. This result encompasses all the constructions and applications of topological optimization in the literature. Finally, we provide associated code, that is easy to handle and to mix with other non-topological methods and constraints, as well as some experiments showcasing the versatility of our approach.

Optimizing persistent homology based functions

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we study a rich homology-based invariant first defined by Dey, Shi, and Wang, which we think of as embedding a metric graph in the barcode space. We prove that this invariant is locally injective on the space of metric graphs and globally injective on a GH-dense subset. Moreover, we show that is globally injective on a full measure subset of metric graphs, in the appropriate sense.

Barcode Embeddings for Metric Graphs

Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.

A Fast and Robust Method for Global Topological Functional Optimization

Stable topological invariants are a cornerstone of persistence theory and applied topology, but their discriminative properties are often poorly-understood. In this paper we investigate the injectivity of a rich homology-based invariant first defined in \cite{dey2015comparing} which we think of as embedding a metric graph in the barcode space.

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

Topological transforms are parametrized families of topological invariants, which, by analogy with transforms in signal processing, are much more discriminative than single measurements The first two topological transforms to be defined were the Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), both of which apply to shapes embedded in Euclidean space The contribution of this paper is to define topological transforms for abstract metric measure spaces Our proposed pipeline is to pre-compose the PHT or ECT with a Euclidean embedding derived from the eigenfunctions and eigenvalues of an integral operator To that end, we define and study an integral operator called the distance kernel operator, and demonstrate that it gives rise to stable and quasi-injective topological transforms We conclude with some numerical experiments, wherein we compute and compare the eigenfunctions and eigenvalues of our operator across a range of standard 2- and 3-manifolds

https://drops.dagstuhl.de/opus/volltexte/2020/12214/pdf/LIPIcs-SoCG-2020-56.pdf/

Elchanan Solomon

Papers

Inverse Problems in Topological Persistence.

Barcode Embeddings for Metric Graphs

A Fast and Robust Method for Global Topological Functional Optimization

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

Intrinsic Topological Transforms via the Distance Kernel Embedding.