Brittany Terese Fasy

Brittany Terese Fasy
Montana State University | MSU · School of Computing and Department of Mathematical Sciences

PhD

About

71
Publications
13,331
Reads
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823
Citations
Introduction
I am a computational topologist. My research develops theoretical foundations for computational data analysis, and applies those foundations to various practical applications.
Additional affiliations
August 2013 - August 2015
Tulane University
Position
  • Fellow
January 2009 - August 2012
Institute of Science and Technology Austria (ISTA)
Position
  • Researcher

Publications

Publications (71)
Article
Full-text available
Persistent homology probes topological properties from point clouds and functions. By looking at multiple scales simultaneously, one can record the births and deaths of topological features as the scale varies. In this paper we use a statistical technique, the empirical bootstrap, to separate topological signal from topological noise. In particular...
Article
Full-text available
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be "topological noise." In this paper, we bring some statistical ideas to persistent h...
Article
Full-text available
Persistent homology is a widely used tool in Topological Data Analysis that encodes multiscale topological information as a multi-set of points in the plane called a persistence diagram. It is difficult to apply statistical theory directly to a random sample of diagrams. Instead, we can summarize the persistent homology with the persistence landsca...
Article
Full-text available
We present a short tutorial and introduction to using the R package TDA, which provides some tools for Topological Data Analysis. In particular, it includes implementations of functions that, given some data, provide topological information about the underlying space, such as the distance function, the distance to a measure, the kNN density estimat...
Preprint
Full-text available
The spatial distribution of galaxies at sufficiently small scales will encode information about the identity of the dark matter. We develop a novel description of the halo distribution using persistent homology summaries, in which collections of points are decomposed into clusters, loops and voids. We apply these methods, together with a set of hyp...
Preprint
Local maxima and minima, or $\textit{extremal events}$, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within...
Article
Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances has been challenging due to the computational cost. In this paper, we propose a persistence diagram hashing f...
Preprint
The purpose of this article is to study directed collapsibility of directed Euclidean cubical complexes. One application of this is in the nontrivial task of verifying the execution of concurrent programs. The classical definition of collapsibility involves certain conditions on a pair of cubes of the complex. The direction of the space can be take...
Preprint
Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances has been challenging due to the computational cost. In this paper, we propose a persistence diagram hashing f...
Preprint
Given a simplicial complex $K$ and an injective function $f$ from the vertices of $K$ to $\mathbb{R}$, we consider algorithms that extend $f$ to a discrete Morse function on $K$. We show that an algorithm of King, Knudson and Mramor can be described on the directed Hasse diagram of $K$. Our description has a faster runtime for high dimensional data...
Article
Full-text available
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology and introduce new invariants to study these equivalence classes. These new invariants are as simple—but more discerning than—existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate an...
Chapter
Full-text available
The Eisenhower Decision Matrix, credited to the task management system of US President Dwight Eisenhower, is a graphical diagram used in strategy and planning for tasks. This matrix, however, only provides four types of priorities. We identify a collection of scenarios in which the traditional matrix provides misleading suggestions and propose an e...
Chapter
In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a single path; we call this the trivial space of directed paths. Directed spaces that are topol...
Article
Full-text available
One of the primary areas of interest in applied algebraic topology is persistent homology, and, more specifically, the persistence diagram. Persistence diagrams have also become objects of interest in topological data analysis. However, persistence diagrams do not naturally lend themselves to statistical goals, such as inferring certain population...
Conference Paper
Full-text available
We study the interplay between the recently-defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves that are the image of the boundary of an immersed disk. Our first contribution is to prove new sufficient combinatorial conditions for a curve to be self-overlapping. We show that a cur...
Preprint
Full-text available
We consider the topological and geometric reconstruction of a geodesic subspace of $\mathbb{R}^N$ both from the Čech and Vietoris-Rips filtrations on a finite, Hausdorff-close, Euclidean sample. Our reconstruction technique leverages the intrinsic length metric induced by the geodesics on the subspace. We consider the distortion and convexity radiu...
Article
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that well-chosen (finite) sets of PDs can differentiate between geometric simplicial compl...
Article
We consider several variants of the map-matching problem, which seeks to find a path Q in graph G that has the smallest distance to a given trajectory P (which is likely not to be exactly on the graph). In a typical application setting, P models a noisy GPS trajectory from a person traveling on a road network, and the desired path Q should ideally...
Preprint
Topological descriptors have been shown to be useful for summarizing and differentiating shapes. Related work uses persistence diagrams and Euler characteristic curves to differentiate between shapes and quantifies the number of descriptors necessary for shape reconstruction, given certain assumptions such as minimum curvature. In this work, we pro...
Preprint
The persistence diagram (PD) is an increasingly popular topological descriptor. By encoding the size and prominence of topological features at varying scales, the PD provides important geometric and topological information about a space. Recent work has shown that particular sets of PDs can differentiate between different shapes. This trait is desi...
Preprint
Full-text available
We propose an algorithm to estimate the topology of an embedded metric graph from a well-sampled finite subset of the underlying graph.
Preprint
Full-text available
Discrete Morse theory has recently been applied in metric graph reconstruction from a given density function concentrated around an (unknown) underlying embedded graph. We propose a new noise model for the density function to reconstruct a connected graph both topologically and geometrically.
Preprint
Full-text available
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning than existing topological invariants, such as persistence barcodes and Reeb graphs. We give a method to relate...
Preprint
Full-text available
Defining homotopy or collapsibility in directed topological spaces is surprisingly difficult. In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a...
Preprint
Defining homotopy or collapsibility in directed topological spaces is surprisingly difficult. In the directed setting, the spaces of directed paths between fixed initial and terminal points are the defining feature for distinguishing different directed spaces. The simplest case is when the space of directed paths is homotopy equivalent to that of a...
Article
Full-text available
The current system for evaluating prostate cancer architecture is the Gleason grading system which divides the morphology of cancer into five distinct architectural patterns, labeled 1 to 5 in increasing levels of cancer aggressiveness, and generates a score by summing the labels of the two most dominant patterns. The Gleason score is currently the...
Preprint
Persistence diagrams are important tools in the field of topological data analysis that describe the presence and magnitude of features in a filtered topological space. However, current approaches for comparing a persistence diagram to a set of other persistence diagrams is linear in the number of diagrams or do not offer performance guarantees. In...
Preprint
Shape recognition and classification is a problem with a wide variety of applications. Several recent works have demonstrated that topological descriptors can be used as summaries of shapes and utilized to compute distances. In this abstract, we explore the use of a finite number of Euler Characteristic Curves (ECC) to reconstruct plane graphs. We...
Conference Paper
Discrete Morse theory has recently been applied in metric graph reconstruction from a given density function concentrated around an (unknown) underlying embedded graph. We propose a new noise model for the density function to reconstruct a connected graph both topologically and geometrically.
Conference Paper
Full-text available
Discrete Morse theory has recently been applied in metric graph reconstruction from a given density function concentrated around an (unknown) underlying embedded graph. We propose a new noise model for the density function to reconstruct a connected graph both topologically and geometrically.
Preprint
Full-text available
We investigate the intrinsic topology and geometry of a Euclidean point cloud by assuming that it is sampled around a much simpler underlying geodesic space. We develop a persistence based algorithm to reconstruct the topology of the unknown underlying space by considering the \v{C}ech and Rips complexes built on the point cloud. Also, in an intere...
Chapter
With the increasing availability of GPS trajectory data, map construction algorithms have been developed that automatically construct road maps from this data. In order to assess the quality of such (constructed) road maps, the need for meaningful road map comparison algorithms becomes increasingly important. Indeed, different approaches for map co...
Preprint
Full-text available
Topological Data Analysis (TDA) studies the shape of data. A common topological descriptor is the persistence diagram, which encodes topological features in a topological space at different scales. Turner, Mukeherjee, and Boyer showed that one can reconstruct a simplicial complex embedded in R^3 using persistence diagrams generated from all possibl...
Conference Paper
We consider several variants of the map-matching problem, which seeks to find a path Q in graph G that has the smallest distance to a given trajectory P (which is likely not to be exactly on the graph). In a typical application setting, P models a noisy GPS trajectory from a person traveling on a road network, and the desired path Q should ideally...
Conference Paper
Montana is home to a large American Indian population and a rich history. The Indian Education for All (IEFA) Act, passed in 1999, reinforces the educational goals stated in Montana's 1972 Constitution that "every Montanan, whether Indian or non-Indian, be encouraged to learn about the distinct and unique heritage of American Indians in a culturall...
Conference Paper
We propose a new approach for constructing the underlying map from trajectory data. Our algorithm is based on the idea that road segments can be identified as stable subtrajectory clusters in the data. For this, we consider how subtrajectory clusters evolve for varying distance values, and choose stable values for these. In doing so we avoid a glob...
Article
Full-text available
In this paper, we study the problem of computing a homotopy from a planar curve $C$ to a point that minimizes the area swept. The existence of such a minimum homotopy is a direct result of the solution of Plateau's problem. Chambers and Wang studied the special case that $C$ is the concatenation of two simple curves, and they gave a polynomial-time...
Article
Full-text available
Tumorigenesis is an evolutionary process by which tumor cells acquire mutations through successive diversification and differentiation. There is much interest in reconstructing this process of evolution due to its relevance to identifying drivers of mutation and predicting future prognosis and drug response. Efforts are challenged by high tumor het...
Conference Paper
Due to the ubiquitous use of various positioning technologies in smart phones and other devices, geospatial tracking data has become a routine data source. One of its uses that has gained recent popularity is the construction of street maps from vehicular tracking data. Due to the inherent noise in the data, many map construction algorithms are bas...
Conference Paper
Full-text available
Several researchers proposed using non-Euclidean metrics on point sets in Euclidean space for clustering noisy data. Almost always, a distance function is desired that recognizes the closeness of the points in the same cluster, even if the Euclidean cluster diameter is large. Therefore, it is preferred to assign smaller costs to the paths that stay...
Article
Full-text available
Let P be a distribution with support S. The salient features of S can be quantified with persistent homology, which summarizes topological features of the sublevel sets of the distance function (the distance of any point x to S). Given a sample from P we can infer the persistent homology using an empirical version of the distance function. However,...
Conference Paper
We define a topology-based distance metric between road networks embedded in the plane. This distance measure is based on local persistent homology, and employs a local distance signature that enables identification and visualization of local differences between the road networks. This paper is motivated by the need to recognize changes in road net...
Article
Full-text available
Persistent homology is a multiscale method for analyzing the shape of sets and functions from point cloud data arising from an unknown distribution supported on those sets. When the size of the sample is large, direct computation of the persistent homology is prohibitive due to the combinatorial nature of the existing algorithms. We propose to comp...
Article
After two decades of progress in hardness of approximation we finally completely understand the extent to which many optimization problems can be approximated in polynomial time. Unfortunately, however, and despite significant efforts, many important ...
Chapter
We present an efficient algorithm for solving a linear system arising from the 1-Laplacian corresponding to a collapsible simplicial complex with a known collapsing sequence. When combined with a result of Chillingworth, our algorithm is applicable to convex simplicial complexes embedded in R3. The running time of our algorithm is nearly-linear in...
Presentation
Full-text available
Road networks are always changing: new streets are built, accidents and floods close roads, etc. Detecting when, and if, a change has occurred is an important question. In this presentation, we highlight recent progress in computing the distance between two road networks.
Article
Full-text available
Comparing two geometric graphs embedded in space is important in the field of transportation network analysis. Given street maps of the same city collected from different sources, researchers often need to know how and where they differ. However, the majority of current graph comparison algorithms are based on structural properties of graphs, such...
Article
The decision tree model, aka the query model, perhaps due to its simplicity and fundamental nature has been extensively studied over decades. Yet there remain some fascinating open questions about it. The purpose of this paper is to revisit three such ...
Article
Full-text available
Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features as one varies a tuning parameter. Features with short lifetimes are informally considered to be "topological noise." In this paper, we bring some statistical ideas to persistent h...
Article
It has been an open question whether the sum of finitely many isotropic Gaussian kernels in n ≥ 2 dimensions can have more modes than kernels, until in 2003 Carreira-Perpinan and Williams exhibited n+1 isotropic Gaussian kernels in Rn with n+2 modes. We give a detailed analysis of this example, showing that it has exponentially many critical points...
Article
Full-text available
We bound the difference in length of two curves in terms of their total curvatures and the Fréchet distance. The bound is independent of the dimension of the ambient Euclidean space, it improves upon a bound by Cohen-Steiner and Edelsbrunner, and it generalizes a result by Fáry and Chakerian.
Article
The last two decades have seen enormous progress in the development of sublinear-time algorithms -- i.e., algorithms that examine/reveal properties of "data" in less time than it would take to read all of the data. A large, and important, subclass of ...
Article
The last two decades have seen enormous progress in the development of sublinear-time algorithms -- i.e., algorithms that examine/reveal properties of "data" in less time than it would take to read all of the data. A large, and important, subclass of ...
Article
Full-text available
Persistence homology is a tool used to measure topological features that are present in data sets and functions. Persistence pairs births and deaths of these features as we iterate through the sublevel sets of the data or function of interest. I am concerned with using persistence to characterize the difference between two functions f, g : M -> R,...
Article
Full-text available
We present a system for creating an edge-unfolding of a trian- gulated simplicial manifold from a vertex-unfolding by gluing edges, while maintaining two rigid geometries. We implemented a known vertex-unfolding algorithm as well as several gluing techniques which we created. The entire project consisted of a total of 11,280 lines of code.