%% Papers Published
@article{fds226384,
Author = {P.L. Bendich and David Cohen-Steiner and Herbert Edelsbrunner and John Harer and Dmitriy Morozov},
Title = {Inferring Local Homology from Sampled Stratified
Spaces},
Journal = {In Proceedings of the 48th Annual IEEE Symposium on
Foundations of Computer Science, pages 536-546,
2007.},
Year = {2007},
Key = {fds226384}
}
@article{fds302432,
Author = {Bendich, P and Mukherjee, S and Wang, B},
Title = {Stratification learning through homology
inference},
Volume = {FS-10-06},
Pages = {10-17},
Year = {2010},
Month = {January},
ISBN = {9781577354888},
Abstract = {We develop a topological approach to stratification
learning. Given point cloud data drawn from a stratified
space, our objective is to infer which points belong to the
same strata. First we define a multi-scale notion of a
stratified space, giving a stratification for each radius
level. We then use methods derived from kernel and cokernel
persistent homology to cluster the data points into
different strata, and we prove a result which guarantees the
correctness of our clustering, given certain topological
conditions. We later give bounds on the minimum number of
sample points required to infer, with probability, which
points belong to the same strata. Finally, we give an
explicit algorithm for the clustering and apply it to some
simulated data. Copyright © 2010, Association for the
Advancement of Artificial Intelligence. All rights
reserved.},
Key = {fds302432}
}
@article{fds315428,
Author = {Bendich, P and Mukherjee, S and Wang, B},
Title = {Towards Stratification Learning through Homology
Inference},
Year = {2010},
Month = {August},
url = {http://arxiv.org/abs/1008.3572v1},
Abstract = {A topological approach to stratification learning is
developed for point cloud data drawn from a stratified
space. Given such data, our objective is to infer which
points belong to the same strata. First we define a
multi-scale notion of a stratified space, giving a
stratification for each radius level. We then use methods
derived from kernel and cokernel persistent homology to
cluster the data points into different strata, and we prove
a result which guarantees the correctness of our clustering,
given certain topological conditions; some geometric
intuition for these topological conditions is also provided.
Our correctness result is then given a probabilistic flavor:
we give bounds on the minimum number of sample points
required to infer, with probability, which points belong to
the same strata. Finally, we give an explicit algorithm for
the clustering, prove its correctness, and apply it to some
simulated data.},
Key = {fds315428}
}
@article{fds302434,
Author = {P.L. Bendich and Bendich, P and Edelsbrunner, H and Kerber, M},
Title = {Computing robustness and persistence for
images.},
Journal = {IEEE transactions on visualization and computer
graphics},
Volume = {16},
Number = {6},
Pages = {1251-1260},
Year = {2010},
Month = {November},
ISSN = {1077-2626},
url = {http://dx.doi.org/10.1109/tvcg.2010.139},
Abstract = {We are interested in 3-dimensional images given as arrays of
voxels with intensity values. Extending these values to a
continuous function, we study the robustness of homology
classes in its level and interlevel sets, that is, the
amount of perturbation needed to destroy these classes. The
structure of the homology classes and their robustness, over
all level and interlevel sets, can be visualized by a
triangular diagram of dots obtained by computing the
extended persistence of the function. We give a fast
hierarchical algorithm using the dual complexes of oct-tree
approximations of the function. In addition, we show that
for balanced oct-trees, the dual complexes are geometrically
realized in R³ and can thus be used to construct level and
interlevel sets. We apply these tools to study 3-dimensional
images of plant root systems.},
Doi = {10.1109/tvcg.2010.139},
Key = {fds302434}
}
@article{fds302431,
Author = {P.L. Bendich and Bendich, P and Edelsbrunner, H and Morozov, D and Patel,
A},
Title = {The robustness of level sets},
Journal = {Lecture Notes in Computer Science (including subseries
Lecture Notes in Artificial Intelligence and Lecture Notes
in Bioinformatics)},
Volume = {6346 LNCS},
Number = {PART 1},
Pages = {1-10},
Publisher = {Springer Berlin Heidelberg},
Year = {2010},
Month = {November},
ISSN = {0302-9743},
url = {http://dx.doi.org/10.1007/978-3-642-15775-2_1},
Abstract = {We define the robustness of a level set homology class of a
function f : double-struck X → ℝ as the magnitude of a
perturbation necessary to kill the class. Casting this
notion into a group theoretic framework, we compute the
robustness for each class, using a connection to extended
persistent homology. The special case double-struck X = ℝ3
has ramifications in medical imaging and scientific
visualization. © 2010 Springer-Verlag.},
Doi = {10.1007/978-3-642-15775-2_1},
Key = {fds302431}
}
@article{fds302433,
Author = {P.L. Bendich and Bendich, P and Edelsbrunner, H and Kerber, M and Patel,
A},
Title = {Persistent homology under non-uniform error},
Journal = {Lecture Notes in Computer Science (including subseries
Lecture Notes in Artificial Intelligence and Lecture Notes
in Bioinformatics)},
Volume = {6281 LNCS},
Pages = {12-23},
Publisher = {Springer Berlin Heidelberg},
Year = {2010},
Month = {November},
ISBN = {9783642151545},
ISSN = {0302-9743},
url = {http://dx.doi.org/10.1007/978-3-642-15155-2_2},
Abstract = {Using ideas from persistent homology, the robustness of a
level set of a real-valued function is defined in terms of
the magnitude of the perturbation necessary to kill the
classes. Prior work has shown that the homology and
robustness information can be read off the extended
persistence diagram of the function. This paper extends
these results to a non-uniform error model in which
perturbations vary in their magnitude across the domain. ©
2010 Springer-Verlag.},
Doi = {10.1007/978-3-642-15155-2_2},
Key = {fds302433}
}
@article{fds243366,
Author = {P.L. Bendich and Bendich, P and Harer, J},
Title = {Persistent Intersection Homology},
Journal = {Foundations of Computational Mathematics},
Volume = {11},
Number = {3},
Pages = {305-336},
Publisher = {Springer Nature},
Year = {2011},
Month = {June},
ISSN = {1615-3375},
url = {http://dx.doi.org/10.1007/s10208-010-9081-1},
Abstract = {The theory of intersection homology was developed to study
the singularities of a topologically stratified space. This
paper incorporates this theory into the already developed
framework of persistent homology. We demonstrate that
persistent intersection homology gives useful information
about the relationship between an embedded stratified space
and its singularities. We give an algorithm for the
computation of the persistent intersection homology groups
of a filtered simplicial complex equipped with a
stratification by subcomplexes, and we prove its
correctness. We also derive, from Poincaré Duality, some
structural results about persistent intersection homology.
© 2010 SFoCM.},
Doi = {10.1007/s10208-010-9081-1},
Key = {fds243366}
}
@article{fds243365,
Author = {P.L. Bendich and Bendich, P and Galkovskyi, T and Harer, J},
Title = {Improving homology estimates with random
walks},
Journal = {Inverse Problems},
Volume = {27},
Number = {12},
Pages = {124002-124002},
Publisher = {IOP Publishing},
Year = {2011},
Month = {December},
ISSN = {0266-5611},
url = {http://dx.doi.org/10.1088/0266-5611/27/12/124002},
Abstract = {This experimental paper makes the case for a new approach to
the use of persistent homology in the study of shape and
feature in datasets. By introducing ideas from diffusion
geometry and random walks, we discover that homological
features can be enhanced and more effectively extracted from
spaces that are sampled densely and evenly, and with a small
amount of noise. This study paves the way for a more
theoretical analysis of how random walk metrics affect
persistence diagrams, and provides evidence that combining
topological data analysis with techniques inspired by
diffusion geometry holds great promise for new analyses of a
wide variety of datasets. © 2011 IOP Publishing
Ltd.},
Doi = {10.1088/0266-5611/27/12/124002},
Key = {fds243365}
}
@article{fds302435,
Author = {P.L. Bendich and Bendich, P and Wang, B and Mukherjee, S},
Title = {Local homology transfer and stratification
learning},
Journal = {Proceedings of the Annual ACM-SIAM Symposium on Discrete
Algorithms},
Pages = {1355-1370},
Year = {2012},
Month = {January},
ISBN = {9781611972108},
url = {http://dx.doi.org/10.1137/1.9781611973099.107},
Abstract = {The objective of this paper is to show that point cloud data
can under certain circumstances be clustered by strata in a
plausible way. For our purposes, we consider a stratified
space to be a collection of manifolds of different
dimensions which are glued together in a locally trivial
manner inside some Euclidean space. To adapt this abstract
definition to the world of noise, we first define a
multi-scale notion of stratified spaces, providing a
stratification at different scales which are indexed by a
radius parameter. We then use methods derived from kernel
and cokernel persistent homology to cluster the data points
into different strata. We prove a correctness guarantee for
this clustering method under certain topological conditions.
We then provide a probabilistic guarantee for the clustering
for the point sample setting - we provide bounds on the
minimum number of sample points required to state with high
probability which points belong to the same strata. Finally,
we give an explicit algorithm for the clustering. Copyright
© SIAM.},
Doi = {10.1137/1.9781611973099.107},
Key = {fds302435}
}
@article{fds302436,
Author = {P.L. Bendich and Bendich, P and Cabello, S and Edelsbrunner, H},
Title = {A point calculus for interlevel set homology},
Journal = {Pattern Recognition Letters},
Volume = {33},
Number = {11},
Pages = {1436-1444},
Publisher = {Elsevier BV},
Year = {2012},
Month = {August},
ISSN = {0167-8655},
url = {http://dx.doi.org/10.1016/j.patrec.2011.10.007},
Abstract = {The theory of persistent homology opens up the possibility
to reason about topological features of a space or a
function quantitatively and in combinatorial terms. We refer
to this new angle at a classical subject within algebraic
topology as a point calculus, which we present for the
family of interlevel sets of a real-valued function. Our
account of the subject is expository, devoid of proofs, and
written for non-experts in algebraic topology. © 2011
Elsevier B.V. All rights reserved.},
Doi = {10.1016/j.patrec.2011.10.007},
Key = {fds302436}
}
@article{fds220713,
Author = {Paul Bendich and Herbert Edelsbrunner and Dmitriy Morozov and Amit Patel},
Title = {Homology and Robustness of Level and Interlevel
Sets},
Journal = {Homology, Homotopy, and Applications},
Volume = {15},
Number = {1},
Pages = {51-72},
Editor = {Gunnar Carlsson},
Year = {2013},
Month = {March},
Abstract = {Given a continuous function f : X → R on a topological
space, we consider the preimages of intervals and their
homol- ogy groups and show how to read the ranks of these
groups from the extended persistence diagram of f. In
addition, we quan- tify the robustness of the homology
classes under perturbations of f using well groups, and we
show how to read the ranks of these groups from the same
extended persistence diagram. The special case X = R^3 has
ramifications in the fields of medical imaging and
scientific visualization.},
Key = {fds220713}
}
@article{fds303523,
Author = {Bendich, P and Edelsbrunner, H and Morozov, D and Patel,
A},
Title = {Homology and robustness of level and interlevel
sets},
Journal = {Homology, Homotopy and Applications},
Volume = {15},
Number = {1},
Pages = {51-72},
Publisher = {International Press of Boston},
Year = {2013},
Month = {April},
url = {http://arxiv.org/abs/1102.3389v1},
Abstract = {Given a continuous function f: X → ℝ on a topological
space, we consider the preimages of intervals and their
homology groups and show how to read the ranks of these
groups from the extended persistence diagram of f. In
addition, we quantify the robustness of the homology classes
under perturbations of f using well groups, and we show how
to read the ranks of these groups from the same extended
persistence diagram. The special case X = ℝ3 has
ramifications in the fields of medical imaging and
scientific visualization. © 2013, International
Press.},
Doi = {10.4310/HHA.2013.v15.n1.a3},
Key = {fds303523}
}
@article{fds227232,
Author = {Christopher J Tralie and Paul Bendich},
Title = {Cover Song Identification with Timbral Shape},
Journal = {Proceedings of the 16th International Society for Music
Information Retrieval},
Pages = {38-44},
Year = {2015},
url = {http://arxiv.org/abs/1507.05143},
Abstract = {We introduce a novel low level feature for identifying cover
songs which quantifies the relative changes in the smoothed
frequency spectrum of a song. Our key insight is that a
sliding window representation of a chunk of audio can be
viewed as a time-ordered point cloud in high dimensions. For
corresponding chunks of audio between different versions of
the same song, these point clouds are approximately rotated,
translated, and scaled copies of each other. If we treat
MFCC embeddings as point clouds and cast the problem as a
relative shape sequence, we are able to correctly identify
42/80 cover songs in the “Covers 80” dataset. By
contrast, all other work to date on cover songs exclusively
relies on matching note sequences from Chroma derived
features.},
Key = {fds227232}
}
@article{fds226628,
Author = {Liz Munch and Paul Bendich and Kate Turner and Sayan Mukherjee and Jonathan Mattingly and John Harer},
Title = {Probabalistic Frechet Means and Statistics on
Vineyards},
Journal = {Electronic Journal of Statistics},
Volume = {9},
Pages = {1173-1204},
Year = {2015},
url = {http://arxiv.org/abs/1307.6530},
Abstract = {In order to use persistence diagrams as a true statistical
tool, it would be very useful to have a good notion of mean
and variance for a set of diagrams. In [21], Mileyko and his
collaborators made the rst study of the properties of the
Frechet mean in (Dp;Wp), the space of persistence diagrams
equipped with the p-th Wasserstein metric. In particular,
they showed that the Frechet mean of a nite set of diagrams
always exists, but is not necessarily unique. As an
unfortunate consequence, one sees that the means of a
continuously-varying set of diagrams do not themselves vary
continuously, which presents obvious problems when trying to
extend the Frechet mean de nition to the realm of
vineyards. We x this problem by altering the original de
nition of Frechet mean so that it now becomes a probability
measure on the set of persistence diagrams; in a nutshell,
the mean of a set of diagrams will be a weighted sum of
atomic measures, where each atom is itself the (Frechet
mean) persistence diagram of a perturbation of the input
diagrams. We show that this new de nition de nes a (Holder)
continuous map, for each k, from (Dp)k ! P(Dp), and we
present several examples to show how it may become a useful
statistic on vineyards.},
Key = {fds226628}
}
@article{fds303522,
Author = {Munch, E and Turner, K and Bendich, P and Mukherjee, S and Mattingly, J and Harer, J},
Title = {Probabilistic Fréchet means for time varying persistence
diagrams},
Journal = {Electronic Journal of Statistics},
Volume = {9},
Number = {1},
Pages = {1173-1204},
Publisher = {Institute of Mathematical Statistics},
Year = {2015},
Month = {January},
url = {http://arxiv.org/abs/1307.6530v3},
Abstract = {In order to use persistence diagrams as a true statistical
tool, it would be very useful to have a good notion of mean
and variance for a set of diagrams. In [23], Mileyko and his
collaborators made the first study of the properties of the
Fréchet mean in (D<inf>p</inf>, W<inf>p</inf>), the space
of persistence diagrams equipped with the p-th Wasserstein
metric. In particular, they showed that the Fréchet mean of
a finite set of diagrams always exists, but is not
necessarily unique. The means of a continuously-varying set
of diagrams do not themselves (necessarily) vary
continuously, which presents obvious problems when trying to
extend the Fréchet mean definition to the realm of
time-varying persistence diagrams, better known as
vineyards. We fix this problem by altering the original
definition of Fréchet mean so that it now becomes a
probability measure on the set of persistence diagrams; in a
nutshell, the mean of a set of diagrams will be a weighted
sum of atomic measures, where each atom is itself a
persistence diagram determined using a perturbation of the
input diagrams. This definition gives for each N a map
(D<inf>p</inf>)<sup>N</sup>→ℙ(D<inf>p</inf>). We show
that this map is Hölder continuous on finite diagrams and
thus can be used to build a useful statistic on
vineyards.},
Doi = {10.1214/15-EJS1030},
Key = {fds303522}
}
@article{fds321987,
Author = {Rouse, D and Watkins, A and Porter, D and Harer, J and Bendich, P and Strawn, N and Munch, E and Desena, J and Clarke, J and Gilbert, J and Chin,
S and Newman, A},
Title = {Feature-aided multiple hypothesis tracking using topological
and statistical behavior classifiers},
Journal = {Proceedings of SPIE - The International Society for Optical
Engineering},
Volume = {9474},
Publisher = {SPIE},
Year = {2015},
Month = {January},
ISBN = {9781628415902},
url = {http://dx.doi.org/10.1117/12.2179555},
Abstract = {This paper introduces a method to integrate target behavior
into the multiple hypothesis tracker (MHT) likelihood ratio.
In particular, a periodic track appraisal based on behavior
is introduced that uses elementary topological data analysis
coupled with basic machine learning techniques. The track
appraisal adjusts the traditional kinematic data association
likelihood (i.e., track score) using an established
formulation for classification-aided data association. The
proposed method is tested and demonstrated on synthetic
vehicular data representing an urban traffic scene generated
by the Simulation of Urban Mobility package. The vehicles in
the scene exhibit different driving behaviors. The proposed
method distinguishes those behaviors and shows improved data
association decisions relative to a conventional, kinematic
MHT.},
Doi = {10.1117/12.2179555},
Key = {fds321987}
}
@article{fds315427,
Author = {Bendich, P and Gasparovic, E and Harer, J and Izmailov, R and Ness,
L},
Title = {Multi-scale local shape analysis and feature selection in
machine learning applications},
Journal = {Proceedings of the International Joint Conference on Neural
Networks},
Volume = {2015-September},
Pages = {1-8},
Publisher = {IEEE},
Year = {2015},
Month = {September},
ISBN = {9781479919604},
url = {http://hdl.handle.net/10161/12014 Duke open
access},
Abstract = {We introduce a method called multi-scale local shape
analysis for extracting features that describe the local
structure of points within a dataset. The method uses both
geometric and topological features at multiple levels of
granularity to capture diverse types of local information
for subsequent machine learning algorithms operating on the
dataset. Using synthetic and real dataset examples, we
demonstrate significant performance improvement of
classification algorithms constructed for these datasets
with correspondingly augmented features.},
Doi = {10.1109/IJCNN.2015.7280428},
Key = {fds315427}
}
@article{fds315425,
Author = {Tralie, CJ and Bendich, P},
Title = {Cover Song Identification with Timbral Shape
Sequences},
Journal = {16th International Society for Music Information Retrieval
(ISMIR)},
Pages = {38-44},
Year = {2015},
Month = {October},
url = {http://arxiv.org/abs/1507.05143v1},
Abstract = {We introduce a novel low level feature for identifying cover
songs which quantifies the relative changes in the smoothed
frequency spectrum of a song. Our key insight is that a
sliding window representation of a chunk of audio can be
viewed as a time-ordered point cloud in high dimensions. For
corresponding chunks of audio between different versions of
the same song, these point clouds are approximately rotated,
translated, and scaled copies of each other. If we treat
MFCC embeddings as point clouds and cast the problem as a
relative shape sequence, we are able to correctly identify
42/80 cover songs in the "Covers 80" dataset. By contrast,
all other work to date on cover songs exclusively relies on
matching note sequences from Chroma derived
features.},
Key = {fds315425}
}
@article{fds315426,
Author = {Bendich, P and Marron, JS and Miller, E and Pieloch, A and Skwerer,
S},
Title = {Persistent homology analysis of brain artery
trees},
Journal = {Annals of Applied Statistics},
Volume = {10},
Number = {1},
Pages = {198-218},
Year = {2016},
ISSN = {1932-6157},
url = {http://hdl.handle.net/10161/11157 Duke open
access},
Abstract = {New representations of tree-structured data objects, using
ideas from topological data analysis, enable improved
statistical analyses of a population of brain artery trees.
A number of representations of each data tree arise from
persistence diagrams that quantify branching and looping of
vessels at multiple scales. Novel approaches to the
statistical analysis, through various summaries of the
persistence diagrams, lead to heightened correlations with
covariates such as age and sex, relative to earlier analyses
of this data set. The correlation with age continues to be
significant even after controlling for correlations from
earlier significant summaries},
Doi = {10.1214/15-AOAS886},
Key = {fds315426}
}
@article{fds330930,
Author = {Bendich, P and Gasparovic, E and Harer, J and Tralie,
C},
Title = {Geometric Models for Musical Audio Data},
Journal = {Proceedings of the 32st International Symposium on
Computational Geometry (SOCG)},
Year = {2016},
Month = {June},
Key = {fds330930}
}
@article{fds321986,
Author = {Bendich, P and Gasparovic, E and Harer, J and Tralie,
C},
Title = {Geometric models for musical audio data},
Journal = {Leibniz International Proceedings in Informatics,
LIPIcs},
Volume = {51},
Pages = {65.1-65.5},
Year = {2016},
Month = {June},
ISBN = {9783959770095},
url = {http://dx.doi.org/10.4230/LIPIcs.SoCG.2016.65},
Abstract = {We study the geometry of sliding window embeddings of audio
features that summarize perceptual information about audio,
including its pitch and timbre. These embeddings can be
viewed as point clouds in high dimensions, and we add
structure to the point clouds using a cover tree with
adaptive thresholds based on multi-scale local principal
component analysis to automatically assign points to
clusters. We connect neighboring clusters in a scaffolding
graph, and we use knowledge of stratified space structure to
refine our estimates of dimension in each cluster,
demonstrating in our music applications that choruses and
verses have higher dimensional structure, while transitions
between them are lower dimensional. We showcase our
technique with an interactive web-based application powered
by Javascript and WebGL which plays music synchronized with
a principal component analysis embedding of the point cloud
down to 3D. We also render the clusters and the scaffolding
on top of this projection to visualize the transitions
between different sections of the music.},
Doi = {10.4230/LIPIcs.SoCG.2016.65},
Key = {fds321986}
}
@article{fds324396,
Author = {Bendich, P and Chin, SP and Clark, J and DeSena, J and Harer, J and Munch,
E and Newman, A and Porter, D and Rouse, D and Strawn, N and Watkins,
A},
Title = {Topological and statistical behavior classifiers for
tracking applications},
Journal = {IEEE Transactions on Aerospace and Electronic
Systems},
Volume = {52},
Number = {6},
Pages = {2644-2661},
Publisher = {Institute of Electrical and Electronics Engineers
(IEEE)},
Year = {2016},
Month = {December},
url = {http://dx.doi.org/10.1109/TAES.2016.160405},
Abstract = {This paper introduces a method to integrate target behavior
into the multiple hypothesis tracker (MHT) likelihood ratio.
In particular, a periodic track appraisal based on behavior
is introduced. The track appraisal uses elementary
topological data analysis coupled with basic
machine-learning techniques, and it adjusts the traditional
kinematic data association likelihood (i.e., track score)
using an established formulation for feature-aided data
association. The proposed method is tested and demonstrated
on synthetic vehicular data representing an urban traffic
scene generated by the Simulation of Urban Mobility package.
The vehicles in the scene exhibit different driving
behaviors. The proposed method distinguishes those behaviors
and shows improved data association decisions relative to a
conventional, kinematic MHT.},
Doi = {10.1109/TAES.2016.160405},
Key = {fds324396}
}
@article{fds346387,
Author = {Bendich, P and Gasparovic, E and Harer, J and Tralie,
CJ},
Title = {Scaffoldings and Spines: Organizing High-Dimensional Data
Using Cover Trees, Local Principal Component Analysis, and
Persistent Homology},
Volume = {13},
Pages = {93-114},
Year = {2018},
Month = {January},
url = {http://dx.doi.org/10.1007/978-3-319-89593-2_6},
Abstract = {We propose a flexible and multi-scale method for organizing,
visualizing, and understanding point cloud datasets sampled
from or near stratified spaces. The first part of the
algorithm produces a cover tree for a dataset using an
adaptive threshold that is based on multi-scale local
principal component analysis. The resulting cover tree nodes
reflect the local geometry of the space and are organized
via a scaffolding graph. In the second part of the
algorithm, the goals are to uncover the strata that make up
the underlying stratified space using a local dimension
estimation procedure and topological data analysis, as well
as to ultimately visualize the results in a simplified spine
graph. We demonstrate our technique on several synthetic
examples and then use it to visualize song structure in
musical audio data.},
Doi = {10.1007/978-3-319-89593-2_6},
Key = {fds346387}
}
@article{fds330929,
Author = {Tralie, CJ and Smith, A and Borggren, N and Hineman, J and Bendich, P and Zulch, P and Harer, J},
Title = {Geometric Cross-Modal Comparison of Heterogeneous Sensor
Data},
Journal = {Proceedings of the 39th IEEE Aerospace Conference},
Volume = {2018-March},
Pages = {1-10},
Publisher = {IEEE},
Year = {2018},
Month = {March},
ISBN = {9781538620144},
url = {http://dx.doi.org/10.1109/AERO.2018.8396789},
Abstract = {In this work, we address the problem of cross-modal
comparison of aerial data streams. A variety of simulated
automobile trajectories are sensed using two different
modalities: full-motion video, and radio-frequency (RF)
signals received by detectors at various locations. The
information represented by the two modalities is compared
using self-similarity matrices (SSMs) corresponding to
time-ordered point clouds in feature spaces of each of these
data sources; we note that these feature spaces can be of
entirely different scale and dimensionality. Several metrics
for comparing SSMs are explored, including a cutting-edge
time-warping technique that can simultaneously handle local
time warping and partial matches, while also controlling for
the change in geometry between feature spaces of the two
modalities. We note that this technique is quite general,
and does not depend on the choice of modalities. In this
particular setting, we demonstrate that the cross-modal
distance between SSMs corresponding to the same trajectory
type is smaller than the cross-modal distance between SSMs
corresponding to distinct trajectory types, and we formalize
this observation via precision-recall metrics in
experiments. Finally, we comment on promising implications
of these ideas for future integration into
multiple-hypothesis tracking systems.},
Doi = {10.1109/AERO.2018.8396789},
Key = {fds330929}
}
@article{fds335533,
Author = {Garagić, D and Peskoe, J and Liu, F and Claffey, MS and Bendich, P and Hineman, J and Borggren, N and Harer, J and Zulch, P and Rhodes,
BJ},
Title = {Upstream fusion of multiple sensing modalities using machine
learning and topological analysis: An initial
exploration},
Journal = {IEEE Aerospace Conference Proceedings},
Volume = {2018-March},
Pages = {1-8},
Publisher = {IEEE},
Year = {2018},
Month = {June},
ISBN = {9781538620144},
url = {http://dx.doi.org/10.1109/AERO.2018.8396737},
Abstract = {This paper presents a processing pipeline for fusing 'raw'
and / or feature-level multi-sensor data - upstream fusion -
and initial results from this pipeline using imagery, radar,
and radio frequency (RF) signals data to determine which
tracked object, among several, hosts an emitter of interest.
Correctly making this determination requires fusing data
across these modalities. Our approach performs better than
standard fusion approaches that make detection /
characterization decisions for each modality individually
and then try to fuse those decisions - downstream (or
post-decision) fusion. Our approach (1) fully exploits the
inter-modality dependencies and phenomenologies inherent in
different sensing modes, (2) automatically discovers
compressive hierarchical representations that integrate
structural and statistical characteristics to enhance target
/ event discriminability, and (3) completely obviates the
need to specify features, manifolds, or model scope a
priori. This approach comprises a unique synthesis of Deep
Learning (DL), topological analysis over probability measure
(TAPM), and hierarchical Bayesian non-parametric (HBNP)
recognition models. Deep Generative Networks (DGNs - a deep
generative statistical form of DL) create probability
measures that provide a basis for calculating homologies
(topological summaries over the probability measures). The
statistics of the resulting persistence diagrams are inputs
to HBNP methods that learn to discriminate between target
types and distinguish emitting targets from non-emitting
targets, for example. HBNP learning obviates batch-mode
off-line learning. This approach overcomes the inadequacy of
pre-defined features as a means for creating efficient,
discriminating, low-dimensional representations from
high-dimensional multi-modality sensor data collected under
difficult, dynamic sensing conditions. The invariant
properties in the resulting compact representations afford
multiple compressive sensing benefits, including concise
information sharing and enhanced performance. Machine
learning makes adaptivity a central feature of our approach.
Adaptivity is critical because it enables flexible
processing that automatically accommodates a broad range of
challenges that non-adaptive, standard fusion approaches
would typically require manual intervention to begin to
address. These include (a) interest in unknown or
unanticipated targets, (b) desire to be rapidly able to fuse
between different combinations of sensor modalities, and (c)
potential need to transfer information between platforms
that host different sensors. This paper presents results
that demonstrate our approach enables accurate, real-time
target detection, tracking, and recognition of known and
unknown moving or stationary targets or events and their
activities evolving over space and time.},
Doi = {10.1109/AERO.2018.8396737},
Key = {fds335533}
}
@article{fds346572,
Author = {Bendich, P},
Title = {Topology, geometry, and machine-learning for tracking and
sensor fusion},
Journal = {Proceedings of SPIE - The International Society for Optical
Engineering},
Volume = {11017},
Pages = {lxxxiii-cii},
Year = {2019},
Month = {January},
ISBN = {9781510627017},
Key = {fds346572}
}
@article{fds345757,
Author = {Tralie, CJ and Bendich, P and Harer, J},
Title = {Multi-Scale Geometric Summaries for Similarity-Based Sensor
Fusion},
Journal = {IEEE Aerospace Conference Proceedings},
Volume = {2019-March},
Year = {2019},
Month = {March},
ISBN = {9781538668542},
url = {http://dx.doi.org/10.1109/AERO.2019.8741399},
Abstract = {In this work, we address fusion of heterogeneous sensor data
using wavelet-based summaries of fused self-similarity
information from each sensor. The technique we develop is
quite general, does not require domain specific knowledge or
physical models, and requires no training. Nonetheless, it
can perform surprisingly well at the general task of
differentiating classes of time-ordered behavior sequences
which are sensed by more than one modality. As a
demonstration of our capabilities in the audio to video
context, we focus on the differentiation of speech
sequences. Data from two or more modalities first are
represented using self-similarity matrices(SSMs)
corresponding to time-ordered point clouds in feature spaces
of each of these data sources; we note that these feature
spaces can be of entirely different scale and
dimensionality. A fused similarity template is then derived
from the modality-specific SSMs using a technique called
similarity network fusion (SNF). We investigate pipelines
using SNF as both an upstream (feature-level) and a
downstream (ranking-level) fusion technique. Multiscale
geometric features of this template are then extracted using
a recently-developed technique called the scattering
transform, and these features are then used to differentiate
speech sequences. This method outperforms unsupervised
techniques which operate directly on the raw data, and it
also outperforms stovepiped methods which operate on SSMs
separately derived from the distinct modalities. The
benefits of this method become even more apparent as the
simulated peak signal to noise ratio decreases.},
Doi = {10.1109/AERO.2019.8741399},
Key = {fds345757}
}
@article{fds347287,
Author = {Bendich, P and Bubenik, P and Wagner, A},
Title = {Stabilizing the unstable output of persistent homology
computations},
Journal = {Journal of Applied and Computational Topology},
Pages = {1-30},
Publisher = {Springer},
Year = {2019},
Month = {November},
Abstract = {We propose a general technique for extracting a larger set
of stable information from persistent homology computations
than is currently done. The persistent homology algorithm is
usually viewed as a procedure which starts with a filtered
complex and ends with a persistence diagram. This procedure
is stable (at least to certain types of perturbations of the
input). This justifies the use of the diagram as a signature
of the input, and the use of features derived from it in
statistics and machine learning. However, these computations
also produce other information of great interest to
practitioners that is unfortunately unstable. For example,
each point in the diagram corresponds to a simplex whose
addition in the filtration results in the birth of the
corresponding persistent homology class, but this
correspondence is unstable. In addition, the persistence
diagram is not stable with respect to other procedures that
are employed in practice, such as thresholding a point cloud
by density. We recast these problems as real-valued
functions which are discontinuous but measurable, and then
observe that convolving such a function with a suitable
function produces a Lipschitz function. The resulting stable
function can be estimated by perturbing the input and
averaging the output. We illustrate this approach with a
number of examples, including a stable localization of a
persistent homology generator from brain imaging
data.},
Key = {fds347287}
}
@article{fds350796,
Author = {Blasch, E and Grewe, LL and Waltz, EL and Bendich, P and Pavlovic, V and Kadar, I and Chong, CY},
Title = {Machine learning in/with information fusion for
infrastructure understanding, panel summary},
Journal = {Proceedings of SPIE - The International Society for Optical
Engineering},
Volume = {11423},
Year = {2020},
Month = {January},
ISBN = {9781510636231},
url = {http://dx.doi.org/10.1117/12.2559416},
Abstract = {During the 2019 SPIE DSS conference, panelists were invited
to highlight the trends and use of artificial intelligence
and machine learning (AI/ML) for information fusion. The
common themes between the panelists include leveraging AI/ML
coordinated with Information Fusion for: (1) knowledge
reasoning, (2) model building, (3) object recognition and
tracking, (4) multimodal learning, and (5) information
processing. The opportunity for machine learning exists
within all the fusion levels of the Data Fusion Information
Group model.},
Doi = {10.1117/12.2559416},
Key = {fds350796}
}
@article{fds352783,
Author = {Yao, L and Bendich, P},
Title = {Graph Spectral Embedding for Parsimonious Transmission of
Multivariate Time Series},
Journal = {IEEE Aerospace Conference Proceedings},
Year = {2020},
Month = {March},
ISBN = {9781728127347},
url = {http://dx.doi.org/10.1109/AERO47225.2020.9172767},
Abstract = {We propose a graph spectral representation of time series
data that 1) is parsimoniously encoded to user-demanded
resolution; 2) is unsupervised and performant in
data-constrained scenarios; 3) captures event and
event-transition structure within the time series; and 4)
has near-linear computational complexity in both signal
length and ambient dimension. This representation, which we
call Laplacian Events Signal Segmentation (LESS), can be
computed on time series of arbitrary dimension and
originating from sensors of arbitrary type. Hence, time
series originating from sensors of heterogeneous type can be
compressed to levels demanded by constrained-communication
environments, before being fused at a common center.
Temporal dynamics of the data is summarized without explicit
partitioning or probabilistic modeling. As a
proof-of-principle, we apply this technique on high
dimensional wavelet coefficients computed from the Free
Spoken Digit Dataset to generate a memory efficient
representation that is interpretable. Due to its
unsupervised and non-parametric nature, LESS representations
remain performant in the digit classification task despite
the absence of labels and limited data.},
Doi = {10.1109/AERO47225.2020.9172767},
Key = {fds352783}
}
@article{fds352782,
Author = {Solomon, E and Bendich, P},
Title = {Geometric fusion via joint delay embeddings},
Journal = {Proceedings of 2020 23rd International Conference on
Information Fusion, FUSION 2020},
Year = {2020},
Month = {July},
url = {http://dx.doi.org/10.23919/FUSION45008.2020.9190173},
Abstract = {We introduce geometric and topological methods to develop a
new framework for fusing multi-sensor time series. This
framework consists of two steps: (1) a joint delay
embedding, which reconstructs a high-dimensional state space
in which our sensors correspond to observation functions,
and (2) a simple orthogonalization scheme, which accounts
for tangencies between such observation functions, and
produces a more diversified geometry on the embedding space.
We conclude with some synthetic and real-world experiments
demonstrating that our framework outperforms traditional
metric fusion methods.},
Doi = {10.23919/FUSION45008.2020.9190173},
Key = {fds352782}
}
@article{fds359984,
Author = {Solomon, E and Wagner, A and Bendich, P},
Title = {A Fast and Robust Method for Global Topological Functional
Optimization},
Journal = {24TH INTERNATIONAL CONFERENCE ON ARTIFICIAL INTELLIGENCE AND
STATISTICS (AISTATS)},
Volume = {130},
Pages = {109-+},
Year = {2021},
Abstract = {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.},
Key = {fds359984}
}
@article{fds376397,
Author = {Solomon, E and Wagner, A and Bendich, P},
Title = {A Fast and Robust Method for Global Topological Functional
Optimization},
Journal = {Proceedings of Machine Learning Research},
Volume = {130},
Pages = {109-117},
Year = {2021},
Month = {January},
Abstract = {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.},
Key = {fds376397}
}
@article{fds365495,
Author = {Smith, AD and Bendich, P and Harer, J},
Title = {PERSISTENT OBSTRUCTION THEORY FOR A MODEL CATEGORY OF
MEASURES WITH APPLICATIONS TO DATA MERGING},
Journal = {Transactions of the American Mathematical Society Series
B},
Volume = {8},
Number = {1},
Pages = {1-38},
Publisher = {American Mathematical Society (AMS)},
Year = {2021},
Month = {February},
url = {http://dx.doi.org/10.1090/btran/56},
Abstract = {Collections of measures on compact metric spaces form a
model category (“data complexes”), whose morphisms are
marginalization integrals. The fibrant objects in this
category represent collections of measures in which there is
a measure on a product space that marginalizes to any
measures on pairs of its factors. The homotopy and homology
for this category allow measurement of obstructions to
finding measures on larger and larger product spaces. The
obstruction theory is compatible with a fibrant filtration
built from the Wasserstein distance on measures. Despite the
abstract tools, this is motivated by a widespread problem in
data science. Data complexes provide a mathematical
foundation for semi-automated data-alignment tools that are
common in commercial database software. Practically
speaking, the theory shows that database JOIN operations are
subject to genuine topological obstructions. Those
obstructions can be detected by an obstruction cocycle and
can be resolved by moving through a filtration. Thus, any
collection of databases has a persistence level, which
measures the difficulty of JOINing those databases. Because
of its general formulation, this persistent obstruction
theory also encompasses multi-modal data fusion problems,
some forms of Bayesian inference, and probability
couplings.},
Doi = {10.1090/btran/56},
Key = {fds365495}
}
@article{fds364276,
Author = {Solomon, E and Wagner, A and Bendich, P},
Title = {From Geometry to Topology: Inverse Theorems for Distributed
Persistence},
Journal = {Leibniz International Proceedings in Informatics,
LIPIcs},
Volume = {224},
Year = {2022},
Month = {June},
ISBN = {9783959772273},
url = {http://dx.doi.org/10.4230/LIPIcs.SoCG.2022.61},
Abstract = {What is the “right” topological invariant of a large
point cloud X? Prior research has focused on estimating the
full persistence diagram of X, a quantity that is very
expensive to compute, unstable to outliers, and far from
injective. We therefore propose that, in many cases, the
collection of persistence diagrams of many small subsets of
X is a better invariant. This invariant, which we call
“distributed persistence,” is perfectly parallelizable,
more stable to outliers, and has a rich inverse theory. The
map from the space of metric spaces (with the quasi-isometry
distance) to the space of distributed persistence invariants
(with the Hausdorff-Bottleneck distance) is globally
bi-Lipschitz. This is a much stronger property than simply
being injective, as it implies that the inverse image of a
small neighborhood is a small neighborhood, and is to our
knowledge the only result of its kind in the TDA literature.
Moreover, the inverse Lipschitz constant depends on the size
of the subsets taken, so that as the size of these subsets
goes from small to large, the invariant interpolates between
a purely geometric one and a topological one. Lastly, we
note that our inverse results do not actually require
considering all subsets of a fixed size (an enormous
collection), but a relatively small collection satisfying
simple covering properties. These theoretical results are
complemented by synthetic experiments demonstrating the use
of distributed persistence in practice.},
Doi = {10.4230/LIPIcs.SoCG.2022.61},
Key = {fds364276}
}
@article{fds367804,
Author = {Voisin, S and Hineman, J and Polly, JB and Koplik, G and Ball, K and Bendich, P and D‘Addezio, J and Jacobs, GA and Özgökmen,
T},
Title = {Topological Feature Tracking for Submesoscale
Eddies},
Journal = {Geophysical Research Letters},
Volume = {49},
Number = {20},
Year = {2022},
Month = {October},
url = {http://dx.doi.org/10.1029/2022GL099416},
Abstract = {Current state-of-the art procedures for studying modeled
submesoscale oceanographic features have made a strong
assumption of independence between features identified at
different times. Therefore, all submesoscale eddies
identified in a time series were studied in aggregate.
Statistics from these methods are illuminating but
oversample identified features and cannot determine the
lifetime evolution of the transient submesoscale processes.
To this end, the authors apply the Topological Feature
Tracking (TFT) algorithm to the problem of identifying and
tracking submesoscale eddies over time. TFT identifies
critical points on a set of time-ordered scalar fields and
associates those points between consecutive timesteps. The
procedure yields tracklets which represent spatio-temporal
displacement of eddies. In this way we study the
time-dependent behavior of submesoscale eddies, which are
generated by a 1-km resolution submesoscale-permitting
model. We summarize the submesoscale eddy data set produced
by TFT, which yields unique, time-varying
statistics.},
Doi = {10.1029/2022GL099416},
Key = {fds367804}
}
@article{fds371114,
Author = {Koplik, G and Borggren, N and Voisin, S and Angeloro, G and Hineman, J and Johnson, T and Bendich, P},
Title = {Topological Simplification of Signals for Inference and
Approximate Reconstruction},
Journal = {IEEE Aerospace Conference Proceedings},
Volume = {2023-March},
Year = {2023},
Month = {January},
ISBN = {9781665490320},
url = {http://dx.doi.org/10.1109/AERO55745.2023.10115654},
Abstract = {As Internet of Things (loT) devices become both cheaper and
more powerful, researchers are increasingly finding
solutions to their scientific curiosities both financially
and com-putationally feasible. When operating with
restricted power or communications budgets, however, devices
can only send highly-compressed data. Such circumstances are
common for devices placed away from electric grids that can
only communicate via satellite, a situation particularly
plausible for environmental sensor networks. These
restrictions can be further complicated by potential
variability in the communications budget, for ex-ample a
solar-powered device needing to expend less energy when
transmitting data on a cloudy day. We propose a novel,
topology-based, lossy compression method well-equipped for
these restrictive yet variable circumstances. This
technique, Topological Signal Compression, allows sending
compressed sig-nals that utilize the entirety of a variable
communications budget. To demonstrate our algorithm's
capabilities, we per-form entropy calculations as well as a
classification exercise on increasingly topologically
simplified signals from the Free-Spoken Digit Dataset and
explore the stability of the resulting performance against
common baselines.},
Doi = {10.1109/AERO55745.2023.10115654},
Key = {fds371114}
}
@article{fds376284,
Author = {Solomon, E and Wagner, A and Bendich, P},
Title = {FROM GEOMETRY TO TOPOLOGY: INVERSE THEOREMS FOR DISTRIBUTED
PERSISTENCE},
Journal = {Journal of Computational Geometry},
Volume = {14},
Number = {2 Special Issue},
Pages = {172-196},
Year = {2023},
Month = {January},
url = {http://dx.doi.org/10.20382/jocg.v14i2a8},
Abstract = {What is the “right” topological invariant of a large
point cloud X? Prior research has focused on estimating the
full persistence diagram of X, a quantity that is very
expensive to compute, unstable to outliers, and far from
injective. We therefore propose that, in many cases, the
collection of persistence diagrams of many small subsets of
X is a better invariant. This invariant, which we call
“distributed persistence,” is perfectly parallelizable,
more stable to outliers, and has a rich inverse theory. The
map from the space of metric spaces (with the quasi-isometry
distance) to the space of distributed persistence invariants
(with the Hausdorff-Bottleneck distance) is globally
bi-Lipschitz. This is a much stronger property than simply
being injective, as it implies that the inverse image of a
small neighborhood is a small neighborhood, and is to our
knowledge the only result of its kind in the TDA literature.
Moreover, the inverse Lipschitz constant depends on the size
of the subsets taken, so that as the size of these subsets
goes from small to large, the invariant interpolates between
a purely geometric one and a topological one. Lastly, we
note that our inverse results do not actually require
considering all subsets of a fixed size (an enormous
collection), but a relatively small collection satisfying
simple covering properties. These theoretical results are
complemented by synthetic experiments demonstrating the use
of distributed persistence in practice.},
Doi = {10.20382/jocg.v14i2a8},
Key = {fds376284}
}
@article{fds376122,
Author = {Solomon, YE and Bendich, P},
Title = {Convolutional persistence transforms},
Journal = {Journal of Applied and Computational Topology},
Year = {2024},
Month = {January},
url = {http://dx.doi.org/10.1007/s41468-024-00164-x},
Abstract = {In this paper, we consider topological featurizations of
data defined over simplicial complexes, like images and
labeled graphs, obtained by convolving this data with
various filters before computing persistence. Viewing a
convolution filter as a local motif, the persistence diagram
of the resulting convolution describes the way the motif is
distributed across the simplicial complex. This pipeline,
which we call convolutional persistence, extends the
capacity of topology to observe patterns in such data.
Moreover, we prove that (generically speaking) for any two
labeled complexes one can find some filter for which they
produce different persistence diagrams, so that the
collection of all possible convolutional persistence
diagrams is an injective invariant. This is proven by
showing convolutional persistence to be a special case of
another topological invariant, the Persistent Homology
Transform. Other advantages of convolutional persistence are
improved stability, greater flexibility for data-dependent
vectorizations, and reduced computational complexity for
certain data types. Additionally, we have a suite of
experiments showing that convolutions greatly improve the
predictive power of persistence on a host of classification
tasks, even if one uses random filters and vectorizes the
resulting diagrams by recording only their total
persistences.},
Doi = {10.1007/s41468-024-00164-x},
Key = {fds376122}
}
%% Papers Submitted
@article{fds292867,
Author = {Paul Bendich and Peter Bubenik},
Title = {Stabilizing the output of persistent homology
computations},
Journal = {Proc. 2016 Symposium on Computational Geometry},
Year = {2015},
url = {http://arxiv.org/abs/1512.01700},
Key = {fds292867}
}
@article{fds311346,
Author = {Paul Bendich and Ellen Gasparovic and John Harer and Christopher
J. Tralie},
Title = {Scaffoldings and Spines: Organizing High-Dimensional Data
Using Cover Trees, Local Principal Component Analysis, and
Persistent Homology},
Year = {2016},
url = {http://arxiv.org/abs/1602.06245},
Key = {fds311346}
}
|
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Mathematics Department