%% Papers Published
@article{fds358805,
Author = {Oudot, S and Solomon, E},
Title = {Barcode embeddings for metric graphs},
Journal = {Algebraic & Geometric Topology},
Volume = {21},
Number = {3},
Pages = {1209-1266},
Publisher = {Mathematical Sciences Publishers},
Year = {2021},
Month = {August},
url = {http://dx.doi.org/10.2140/agt.2021.21.1209},
Doi = {10.2140/agt.2021.21.1209},
Key = {fds358805}
}
@article{fds361654,
Author = {Wagner, A and Solomon, E and Bendich, P},
Title = {Improving Metric Dimensionality Reduction with Distributed
Topology},
Year = {2021},
Month = {June},
Abstract = {We propose a novel approach to dimensionality reduction
combining techniques of metric geometry and distributed
persistent homology, in the form of a gradient-descent based
method called DIPOLE. DIPOLE is a dimensionality-reduction
post-processing step that corrects an initial embedding by
minimizing a loss functional with both a local, metric term
and a global, topological term. By fixing an initial
embedding method (we use Isomap), DIPOLE can also be viewed
as a full dimensionality-reduction pipeline. This framework
is based on the strong theoretical and computational
properties of distributed persistent homology and comes with
the guarantee of almost sure convergence. We observe that
DIPOLE outperforms popular methods like UMAP, t-SNE, and
Isomap on a number of popular datasets, both visually and in
terms of precise quantitative metrics.},
Key = {fds361654}
}
@article{fds361352,
Author = {Solomon, E and Wagner, A and Bendich, P},
Title = {From Geometry to Topology: Inverse Theorems for Distributed
Persistence},
Year = {2021},
Month = {January},
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 a sufficient
statistic. We therefore propose that the correct invariant
is not the persistence diagram of X, but rather the
collection of persistence diagrams of many small subsets.
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 point clouds
(with the quasi-isometry metric) to the space of distributed
persistence invariants (with the Hausdorff-Bottleneck
distance) is a global quasi-isometry. This is a much
stronger property than simply being injective, as it implies
that the inverse 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 quasi-isometry
bounds depend 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 certain covering properties that arise with high
probability when randomly sampling subsets. These
theoretical results are complemented by two synthetic
experiments demonstrating the use of distributed persistence
in practice.},
Key = {fds361352}
}
@article{fds358753,
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},
Key = {fds358753}
}
@article{fds361423,
Author = {Maria, C and Oudot, S and Solomon, E},
Title = {Intrinsic Topological Transforms via the Distance Kernel
Embedding},
Year = {2019},
Month = {December},
Abstract = {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 and Euler
Characteristic Transform, both of which apply to shapes
embedded in Euclidean space. The contribution of this paper
is to define topological transforms that depend only on the
intrinsic geometry of a shape, and hence are invariant to
the choice of embedding. To that end, given an abstract
metric measure space, we define an integral operator whose
eigenfunctions are used to compute sublevel set persistent
homology. We demonstrate that this operator, which we call
the distance kernel operator, enjoys desirable stability
properties, and that its spectrum and eigenfunctions
concisely encode the large-scale geometry of our metric
measure space. We then define a number of topological
transforms using the eigenfunctions of this operator, and
observe that these transforms inherit many of the stability
and injectivity properties of the distance kernel
operator.},
Key = {fds361423}
}
@article{fds361424,
Author = {Oudot, S and Solomon, E},
Title = {Inverse Problems in Topological Persistence},
Year = {2018},
Month = {October},
Abstract = {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.},
Key = {fds361424}
}
@article{fds361425,
Author = {Solomon, E},
Title = {Stability of Extended Functional Persistence in Dimensions
Zero and One},
Year = {2018},
Month = {April},
Abstract = {The stability result, as stated, is incorrect. In
particular, the 1-dimensional extended persistence diagrams
of a finely-triangulated simplicial complex X equipped with
a continuous real-valued function f, and its one-skeleton
graph G (also equipped with f), need not be close. To take
an example, let X be a finely-triangulated disc of radius r
and let f be the distance-to -the-boundary function, which
increases radially from the boundary circle of X and is
maximized at the center. The extended persistence diagram of
(X,f) contains a point in 1-dimensional persistence of the
form (0,r), whereas the 1-dimensional extended persistence
diagram of its one-skeleton only has points near the
diagonal.},
Key = {fds361425}
}
@article{fds345762,
Author = {Carter, P and Solomon, YE},
Title = {Relaxing the integral test: A challenge for the advanced
calculus student},
Journal = {The College Mathematics Journal},
Volume = {48},
Number = {4},
Pages = {290-291},
Year = {2017},
Month = {September},
url = {http://dx.doi.org/10.4169/college.math.j.48.4.290},
Abstract = {Illustrative, elementary counterexamples are hard to come
by. In this note we propose an elementary, closed-form
counterexample to a generalization of the classic integral
test where the condition of monotonicity is relaxed. The
analysis only uses techniques accessible to a
second-semester calculus student.},
Doi = {10.4169/college.math.j.48.4.290},
Key = {fds345762}
}
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Mathematics Department