911 research outputs found
Statistical topological data analysis using persistence landscapes
We define a new topological summary for data that we call the persistence
landscape. Since this summary lies in a vector space, it is easy to combine
with tools from statistics and machine learning, in contrast to the standard
topological summaries. Viewed as a random variable with values in a Banach
space, this summary obeys a strong law of large numbers and a central limit
theorem. We show how a number of standard statistical tests can be used for
statistical inference using this summary. We also prove that this summary is
stable and that it can be used to provide lower bounds for the bottleneck and
Wasserstein distances.Comment: 26 pages, final version, to appear in Journal of Machine Learning
Research, includes two additional examples not in the journal version: random
geometric complexes and Erdos-Renyi random clique complexe
A Stable Multi-Scale Kernel for Topological Machine Learning
Topological data analysis offers a rich source of valuable information to
study vision problems. Yet, so far we lack a theoretically sound connection to
popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In
this work, we establish such a connection by designing a multi-scale kernel for
persistence diagrams, a stable summary representation of topological features
in data. We show that this kernel is positive definite and prove its stability
with respect to the 1-Wasserstein distance. Experiments on two benchmark
datasets for 3D shape classification/retrieval and texture recognition show
considerable performance gains of the proposed method compared to an
alternative approach that is based on the recently introduced persistence
landscapes
Linearization of Hyperbolic Finite-Time Processes
We adapt the notion of processes to introduce an abstract framework for
dynamics in finite time, i.e.\ on compact time sets. For linear finite-time
processes a notion of hyperbolicity namely exponential monotonicity dichotomy
(EMD) is introduced, thereby generalizing and unifying several existing
approaches. We present a spectral theory for linear processes in a coherent
way, based only on a logarithmic difference quotient, prove robustness of EMD
with respect to a suitable (semi-)metric and provide exact perturbation bounds.
Furthermore, we give a complete description of the local geometry around
hyperbolic trajectories, including a direct and intrinsic proof of finite-time
analogues of the local (un)stable manifold theorem and theorem of linearized
asymptotic stability. As an application, we discuss our results for ordinary
differential equations on a compact time-interval.Comment: 32 page
Persistence barcodes and Laplace eigenfunctions on surfaces
We obtain restrictions on the persistence barcodes of Laplace-Beltrami
eigenfunctions and their linear combinations on compact surfaces with
Riemannian metrics. Some applications to uniform approximation by linear
combinations of Laplace eigenfunctions are also discussed.Comment: Revised version; some references adde
Critical Transitions In a Model of a Genetic Regulatory System
We consider a model for substrate-depletion oscillations in genetic systems,
based on a stochastic differential equation with a slowly evolving external
signal. We show the existence of critical transitions in the system. We apply
two methods to numerically test the synthetic time series generated by the
system for early indicators of critical transitions: a detrended fluctuation
analysis method, and a novel method based on topological data analysis
(persistence diagrams).Comment: 19 pages, 8 figure
Exact Computation of a Manifold Metric, via Lipschitz Embeddings and Shortest Paths on a Graph
Data-sensitive metrics adapt distances locally based the density of data
points with the goal of aligning distances and some notion of similarity. In
this paper, we give the first exact algorithm for computing a data-sensitive
metric called the nearest neighbor metric. In fact, we prove the surprising
result that a previously published -approximation is an exact algorithm.
The nearest neighbor metric can be viewed as a special case of a
density-based distance used in machine learning, or it can be seen as an
example of a manifold metric. Previous computational research on such metrics
despaired of computing exact distances on account of the apparent difficulty of
minimizing over all continuous paths between a pair of points. We leverage the
exact computation of the nearest neighbor metric to compute sparse spanners and
persistent homology. We also explore the behavior of the metric built from
point sets drawn from an underlying distribution and consider the more general
case of inputs that are finite collections of path-connected compact sets.
The main results connect several classical theories such as the conformal
change of Riemannian metrics, the theory of positive definite functions of
Schoenberg, and screw function theory of Schoenberg and Von Neumann. We develop
novel proof techniques based on the combination of screw functions and
Lipschitz extensions that may be of independent interest.Comment: 15 page
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