86 research outputs found
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
Sliced Wasserstein Kernel for Persistence Diagrams
Persistence diagrams (PDs) play a key role in topological data analysis
(TDA), in which they are routinely used to describe topological properties of
complicated shapes. PDs enjoy strong stability properties and have proven their
utility in various learning contexts. They do not, however, live in a space
naturally endowed with a Hilbert structure and are usually compared with
specific distances, such as the bottleneck distance. To incorporate PDs in a
learning pipeline, several kernels have been proposed for PDs with a strong
emphasis on the stability of the RKHS distance w.r.t. perturbations of the PDs.
In this article, we use the Sliced Wasserstein approximation SW of the
Wasserstein distance to define a new kernel for PDs, which is not only provably
stable but also provably discriminative (depending on the number of points in
the PDs) w.r.t. the Wasserstein distance between PDs. We also demonstrate
its practicality, by developing an approximation technique to reduce kernel
computation time, and show that our proposal compares favorably to existing
kernels for PDs on several benchmarks.Comment: Minor modification
Persistent Homology of Attractors For Action Recognition
In this paper, we propose a novel framework for dynamical analysis of human
actions from 3D motion capture data using topological data analysis. We model
human actions using the topological features of the attractor of the dynamical
system. We reconstruct the phase-space of time series corresponding to actions
using time-delay embedding, and compute the persistent homology of the
phase-space reconstruction. In order to better represent the topological
properties of the phase-space, we incorporate the temporal adjacency
information when computing the homology groups. The persistence of these
homology groups encoded using persistence diagrams are used as features for the
actions. Our experiments with action recognition using these features
demonstrate that the proposed approach outperforms other baseline methods.Comment: 5 pages, Under review in International Conference on Image Processin
Topological descriptors for 3D surface analysis
We investigate topological descriptors for 3D surface analysis, i.e. the
classification of surfaces according to their geometric fine structure. On a
dataset of high-resolution 3D surface reconstructions we compute persistence
diagrams for a 2D cubical filtration. In the next step we investigate different
topological descriptors and measure their ability to discriminate structurally
different 3D surface patches. We evaluate their sensitivity to different
parameters and compare the performance of the resulting topological descriptors
to alternative (non-topological) descriptors. We present a comprehensive
evaluation that shows that topological descriptors are (i) robust, (ii) yield
state-of-the-art performance for the task of 3D surface analysis and (iii)
improve classification performance when combined with non-topological
descriptors.Comment: 12 pages, 3 figures, CTIC 201
Topological exploration of artificial neuronal network dynamics
One of the paramount challenges in neuroscience is to understand the dynamics
of individual neurons and how they give rise to network dynamics when
interconnected. Historically, researchers have resorted to graph theory,
statistics, and statistical mechanics to describe the spatiotemporal structure
of such network dynamics. Our novel approach employs tools from algebraic
topology to characterize the global properties of network structure and
dynamics.
We propose a method based on persistent homology to automatically classify
network dynamics using topological features of spaces built from various
spike-train distances. We investigate the efficacy of our method by simulating
activity in three small artificial neural networks with different sets of
parameters, giving rise to dynamics that can be classified into four regimes.
We then compute three measures of spike train similarity and use persistent
homology to extract topological features that are fundamentally different from
those used in traditional methods. Our results show that a machine learning
classifier trained on these features can accurately predict the regime of the
network it was trained on and also generalize to other networks that were not
presented during training. Moreover, we demonstrate that using features
extracted from multiple spike-train distances systematically improves the
performance of our method
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