143 research outputs found
Induced Matchings and the Algebraic Stability of Persistence Barcodes
We define a simple, explicit map sending a morphism of
pointwise finite dimensional persistence modules to a matching between the
barcodes of and . Our main result is that, in a precise sense, the
quality of this matching is tightly controlled by the lengths of the longest
intervals in the barcodes of and . As an
immediate corollary, we obtain a new proof of the algebraic stability of
persistence, a fundamental result in the theory of persistent homology. In
contrast to previous proofs, ours shows explicitly how a -interleaving
morphism between two persistence modules induces a -matching between
the barcodes of the two modules. Our main result also specializes to a
structure theorem for submodules and quotients of persistence modules, and
yields a novel "single-morphism" characterization of the interleaving relation
on persistence modules.Comment: Expanded journal version, to appear in Journal of Computational
Geometry. Includes a proof that no definition of induced matching can be
fully functorial (Proposition 5.10), and an extension of our single-morphism
characterization of the interleaving relation to multidimensional persistence
modules (Remark 6.7). Exposition is improved throughout. 11 Figures adde
Multidimensional Interleavings and Applications to Topological Inference
This work concerns the theoretical foundations of persistence-based
topological data analysis. We develop theory of topological inference in the
multidimensional persistence setting, and directly at the (topological) level
of filtrations rather than only at the (algebraic) level of persistent homology
modules.
Our main mathematical objects of study are interleavings. These are tools for
quantifying the similarity between two multidimensional filtrations or
persistence modules. They were introduced for 1-D filtrations and persistence
modules by Chazal, Cohen-Steiner, Glisse, Guibas, and Oudot. We introduce
generalizations of the definitions of interleavings given by Chazal et al. and
use these to define pseudometrics, called interleaving distances, on
multidimensional filtrations and multidimensional persistence modules.
We present an in-depth study of interleavings and interleaving distances. We
then use them to formulate and prove several multidimensional analogues of a
topological inference theorem of Chazal, Guibas, Oudot, and Skraba. These
results hold directly at the level of filtrations; they yield as corollaries
corresponding results at the module level.Comment: Late stage draft of Ph.D. thesis. 176 pages. Expands upon content in
arXiv:1106.530
The structure and stability of persistence modules
We give a self-contained treatment of the theory of persistence modules
indexed over the real line. We give new proofs of the standard results.
Persistence diagrams are constructed using measure theory. Linear algebra
lemmas are simplified using a new notation for calculations on quiver
representations. We show that the stringent finiteness conditions required by
traditional methods are not necessary to prove the existence and stability of
the persistence diagram. We introduce weaker hypotheses for taming persistence
modules, which are met in practice and are strong enough for the theory still
to work. The constructions and proofs enabled by our framework are, we claim,
cleaner and simpler.Comment: New version. We discuss in greater depth the interpolation lemma for
persistence module
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