
Zigzag Zoology: Rips Zigzags for Homology Inference
Steve Y. Oudot and Donald R. Sheehy
Proc. 29th Annual Symposium on Computational Geometry,
June 2013. Full version available as INRIA research report RR8141.
Abstract:
For points sampled near a compact set X,
the persistence barcode of the Rips filtration built from the sample
contains information about the homology of X as long as X
satisfies some geometric assumptions. The Rips filtration is
prohibitively large, however zigzag persistence can be used to keep
the size linear. We present several species of Ripslike zigzags
and compare them with respect to the signaltonoise ratio, a
measure of how well the underlying homology is represented in the
persistence barcode relative to the noise in the barcode at the
relevant scales. Some of these Ripslike zigzags have been
available as part of the Dionysus library for several years while
others are new. Interestingly, we show that some species of Rips
zigzags will exhibit less noise than the (nonzigzag) Rips
filtration itself. Thus, the Rips zigzag can offer improvements in
both size complexity and signaltonoise ratio.
Along the way, we develop new techniques for manipulating and
comparing persistence barcodes from zigzag modules. We give methods
for reversing arrows and removing spaces from a zigzag. We also
discuss factoring zigzags and a kind of interleaving of two zigzags
that allows their barcodes to be compared. These techniques were
developed to provide our theoretical analysis of the signaltonoise
ratio of Ripslike zigzags, but they are of independent interest as
they apply to zigzag modules generally.
Bibtex:
@techreport{oszz12rr
, author = "Steve Y. Oudot and Donald R. Sheehy"
, title = "{Z}igzag {Z}oology: {R}ips {Z}igzags for {H}omology {I}nference"
, institution = "INRIA"
, type = "Research Report"
, number = "8141"
, month = "November"
, year = "2012"
, url = "http://hal.inria.fr/hal00755280/en"
}
@inproceedings{oszz13conf
, author = "Steve Y. Oudot and Donald R. Sheehy"
, title = "{Z}igzag {Z}oology: {R}ips {Z}igzags for {H}omology {I}nference"
, booktitle = "Proc. 29th Annual Symposium on Computational Geometry"
, pages = "387396"
, month = "June"
, year = "2013"
}

