28 research outputs found
Dimensionality Reduction for Persistent Homology with Gaussian Kernels
Computing persistent homology using Gaussian kernels is useful in the domains of topological data analysis and machine learning as shown by Phillips, Wang and Zheng [SoCG 2015]. However, contrary to the case of computing persistent homology using the Euclidean distance or even the k-distance, it is not known how to compute the persistent homology of high dimensional data using Gaussian kernels. In this paper, we consider a power distance version of the Gaussian kernel distance (GKPD) given by Phillips, Wang and Zheng, and show that the persistent homology of the Čech filtration of P computed using the GKPD is approximately preserved. For datasets in R D , under a relative error bound of ε ∈ (0, 1], we obtain a dimensionality of (i) O(ε −2 log 2 n) for n-point datasets and (ii) O(Dε −2 log(Dr/ε)) for datasets having diameter r (up to a scaling factor). We use two main ingredients. The first one is a new decomposition of the squared radii of Čech simplices using the kernel power distance, in terms of the pairwise GKPDs between the vertices, which we state and prove. The second one is the Random Fourier Features (RFF) map of Rahimi and Recht [NeurIPS 2007], as used by Chen and Phillips [ALT 2017]
Parameterized complexity of quantum knot invariants
We give a general fixed parameter tractable algorithm to compute quantum invariants of links presented by diagrams, whose complexity is singly exponential in the carving-width (or the tree-width) of the diagram. In particular, we get a O(N^{3/2 cw} poly(n)) time algorithm to compute any Reshetikhin-Turaev invariant-derived from a simple Lie algebra g-of a link presented by a planar diagram with n crossings and carving-width cw, and whose components are coloured with g-modules of dimension at most N. For example, this includes the N th-coloured Jones polynomial and the N th-coloured HOMFLYPT polynomial
Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT
Topological data analysis (TDA) delivers invaluable and complementary
information on the intrinsic properties of data inaccessible to conventional
methods. However, high computational costs remain the primary roadblock
hindering the successful application of TDA in real-world studies, particularly
with machine learning on large complex networks.
Indeed, most modern networks such as citation, blockchain, and online social
networks often have hundreds of thousands of vertices, making the application
of existing TDA methods infeasible. We develop two new, remarkably simple but
effective algorithms to compute the exact persistence diagrams of large graphs
to address this major TDA limitation. First, we prove that -core of a
graph suffices to compute its persistence diagram,
. Second, we introduce a pruning algorithm for graphs to
compute their persistence diagrams by removing the dominated vertices. Our
experiments on large networks show that our novel approach can achieve
computational gains up to 95%.
The developed framework provides the first bridge between the graph theory
and TDA, with applications in machine learning of large complex networks. Our
implementation is available at
https://github.com/cakcora/PersistentHomologyWithCoralPrunitComment: Spotlight paper at NeurIPS 202
On Complexity of 1-Center in Various Metrics
We consider the classic 1-center problem: Given a set P of n points in a
metric space find the point in P that minimizes the maximum distance to the
other points of P. We study the complexity of this problem in d-dimensional
-metrics and in edit and Ulam metrics over strings of length d. Our
results for the 1-center problem may be classified based on d as follows.
Small d: We provide the first linear-time algorithm for 1-center
problem in fixed-dimensional metrics. On the other hand, assuming the
hitting set conjecture (HSC), we show that when , no
subquadratic algorithm can solve 1-center problem in any of the
-metrics, or in edit or Ulam metrics.
Large d. When , we extend our conditional lower bound
to rule out sub quartic algorithms for 1-center problem in edit metric
(assuming Quantified SETH). On the other hand, we give a
-approximation for 1-center in Ulam metric with running time
.
We also strengthen some of the above lower bounds by allowing approximations
or by reducing the dimension d, but only against a weaker class of algorithms
which list all requisite solutions. Moreover, we extend one of our hardness
results to rule out subquartic algorithms for the well-studied 1-median problem
in the edit metric, where given a set of n strings each of length n, the goal
is to find a string in the set that minimizes the sum of the edit distances to
the rest of the strings in the set
Labeled Nearest Neighbor Search and Metric Spanners via Locality Sensitive Orderings
Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive
orderings (LSO) for Euclidean space. A -LSO is a collection
of orderings such that for every there is an
ordering , where all the points between and w.r.t.
are in the -neighborhood of either or . In essence, LSO
allow one to reduce problems to the -dimensional line. Later, Filtser and Le
[STOC 2022] developed LSO's for doubling metrics, general metric spaces, and
minor free graphs.
For Euclidean and doubling spaces, the number of orderings in the LSO is
exponential in the dimension, which made them mainly useful for the low
dimensional regime. In this paper, we develop new LSO's for Euclidean,
, and doubling spaces that allow us to trade larger stretch for a much
smaller number of orderings. We then use our new LSO's (as well as the previous
ones) to construct path reporting low hop spanners, fault tolerant spanners,
reliable spanners, and light spanners for different metric spaces.
While many nearest neighbor search (NNS) data structures were constructed for
metric spaces with implicit distance representations (where the distance
between two metric points can be computed using their names, e.g. Euclidean
space), for other spaces almost nothing is known. In this paper we initiate the
study of the labeled NNS problem, where one is allowed to artificially assign
labels (short names) to metric points. We use LSO's to construct efficient
labeled NNS data structures in this model
LIPIcs, Volume 258, SoCG 2023, Complete Volume
LIPIcs, Volume 258, SoCG 2023, Complete Volum