## 8 Citations

Combining Persistent Homology and Invariance Groups for Shape Comparison

- MathematicsDiscret. Comput. Geom.
- 2016

A way to combine persistent homology with the use of G-invariant non-expansive operators defined on varPhi, where G is a group of self-homeomorphisms of X, is proposed.

Persistent Homology Lower Bounds on High-Order Network Distances

- Computer ScienceIEEE Transactions on Signal Processing
- 2017

The paper does so by mapping high-order networks to filtrations of simplicial complexes and showing that the distance between networks can be lower bounded by the difference between the homological features of their respective filtration.

Necessary conditions for discontinuities of multidimensional persistent Betti numbers

- Mathematics
- 2015

Topological persistence has proven to be a promising framework for dealing with problems concerning the analysis of data. In this context, it was originally introduced by taking into account…

Towards a topological–geometrical theory of group equivariant non-expansive operators for data analysis and machine learning

- Computer Science, MathematicsNat. Mach. Intell.
- 2019

Bergomi et al introduce a mathematical framework in which the space of possible operators representing the data is constrained by using symmetries and is still suitable for machine learning: operators can be efficiently computed, approximated and parameterized for optimization.

P-persistent homology of finite topological spaces

- MathematicsArXiv
- 2015

It is shown that for any reasonable P-persistent object X in the category of finite topological spaces, there is a P− weighted graph, whose clique complex has the same P-Persistent homology as X.

Bongard Problems: A Topological Data Analysis Approach

- Computer ScienceWSEAS TRANSACTIONS ON SYSTEMS AND CONTROL
- 2020

An algorithm is presented and it is shown that it can solve problems involving a much larger set of differences provided the right G-equivariant operators.

Dynamical and Topological Tools for (Modern) Music Analysis. (Outils dynamiques et topologiques pour l'analyse musicale)

- Computer Science
- 2015

A model at the crossroad between the signal and symbolic analysis of music uses multiple sequences alignment to provide an encompassing, novel viewpoint on the musical inspiration transfer among compositions belonging to different artists, genres and time.

Networked Data Analytics: Network Comparison And Applied Graph Signal Processing

- Computer Science
- 2018

This thesis constructs models to compare and cluster networked data, to simplify a complicated networked structure, and to formalize the notion of smoothness and variation for domain-specific signals on a network to suggest that the intuition in analyzing huge data can be transformed into rigorous algorithms.

## References

SHOWING 1-10 OF 23 REFERENCES

Uniqueness of models in persistent homology: the case of curves

- MathematicsArXiv
- 2010

We consider generic curves in , i.e. generic C1 functions . We analyze these curves through the persistent homology groups of a filtration induced on S1 by f. In particular, we consider the question…

The theory of multidimensional persistence

- MathematicsSCG '07
- 2007

This paper proposes the rank invariant, a discrete invariant for the robust estimation of Betti numbers in a multifiltration, and proves its completeness in one dimension.

Natural pseudo-distances between closed curves

- Mathematics
- 2009

Abstract Let us consider two closed curves ℳ, of class C 1 and two functions of class C 1, called measuring functions. The natural pseudo-distance d between the pairs (ℳ, φ), (, ψ) is defined as the…

Betti numbers in multidimensional persistent homology are stable functions

- Mathematics
- 2013

Multidimensional persistence mostly studies topological features of shapes by analyzing the lower level sets of vector‐valued functions, called filtering functions. As is well known, in the case of…

A Mayer–Vietoris Formula for Persistent Homology with an Application to Shape Recognition in the Presence of Occlusions

- MathematicsFound. Comput. Math.
- 2011

It is shown that persistence diagrams are able to recognize an occluded shape by showing a common subset of points and a Mayer–Vietoris formula involving the ranks of the persistent homology groups of X, A, B, and A∩B plus three extra terms is obtained.

REEB GRAPHS OF CURVES ARE STABLE UNDER FUNCTION PERTURBATIONS

- Mathematics
- 2011

Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function. One question deserving attention is whether Reeb graphs are robust against function…

Reeb graphs of curves are stable under function perturbations

- Mathematics
- 2012

Reeb graphs provide a method to combinatorially describe the shape of a manifold endowed with a Morse function. One question deserving attention is whether Reeb graphs are robust against function…

Stability of persistence diagrams

- MathematicsSCG
- 2005

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram…

Natural pseudodistances between closed manifolds

- Mathematics
- 2004

Let us consider two closed homeomorphic manifolds M;N of class C 1 and two functions j : M ! R, c : N ! R of class C . The natural pseudodistance d between the pairs ðM; jÞ; ðN;cÞ is defined as the…

Size Functions from a Categorical Viewpoint

- Mathematics
- 2001

A new categorical approach to size functions is given. Using this point of view, it is shown that size functions of a Morse map, f: M→ℜ can be computed through the 0-dimensional homology. This result…