library to compute fermat distance
Project description
Fermat distance
Fermat is a Python library that computes the Fermat distance estimator (also called ddistance estimator) proposed in
 Weighted Geodesic Distance Following Fermat's Principle (see https://openreview.net/pdf?id=BJfaMIJwG).
 Nonhomogeneous Euclidean firstpassage percolation and distance learning (see https://arxiv.org/abs/1810.09398)
Table of contents
Introduction
A densitybased estimator for weighted geodesic distances is proposed. Let M be a Ddimensional manifold and consider a sample of N points X_n living in M. Let l(.,.) be a distance defined in M (a typical choice could be Euclidean distance). For d>=1 and given two points p and q in M we define the Fermat distance estimator as
The minimization is done over all K>=2 and all finite sequences of data points with x1= argmin l(x,p) and xK = argmin l(x,q).
When d=1, we recover the distance l(.,.) but if d>1, the Fermat distance tends to follow more closely the manifold structure and regions with high density values.
This distance is useful for clustering pourposes. Replacing Euclidean distance with Fermat distance in standard clustering alorithms leads to much better results.
Installation
pip install fermat
Implementation
The optimization performed to compute the Fermat distance estimator runs all over the possible paths of points between each pair of points. We implement an algorithm that computes the exact Fermat distance and two that compute approximations.

Exact: FloydWarshall
Permorf the FloydWarshall algorithm that gives the exact Fermat distance estimator in O( n^3 )
operations between all possible paths that conects each pair of points.

Aprox: Dijsktra + knearest neighbours
With probability arbitrary high we can restrict the minimum path search to paths where each consecutive pair of points are knearest neighbours, with k = O(log n)
. Then, we use Dijkstra algorithm on the graph of knearest neighbours from each point. The complexity is O( n * ( k * n * log n ) )
.

Aprox: Landmarks
If the number of points n is too high and neither FloydWarshall and Dijkstra run in appropiate times, we implemente a gready version based on landmarks. Let consider a set of l of point in the data set (the landmarks) and denote s_j
the distance of the point s
to the landmark j
. Then, we can bound the distance d(s,t)
between any two points s
and t
as
lower = max_j {  s_j  t_j  } <= d(s,t) <= min_j { s_j + t_j } = upper
and estimate d(s,t)
as a function of lower
and upper
(for example, d(s,t) ~ (_lower + upper_) / 2
). The complexity is O( l * ( k * n * log n ) )
.
Clustering
The tool FermatKmeans implements the Kmedoids algorithm with Fermat distance as an input. Here is an example explaining how to use it to compute clusters in MNIST dataset
Features
 Exact and approximate algorithms to compute the Fermat distance.
 Examples explaining how to use this package.
 Quick start
 MNIST data set
 [Clustering using Fermat]
 Documentation
Support
If you have an openended or a research question:
'support@aristas.com.ar'
Fermat distance citation
Fermat distance has been introduced and studied in the following papers
@inproceedings{SGJ2018,
title={Weighted Geodesic Distance Following Fermat's Principle},
author={Facundo Sapienza and Pablo Groisman and Matthieu Jonckheere},
year={2018},
url={https://openreview.net/forum?id=BJfaMIJwG}
}
@article{GJS2018,
title={Nonhomogeneous Euclidean firstpassage percolation and distance learning},
author={Groisman, Pablo and Jonckheere, Matthieu and Sapienza, Facundo},
journal={arXiv preprint arXiv:1810.09398},
year={2018}
}
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