# Metric recovery from directed unweighted graphs

@article{Hashimoto2015MetricRF, title={Metric recovery from directed unweighted graphs}, author={Tatsunori B. Hashimoto and Yi Sun and T. Jaakkola}, journal={ArXiv}, year={2015}, volume={abs/1411.5720} }

We analyze directed, unweighted graphs obtained from $x_i\in \mathbb{R}^d$ by connecting vertex $i$ to $j$ iff $|x_i - x_j| < \epsilon(x_i)$. Examples of such graphs include $k$-nearest neighbor graphs, where $\epsilon(x_i)$ varies from point to point, and, arguably, many real world graphs such as co-purchasing graphs. We ask whether we can recover the underlying Euclidean metric $\epsilon(x_i)$ and the associated density $p(x_i)$ given only the directed graph and $d$.
We show that consistent…

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## References

SHOWING 1-10 OF 20 REFERENCES

Density estimation from unweighted k-nearest neighbor graphs: a roadmap

- MathematicsNIPS
- 2013

It is proved how one can estimate the density p just from the unweighted adjacency matrix of the graph, without knowing the points themselves or any distance or similarity scores.

Density-preserving quantization with application to graph downsampling

- Computer ScienceCOLT
- 2014

This work provides a solution to the problem of vector quantization of i.i.d. samples drawn from a densityp on R d that takes the unweighted k-nearest neighbor graph on the sample as input and generates quantization centers that are “evenly spaced”.

Shortest path distance in random k-nearest neighbor graphs

- Computer Science, MathematicsICML
- 2012

It is proved that for unweighted kNN graphs, this distance converges to an unpleasant distance function on the underlying space whose properties are detrimental to machine learning.

Random Walks on Infinite Graphs and Groups — a Survey on Selected topics

- Mathematics
- 1994

Contents 1. Introduction 2 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 C. Random walks on groups 5 D. Group-invariant random walks…

An Analysis of the Convergence of Graph Laplacians

- Computer Science, MathematicsICML
- 2010

A kernel-free framework is introduced to analyze graph constructions with shrinking neighborhoods in general and apply it to analyze locally linear embedding (LLE) and how desirable properties such as a convergent spectrum and sparseness can be achieved by choosing the appropriate graph construction.

The PageRank Citation Ranking : Bringing Order to the Web

- Computer Science, MathematicsWWW 1999
- 1999

This paper describes PageRank, a mathod for rating Web pages objectively and mechanically, effectively measuring the human interest and attention devoted to them, and shows how to efficiently compute PageRank for large numbers of pages.

The dynamics of viral marketing

- Computer ScienceTWEB
- 2007

While on average recommendations are not very effective at inducing purchases and do not spread very far, this work presents a model that successfully identifies communities, product, and pricing categories for which viral marketing seems to be very effective.

The igraph software package for complex network research

- Computer Science
- 2006

Platform-independent and open source igraph aims to satisfy all the requirements of a graph package while possibly remaining easy to use in interactive mode as well.

Some Useful Functions for Functional Limit Theorems

- MathematicsMath. Oper. Res.
- 1980

This paper facilitates applications of the continuous mapping theorem by determining when several important functions and sequences of functions preserve convergence.

A Database for Handwritten Text Recognition Research

- Computer ScienceIEEE Trans. Pattern Anal. Mach. Intell.
- 1994

An image database for handwritten text recognition research is described that contains digital images of approximately 5000 city names, 5000 state names, 10000 ZIP Codes, and 50000 alphanumeric characters to overcome the limitations of earlier databases.