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- Ronald R. Coifman, Matthew J. Hirn
- ArXiv
- 2012

Graph Laplacians and related nonlinear mappings into low dimensional spaces have been shown to be powerful tools for organizing high dimensional data. Here we consider a data set X in which the graph associated with it changes depending on some set of parameters. We analyze this type of data in terms of the diffusion distance and the corresponding diffusion… (More)

- Ronald R. Coifman, Matthew J. Hirn
- ArXiv
- 2012

In this short letter we present the construction of a bi-stochastic kernel p for an arbitrary data set X that is derived from an asymmetric affinity function α. The affinity function α measures the similarity between points in X and some reference set Y. Unlike other methods that construct bi-stochastic kernels via some convergent iteration process or… (More)

- Matthew J. Hirn, Erwan Le Gruyer
- 2013

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier… (More)

- Matthew J. Hirn
- 2012

Harmonic frames of prime order are investigated. The primary focus is the enumeration of inequivalent harmonic frames, with the exact number given by a recursive formula. The key to this result is a one-to-one correspondence developed between inequivalent harmonic frames and the orbits of a particular set. Secondarily, the symmetry group of prime order… (More)

We consider the following interpolation problem. Suppose one is given a finite set E ⊂ R, a function f : E → R, and possibly the gradients of f at the points of E as well. We want to interpolate the given information with a function F ∈ C(R) with the minimum possible value of Lip(∇F ). We present practical, efficient algorithms for constructing an F such… (More)

- Ariel Herbert-Voss, Matthew J. Hirn, Frederick McCollum
- ArXiv
- 2014

We consider the following interpolation problem. Suppose one is given a finite set E ⊂ R, a function f : E → R, and possibly the gradients of f at the points of E. We want to interpolate the given information with a function F ∈ C(R) with the minimum possible value of Lip(∇F ). We present practical, efficient algorithms for constructing an F such that… (More)

- Nicholas F. Marshall, Matthew J. Hirn
- ArXiv
- 2014

We consider a collection of n points in R measured at m times, which are encoded in an n × d × m data tensor. Our objective is to define a single embedding of the n points into Euclidean space which summarizes the geometry as described by the data tensor. In the case of a fixed data set, diffusion maps (and related graph Laplacian methods) define such an… (More)

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