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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)

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)

- 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 d , 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 1,1 (R d) with the minimum possible value of Lip(∇F). We present practical, efficient algorithms for constructing an… (More)

We consider a collection of n points in R d 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)

- Matthew J. Hirn, Stéphane Mallat, Ronald R. Coifman, John J. Benedetto, Kasso Okoudjou, Robert R. Strichartz +3 others
- 2015

Applied mathematics and data analysis, with particular emphasis on: • Applied harmonic analysis • Wavelet theory and deep learning • Quantum chemistry • Manifold learning • Smooth extensions and interpolations

We consider the following interpolation problem. Suppose one is given a finite set E ⊂ R d , 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 1,1 (R d) with the minimum possible value of Lip(∇F). We present practical, efficient algorithms for constructing an F such… (More)

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