Harmonic Analysis of Digital Data Bases

“…Given a Graph G = (V, E), define, if possible, an analogous geometry on its eigenvectors. This is an absolutely fundamental problem, we refer to [1,2,5,7,8,9,10,12,15,17,18,23,25,28,29,30] for recent examples. Moreover, it is not expected that this is always (or even generically) possible -even in Euclidean space, one would expect that eigenfunctions on generic domains do not have any distinguishing features except for their eigenvalue; this vague statement is made precise in different ways in the study of quantum chaos [16,21].…”

Section: Graph Signalmentioning

confidence: 99%

On the Dual Geometry of Laplacian Eigenfunctions

Cloninger

Steinerberger

2018

Experimental Mathematics

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We discuss the geometry of Laplacian eigenfunctions −∆φ = λφ on compact manifolds (M, g) and combinatorial graphs G = (V, E). The 'dual' geometry of Laplacian eigenfunctions is well understood on T d (identified with Z d ) and R n (which is self-dual). The dual geometry is of tremendous role in various fields of pure and applied mathematics. The purpose of our paper is to point out a notion of similarity between eigenfunctions that allows to reconstruct that geometry. Our measure of 'similarity' α(φ λ , φµ) between eigenfunctions φ λ and φµ is given by a global average of local correlationswhere p(t, x, y) is the classical heat kernel and e −tλ + e −tµ = 1. This notion recovers all classical notions of duality but is equally applicable to other (rough) geometries and graphs; many numerical examples in different continuous and discrete settings illustrate the result.2010 Mathematics Subject Classification. 35J05, 35P05, 42C10, 65T60, 81Q50, 94A11.

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“…The approach is based on a new formulation for harmonic analysis [23] with application to useritem matrix by deriving a multiresolution transformation of the matrix similar to a wavelet transform. This transformation inherently captures the interplay between users and items at multiple levels of granularity, which enables similarity evaluation at both individual and group levels.…”

Section: Multiresolution Approach For Rating Predictionmentioning

confidence: 99%

“…The process initially starts by deriving a multiresolution partition tree for items as described in [23] by clustering similar groups of items (columns) together at different granularity levels. The clustering is performed using diffusion distances [24].…”

Section: Multiresolution Approach For Rating Predictionmentioning

confidence: 99%

A Multiresolution Approach to Recommender Systems

Badaro

Hajj

Haddad

et al. 2014

Proceedings of the 8th Workshop on Social Network Mining and Analysis

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Recommender systems face performance challenges when dealing with sparse data. This paper addresses these challenges and proposes the use of Harmonic Analysis. The method provides a novel approach to the user-item matrix and extracts the interplay between users and items at multiple resolution levels. New affinity matrices are defined to measure similarities among users, among items, and across items and users. Furthermore, the similarities are assessed at multiple levels of granularity allowing individual and group level similarities. These affinity matrices thus produce multiresolution groupings of items and users, and in turn lead to higher accuracy in matching similar context for ratings, and more accurate prediction of new ratings. Evaluation results show superiority of the approach compared to state of the art solutions.

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Harmonic Analysis of Digital Data Bases

Cited by 33 publications

References 13 publications

Wavelets

Wavelets

On the Dual Geometry of Laplacian Eigenfunctions

A Multiresolution Approach to Recommender Systems

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