Upload File

harmonic analysis on graphs

41
Harmonic Analysis On Graphs Sarah Constantin December 1, 2011 1 Lecture 2 1.1 Mul tidime nsi ona l scaling x i ∈ R n i = 1 ...m d 2 ij = ||x i − x  j || 2 We can do an embedding x i → ˜ X i . We can compute the matrix C ij = x i · x  j inner products and diagonalize it. O t Λ 2 O = (ΛO) t ΛO = ˜ X  j · ΛO Another way to think of it: C ij = λ 2 l v l (i)v l (  j ) inner products between eigenvectors. The vector associate d with x i is ˜ x i = {λ l v l (i),l = 1 ...m}. If the x i lie in a low-rank space, then only few λ l will be nonzero. Here we associated x i and x  j with the kernel < x i ,x  j >. But we could have had a different kernel k(x i ,x  j ). Abstract metric space X, distance d(x, y), want to define a mapping φ : X → R n such that ||φ(x) − φ(y)|| d(x, y). The re is a lot of literature on the subject. Y ou hav e some estimate on the ratio betw een the distance s. If |X | = 2 L , then there is a map into R cL . An y metric spa ce. Thi s is a ”coding ” theorem. In reality L is never reall y bigger tha n 50. Notic e that in the multid imens ional scaling example it didn’t really matter if N was big. 1

Upload: srconstantin

Post on 06-Apr-2018

226 views

Category:

Documents


0 download

Report

TRANSCRIPT

Page 1: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 1/41

Page 2: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 2/41

Page 3: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 3/41

Page 4: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 4/41

Page 5: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 5/41

Page 6: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 6/41

Page 7: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 7/41

Page 8: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 8/41

Page 9: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 9/41

Page 10: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 10/41

Page 11: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 11/41

Page 12: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 12/41

Page 13: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 13/41

Page 14: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 14/41

Page 15: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 15/41

Page 16: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 16/41

Page 17: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 17/41

Page 18: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 18/41

Page 19: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 19/41

Page 20: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 20/41

Page 21: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 21/41

Page 22: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 22/41

Page 23: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 23/41

Page 24: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 24/41

Page 25: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 25/41

Page 26: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 26/41

Page 27: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 27/41

Page 28: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 28/41

Page 29: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 29/41

Page 30: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 30/41

Page 31: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 31/41

Page 32: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 32/41

Page 33: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 33/41

Page 34: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 34/41

Page 35: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 35/41

Page 36: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 36/41

Page 37: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 37/41

Page 38: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 38/41

Page 39: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 39/41

Page 40: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 40/41

Page 41: Harmonic Analysis on Graphs

8/3/2019 Harmonic Analysis on Graphs

http://slidepdf.com/reader/full/harmonic-analysis-on-graphs 41/41


Semiclassical analysis of high harmonic generation … · Semiclassical analysis of high harmonic generation in bulk crystals ... mechanisms dominate the perturbative ... harmonic

MAT 280: Harmonic Analysis on Graphs & Networks …saito/courses/HarmGraph/lecture16.pdf · MAT 280: Harmonic Analysis on Graphs & Networks Lecture 16: Wavelets on Graphs Naoki Saito

Transint and Harmonic Analysis

Harmonic Analysis - maths.tcd.ieormondca/notes/Harmonic Analysis.pdf · Harmonic Analysis Cathal Ormond 1. Contents 1 Fourier Analysis 4 1.1 Introduction . . . . . . . . . . . .

Harmonic Analysis Report - Power Quality Specialists | Harmonic

Harmonic Analysis report

Harmonic Analysis - Comparative

Piping Harmonic Analysis

Commutative Harmonic Analysis IV: Harmonic Analysis in IRn

Automatic Functional Harmonic Analysis

Harmonic Analysis on Graphs Global vs. Multiscale Approaches

Computation Harmonic Analysis

Harmonic Analysis Report -1

Pre-Viva Talk: Parsing Jazz: Harmonic Analysis of Music ...jazzparser.granroth-wilding.co.uk/attachments/PreVivaTalk/slides.pdf · Introduction Harmonic Analysis Harmonic CCG Statistical

Noise and Harmonic Analysis

Hyperelliptic Curves and Harmonic Analysis · Contemporary Mathematics Volume 00, 0000 Hyperelliptic Curves and Harmonic Analysis (Why harmonic analysis on reductive p-adic groups

Harmonic Analysis (1910)

Harmonic Analysis Example

Representation Theory and Harmonic Analysis on Symmetric ... · Harmonic Analysis on Symmetric Spaces AMS Special Session in Honor of Gestur Ólafsson’s 65th Birthday Harmonic Analysis

Towards Multiscale Harmonic Analysis of Graphs and ...helper.ipam.ucla.edu/publications/ds2006/ds2006_5859.pdf · 26.01.2006  · Towards Multiscale Harmonic Analysis of Graphs and

Harmonic Analysis

Multiscale Wavelets on Trees, Graphs and High Dimensional Dataweb.stanford.edu/~gavish/documents/ICML_Haar_like_slides.pdf · Harmonic analysis wisdom in low dim Euclidean space:

GEbIERALIZED HARMONIC ANALYSIS. 1

Harmonic Analysis and Hypergroups

Viola string harmonic analysis

Data Compression and Harmonic Analysis - Information ...ingrid/publications/IEEE_Inf_Th_44_2435.pdfof harmonic analysis. Recent harmonic analysis constructions, such as wavelet transforms

Diffusion Geometries, and multiscale Harmonic Analysis on graphs and complex data sets. Multiscale diffusion geometries, “Ontologies and knowledge building”

Convergence of discrete harmonic functions and the ...dchelkak/Slides/Chelkak_Obergurgl.pdf · D. Chelkak, S. Smirnov: Discrete complex analysis on isoradial graphs. arXiv:0810.2188

Laporan Praktik - Harmonic Analysis

Harmonic Analysis Notes

CE31A06 - Harmonic Analysis

Harmonic Analysis IN ANSYS

Harmonic Analysis Lecture Notes

Lecture notes: harmonic analysis

Harmonic Analysis - RL Laster | Power and Energy ... · Analysis With ETAP’s Harmonic Analysis module, you can identify harmonic problems, reduce nuisance trips, design and test