Kernel Normalized Cut: a Theoretical Revisit * Yoshikazu Terada 1,3 - - PowerPoint PPT Presentation

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Dec 09, 2023 346 likes •415 views

Kernel Normalized Cut: a Theoretical Revisit * Yoshikazu Terada 1,3 & Michio Yamamoto 2,3 1 Graduate School of Engineering Science, Osaka University 2 Graduate School of Environmental and Life Science, Okayama University 3 RIKEN Center for

SLIDE 1

Kernel Normalized Cut: a Theoretical Revisit

*Yoshikazu Terada1,3 & Michio Yamamoto2,3

1Graduate School of Engineering Science, Osaka University 2Graduate School of Environmental and Life Science, Okayama University 3RIKEN Center for Advanced Intelligence Project (AIP)

Unsupervised Learning (Room 103) 12:05 - 12:10, Jun 13, 2019 (Thu) ICML2019@Long Beach

1 2 3 4 5 −2 −1 1 2 1 2 3 4 5 −2 −1 1 2

SLIDE 2

Normalized cut (Ncut; Shi and Malik, 2000)

Ncut = Graph partitioning method Goal = To find “clusters” in the graph: Many edges inside the cluster Fewer edges between different clusters Ncut = Balanced cut Each cluster is “reasonably large”! Cut between different clusters is small. Objective function of Ncut (Number of clusters = 2)

: Similarity matrix,
Min cut:

What is Normalized cut?

Mcut

Cluster 1 Cluster 2

Ncut Ncut

Balancing term!

SLIDE 3

Normalized cut, Spectral clustering, Weighted kernel k-means

Ncut is an NP hard problem Normalized Spectral clustering (SC) = Continuous relaxation of Ncut Ncut and Weighted Kernel K-Means (WKKM) (Dhillon et al., 2007)

WKKM with kernel h and weight :
Ncut = WKKM with

Setting

Normalized cut and its related methods

1 2 3 4 −2 −1 1 2 20 40 60 80 100 20 40 60 80 100 1 2 3 4 −2 −1 1 2

Data points Similarity matrix Clustering result! Kernel function Graph cut

SLIDE 4

Overview of this study

We study theoretical properties of clustering based on Ncut! We also derive the fast rate of convergence of the normalized cut!

Theoretical properties of Ncut

=

Weighted KM in n-dim. space Weighted KM in RKHS

Norm. graph

Laplacian (eigenvector) Limit operator in func. space (eigenfunction)

von Luxburg et al. (2008, AoS)

Ncut for population distribution Ncut for data points

Norm. SC for

data points Optimality

f the partition

is not clear

Dhillon et al. (2007, IEEE PAMI) Shi and Malik (2000, IEEE PAMI)

6=

Empirical Population

This study

SLIDE 5

Numerical experiments

! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " # ! " # $ % & −2 −1 ! " #

Normalized SC Nromalized Cut

500 1000 1500 2000 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12

Normalized SC Normalized cut