Random Swap EM algorithm for GMM and Image Segmentation

Qinpei Zhao, Ville Hautamäki, Ismo Kärkkäinen, Pasi Fränti

Speech & Image Processing UnitDepartment of Computer Science, University of Joensuu

Box 111, Fin-80101 JoensuuFINLAND

zhao@cs.joensuu.fi

Outline

Background & StatusRS-EMApplication

Background: Mixture Model

Background: EM algorithm

EM algorithm -> {α, Θ} E-step (Expectation):

M-step (Maximization):

Iterate E,M step until convergence α- mixing coefficient

Θ- model parameters, eg. {μ,∑}

( 1) ( 1)( , ) [log ( , | ) | , ]i iQ E p X Y X

( ) ( 1)argmax ( , )i iQ

Local MaximaLet’s describe it as mountain climbing……

600km

2160m 3099m

Initialization Effect

Initialization and Result(1) Initialization and Result(2)

Sub-optimal Example

The situation of local maxima trap

Status

Standard EM for Mixture Models(1977) Deterministic Annealing EM (DAEM) (1998) Split-Merge EM (SMEM) (2000) Greedy EM (2002) RS-EM coming…

Outline

Background & StatusRS-EM (Random Swap)Application

RSEM: Motivations Random manner Prevent from staying near the unstable or

hyperbolic fixed points of EM. Prevent from its stable fixed points corresponding

to insignificant local maxima of the likelihood function

Avoid the slow convergence of EM algorithm Less sensitive to its initialization

Formulas SMEM

Greedy EM

RSEM

Random Swap EMAfter EM

Afte

r Sw

ap

After EM

Comparisons(1)

Comparisons(2)

Q1 Q2 S1 S4

Outline

Background & StatusRS-EMApplication

Application Image Segmentation Color Quantization Image Retrieval ……

Conclusion Introduce Randomization into algorithm Performs better Without heavy time complexity Wider applications

Thanks!☺