Manifold Learning on Probabilistic Graphical Models 概率图上的流形学习 答辩人 :...
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Transcript of Manifold Learning on Probabilistic Graphical Models 概率图上的流形学习 答辩人 :...
Manifold Learning on Probabilistic Graphical Models概率图上的流形学习
答辩人 : 邵元龙导师 : 鲍虎军 教授 & 何晓飞 教授
浙江大学 CAD&CG 国家重点实验室2010 年 3 月 5 日
Outline
Background & Motivation Function Learning v.s. Statistical Modeling
Manifold Regularized Variational Inference Algorithm Design & Examples In Depth Analysis Implementation Experimental Results
Function Learning Given data points , and a function space , find the optimal function , such that
Regularization is Important!!
N 1
,N
i i ix y
2*
1
arg min ,N
i if i
f V y f x f
F*f F
Statistical Modeling All quantities, no matter given or to be
estimated, are random variables. Then we model the joint distribution.
, ,,|
, ,
p dpp
p p d d
H V Θ θ θH V
H VV H h V Θ θ h θ
Manifold Assumption Y changes smoothly with X, and we have
so should be small over manifold Minimizing it over the manifold,
f x x f x
f xx
f x
2min
xf dx
MM
Manifold Regularization
2
2
2,
2
2,
min
1
1
x
iji
i
j
ij j
i
j
j
i
f dx
f x f xSN
S yN
y
S
S
MM
Transductive Learning
Variational Inference For , define , a var. dist.
Approximate the true posterior with it
by minimizing the KL divergence
H H q H
|H
q q H p
H
H H V
* arg min || |q
q KL q pH
H H H V
Manifold Regularized Variational Inference
* arg minq
q H
H F
2
,
|| |
,ij i ji j
KL q p
S d q Z q Z
H H V
S
F
Optimization Algorithm
* arg mini
iii
q ZZ Zq Z F
2
,: ,
|| | , 2 ,ii iZ Zi i Z ij i j
j i j j i
KL q Z p Z S d q Z q Z
MB S
F
1
*
for ,...,
arg min || |
N
Hq H
H Z Z
q H KL q H p H
MB
Works Done
Example Distribution Types Convergence Proof Convexity Analysis (More TODO) Computational Complexity Numerical Stability A Flexible Inference Engine
YASIE (Yet Another Statistical Inference Engine)
Interface Design Inference Scheduling Type-Free Mixture Model Design Issues (e.g. Balance of Memory & Comp.
Time)