Physical Fluctuomatics 1st Review of probabilistic information

Physical Fuctuomatics (Tohoku

University) 1

Physical Fluctuomatics

1st Review of probabilistic information processing

Kazuyuki Tanaka

Graduate School of Information Sciences

[email protected]

http://www.smapip.is.tohoku.ac.jp/~kazu/

Webpage:

http://www.smapip.is.tohoku.ac.jp/~kazu/PhysicalFluctuomatics/2013/


University) 2

Textbooks

Kazuyuki Tanaka: Introduction of Image

Processing by Probabilistic Models, Morikita

Publishing Co., Ltd., 2006 (in Japanese) .

Kazuyuki Tanaka: Mathematics of Statistical

Inference by Bayesian Network, Corona

Publishing Co., Ltd., 2009 (in Japanese).


University) 3

References of the present lecture

K. Tanaka: Statistical-mechanical approach to image processing (Topical Review), Journal of Physics A: Mathematical and General, vol.35, no.37, pp.R81-R150, 2002. Y. Kabashima and D. Saad: Statistical mechanics of low-density parity-check codes (Topical Review), J. Phys. A, vol.37, no.6, pp.R1-R43, 2004.

H. Nishimori: Statistical Physics of Spin Glasses and Information Processing, ---An Introduction, Oxford University Press, 2001.

M. Opper and D. Saad D (eds): Advanced Mean Field Methods --- Theory and Practice, MIT Press, 2001.

C. M. Bishop: Pattern Recognition and Machine Learning, Springer, 2006.

M. J. Wainwright and M. I. Jordan: Graphical Models, Exponential Families, and Variational Inference, now Publishing Inc, 2008.

M. Mezard, A. Montanari: Information, Physics, and Computation, Oxford University Press, 2009.


University) 4

Benefit of Information &

Communications Technology

Ubiquitous Computing

Ubiquitous Internet

Benefit of Information & Communications Technology

Demand for Intelligence

It cannot be satisfied only with it being only cheap

and being quick.


University) 5

Field of Information Processing

Information processing according to theories

Inference from propositions

Realization by progress of computational processing capacity

Information processing in real world

Diversity of reason in phenomenon

Compete data is not necessarily obtained.

It is difficult to extract and select some important

information from a lot of data.

Uncertainty caused by the gap of knowing simply and

understanding actually.

We hope to deal successfully with such uncertainty.

Information processing for numerical calculations

Definite Procedure has been given for each calculation.


University) 6

Computer for next generations

Required Capacity

Capability to sympathize with a user （Knowledge)

Capability to put failure and experience to account in the next chance （Learning）

How should we deal successfully with the uncertainty caused

by the gap of knowing simply and understanding actually?

Formulation of knowledge and uncertainty

Realization of information processing data with

uncertainty


University) 7

Computational model for information

processing in data with uncertainty

Probabilistic Inference Probabilistic model

with graphical structure

（Bayesian network） Medical diagnosis

Failure diagnosis

Risk Management

Probabilistic information processing can give us unexpected

capacity in a system constructed from many cooperating

elements with randomness.

Inference system for data with uncertainty

modeling

Node is random variable.

Arrow is conditional probability.

Mathematical expression of uncertainty

=>Probability and Statistics

Graph with cycles

Important aspect


University) 8

Computational Model

for Probabilistic Information Processing

Probabilistic

Information Processing Probabilistic Model

Bayes Formula

Algorithm

Monte Carlo Method

Markov Chain Monte Carlo Method

Randomized Algorithm

Genetic Algorithm

Approximate Method

Belief Propagation

Mean Field Method

Randomness and

Approximation


University) 9

Probabilistic Image Processing

Noise Reduction by

Probabilistic Image

Processing

K. Tanaka: J. Phys. A, vol.35, 2002.

A. S. Willsky: Proceedings of IEEE, vol.90, 2002.

173

110218100

120219202

190202192

Average

192 202 190

202 219 120

100 218 110

192 202 190

202 173 120

100 218 110

Modeling of Probabilistic Image Processing based on Conventional Filters

Markov Random Field Model Probabilistic Image

Processing

The elements of such a

digital array are called pixels.

At each point, the intensity

of light is represented as an

integer number or a real

number in the digital image

data.

Algorithm

Conventional Filter


University) 10

Probabilistic Image Processing

Degraded Image (Gaussian Noise） Probabilistic Image Processing

Lowpass Filter Wiener Filter Median Filter

MSE:520 MSE: 2137

MSE:860 MSE:767 MSE:1040

K. Tanaka: J. Phys. A, vol.35, 2002.

A. S. Willsky: Proceedings of IEEE, vol.90, 2002.


University) 11

Error Correcting Code

Y. Kabashima and D. Saad: J. Phys. A, vol.37, 2004.

High Performance Decoding Algorithm

010 000001111100000 001001011100001

0 1 0

code

010

error

decode

Parity Check Code

Turbo Code, Low Density Parity Check (LDPC) Code

majority rule Error Correcting Codes

14 January, 2010

Hokkaido University GCOE Tutorial

(Sapporo） 12

Error Correcting Codes and

Belief Propagation

)2 (mod

)2 (mod

)2 (mod

5329

6438

4217

XXXX

XXXX

XXXX

1

1

0

0

1

1

6

5

4

3

2

1

x

x

x

x

x

x

0

1

0

1

1

0

0

1

1

9

8

7

6

5

4

3

2

1

x

x

x

x

x

x

x

x

x

0

1

1

1

1

0

1

1

1

9

8

7

6

5

4

3

2

1

y

y

y

y

y

y

y

y

y

0)2 (mod 101

1)2 (mod 100

0)2 (mod 011

9

8

7

X

X

X

1 1

0

p1

1

0

0

p1

p

p

Received Word

Code Word

Binary Symmetric

Channel

14 January, 2010

Hokkaido University GCOE Tutorial

(Sapporo） 13

Error Correcting Codes and

Belief Propagation

)2 (mod

)2 (mod

)2 (mod

5329

6438

4217

XXXX

XXXX

XXXX

Fundamental Concept for Turbo Codes and LDPC Codes


University) 14

CDMA Multiuser Detectors in Mobile Phone Communication

Relationship between mobile phone communication

and spin glass theory

T. Tanaka, IEEE Trans. on Information Theory, vol.48, 2002

Signals of

User A

Spreading Code

Sequence

Wireless

Communication

Received Data

Decode

Spreading Code

Sequence

Probabilistic model for

decoding can be

expressed in terms of a

physical model for spin

glass phenomena

Noise

Coded Signals

of Other Users

Coded

Signals of

User A


University) 15

Artificial Intelligence

Bayesian Network

CA

SA RA

WA

Probabilistic inference system

Practical algorithms

by means of belief

propagation

J. Pearl: Probabilistic Reasoning in Intelligent Systems: Networks of

Plausible Inference (Morgan Kaufmann, 1988).


University) 16

Main Interests

Information Processing:

Data

Physics:

Material,

Natural Phenomena

System of a lot of elements with mutual relation

Common Concept between Information Sciences and Physics

Material Molecule

Materials are constructed from a lot of molecules.

Molecules have interactions of each other.

０,1 101101

110001

01001110111010

10001111100001

10000101000000

11101010111010

1010 Bit

Data

Data is constructed from many bits

A sequence is formed by deciding the arrangement of bits.

A lot of elements have mutual relation of each other Some physical concepts

in Physical models are

useful for the design of

computational models in

probabilistic

information processing.


University) 17

Horizon of Computation

in Probabilistic Information Processing

Compensation of expressing uncertainty

using probability and statistics

It must be calculated by taking account of both events

with high probability and events with low probability.

Computational Complexity

It is expected to break throw the computational

complexity by introducing approximation algorithms.


University) 18

What is an important point in

computational complexity?

How should we treat the calculation of the summation over 2N configuration?

FT, FT, FT,

21

1 2

,,,x x x

N

N

xxxf

}

}

}

;,,,

F){or Tfor(

F){or Tfor(

F){or Tfor(

;0

21

2

1

L

N

xxxfaa

x

x

x

a

N fold loops

If it takes 1 second in the case of

N=10, it takes 17 minutes in

N=20, 12 days in N=30 and 34

years in N=40.


University) 19

Why is a physical viewpoint effective in probabilistic

information processing?

Matrials are constructed from a lot of molecules.

(1023

molecules exist in 1 mol.)

Molecules have intermolecular forces of each other

1 2

,,, 21

x x x

N

N

xxxf

Theoretical physicists always have to

treat such multiple summation.

Development of Approximate Methods

Probabilistic information processing is also usually reduced

to multiple summations or integrations.

Application of physical approximate methods

to probabilistic information processing


University) 20

Academic

Circulation

Academic

Circulation

Academic Circulation between

Physics and Information Sciences

Physics Information Sciences

Understanding and prediction

of properties of materials and

natural phenomena

Extraction and processing of

information in data

Common Concept Statistical

Mechanical

Informatics

Probabilistic

Information

Processing

Statistical

Sciences


University) 21

Summary of the present lecture

Probabilistic information processing

Examples of probabilistic information processing

Common concept in physics and information sciences

Application of physical modeling and approximations

Future Lectures

Fundamental theory of probability and statistics

Linear model

Graphical model

.

.

.

Documents

Physical Fluctuomatics 1st Review of probabilistic information