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Sensor Networks, Rate Distortion Codes, and Spin Glasses

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## Sensor Networks, Rate Distortion Codes, and Spin Glasses

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### Sensor Networks, Rate Distortion Codes, and Spin Glasses

NTT Communication Science Laboratories

Tatsuto Murayama

murayama@cslab.kecl.ntt.co.jp

In collaboration with Peter Davis

March 7th, 2008 at the Chinese Academy of Sciences

Sensor Networks

Sensor

Sensors transmit their noisy observations independently.

0100110

Computer

Computer estimates the quantity of interest from sensor information.

1100101

100110101011001010110

Network

Network has a limited bandwidth constraint.

A Pessimistic Forecast

《Supply Side Economics》Semiconductors are going to be very small and also cheap, so they’d like to sell them a lot!

Sensor

Networks

Large-scale information integration

Smartdusts,

IC tags…

Central Unit

Network Capacity is limited

Target Source

Information loss via sensing

Information loss via communications

VS

High Noise RegionNetwork is going to be large and dense!

Finite Network CapacityEfficient use of the given bandwidth is required!

Need a new information integration theory!

What to look for?

- Given a combined data rate, we examine the optimal aggregation level for sensor networks.

Saturate Strategy (SS)

Transmit as much sensor information as possiblewithout data compression.

Which strategy is outperforming?

A small quantity of high quality statistics

A large quantity of low quality statistics

Large System Strategy (LSS)

Transmit the overwhelming majority of compressed sensor information.

What to Evaluate?

- It is natural to introduce the following indicator function in decibel manner.

- Which Strategy is Outperforming to the Other?
- The large system strategy is outperforming when the indicator function is negative.
- The saturate strategy is outperforming when the indicator function is positive.
- The zero level corresponds to the strategic transition point if available.

What to Expect?

- Conjecture on the existence of the strategic transition point.

Strategic Transition Point.

- Some Evidences
- At the low noise level, the indicator function should diverge to infinity.
- At the high noise level, the indicator function should converge to zero.

Sensing Model

- Target Information is a Bernoulli(1/2) Source.
- Environmental Noise is modeled by the Binary Symmetric Channel.

Source

Observations

- Binary Symmetric Channel (BSC)
- The input alphabet is `flipped’ with a given probability.

Communication Model

- To satisfy the bandwidth constraints, each sensor encodes its observation independently.

Codewords

Reproductions

- Nature of Bandwidth-Given Communication
- If the bandwidth is bigger than the entropy rate, revertible coding can be possible.
- If the bandwidth is smaller than the entropy rate, only non-revertible coding can be possible.

Estimation Model

- Collective estimation is done by applying the majority vote algorithm to the reproductions.

Estimation

In case of the `Ising’ alphabet

- Majority Vote
- Estimation is calculated from the reproductions by sequentially applying the following algorithm.

System Model

Assume purely random Source is observed

Sensing Model

Encoding Model

Independent decoding process is forced

Bitwise majority vote is concerned

Estimation Model

Case of Saturate Strategy

Sensing

Encoding

Decoding

2 messages saturate network.

Estimation

Cost of comm.= # of sensors ( bits of info.)

Moderate aggregation levels are possible.

Case of Large System Strategy

Still 2 messages saturate network.

Sensing

Encoding

Decoding

Estimation

Cost of comm.＝ # of sensors data rate

We can make system as large as we want!

Rate Distortion Tradeoff

- Variety of communication reduces to a simple rate distortion tradeoff.

Black Box

- Rate Distortion Tradeoff
- Each observation bit is flipped with the same probability.

Effective Distortion

- Under the stochastic description of the tradeoff, we introduce the effective distortion as follows.
- Then, our sensing and communications tasks reduces to a channel.

- The Channel Model
- The channel is labeled by effective distortion.

Formula for Finite Sensors

- Finite-scale Sensor Networks
- Given the number of sensors, we get
- with
- where

A Glimpse at Statistics

- In the large system limit, binomial distribution converges to normal distribution.

Formula for Infinite Sensors

- Infinite-scale Sensor Networks
- Given only the noise and bandwidth, we get
- with
- where we naturally expect that

Lossy Data Compression

- There exists tradeoff between compression rate and the resulting quality of reproduction.

《Encoding》

《Storage》

《Decoding》

What is the best bound for the lossy compression?

Rate Distortion Theory

- Theory for compression beyond entropy rate.

Compression

Rate

○

×

Hamming

Distortion

Best bound is the rate distortion function.

Can the CEO be informed?

- Rate Distortion Function gives the best bound.
- Large System Strategy by optimal codes

Leading Contribution

Taylor Expansion

Non-trivial regions are feasible

The CEO can be informed!

Does LSS have any advantage over SS?

Indicator Function

- In what condition the large system strategy outperforms the saturate strategy?
- Saturate Strategy is used as the `reference’ in the decibel measure.

LSS

SS

Which is outperforming?

LSS is outperforming when measure is negative.

SS is outperforming when measure is positive.

Theoretical System Gain

- In the noisy environment, LSS is superior to SS!

Existence of comparative advantage gives a strong motivation for making large systems.

Definition of VQ

- Any information bit belongs to the Voronoi region, and is replaced by its representative bit.
- Index map specifies the representative bits.
- Voronoi region is labeled by an index.

Gauge of Representative Bit

- Information is first divided into Voronoi regions, and then representative gauge is chosen.

Isolated Free Energy

- Free energy can be decoupled.
- Hamming Distortion can be derived.

Exact Solution

Cost Function

（Energy）

Random Walk Statistics

Isolated Model Reduces to Random Walk Statistics.

Bit Error Probability

- Substitute exact solution into general formula.
- Theoretical Performance

Large System Gain

- Bit error probability in decibel measure

Large system strategy is not so outperforming

Rate Distortion Theory

- N bit sequence is encoded into M bit codeword.
- M bit codeword is decoded to reproduce N bit sequence, but not perfectly.
- Tradeoff relation between the rate R=M/N and the Hamming distortion D.
- Rate distortion function for random sequences

Sparse Matrix Coding

- Find a codeword sequence that satisfies:

where the fidelity criterion:

- Boolean matrix A is characterized by K ones per row and C per column; an LDPC matrix.
- Bit wise reproduction errors are considered; the Hamming distortion measure D is selected.

Example: 4 bit sequence

- Set an LDPC matrix.
- Given a sequence:
- Find a codeword:
- Reproduce the original sequence.

Design Principle

- Algebraic constraints are represented in a graph.
- Probabilistic constraint is considered as a prior.

Microscopic consistency might induce the macroscopic order of the frustrated system.

Easy

Low-resource Computation- Introduce the mean field to avoid complex tasks.
- Eliminate many candidates of the solution by dynamical techniques.

TAP Approach

- A codeword bit is calculated by its marginal.
- Marginal probability is evaluated by heuristics.

Empirical Performance

- Message passing algorithm works very well.

Example of Saturate Strategy

- Six sensors transmit their original datawords.

5.4k bps

BER 20.0%

Sensing

32.4k bps

Transmission

BER 20.0%

BER 9%

Estimation

Example of Large System Strategy

- Nine sensors transmit their codewords.

5.4k bps

BER 20.0%

Sensing

Encoding

&

Transmission

&

Decoding

32.4k bps

BER 24.7%

BER 5%

Estimation

Frustrated Free Energy

- Free energy cannot be decoupled.
- General formula for Hamming Distortion

Approximation

Cost Function

（Energy）

Replica Method

Saddle Point of Free Energy

Frustrated model reduces to spin glass statistics.

Bit Error Probability

- Substitute replica solution into general formula.
- Theoretical Performance

Scaling Evaluation

for Replica Solution

Characteristic Constant

- Constant:
- Saddle Point Equations
- Variance of order parameter:
- Non-negative entropy condition:
- Measure:

Large System Gain: K=2

- Bit error probability in decibel measure

Similar to the case of optimal random coding.

Concluding Remarks

- We consider the problem of distributed sensing in a noisy environment.
- Limited bandwidth constraint induces tradeoff between reducing errors due to environmental noise and increasing errors due to lossy coding as number of sensors increases.
- Analysis shows threshold behavior for optimal number of sensors.

Analysis

- TM and M. Okada: `Rate Distortion Function in the Spin Glass State: A Toy Model’, Advances in Neural Information Processing Systems 15, 423-430, MIT Press (2003).
- Available at http://books.nips.cc/nips15.html
- TM and P. Davis: `Rate Distortion Codes in Sensor Networks: A System-level Analysis’, Advances in Neural Information Processing Systems 18, 931-938, MIT Press (2006).
- Available at http://books.nips.cc/nips18.html

Algorithms

- TM: `Statistical mechanics of the data compression theorem’, Journal of Physics A 35, L95-L100 (2002).
- Available at http://www.iop.org/EJ/article/0305-4470/35/8/101/a208l1.html
- TM: `Thouless-Anderson-Palmer Approach for Lossy Compression’, Physical Review E 69, 035105(R) (2004).
- Available at http://prola.aps.org/abstract/PRE/v69/i3/e035105

Reviews

- TM and P. Davis: `Statistical mechanics of sensing and communications: Insights and techniques’, Journal of Physics: Conference Series 95, 012010 (2007).
- Available at http://www.iop.org/EJ/toc/1742-6596/95/1
- For more information, please google “tatsuto murayama” or “村山立人”.

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