Sensor Networks, Rate Distortion Codes, and Spin Glasses NTT Communication Science Laboratories Tatsuto Murayama In collaboration with Peter Davis March 7 th, 2008 at the Chinese Academy of Sciences

Problem Statement

Sensor Networks Sensor Sensors transmit their noisy observations independently. Computer Computer estimates the quantity of interest from sensor information. Network Network has a limited bandwidth constraint.

A Pessimistic Forecast Sensor Networks Central Unit Target Source Information loss via communications Information loss via sensing VS 《 Supply Side Economics 》 Semiconductors are going to be very small and also cheap, so they’d like to sell them a lot! Smartdusts, IC tags… Large-scale information integration Finite Network Capacity Finite Network Capacity Efficient use of the given bandwidth is required! Need a new information integration theory! High Noise Region High Noise Region Network is going to be large and dense! Network Capacity is limited

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 possible without data compression. Which strategy is outperforming? A small quantity of high quality statistics Large System Strategy (LSS) Transmit the overwhelming majority of compressed sensor information. A large quantity of low quality statistics

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. 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. Strategic Transition Point.

System Model

Target Information is a Bernoulli(1/2) Source. Environmental Noise is modeled by the Binary Symmetric Channel. Sensing Model Binary Symmetric Channel (BSC) The input alphabet is `flipped’ with a given probability. ObservationsSource

Communication Model To satisfy the bandwidth constraints, each sensor encodes its observation independently. CodewordsReproductions 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 Majority Vote Estimation is calculated from the reproductions by sequentially applying the following algorithm. In case of the `Ising’ alphabet

System Model Sensing Model Estimation Model Encoding Model Independent decoding process is forced Bitwise majority vote is concerned Assume purely random Source is observed

Case of Saturate Strategy Cost of comm.= # of sensors ( bits of info.) Moderate aggregation levels are possible. 2 messages saturate network. Encoding Decoding Estimation Sensing

Case of Large System Strategy Cost of comm. ＝ # of sensors data rate We can make system as large as we want! Sensing Encoding Decoding Estimation Still 2 messages saturate network.

Collective Estimation

Rate Distortion Tradeoff Variety of communication reduces to a simple rate distortion tradeoff. Rate Distortion Tradeoff Each observation bit is flipped with the same probability. Black Box

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.

Changing Variables By the change of variables we have the following result.

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

The Shannon Limit

《 Decoding 》 《 Encoding 》 Lossy Data Compression There exists tradeoff between compression rate and the resulting quality of reproduction. What is the best bound for the lossy compression? 《 Storage 》

Rate Distortion Theory Theory for compression beyond entropy rate. Best bound is the rate distortion function. Compression Rate Hamming Distortion ○ ×

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 Does LSS have any advantage over SS? The CEO can be informed!

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.

Vector Quantization

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. Isolated Model Reduces to Random Walk Statistics. Random Walk Statistics Cost Function （ Energy ） Exact Solution

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

Near Shannon Limit Algorithm

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.

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

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. Sensing Transmission Estimation BER 9% 32.4k bps 5.4k bps BER 20.0%

Example of Large System Strategy Nine sensors transmit their codewords. Sensing Encoding & Transmission & Decoding Estimation BER 20.0% BER 24.7% BER 5% 5.4k bps 32.4k bps

Statistical Mechanics

Frustrated Free Energy Free energy cannot be decoupled. General formula for Hamming Distortion Frustrated model reduces to spin glass statistics. Saddle Point of Free Energy Cost Function （ Energy ） Approximation Replica Method

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.

Large System Gain: K→∞ Bit error probability in decibel measure Coincides with 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.

References

Analysis TM and M. Okada: `Rate Distortion Function in the Spin Glass State: A Toy Model’, Advances in Neural Information Processing Systems 15, , MIT Press (2003). Available at TM and P. Davis: `Rate Distortion Codes in Sensor Networks: A System-level Analysis’, Advances in Neural Information Processing Systems 18, , MIT Press (2006). Available at

Algorithms TM: `Statistical mechanics of the data compression theorem’, Journal of Physics A 35, L95-L100 (2002). Available at /35/8/101/a208l1.htmlhttp:// 4470/35/8/101/a208l1.html TM: `Thouless-Anderson-Palmer Approach for Lossy Compression’, Physical Review E 69, (R) (2004). Available at 05

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