Lattices for Distributed Source Coding - Reconstruction of a Linear function of Jointly Gaussian Sources -D. Krithivasan and S. Sandeep Pradhan - University of Michigan, Ann Arbor TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A A A AAA A A

Presentation Overview Problem Formulation A Straightforward coding scheme using random coding A new coding scheme using lattice coding –Motivation for the coding scheme –An overview of lattices and lattice coding –A lattice based coding scheme Results and Extensions Conclusion

Problem Formulation Source (X 1,X 2 ) is a bivariate Gaussian. Encoders observe X 1,X 2 separately and quantize them at rates R 1,R 2 Decoder interested in a linear function Z, X 1 -cX 2 Lossy reconstruction to within mean square distortion D Objective: Achievable rates (R 1,R 2 ) at distortion D

Pictorial Representation Reconstruct, the best estimate of Z = F(X,Y)

Features of the proposed scheme Multi-terminal source coding problem. Random coding techniques typically used to tackle such problems. Our approach based on “structured” lattice codes. –Previously, structured codes used only to achieve known bounds with low complexity. –Our coding scheme relies critically on the structure of the code. –Structured codes give performance gains not attainable by random codes.

Berger-Tung based Coding Scheme Encoders: Quantize X 1 to W 1, X 2 to W 2. Transmit W 1 and W 2 Decoder: Reconstruct Can use Gaussian test channels for P(W 1 j X 1 ) and P(W 2 j X 2 ) to derive achievable rates and distortion Known to be optimal for c<0 with Gaussian test channels

Motivation for Our Coding Scheme Korner and Marton considered lossless coding of Z = X 1 © X 2 Coding scheme if the encoding is centralized? –Compute Z = X 1 © X 2. Compress it to f(Z) and transmit. –f( ¢ ) is any good source coding scheme. Suppose f( ¢ ) distributes over © ? –Compress X 1, X 2 as f(X 1 ) and f(X 2 ). –Decoder computes f(X 1 ) © f(X 2 ) = f(X 1 © X 2 ) = f(Z). No difference between centralized and distributed coding. Choosing f( ¢ ) as a linear code will work.

An Overview of Lattices An n dimensional lattice ¤ – collection of integer combinations of columns of a n £ n generator matrix G. Nearest neighbor quantizer Quantization error : x mod ¤, x – Q ¤ (x) Voronoi region Second moment ¾ 2 ( ¤ ) of lattice ¤ – expected value per dimension of a random vector uniformly distributed in Normalized second moment G( ¤ ) = ¾ 2 ( ¤ ) / V 2/n`

Introduction to Lattice Codes Lattice code – a subset of the lattice points are used as codewords. Have been used for both source and channel coding problems. Various notions of goodness: –Good source D-code if log (2 ¼ eG( ¤ )) · ² and ¾ 2 ( ¤ )=D –Good channel ¾ 2 Z ( ¤ ) if it achieves the Poltyrev exponent on the unconstrained AWGN channel of variance Why Lattice codes? –Lattices achieve the Gaussian rate distortion bound –Lattice encoding distributes over Z = X 1 – c X 2

Nested Lattice Codes ( ¤ 1, ¤ 2 ) is a nested lattice if ¤ 1 ½ ¤ 2 Nested lattice codes widely used in multiterminal communications. Need nested lattice codes such that ¤ 1 and ¤ 2 are good source and channel codes. Such nested lattices termed “good nested lattice codes”. Known that such nested lattices do exist in sufficiently high dimension.

The Coding Scheme Good nested lattice code ( ¤ 11, ¤ 12, ¤ 2 ), ¤ 2 ½ ¤ 11, ¤ 12 Dithers Note: second encoder scales the input before encoding.

The Coding Scheme contd. Lattice parameters Rate of a nested lattice ( ¤ 1, ¤ 2 ) is Encoder rates are Sum rate R 1 + R 2 = log achievable at distortion D.

The Coding Theorem The set of all rate-distortion tuples (R 1,R 2,D) that satisfy are achievable.

Proof Outline Key Idea: Distributive property of lattice mod operation Using this, one of the mod- ¤ 2 can be removed from the signal path. Simplified but equivalent coding scheme is

Proof Outline contd. e q 1, e q 2 - subtractive dither quantization noises independent of the sources. Decoder operation But ¤ 2 is a good ¾ 4 Z /( ¾ 2 Z -D) channel code. Implies ((Z+e q ) mod ¤ 2 ) = (Z+e q ) with high probability Decoder reconstruction Can be checked that

Comments about the Coding Scheme Larger rate region than the random coding scheme for some source statistics and some range of D. Coding scheme is lattice vector quantization followed by “correlated” lattice-structured binning. Previously lattice coding used only to achieve known performance bounds. Our theorem – first instance of lattice coding being central to improving the rate region.

Extensions to more than Two Sources Reconstructing –Some sources coded together using the “correlated” binning strategy of the lattice coding scheme. –Others coded independently using the Berger-Tung coding scheme. –Decoder has some side information – previously decoded sources. –Possible to present a unified rate region combining all such strategies.

Comparison of the Rate Regions Compare sum rates for ½ = 0.8 and c = 0.8 Lattice based scheme – lower sum rates for small distortions. Time sharing between the two schemes – Better rate region than either scheme alone.

Range of values for Lower Sum Rate Shows where ( ½,c) should lie for lattice sum rate to be lower than Berger- Tung sum rate for some distorion D. Contour marked R – lattice sum rate lower by R units Improvement only for c > 0.

Conclusion Considered lossy reconstruction of a function of the sources in a distributed setting. Presented a coding scheme of vector quantization followed by “correlated” binning using lattices. Improves upon the rate region of the natural random coding scheme. Currently working on extending the scheme to discrete sources and arbitrary functions.