Lattices for Distributed Source Coding Reconstruction of a
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
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 -c. X 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 (2¼e. G(¤)) �² and ¾ 2(¤)=D – Good channel ¾ 2 –code if it achieves the Poltyrev exponent on the unconstrained AWGN channel of variance ¾ 2 • 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 D. achievable at distortion
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. • eq 1, eq 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+eq) mod ¤ 2) = (Z+eq) 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.
- Slides: 20