Polar Lattices for Lossy Compression

In this work, we propose a new construction of polar lattices to achieve the rate-distortion bound of a memoryless Gaussian source. The structure of the proposed polar lattices allows to integrate entropy coding into the lattice quantizer, which greatly simplifies the compression process. The overall complexity of encoding and decoding is
$O(N \log ^{2} N)$
for any target distortion and fixed rate larger than the rate-distortion bound. Moreover, the nesting structure of polar lattices provides solutions to various multi-terminal coding problems. The Wyner-Ziv coding problem for a Gaussian source can be solved by using a capacity-achieving polar lattice for the Gaussian channel, nested with a rate-distortion bound achieving lattice, while the Gelfand-Pinsker problem can be solved in a reversed manner. The polar lattice quantizer is further extended to extract Wyner’s common information of a pair of Gaussian sources or multiple Gaussian sources.