Polar lattices are good for lossy compression

Abstract

Polar lattices, which are constructed from polar codes, have recently been proved to be able to achieve the capacity of the additive white Gaussian noise (AWGN) channel. In this work, we show that polar lattices can also solve the dual problem, i.e., achieving the rate-distortion bound of a memoryless Gaussian source, which means that polar lattices can also be good for the lossy compression of continuous sources. The structure of the proposed polar lattices enables us to integrate the post-entropy coding process into the lattice quantizer, which simplifies the quantization process. Moreover, the nesting structure of polar lattices further provides solutions for some multi-terminal coding problems. The Wyner-Ziv coding problem for a Gaussian source can be solved by an AWGN capacity achieving polar lattice nested in a rate-distortion bound achieving one, and the Gelfand-Pinsker problem can be solved in a reversed manner.

Polar Codes and Polar Lattices for the Heegard–Berger Problem

“…requires no coding. Polar lattice codes that meet the classical and Wyner-Ziv ratedistortion functions for Gaussian sources, introduced in [6], can be used to achieve the HBRDF in these degenerate regions. The only non-degenerate distortion region is specified by…”

“…It is shown in [6] that polar lattices achieve the optimal rate-distortion performance for both the standard and the Wyner-Ziv compression of Gaussian sources under squared-error distortion.…”

“…A nested construction of polar codes is presented in [5] to achieve the binary Wyner-Ziv rate-distortion function. For Gaussian sources, a polar lattice to achieve both the standard and Wyner-Ziv rate-distortion functions is proposed in [6]. Different from the solutions of the Wyner-Ziv problem in [5] and [6], we need to consider the requirements for the two decoders jointly.…”

“…For Gaussian sources, a polar lattice to achieve both the standard and Wyner-Ziv rate-distortion functions is proposed in [6]. Different from the solutions of the Wyner-Ziv problem in [5] and [6], we need to consider the requirements for the two decoders jointly. Therefore, we make use of the low-fidelity reconstruction at Decoder 1, and combine it with the original source and the side information to form the nested structure that achieves the optimal distortion for Decoder 2.…”

“…Finally, we prove that polar codes achieve an exponentially decaying block error probability and excess distortion at both decoders for the binary Heegard-Berger problem.• We then consider the Gaussian Heegard-Berger problem, and propose a polar lattice construction that consists of two nested polar lattices, one of which is additive white Gaussian noise (AWGN) capacity-achieving while the other is Gaussian rate-distortion function achieving. This construction is similar to the one proposed for the Gaussian Wyner-Ziv problem in [6]. However, in the Heegard-Berger problem setting, we need to treat the difference between the original source and its reconstruction at Decoder 1 as a new source, and the difference between the original side information and the reconstruction at Decoder 1 as a new side information.…”