Entropy-Constrained One-Shot Wyner-Ziv Compression of a Uniform Source with Side Information

Core Concepts

The authors study the entropy-distortion trade-off for one-shot lossy compression of a uniform source with side information available only at the decoder. They provide upper and lower bounds for the entropy-distortion functions under two different side information models: quantized side information and noisy side information.

Abstract

The authors consider the one-shot version of the classical Wyner-Ziv problem, where a source is compressed in a lossy fashion when only the decoder has access to a correlated side information. They focus on compressing a uniform source, motivated by its role in the compression of processes with low-dimensional features embedded within a high-dimensional ambient space.
For the quantized side information model, the authors show that the entropy-distortion function with decoder-only side information is equal to the conditional entropy-distortion function, and provide an optimization problem to characterize it. They also derive easy-to-compute upper and lower bounds for this function.
For the noisy side information model, the authors provide an achievability result by dividing the unit interval into many intervals and grouping them into a single encoder output. They also derive a converse bound by assuming the side information is available at the encoder as well.
The authors show that the bounds get tighter at higher compression rates, and the difference between the entropy-distortion functions with and without side information decreases as the correlation between the source and side information increases.

Stats

The uniform source X is distributed on the interval [0, 1].
The distortion metric is the L1 distance, d(x, x̂) = |x - x̂|.

Quotes

"Motivated by practical compression techniques that operate in the finite blocklength regime, in this paper, we are interested in the one-shot version of the Wyner–Ziv problem where the encoder first quantizes a single realization of the source into a countable set and then uses variable length lossless coding to turn it into a bitstream."
"Motivated by this observation and the emergence of neural compressors in decoder-only SI settings [15]–[17], we study lossy one-shot compression of a uniform source with SI."

Key Insights Distilled From

One-Shot Wyner-Ziv Compression of a Uniform Source

by Oğuz... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.01774.pdf

Deeper Inquiries

How can the results be extended to other source distributions beyond the uniform distribution

The results obtained in this study for the one-shot Wyner-Ziv problem with a uniform source can be extended to other source distributions beyond the uniform distribution by considering the characteristics of the specific distribution. The key lies in understanding how the source distribution impacts the entropy and distortion trade-offs in the compression process. By analyzing the properties of different source distributions, such as their probability density functions and support intervals, one can adapt the optimization framework used in this work to derive entropy-distortion functions for those distributions. For instance, for non-uniform sources, the quantization intervals and encoder-decoder strategies may need to be adjusted to account for the varying probabilities associated with different source values. Additionally, the distortion metric may need to be tailored to suit the specific characteristics of the source distribution. By appropriately modifying the quantization and encoding schemes based on the source distribution, one can establish entropy-distortion bounds for a wide range of source distributions.

What are the implications of these entropy-distortion bounds on the design of distributed neural image compressors

The entropy-distortion bounds derived in this study have significant implications for the design of distributed neural image compressors. Neural image compressors aim to efficiently represent images using neural networks while minimizing distortion. By leveraging the entropy-distortion bounds obtained in this work, designers of neural image compressors can gain insights into the fundamental limits of compression for images with low-dimensional features embedded within high-dimensional ambient spaces. These bounds provide a benchmark for evaluating the performance of neural image compression algorithms and can guide the development of more effective compression techniques. Furthermore, the study's focus on decoder-only side information models can inform the design of neural image compressors that utilize correlated side information at the decoder to enhance compression efficiency. By aligning the design of neural image compressors with the entropy-distortion bounds established in this work, researchers can optimize the performance of these systems and potentially achieve better compression results for natural sources like images.

How can the techniques used in this work be applied to study the fundamental limits of compression for more complex data structures, such as manifold-structured data

The techniques employed in this work can be applied to study the fundamental limits of compression for more complex data structures, such as manifold-structured data, by adapting the framework to capture the unique characteristics of these data types. Manifold-structured data, which often exhibit low-dimensional features within high-dimensional spaces, present challenges and opportunities for compression due to their intrinsic structure. By extending the analysis to encompass manifold-structured data, researchers can investigate how the entropy-distortion trade-offs are influenced by the underlying manifold geometry and dimensionality. This extension may involve developing specialized quantization and encoding strategies tailored to the manifold structure, as well as defining distortion metrics that capture the geometric properties of the data. By applying the techniques used in this work to manifold-structured data, researchers can uncover the compression capabilities and limitations for these complex data structures, leading to advancements in compression algorithms for manifold-structured data in various applications.