Lossy Compression of Decimated Gaussian Random Walks

TL;DR

This paper analyzes the lossy compression of decimated Gaussian random walks, deriving minimal distortion formulas for different encoding schemes and highlighting the impact of encoder-side decimation knowledge.

Contribution

It provides a closed-form expression for minimal distortion in estimate-and-compress schemes and compares it with compress-and-estimate schemes, revealing the importance of encoder-decimation awareness.

Findings

Closed-form minimal distortion formula for estimate-and-compress scheme

Explicit distortion evaluation for compress-and-estimate scheme

Nonzero performance gap demonstrating encoder decimation importance

Abstract

We consider the problem of estimating a Gaussian random walk from a lossy compression of its decimated version. Hence, the encoder operates on the decimated random walk, and the decoder estimates the original random walk from its encoded version under a mean squared error (MSE) criterion. It is well-known that the minimal distortion in this problem is attained by an estimate-and-compress (EC) source coding strategy, in which the encoder first estimates the original random walk and then compresses this estimate subject to the bit constraint. In this work, we derive a closed-form expression for this minimal distortion as a function of the bitrate and the decimation factor. Next, we consider a compress-and-estimate (CE) source coding scheme, in which the encoder first compresses the decimated sequence subject to an MSE criterion (with respect to the decimated sequence), and the original random walk is estimated only at the decoder. We evaluate the distortion under CE in a closed form and show that there exists a nonzero gap between the distortion under the two schemes. This difference in performance illustrates the importance of having the decimation factor at the encoder.