Random encoding of quantized finite frame expansions

Frames, which in finite dimensions are spanning sets of vectors, generalize the notion of bases and provide a useful tool for modeling the measurement (or sampling) process in several modern signal processing applications. In the digital era, the measurement process is typically followed by a quantization, or digitization step that allows for storage, transmission, and processing using digital devices. One family of quantization methods, popular for its robustness to errors caused by circuit imperfections and for its ability to act on the measurements progressively, is Sigma-Delta quantization. In the finite frame setting, Sigma-Delta quantization, unlike scalar quantization, has recently been shown to exploit the redundancy in the measurement process leading to a more efficient rate-distortion performance. Nevertheless, on its own, it is not known whether Sigma-Delta quantization can provide optimal rate-distortion performance. In this note, we show that a simple post-processing step consisting of a discrete, random Johnson-Lindenstrauss embedding of the resulting bit-stream yields near-optimal rate distortion performance, with high probability. In other words, it near optimally compresses the resulting bit-stream. Our result holds for a wide variety of frames, including smooth frames and random frames.