Mathematics of Information and Applications Seminar: Rayan Saab

Frames 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, such a measurement process is typically followed by quantization, or digitization.

We discuss the quantization of frame coefficients using a scheme known as Sigma-Delta quantization. We show that a simple encoding via a discrete random Johnson-Lindenstrauss embedding of the integrated bit-stream yields near-optimal approximation error (as a function of the number of bits used). The result holds with high probability on the draw of the embedding, allows efficient reconstruction, and holds for a wide class of frames including random frames and deterministic smooth frames.

In addition, we show that if the same encoding scheme is applied to quantized compressed sensing measurements (albeit with a different reconstruction scheme), it also yields near-optimal approximation accuracy as a function of the bit-rate. Our results hold for Gaussian and sub-Gaussian compressed sensing matrices.