Frames for the finite world: Sampling, coding and quantization

This workshop, sponsored by AIM and the NSF, will be devoted to frame theory in finite dimensions with an emphasis on compressive sensing and quantization theory for frames.

Recent years have seen significant advances in a number of subjects in signal processing and information theory in which frame theory has played a central role. Some of the most important of these "finite world" applications are analog-to-digital (A/D) conversion, coding theory, and compressive sensing. While these subjects fit well within electrical engineering, several key contributions have been made by mathematicians who have strong interest in real-world applications, resulting in fruitful interdisciplinary research collaborations.

A frame is a collection of vectors in a Hilbert space allowing for a stable linear decomposition of any element in the space, similar to a basis expansion. However, a frame is not required to be linearly independent, and consequently, expansion coefficients can be highly non-unique. This redundancy is advantageous for applications in which additional constraints need to be imposed, such as the set of coefficients being sparse or quantized. The nature of such constraints also establishes a venue for the design of specific frames. This workshop will be concerned with the construction, properties, and applications of frames in finite dimensions.

The workshop will focus on the following specific topics:

• Construction of good deterministic frames (measurement matrices) for compressive sensing.
• Quantization theory for frames: rate-distortion theory, coarse quantization, redundancy vs. robustness, A/D conversion with compressive sensing.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

For more information email workshops@aimath.org

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