Another set of Lecture notes for my course, this time about Compressed Sensing and Sparse Recovery, is available here. As usual, I will document the open problems here, while referring to a much more detailed description of the problems on the notes, including description of partial progress.

**Open Problem 6.1. (Random Partial Discrete Fourier Transform)**

Consider a obtained by sampling random rows of a Discrete Fourier Tranform. How large does need to be in order for, with high probability, to satisfy the -RIP?

**Open Problem 6.2. (Mutually Unbiased Bases)**

How many mutually unbiased bases are there in 6 dimensions?

**Open Problem 6.3. (Zauner’s Conjecture)**

Prove or disprove the SIC-POVM / Zauner’s conjecture: For any , there exists an Equiangular tight frame with vectors in dimensions. (or, there exist equiangular lines in ).

**Open Problem 6.4. (The Paley ETF Conjecture)**

Does the Paley Equiangular tight frame satisfy the Restricted Isometry Property pass the square root bottleneck? (even by logarithmic factors?).

**Open Problem 6.5. (Constructive Kadison-Singer)**

Give a (polynomial time) construction of the tight frame partition satisfying the properties required in the Kadison-Singer problem (or the related Weaver’s conjecture). These partitions were proven to exist (with a non-constructive proof) in the recent breakthrough of Marcus, Spielman, and Srivastava.

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