For anyone who finds the post in the future, just thought I’d point out that what’s really going on is that you’re sketching with an incoherent matrix A. That is, the columns A_1,…,A_n of A have unit norm and || < eps for i != j. To recover x from y = Ax, you set x~ = A^T y. Then the incoherence property easily implies ||x~ – x||_infty < eps * ||x||_1, which is exactly the l1 point query guarantee. See http://arxiv.org/abs/1206.5725 for details.

You can form such a matrix from codes, or via other ways. It turns out that you can replace this "divisor code" in CR-Precis with Reed-Solomon codes and get a strictly better space bound. The main goal is to minimize (block length) x (alphabet size) (which will be the # of rows of A) subject to the code needing to have at least n codewords and relative distance 1 – eps (essentially you break the rows of A into (block length) blocks each of length (alphabet size), then each column of A is a codeword written in unary then normalized by 1/sqrt(block length) to get unit norm).

]]>Also, let me point out that these prime number based constructions have been used in “Improved Bounds for a Deterministic Sublinear-Time Sparse Fourier Algorithm” by Mark Iwen and C.V. Spencer. http://www.ima.umn.edu/~iwen/WebPage/improved08.pdf and other places.

Finally, when I was thinking about that Pg31 result, I had in mind a bigger technical problem: coding and decoding in sublinear space deterministically for suitable codes (Group testing, k-selectors, Reed-Muller, ….). Progress on that problem is here: Efficiently Decodable Non-adaptive Group Testing. Piotr Indyk, Hung Q. Ngo and Atri Rudra

To Appear in SODA 2010.

– Metoo

]]>Graham: Should I replace the link to your VLDB paper with a link to your CACM paper?

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