In celebration of the recent resolution of the Kadison-Singer problem by Marcus, Spielman, and Srivastava, here is a question on isotropic point sets on which Kadison-Singer does not (seem to) shed any light. A positive resolution would likely have strong implications for the Sparsest Cut problem and SDP hierarchies. The question arose in discussions with Shayan Oveis Gharan, Prasad Raghavendra, and David Steurer.

**Open Question:** Do there exist constants such that for any , the following holds? Let be a collection of orthonormal bases and define . Then there are subsets with and .

[Some additional notes: One piece of intuition for why the question should have a positive resolution is that these orthonormal bases which together comprise at most vectors cannot possibly "fill" -dimensional space in a way that achieves -dimensional isoperimetry. One would seem to need points for this.

One can state an equivalent question in terms of vertex expansion. Say that a graph on vertices is a *vertex expander* if for all subsets with . Here, denotes all the nodes that are in or are adjacent to . Then one can ask whether there exists a 1-1 mapping from to for some orthonormal bases such that the endpoints of every edge are mapped at most apart (as ).]

What would be the implication for sparsest cut?

Comment by minilek — June 27, 2013 @ 5:22 pm

Strictly speaking, there is no direct implication because this question is the “hard part” of a more general question. The more general question is to find such separated sets given any set of unit vectors in isotropic position.

The implication is as follows: For every , there is a number such that Sparsest Cut admits a -approximation in time , i.e. sub-exponential time constant factor approximations to (uniform) Sparsest Cut.

Comment by James L. — June 27, 2013 @ 5:26 pm

What is your precise definition of isotropic position? Is it that the covariance matrix (the sum of uu^T over vectors u in the set) is a multiple of the identity or is it exactly the identity? Seems like it’s the former, but just want to make sure.

Comment by NG — June 28, 2013 @ 8:25 pm

More specifically, the general condition is as follows: Given unit vectors , we assume that . Of course, a special case is when the vectors can be partitioned into orthonormal bases.

Comment by James L. — June 28, 2013 @ 10:25 pm

Is it important that the bases are over the reals? I think I can see a construction of such sets over complex vector spaces.

Comment by sflammia — July 5, 2013 @ 1:06 am

A general construction of such A,B over would be equally remarkable (and useful).

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