Semialgebraic Geometry of Nonnegative Tensor Rank

Semialgebraic Geometry of Nonnegative Tensor Rank

We study the semialgebraic structure of Dr, the set of nonnegative tensors of nonnegative rank not more than r, and use the results to infer various properties of nonnegative tensor rank. We determine all nonnegative typical ranks for cubical nonnegative tensors and show that the direct sum conjecture is true for nonnegative tensor rank. Under some mild condition (non-defectivity), we show that...

The Geometry of Rank-one Tensor Completion

The geometry of the set of restrictions of rank-one tensors to some of their coordinates is studied. This gives insight into the problem of rank-one completion of partial tensors. Particular emphasis is put on the semialgebraic nature of the problem, which arises for real tensors with constraints on the parameters. The algebraic boundary of the completable region is described for tensors parame...

An Introduction to Semialgebraic Geometry

Entropy, homology and semialgebraic geometry

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Nonnegative Rank vs. Binary Rank

Motivated by (and using tools from) communication complexity, we investigate the relationship between the following two ranks of a 0-1 matrix: its nonnegative rank and its binary rank (the log of the latter being the unambiguous nondeterministic communication complexity). We prove that for partial 0-1 matrices, there can be an exponential separation. For total 0-1 matrices, we show that if the ...