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Electronic Colloquium on Computational Complexity

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Reports tagged with margin complexity:
TR18-143 | 16th August 2018
Mark Bun, Justin Thaler

The Large-Error Approximate Degree of AC$^0$

We prove two new results about the inability of low-degree polynomials to uniformly approximate constant-depth circuits, even to slightly-better-than-trivial error. First, we prove a tight $\tilde{\Omega}(n^{1/2})$ lower bound on the threshold degree of the Surjectivity function on $n$ variables. This matches the best known threshold degree bound for any AC$^0$ ... more >>>

TR22-079 | 25th May 2022
Hamed Hatami, Pooya Hatami, William Pires, Ran Tao, Rosie Zhao

Lower Bound Methods for Sign-rank and their Limitations

The sign-rank of a matrix $A$ with $\pm 1$ entries is the smallest rank of a real matrix with the same sign pattern as $A$. To the best of our knowledge, there are only three known methods for proving lower bounds on the sign-rank of explicit matrices: (i) Sign-rank is ... more >>>

TR22-130 | 15th September 2022
Hamed Hatami, Kaave Hosseini, Xiang Meng

A Borsuk-Ulam lower bound for sign-rank and its application

We introduce a new topological argument based on the Borsuk-Ulam theorem to prove a lower bound on sign-rank.

This result implies the strongest possible separation between randomized and unbounded-error communication complexity. More precisely, we show that for a particular range of parameters, the randomized communication complexity of ... more >>>

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