
TL;DR
This paper disproves the Gotsman-Linial conjecture by showing that the proposed symmetric polynomial does not always maximize average sensitivity of degree-d polynomial threshold functions, for most values of n and d.
Contribution
The paper provides a counterexample to the Gotsman-Linial conjecture and clarifies the conditions under which the conjecture does or does not hold.
Findings
The conjecture is false for almost all d and n.
The conjecture holds in many remaining cases.
Counterexamples demonstrate the limits of the conjecture's validity.
Abstract
In 1991, Craig Gotsman and Nathan Linial conjectured that for all and , the average sensitivity of a degree- polynomial threshold function on variables is maximized by the degree- symmetric polynomial which computes the parity function on the layers of the hypercube with Hamming weight closest to . We refute the conjecture for almost all and for almost all , and we confirm the conjecture in many of the remaining cases.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
