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## Monday, December 07, 2015

### Nonlinearities and Random Projections

Suresh just mentioned the following:
If you want any hope of understanding the complexity of deep learning, you need to confront threshold circuits (circuits made from threshold gates). These are notoriously hard to deal with, and in many ways represent the limit of what's learnable. Daniel Kane and have a new preprint out with lower bounds for depth 2 and depth 3 threshold circuits.

At about the same time, on twitter, Laurent asked the following:

He added: "Rand. Proj. *with* non-linear operation, eg ". I usually list these under the nonlinearCS tag but I am not consistent. Laurent then mentioned the ones he had in mind:
I mentioned that

Petros pointed out to his recent work

Nikhil pointed out the

Here is the paper mentioned by Suresh at the beginning of the post:

Super-Linear Gate and Super-Quadratic Wire Lower Bounds for Depth-Two and Depth-Three Threshold Circuits by Daniel M. Kane, Ryan Williams

In order to formally understand the power of neural computing, we first need to crack the frontier of threshold circuits with two and three layers, a regime that has been surprisingly intractable to analyze. We prove the first super-linear gate lower bounds and the first super-quadratic wire lower bounds for depth-two linear threshold circuits with arbitrary weights, and depth-three majority circuits computing an explicit function.
We prove that for all ϵlog(n)/n, the linear-time computable Andreev's function cannot be computed on a (1/2+ϵ)-fraction of n-bit inputs by depth-two linear threshold circuits of o(ϵ3n3/2/log3n) gates, nor can it be computed with o(ϵ3n5/2/log7/2n) wires. This establishes an average-case size hierarchy'' for threshold circuits, as Andreev's function is computable by uniform depth-two circuits of o(n3) linear threshold gates, and by uniform depth-three circuits of O(n) majority gates.
We present a new function in P based on small-biased sets, which we prove cannot be computed by a majority vote of depth-two linear threshold circuits with o(n3/2/log3n) gates, nor with o(n5/2/log7/2n) wires.
We give tight average-case (gate and wire) complexity results for computing PARITY with depth-two threshold circuits; the answer turns out to be the same as for depth-two majority circuits.
The key is a new random restriction lemma for linear threshold functions. Our main analytical tool is the Littlewood-Offord Lemma from additive combinatorics.

Credit: NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute
The Mountainous Shoreline of Sputnik Planum
Release Date: December 4, 2015
Keywords: LORRI, Pluto
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