Toward Super-Polynomial Size Lower Bounds for Depth-Two Threshold Circuits

TL;DR

This paper links faster algorithms for fundamental geometric and algebraic problems to proving super-polynomial lower bounds for depth-two threshold circuits, advancing the understanding of circuit complexity.

Contribution

It demonstrates that slight improvements in algorithms for specific problems imply significant circuit lower bounds, following Williams' framework.

Findings

Faster algorithms for $ extsf{$ extbf{ extit{ extlnot}}$-Furthest-Pair} $ imply $ extsf{NEXP}$ lower bounds.

Approximate $ extsf{Bichrom.-$ extbf{ extit{ extlnot}}$-Closest-Pair} algorithms imply $ extsf{NEXP}$ lower bounds.

Algorithms for Max-IP with small dimension imply circuit lower bounds.

Abstract

Proving super-polynomial size lower bounds for $\textsf{TC}^0$, the class of constant-depth, polynomial-size circuits of Majority gates, is a notorious open problem in complexity theory. A major frontier is to prove that $\textsf{NEXP}$ does not have poly-size $\textsf{THR} \circ \textsf{THR}$ circuit (depth-two circuits with linear threshold gates). In recent years, R.~Williams proposed a program to prove circuit lower bounds via improved algorithms. In this paper, following Williams' framework, we show that the above frontier question can be resolved by devising slightly faster algorithms for several fundamental problems: 1. Shaving Logs for $\textsf{$\ell_2$-Furthest-Pair}$. An $n^2 \textrm{poly}(d) / \log^{\omega(1)} n$ time algorithm for $\textsf{$\ell_2$-Furthest-Pair}$ in $\mathbb{R}^d$ for polylogarithmic $d$ implies $\textsf{NEXP}$ has no polynomial size $\textsf{THR} \circ \textsf{THR}$ circuits. The same holds for Hopcroft's problem, $\textsf{Bichrom.-$\ell_2$-Closest-Pair}$ and Integer $\textsf{Max-IP}$. 2. Shaving Logs for Approximate $\textsf{Bichrom.-$\ell_2$-Closest-Pair}$. An $n^2 \textrm(d) / \log^{\omega(1)} n$ time algorithm for $(1+1/\log^{\omega(1)} n)$-approximation to $\textsf{Bichrom.-$\ell_2$-Closest-Pair}$ or $\textsf{Bichrom.-$\ell_1$-Closest-Pair}$ for polylogarithmic $d$ implies $\textsf{NEXP}$ has no polynomial size $\textsf{SYM}\circ\textsf{THR}$ circuits. 3. Shaving Logs for Modest Dimension Boolean $\textsf{Max-IP}$. An $n^2 / \log^{\omega(1)} n$ time algorithm for Bichromatic Maximum Inner Product with vector dimension $d = n^\epsilon$ for any small constant $\epsilon$ would imply $\textsf{NEXP}$ has no polynomial size $\textsf{THR} \circ \textsf{THR}$ circuits. Note there is an $n^2\textrm{polylog}(n)$ time algorithm via fast rectangle matrix multiplication. Our results build on two structure lemmas for threshold circuits.