Toward Super-Polynomial Size Lower Bounds for Depth-Two Threshold Circuits
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Proving super-polynomial size lower bounds for 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 NEXP does not have poly-size THR∘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 ℓ_2-Furthest-Pair. An n^2 poly(d) / ^ω(1) n time algorithm for ℓ_2-Furthest-Pair in R^d for polylogarithmic d implies NEXP has no polynomial size THR∘THR circuits. The same holds for Hopcroft's problem, Bichrom.-ℓ_2-Closest-Pair and Integer Max-IP. 2. Shaving Logs for Approximate Bichrom.-ℓ_2-Closest-Pair. An n^2 (d) / ^ω(1) n time algorithm for (1+1/^ω(1) n)-approximation to Bichrom.-ℓ_2-Closest-Pair or Bichrom.-ℓ_1-Closest-Pair for polylogarithmic d implies NEXP has no polynomial size SYM∘THR circuits. 3. Shaving Logs for Modest Dimension Boolean Max-IP. An n^2 / ^ω(1) n time algorithm for Bichromatic Maximum Inner Product with vector dimension d = n^ϵ for any small constant ϵ would imply NEXP has no polynomial size THR∘THR circuits. Note there is an n^2polylog(n) time algorithm via fast rectangle matrix multiplication. Our results build on two structure lemmas for threshold circuits.
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