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Smaller Depth-2 Linear Circuits for Disjointness Matrices

Lixi Ye

Abstract

We prove two new upper bounds for depth-2 linear circuits computing the $N$th disjointness matrix $D^{\otimes N}$. First, we obtain a circuit of size $O\big(2^{1.24485N}\big)$ over $\{0,1\}$. Second, we obtain a circuit of degree $O\big(2^{0.3199N}\big)$ over $\{0,\pm 1\}$. These improve the previous bounds of Alman and Li, namely size $O\big(2^{1.249424N}\big)$ and degree $O\big(2^{N/3}\big)$. Our starting point is the rebalancing framework developed in a line of works by Jukna and Sergeev, Alman, Sergeev, and Alman-Guan-Padaki, culminating in Alman and Li. We sharpen that framework in two ways. First, we replace the earlier "wild" rebalancing process by a tame, discretized process whose geometric-average behavior is governed by the quenched top Lyapunov exponent of a random matrix product. This allows us to invoke the convex-optimization upper bound of Gharavi and Anantharam. Second, for the degree bound we work explicitly with a cost landscape on the $(p,q)$-plane and show that different circuit families are dominant on different regions, so that the global maximum remains below $0.3199$.

Smaller Depth-2 Linear Circuits for Disjointness Matrices

Abstract

We prove two new upper bounds for depth-2 linear circuits computing the th disjointness matrix . First, we obtain a circuit of size over . Second, we obtain a circuit of degree over . These improve the previous bounds of Alman and Li, namely size and degree . Our starting point is the rebalancing framework developed in a line of works by Jukna and Sergeev, Alman, Sergeev, and Alman-Guan-Padaki, culminating in Alman and Li. We sharpen that framework in two ways. First, we replace the earlier "wild" rebalancing process by a tame, discretized process whose geometric-average behavior is governed by the quenched top Lyapunov exponent of a random matrix product. This allows us to invoke the convex-optimization upper bound of Gharavi and Anantharam. Second, for the degree bound we work explicitly with a cost landscape on the -plane and show that different circuit families are dominant on different regions, so that the global maximum remains below .
Paper Structure (18 sections, 2 theorems, 32 equations)

This paper contains 18 sections, 2 theorems, 32 equations.

Table of Contents

  1. The Three Circuits
  2. The Tame Rebalancing Process
  3. Vertin, Sonetto, and Regulus
  4. Vertin.
  5. Sonetto.
  6. Regulus.
  7. Regulus$^\mathsf{T}$.
  8. The Size and Degree Bounds
  9. The Lyapunov Exponent
  10. Reducing to the Lyapunov Exponent
  11. Upper Bound via Convex Optimization
  12. Machine Verification of the Bounds
  13. Forcing a Representable Ceiling
  14. Verification of the Size Bound
  15. Verification of the Degree Bound
  16. ...and 3 more sections

Key Result

Theorem 2

For any finite circuit $C$ with degree polynomials $f,g$ and any fixed $\varepsilon>0$, there exists a family of depth-2 linear circuits computing $D^{\otimes N}$ for all $N\geq 0$ whose size is

Theorems & Definitions (8)

  • Definition 1
  • Theorem 2
  • proof : Proof sketch
  • Definition 3
  • Theorem 4
  • proof : Proof sketch
  • Claim 5
  • proof : Proof sketch