Smaller Depth-2 Linear Circuits for Disjointness Matrices
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917347681370112 |
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| author | Ye, Lixi |
| author_facet | Ye, Lixi |
| contents | 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 $0.3199$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_15565 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Smaller Depth-2 Linear Circuits for Disjointness Matrices Ye, Lixi Computational Complexity 68Q17, 68Q25 F.2.0; F.1.1 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 $0.3199$. |
| title | Smaller Depth-2 Linear Circuits for Disjointness Matrices |
| topic | Computational Complexity 68Q17, 68Q25 F.2.0; F.1.1 |
| url | https://arxiv.org/abs/2603.15565 |