Fitting Multilinear Polynomials for Logic Gate Networks
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910204572991488 |
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| author | Kim, Youngsung |
| author_facet | Kim, Youngsung |
| contents | We study learnable logic gate networks that stack layers of 2-input Boolean gates to build combinational circuits. Every 2-input gate has a unique multilinear polynomial with 4 coefficients, so the 16 Boolean gates form a codebook of prototypes in a 4-dimensional space, reducing training to a vector-quantization problem. The baseline method, Soft-Mix, learns a 16-dimensional softmax over gate identities, but the codebook has rank~4: 11 of 15 simplex directions carry nullspace gradient, and at uniform initialization the backward signal vanishes exactly. We prove that no affine product reparameterization fixes the resulting interaction-coefficient starvation under STE, and show that the covariance Jacobian of soft-VQ selection bypasses it by coupling the starved coefficient to the always-active constant channel. Working in the 4-dimensional polynomial space reduces each neuron from 16 to 4 parameters. On seven datasets, at least one 4-parameter method matches or exceeds Soft-Mix on every dataset; the CovJac advantage over STE grows monotonically with interaction demand across all seven datasets. At depth, Soft-Mix collapses ($-37.3$pp on CIFAR-10 at 12 layers) while CovJac holds ($-0.5$pp on CIFAR-10, stable on MNIST). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08657 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Fitting Multilinear Polynomials for Logic Gate Networks Kim, Youngsung Machine Learning Artificial Intelligence We study learnable logic gate networks that stack layers of 2-input Boolean gates to build combinational circuits. Every 2-input gate has a unique multilinear polynomial with 4 coefficients, so the 16 Boolean gates form a codebook of prototypes in a 4-dimensional space, reducing training to a vector-quantization problem. The baseline method, Soft-Mix, learns a 16-dimensional softmax over gate identities, but the codebook has rank~4: 11 of 15 simplex directions carry nullspace gradient, and at uniform initialization the backward signal vanishes exactly. We prove that no affine product reparameterization fixes the resulting interaction-coefficient starvation under STE, and show that the covariance Jacobian of soft-VQ selection bypasses it by coupling the starved coefficient to the always-active constant channel. Working in the 4-dimensional polynomial space reduces each neuron from 16 to 4 parameters. On seven datasets, at least one 4-parameter method matches or exceeds Soft-Mix on every dataset; the CovJac advantage over STE grows monotonically with interaction demand across all seven datasets. At depth, Soft-Mix collapses ($-37.3$pp on CIFAR-10 at 12 layers) while CovJac holds ($-0.5$pp on CIFAR-10, stable on MNIST). |
| title | Fitting Multilinear Polynomials for Logic Gate Networks |
| topic | Machine Learning Artificial Intelligence |
| url | https://arxiv.org/abs/2605.08657 |