# A note on the capacity of the binary perceptron

arXiv: https://arxiv.org/abs/2401.15092 · HTML: https://mlsanity.com/paper/2401.15092/

## Key facts

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- First posted on arXiv 22 Jan 2024.

## Abstract (arXiv)

Determining the capacity αc of the Binary Perceptron is a long-standing problem. Krauth and Mezard (1989) conjectured an explicit value of αc, approximately equal to .833, and a rigorous lower bound matching this prediction was recently established by Ding and Sun (2019). Regarding the upper bound, Kim and Roche (1998) and Talagrand (1999) independently showed that αc < .996, while Krauth and Mezard outlined an argument which can be used to show that αc < .847. The purpose of this expository note is to record a complete proof of the bound αc < .847. The proof is a conditional first moment method combined with known results on the spherical perceptron