Capacity of single neurons and threshold functions precisely quantified
Boolean threshold functions, neuron capacity, and memory retrieval
Discrete MathematicsNeural and Evolutionary Computing
Summary
This paper addresses how much a single neuron can remember and how many memories neural networks can retrieve without errors. The authors precisely count the number of Boolean threshold functions a neuron can implement, improving previous estimates with much smaller error margins. They also find a sharp boundary where collections of memories can be perfectly recalled without false states, confirming long-standing conjectures. Their work improves understanding of neuron memory capacity and helps clarify mathematical properties of neural networks.
What this means in practice
- •For neural network architects: Improve design limits and memory capacity estimates for single-threshold neurons in artificial neural networks using precise counts of Boolean threshold functions.
- •For memory system engineers: Use sharp thresholds for errorless memory retrieval in associative memory models informed by probabilistic properties of neuron state configurations.
A theory result. No direct application yet.
Authors
Xinyuan Xie
Abstract
How much information can a single neuron remember? How many memories can neural networks retrieve without creating false memories? These questions are related to a basic question: how many Boolean threshold functions $f(x)=\operatorname{sgn}(a_0+\langle a,x\rangle)$, $x\in\{-1,1\}^n$, are there? In this paper, we show that the number $T_n$ of distinct Boolean threshold functions is \[ T_n=2\binom{2^n-1}{n}\bigl(1+O(n^{-99})\bigr). \] Equivalently, the capacity of a single threshold neuron is $n^2-\log_2(n!)+1+O(n^{-99})$ bits, improving the $O(n)$ error term in the result of Kahn--Komlós--Szemerédi to $O(n^{-99})$. To prove this, we show that, for $1\le r\le n-1$, and $v_1,\ldots,v_r$ are chosen at random from $\{-1,1\}^n$, \[ \mathbb P\!\left\{ \langle v_1,\ldots,v_r\rangle\cap\{-1,1\}^n =\{\pm v_1,\ldots,\pm v_r\} \right\} =1-O(n^{-99}). \] In the context of the Kanter--Sompolinsky Hamiltonian for memory retrieval, this identifies $r=n-1$ as a sharp threshold, at which, for almost every collection of $r$ memories, the only ground states are these memories and their negatives, confirming a weaker form of the Kalai--Linial--Odlyzko conjecture. It also settles a recent open problem posed by M. Anthony on the specification number of Boolean threshold functions. In addition, we show that, for every $1\le r\le n-1$, \[ \mathbb P\{v_1,\ldots,v_r\text{ are linearly dependent}\} =2\binom r2\,2^{-n}+O\!\left(2^{-n}e^{-cn}\right), \] confirming a conjecture of Kahn--Komlós--Szemerédi.