# Understanding the computational difficulty of a binary-weight perceptron and the advantage of input sparseness

## TL;DR

This study investigates how low-precision binary weights and input sparseness affect the computational difficulty of training perceptrons, revealing that input sparseness simplifies the solution-finding process by reducing weight correlations.

## Contribution

It demonstrates that input sparseness alleviates computational difficulty in binary perceptrons by decreasing cross-correlations among weights during decimation, providing insights into neural and artificial learning processes.

## Key findings

- Input sparseness reduces the number of steps needed to fix late-decimation weights.
- Decimation approaches the dense solution region in weight configuration space.
- Computational difficulty stems from the subspace of late-decimation-fixed variables.

## Abstract

Limited precision of synaptic weights is a key aspect of both biological and hardware implementation of neural networks. To assign low-precise weights during learning is a non-trivial task, but may benefit from representing to-be-learned items using sparse code. However, the computational difficulty resulting from low weight precision and the advantage of sparse coding remain not fully understood. Here, we study a perceptron model, which associates binary (0 or 1) input patterns with outputs using binary (0 or 1) weights, modeling a single neuron receiving excitatory inputs. We considered a decimation process, where every time step, marginal probabilities of unfixed weights were evaluated, then the most polarized weight was fixed at its preferred value. We showed that decimation is a process approaching the dense solution region in weight configuration space. In two efficient algorithms (SBPI and rBP) for solving binary-weight perceptron, most time steps are spent on determining values of the weights fixed late in decimation. This algorithmic difficult point may result from strong cross-correlation between late-decimation-fixed weights in the solution subspace where early-decimation-fixed weights take their fixed values, and is related to solution condensation in this subspace during decimation. Input sparseness reduces the time steps that SBPI and rBP need to find solutions, by reducing time steps used to assign values to late-decimation-fixed weights, due to the reduction of cross-correlation between late-decimation-fixed weights. Our work suggests that the computational difficulty of constraint satisfaction problem originates from the subspace of late-decimation-fixed variables. Our work highlights the heterogeneity of learning dynamics of weights, which may help understand axonal pruning in brain development, and inspire more efficient algorithms to train artificial neural networks.