When understood and used properly, the constraint W = tanh(Ŵ) ⊙ σ(M̂) (introduced in NALU by Trask et al. 2018 ) creates a unique parameter topology where optimal weights for discrete operations can be calculated rather than learned . During training, they're able to converge with extreme speed and reliability towards the optimal solution.
Most neural networks struggle with basic arithmetic. They approximate, they fail on extrapolation, and they're inconsistent. But what if there was a way to make them systematically reliable at discrete selection tasks? Is Neural Arithmetic as we know it a discrete selection task?
How four fundamental trigonometric products enable trigonometric operations
How projecting inputs onto the unit circle allows for trigonometric operations
How exponential primitives with specific weights perform operations
How matrix multiplication with specific weights performs mathematical operations
It's difficult to imagine that neural arithmetic has such a simple solution. Play with these widgets to see how setting just a few weights to specific values creates reliable mathematical operations. Each primitive demonstrates machine-precision mathematics through discrete selection.
The Problem: Optimization vs. Discrete Selection
Now that you've seen discrete weight configurations producing perfect mathematics, let's understand why this is remarkable. There's a fundamental tension between what neural network optimizers do naturally and what discrete selection requires.
🎯 Discrete Selection Needs Mathematical operations require specific weight values: [1, 1] → Addition
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