Piecewise-linear interpolation
The leietanic function is a learned piecewise-linear interpolator: how it relates to polynomial regression, the hat function and wavelets.
In plain words
Any curve can be approximated by connecting dots with straight segments; use enough segments and the approximation is as good as you like. This is piecewise-linear interpolation, a method that dates back to Babylonian times. Fitting a single high-degree polynomial instead can also follow curves, but it produces huge numbers and unstable behaviour.
The map-drawing formula of the logic replicant is a learned version of connecting the dots: each folded term is one corner, and both the position and the slope of every corner are learned from the data. Two differences from the textbook version: the corners are not evenly spaced, and the segments keep going at the ends instead of dropping to zero. The vortices stop the formula from using more corners than the logic really needs, which is what keeps it from overfitting.
From polynomial regression to hat functions
Curve fitting defines a mathematical object, normally a function, that best fits a series of data points, by interpolation, smoothing or regression. Polynomial regression models a curvilinear relationship as
estimated with the mean square error
A degree-\(k\) polynomial can have \(k-1\) inflection points, a powerful property for fitting curves, but it involves very large numbers when \(|x_i|>1\) and \(k\) grows. Piecewise-linear interpolation, a method dating back to Babylonian times, avoids this. Given samples \(y_n=f(nT)\) with step \(T\), the interpolated function is built as
where the linear B-spline, or “hat” function, is
The scalar case of the leietanic function
This approach directly influences the leietanic function. When \(L\) works in a feature space of dimension \(F=1\) and \(S=1\), it takes the form
with scalars \((\lambda_a,\lambda_s)\) and \(R\) pairs \((\xi_r,\beta_r)\). This is a linear interpolator with \(R\) segments that change slope at every point given by \(\beta_r\). The terms \(|\lambda_s x+\beta_r|\) resemble the hat function, with two differences: the coefficients \(\beta_r\) are not constrained to any spacing, and the absolute value keeps decreasing or increasing beyond \(|x|>1\) instead of vanishing. Function (23) is differentiable, although its derivative is not continuous at the \(R\) points where \(\lambda_s x+\beta_r=0\). The piece limits \(\beta_r\) are learnt during training like any other parameter, and this applies to the general function (14) as well.
The intention is not complete freedom for \(L\) to capture every logic and all its exceptions, but the smooth middle point that balances a compact representation of the logic with the avoidance of overfitting. A second reading of (23) is as a linear combination of \(R\) non-static, parametric wavelets in a discrete wavelet transform: an intuitive analogy, since the differentiable base \(|\lambda_s x+\beta_r|\) imitates an orthonormal basis without being complete or satisfying the requirements of a canonical transform.
Several variables and the role of the vortices
The leietanic function is not limited to one dimension; with matrices \(\boldsymbol{\Lambda}_s\) it approximates functions of several real variables, and the more parameters \(S\) and \(R\) it has, the more expressive it becomes. That flexibility is what the vortices constrain: by admitting only a few isolated regions per class, they oblige \(L\) to map instances that may be very different in \(X\) into restricted areas of \(Q\), producing a compact logic. Low-entropy isolations discusses this balance.