[Paper Review] Real-valued continued fraction of straight lines

Vijay Prakash S|arXiv (Cornell University)|Dec 16, 2024

Meromorphic and Entire Functions4 citations

TL;DR

This paper introduces a real-valued continued fraction representation of straight lines by embedding a nonlinear bounding term, transforming unbounded linear models into bounded nonlinear curves. The method uses a parametric cubic equation solvable via continued fraction expansion, enabling faster, more stable, and accurate regression in image classification—demonstrated on Fashion-MNIST with reduced variance and monotonic convergence compared to linear models.

ABSTRACT

In an unbounded plane, straight lines are used extensively for mathematical analysis. They are tools of convenience. However, those with high slope values become unbounded at a faster rate than the independent variable. So, straight lines, in this work, are made to be bounded by introducing a parametric nonlinear term that is positive. The straight lines are transformed into bounded nonlinear curves that become unbounded at a much slower rate than the independent variable. This transforming equation can be expressed as a continued fraction of straight lines. The continued fraction is real-valued and converges to the solutions of the transforming equation. Following Euler's method, the continued fraction has been reduced into an infinite series. The usefulness of the bounding nature of continued fraction is demonstrated by solving the problem of image classification. Parameters estimated on the Fashion-MNIST dataset of greyscale images using continued fraction of regression lines have less variance, converge quickly and are more accurate than the linear counterpart. Moreover, this multi-dimensional parametric estimation problem can be expressed on $xy-$ plane using the parameters of the continued fraction and patterns emerge on planar plots.

Motivation & Objective

• To address the unbounded growth of high-slope linear models in regression by introducing a nonlinear bounding mechanism.
• To reformulate the resulting nonlinear equation as a real-valued continued fraction that converges to the true solution.
• To demonstrate the method’s superiority in parameter estimation for machine learning tasks, particularly in reducing variance and improving convergence speed.
• To enable multi-dimensional parameter estimation to be visualized on 2D planar plots using the parameters $a$ and $m$.
• To establish a bounded nonlinear coordinate system where straight lines emerge as asymptotic limits.

Proposed method

• The core equation $ y = \frac{mx}{1 + a y^2} $ is derived to constrain $ y $-values, ensuring boundedness for all $ x $.
• The equation is rewritten as a continued fraction: $ y = \frac{mx}{1 + a \frac{m^2 x^2}{(1 + a (\cdots)^2)^2}} $, which is real-valued and convergent.
• Euler’s method is applied to reduce the continued fraction into an infinite series for analytical and numerical analysis.
• The solution is decomposed into two components: $ i = -\frac{1}{3} t^{1/3} $ and $ ii = \frac{1}{a} t^{-1/3} $, which serve as orthogonal axes in a transformed plane.
• Parameter estimation for image classification uses iterative optimization with $ a $ and $ m $ as primary variables, where convergence of $ a $ ensures convergence of all other parameters.
• Final model parameters are visualized on $ xy $-plane and $ (i,ii) $-plane plots to reveal categorical patterns and sensitivity to initial conditions.

Figure 1 : $y$ for different $a$ and $m$ values. As $a\rightarrow\infty$ we approach the $x-$ axis.

Experimental results

Research questions

• RQ1 Can a nonlinear transformation of linear models yield bounded, stable, and convergent regression curves?
• RQ2 Does expressing the transformed model as a real-valued continued fraction improve convergence speed and parameter stability in machine learning tasks?
• RQ3 Can multi-dimensional parameter estimation in regression be effectively visualized on 2D planar plots using the parameters $ a $ and $ m $?
• RQ4 How do the parameters $ a $ and $ m $ influence the shape and boundedness of the resulting nonlinear curves?
• RQ5 What is the impact of the continued fraction representation on variance and accuracy in image classification compared to linear regression?

Key findings

• Regression parameters estimated using the continued fraction method showed significantly less variance than linear regression counterparts on the Fashion-MNIST dataset.
• The method achieved faster and more stable convergence, with most parameters converging monotonically within 50 iterations.
• For the 1st category, $ a^{(50)} = 0.03415 $ and $ m^{(50)} = 3.28946 $, while for the 2nd category, $ a^{(50)} = 0.00873 $ and $ m^{(50)} = 0.55568 $, indicating stable convergence.
• The $ (i,ii) $-plane representation revealed distinct patterns for different categories, with sensitivity to initial conditions observable in the i-ii plots.
• The final values of $ a $ and $ m $ were unique per class, enabling effective classification and visualization of model behavior.
• The continued fraction representation allowed for adaptive step size and bounded parameter updates, contributing to improved model stability.

(a) $a\in[1e-1,1]$ and $m\in$ interval [-200,200].