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Real-valued continued fraction of straight lines

Vijay Prakash S

TL;DR

The paper tackles the issue of unbounded responses from high-slope lines by introducing a positive nonlinear term, yielding a real-valued continued fraction representation that bounds $y$ relative to $x$ and converges to the real root of a cubic in $y$ via an Euler-based reduction. It derives the real solution, analyzes its properties, and reformulates the problem as a continued fraction, enabling a bounded, data-driven parametric regression framework. The method is demonstrated on Fashion-MNIST classification, where estimating the nonlinear coordinate $(a,m)$ together with regression weights yields lower variance, faster convergence, and higher accuracy than the linear baseline, with additional interpretability from $xy$-plane and $i$-$ii$ plane representations. This approach offers a robust, bounded alternative for high-dimensional parametric regression and provides practical insights for pattern discovery in learned 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.

Real-valued continued fraction of straight lines

TL;DR

The paper tackles the issue of unbounded responses from high-slope lines by introducing a positive nonlinear term, yielding a real-valued continued fraction representation that bounds relative to and converges to the real root of a cubic in via an Euler-based reduction. It derives the real solution, analyzes its properties, and reformulates the problem as a continued fraction, enabling a bounded, data-driven parametric regression framework. The method is demonstrated on Fashion-MNIST classification, where estimating the nonlinear coordinate together with regression weights yields lower variance, faster convergence, and higher accuracy than the linear baseline, with additional interpretability from -plane and - plane representations. This approach offers a robust, bounded alternative for high-dimensional parametric regression and provides practical insights for pattern discovery in learned 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 plane using the parameters of the continued fraction and patterns emerge on planar plots.

Paper Structure

This paper contains 11 sections, 12 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. Real solution and its properties
  3. Properties:
  4. Image classification problem
  5. Parametric representation:
  6. Implementation:
  7. Discussions
  8. Boundedness
  9. Convergence
  10. $xy-$plane representation
  11. Conclusions

Figures (9)

  • Figure 1: $y$ for different $a$ and $m$ values. As $a\rightarrow\infty$ we approach the $x-$ axis.
  • Figure 2: Polar plots of Eqn. (\ref{['eq1']}) with $r=1/\sqrt{a}$
  • Figure 3: Plots of individual components i and ii of $y$ of Eqn. (\ref{['eq3']}).
  • Figure 4: (a) shows i and ii for a fixed $a$. (b) With i and ii as axes, we get unique curves for each $a\in[1e-3,5e-3,1e-2,0.1,1]$ and for various values of $m$.
  • Figure 6: (a) One of the sample images of Fashion-MNIST dataset, (b) A section of the sample image with pixel values shown and (c) Normalized values of (b)
  • ...and 4 more figures