Real-valued continued fraction of straight lines
This work addresses the issue of unbounded behavior in straight lines for mathematical modeling, with an incremental application to image classification tasks.
The authors tackled the problem of unbounded straight lines in mathematical analysis by introducing a parametric nonlinear term to transform them into bounded curves, which converge more slowly. They demonstrated the approach on the Fashion-MNIST dataset, achieving parameters with less variance, faster convergence, and higher accuracy compared to linear methods.
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.