用连续分数改造直线，使其变稳收敛，提升图像分类精度
Real-valued continued fraction of straight lines
- 用正非线性参数将直线转为有界曲线，形成实值连续分数
- 在Fashion-MNIST上参数方差更小，收敛更快，准确率更高
- 适合关注模型稳定性与参数可视化的新手或工程应用者
在无界平面上，直线常用于数学分析，但高斜率直线随自变量增长过快。本文引入正的参数化非线性项，使直线变为有界非线性曲线，其增长速率远慢于自变量。该变换可表示为直线的实值连续分数，且收敛于变换方程的解。通过欧拉方法，连续分数被化为无穷级数。在图像分类任务中，使用连续分数回归线在Fashion-MNIST数据集上估计的参数方差更小、收敛更快、精度更高。此外，多维参数估计问题可在xy平面以连续分数参数作图，展现出明显模式。
原文摘要 · Abstract (English)
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.
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