Accuracy and performance of Kolmogorov-Arnold model

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Kolmogorov-Arnold representation is an adequate replacement of continuous multivariate function by hierarchical tree of the functions of one variable

$$y = G(x_1, x_2, x_3, ... , x_n) = \sum_{i=0}^{2n} \Phi_i \left(\sum_{j=1}^{n} f_{i,j}(x_{j})\right).$$

This article shows the accuracy test with comparison to other popular AI models. The identification technique used in test is published in the article A deep machine learning algorithm for construction of the Kolmogorov–Arnold representation and video.

Making challenging tests for AI appeared the challenging task itself. For example, the article A Scalable Continuous Unbounded Optimisation Benchmark Suite from Neural Network Regression offers 54 synthetic data sets, which did not challenge the tested model.

On that reason I suggested my own test. The modelled (target) value is the area of triangle built on three randomly selected points within the square with the side [0, 100]. The minimum area is 0, maximum possible 5000. The inputs are 6 figures which are X,Y positions of three points forming a triangle. The challenging feature is non-correlated inputs. The large coordinate values does not mean that target is large value as well and vice versa. The dataset is 10 000 records, 80% is used for training and 20% for validation. The Kolmogorov-Arnold model was compared to MATLAB Regression Learner. The test results are shown in the table.

# Model name Activation RMSE training time in sec
1 Narrow Neural Network Sigmoid 624.66 17.12
2 Medium Neural Network Sigmoid 516.71 43.48
3 Wide Neural Network Sigmoid 338.74 6.73
4 Bilayered Neural Network Sigmoid 654.43 13.58
5 Trilayered Neural Network Sigmoid 663.11 11.60
6 Narrow Neural Network ReLU 392.62 13.04
7 Medium Neural Network ReLU 262.07 32.54
8 Wide Neural Network ReLU 141.52 81.89
9 Bilayered Neural Network ReLU 293.77 18.90
10 Trilayered Neural Network ReLU 166.82 24.96
11 Kolmogorov-Arnold quick 220.34 0.94
12 Kolmogorov-Arnold long 140.83 21.47

The difference between QUICK and LONG is in number of iterations through data, they are 10 and 300 accordingly. RMSE near 140 is an error near 3% of the target range. The errors near 600 is very low accuracy, it is about 12% of target range for an algebraic formula.