source code 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 Kolmogorovâ€“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.
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. |