基于截断ANOVA分解的快速可解释支持向量分类
Fast and interpretable Support Vector Classification based on the truncated ANOVA decomposition
AI总结：
本文提出在原始形式下使用仅依赖少量维度的多元基函数求解SVM，并利用截断ANOVA分解和$\ell_1$范数正则化实现快速、可解释的分类，在人工与真实数据集上取得更好结果。
摘要：
支持向量机（SVM）是对散乱数据进行分类的重要工具，通常需要在高维空间中处理大量数据点。我们提出在原始形式下使用基于三角函数或小波的特征映射来求解SVM。在小维度设置下，快速傅里叶变换（FFT）及相关方法是处理所考虑基函数的强大工具。对于不断增长的维度，经典的基于FFT的方法由于维数灾难而变得低效。因此，我们将自己限制在多元基函数上，每个基函数仅依赖于少量维度。这一做法受到众所周知的效应稀疏性以及关于从散乱数据以截断方差分析（ANOVA）分解形式重构函数的近期结果的启发，这使得所得模型在特征重要性及其耦合方面甚至具有可解释性。使用小叠加维度的结果是，计算量不再随维度呈指数增长，而仅呈多项式增长。为了在基系数方面增强稀疏性，我们使用了常用的$\ell_2$范数，并额外使用了$\ell_1$范数正则化。所找到的分类函数是基函数的线性组合，其方差随后可以按照经典的ANOVA函数分解进行分析。基于数值示例，我们表明能够恢复出与模型假设完全吻合的函数的符号。此外，我们在不同的人工数据集和真实世界数据集上执行了分类。在准确性和可解释性清晰度方面，使用$\ell_1$范数正则化均获得了更好的结果。
英文摘要：
Support Vector Machines (SVMs) are an important tool for performing classification on scattered data, where one usually has to deal with many data points in high-dimensional spaces. We propose solving SVMs in primal form using feature maps based on trigonometric functions or wavelets. In small dimensional settings the Fast Fourier Transform (FFT) and related methods are a powerful tool in order to deal with the considered basis functions. For growing dimensions the classical FFT-based methods become inefficient due to the curse of dimensionality. Therefore, we restrict ourselves to multivariate basis functions, each of which only depends on a small number of dimensions. This is motivated by the well-known sparsity of effects and recent results regarding the reconstruction of functions from scattered data in terms of truncated analysis of variance (ANOVA) decompositions, which makes the resulting model even interpretable in terms of importance of the features as well as their couplings. The usage of small superposition dimensions has the consequence that the computational effort no longer grows exponentially but only polynomially with respect to the dimension. In order to enforce sparsity regarding the basis coefficients, we use the frequently applied $\ell_2$-norm and, in addition, $\ell_1$-norm regularization. The found classifying function, which is the linear combination of basis functions, and its variance can then be analyzed in terms of the classical ANOVA decomposition of functions. Based on numerical examples we show that we are able to recover the signum of a function that perfectly fits our model assumptions. Furthermore, we perform classification on different artificial and real-world data sets. We obtain better results with $\ell_1$-norm regularization, both in terms of accuracy and clarity of interpretability.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
