Hamsici Onur C, Martinez Aleix M
Department of Electrical and Computer Engineering, Ohio State University, 2015 Neil Avenue, Columbus, OH 43210, USA.
IEEE Trans Pattern Anal Mach Intell. 2008 Apr;30(4):647-57. doi: 10.1109/TPAMI.2007.70717.
We present an algorithm which provides the one-dimensional subspace where the Bayes error is minimized for the C class problem with homoscedastic Gaussian distributions. Our main result shows that the set of possible one-dimensional spaces v, for which the order of the projected class means is identical, defines a convex region with associated convex Bayes error function g(v). This allows for the minimization of the error function using standard convex optimization algorithms. Our algorithm is then extended to the minimization of the Bayes error in the more general case of heteroscedastic distributions. This is done by means of an appropriate kernel mapping function. This result is further extended to obtain the d-dimensional solution for any given d, by iteratively applying our algorithm to the null space of the (d - 1)-dimensional solution. We also show how this result can be used to improve up on the outcomes provided by existing algorithms, and derive a low-computational cost, linear approximation. Extensive experimental validations are provided to demonstrate the use of these algorithms in classification, data analysis and visualization.
我们提出了一种算法，该算法为具有同方差高斯分布的C类问题提供了使贝叶斯误差最小化的一维子空间。我们的主要结果表明，对于投影类均值顺序相同的可能一维空间v的集合，定义了一个具有相关凸贝叶斯误差函数g(v)的凸区域。这允许使用标准凸优化算法来最小化误差函数。然后，我们的算法扩展到在更一般的异方差分布情况下最小化贝叶斯误差。这是通过适当的核映射函数来完成的。通过将我们的算法迭代应用于(d - 1)维解的零空间，进一步扩展该结果以获得任意给定d的d维解。我们还展示了如何利用这一结果改进现有算法提供的结果，并推导了一种低计算成本的线性近似。提供了广泛的实验验证，以证明这些算法在分类、数据分析和可视化中的应用。
