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/* * Copyright (c) 2010-2021 Haifeng Li. All rights reserved. * * Smile is free software: you can redistribute it and/or modify * it under the terms of the GNU General Public License as published by * the Free Software Foundation, either version 3 of the License, or * (at your option) any later version. * * Smile is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU General Public License for more details. * * You should have received a copy of the GNU General Public License * along with Smile. If not, see . */ /** * Mercer kernels. A Mercer kernel is a kernel that is positive * semi-definite. When a kernel is positive semi-definite, one may exploit * the kernel trick, the idea of implicitly mapping data to a high-dimensional * feature space where some linear algorithm is applied that works exclusively * with inner products. Assume we have some mapping Φ from an input * space X to a feature space H, then a kernel {@code k(u, v) = <Φ(u), Φ(v)>} * may be used to define the inner product in feature space H. *

* Positive definiteness in the context of kernel functions also implies that * a kernel matrix created using a particular kernel is positive semi-definite. * A matrix is positive semi-definite if its associated eigenvalues are * non-negative. * * @author Haifeng Li */ package smile.math.kernel;