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/*******************************************************************************
 * Copyright (c) 2010 Haifeng Li
 *   
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *  
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 *******************************************************************************/

package smile.math.kernel;

import java.io.Serializable;

/**
 * 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 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 nonnegative.
 * 
 * @author Haifeng Li
 */
public interface MercerKernel extends Serializable {

    /**
     * Kernel function.
     */
    double k(T x, T y);
}