content: /*******************************************************************************
   * 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 &#934; from an input space X to a feature space H,
   * then a kernel k(u, v) = &lt;&#934;(u), &#934;(v)&gt; may be used to define the
   * inner product in feature space H.
   * <p>
   * 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<T> extends Serializable {
  
      /**
       * Kernel function.
       */
      double k(T x, T y);
  }
  