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/*
* File: KernelPrincipalComponentsAnalysis.java
* Authors: Justin Basilico
* Company: Sandia National Laboratories
* Project: Cognitive Foundry Learning Core
*
* Copyright December 17, 2010, Sandia Corporation.
* Under the terms of Contract DE-AC04-94AL85000, there is a non-exclusive
* license for use of this work by or on behalf of the U.S. Government. Export
* of this program may require a license from the United States Government.
*
*/
package gov.sandia.cognition.learning.algorithm.pca;
import gov.sandia.cognition.annotation.PublicationReference;
import gov.sandia.cognition.annotation.PublicationReferences;
import gov.sandia.cognition.annotation.PublicationType;
import gov.sandia.cognition.collection.CollectionUtil;
import gov.sandia.cognition.learning.algorithm.BatchLearner;
import gov.sandia.cognition.learning.function.kernel.DefaultKernelContainer;
import gov.sandia.cognition.learning.function.kernel.Kernel;
import gov.sandia.cognition.math.matrix.Matrix;
import gov.sandia.cognition.math.matrix.MatrixFactory;
import gov.sandia.cognition.math.matrix.Vector;
import gov.sandia.cognition.math.matrix.VectorFactory;
import gov.sandia.cognition.math.matrix.VectorOutputEvaluator;
import gov.sandia.cognition.math.matrix.decomposition.EigenDecomposition;
import gov.sandia.cognition.math.matrix.mtj.DenseMatrix;
import gov.sandia.cognition.math.matrix.mtj.DenseMatrixFactoryMTJ;
import gov.sandia.cognition.math.matrix.mtj.decomposition.EigenDecompositionRightMTJ;
import gov.sandia.cognition.util.ArgumentChecker;
import java.util.ArrayList;
import java.util.Collection;
import java.util.List;
/**
* An implementation of the Kernel Principal Components Analysis (KPCA)
* algorithm. KPCA generalizes the standard PCA for use with a Mercer kernel.
* Thus, it can take a kernel function and data and come up with vector
* principal components for it. This allows a transform to be made for the
* arbitrary data from the kernel to some vector space.
*
* The implementation uses a closed-form solution based on an
* eigen-decomposition of the (centered) kernel matrix. Doing so does require
* computing the whole kernel matrix, which means that it is a computationally
* intensive algorithm, that scales in O(n^2) where n is the size of the data.
* Thus, this analysis may not scale well to large datasets.
*
* @param
* The type of data that the analysis is to be done over. It must match
* the input type of the kernel function that is given.
* @author Justin Basilico
* @since 3.1
*/
@PublicationReferences(references={
@PublicationReference(
author={"Bernard Scholkopf", "Alexander Smola", "Klaus-Robert Muller"},
title="Nonlinear Component Analysis as a Kernel Eigenvalue Problem",
year=1996,
type=PublicationType.TechnicalReport,
url="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.29.1366"),
@PublicationReference(
author={"John Shawe-Taylor", "Nello Christianini"},
title="Kernel Methods for Pattern Analysis",
year=2004,
type=PublicationType.Book,
pages={150, 153})})
public class KernelPrincipalComponentsAnalysis
extends DefaultKernelContainer
implements BatchLearner, KernelPrincipalComponentsAnalysis.Function>
{
/** The default number of components to create is {@value}. */
public static final int DEFAULT_COMPONENT_COUNT = 10;
/** The default setting for centering data is {@value}. */
public static final boolean DEFAULT_CENTER_DATA = true;
/** The number of components to create from the analysis. Must be positive.
*/
protected int componentCount;
/** Whether or not the data should be centered before doing KPCA. */
protected boolean centerData;
/**
* Creates a new Kernel Principal Components Analysis with a null kernel
* and a default component count.
*/
public KernelPrincipalComponentsAnalysis()
{
this(null, DEFAULT_COMPONENT_COUNT);
}
/**
* Creates a new Kernel Principal Components Analysis with the given
* kernel and component count. It will perform centering.
*
* @param kernel
* The kernel to use in the analysis.
* @param componentCount
* The number of components for the analysis to create.
* Must be positive.
*/
public KernelPrincipalComponentsAnalysis(
final Kernel kernel,
final int componentCount)
{
this(kernel, componentCount, DEFAULT_CENTER_DATA);
}
/**
* Creates a new Kernel Principal Components Analysis with the given
* kernel and component count.
*
* @param kernel
* The kernel to use in the analysis.
* @param componentCount
* The number of components for the analysis to create.
* Must be positive.
* @param centerData
* True to center the data in the input space before applying the
* analysis. Only set this to false if the data is pre-centered. If
* in doubt, set to true.
*/
public KernelPrincipalComponentsAnalysis(
final Kernel kernel,
final int componentCount,
final boolean centerData)
{
super(kernel);
this.setComponentCount(componentCount);
this.setCenterData(centerData);
}
@Override
public Function learn(
final Collection data)
{
final int dataSize = data.size();
final ArrayList dataList =
CollectionUtil.asArrayList(data);
final DenseMatrix kernelMatrix =
new DenseMatrixFactoryMTJ().createMatrix(dataSize, dataSize);
for (int i = 0; i < dataSize; i++)
{
final DataType x = dataList.get(i);
kernelMatrix.setElement(i, i, this.kernel.evaluate(x, x));
for (int j = i + 1; j < dataSize; j++)
{
final DataType y = dataList.get(j);
final double value = this.kernel.evaluate(x, y);
kernelMatrix.setElement(i, j, value);
kernelMatrix.setElement(j, i, value);
}
}
// Center the data, if needed.
final DenseMatrix k;
if (!this.centerData)
{
// When not centering, just use the kernel matrix.
k = kernelMatrix;
}
else
{
// Center the data before applying the analysis.
// Khat = K - 1_m K - K 1_m + 1_m K 1_m
// Where m is the data size
// and 1_m is a m x m with 1/m on the diagonal
// and K is the kernel m x m matrix
final Matrix centeringTerm =
MatrixFactory.getDiagonalDefault().createIdentity(
dataSize, dataSize);
centeringTerm.scaleEquals(1.0 / dataSize);
k = kernelMatrix.clone();
k.minusEquals(centeringTerm.times(kernelMatrix));
k.minusEquals(kernelMatrix.times(centeringTerm));
k.plusEquals(centeringTerm.times(kernelMatrix.times(centeringTerm)));
}
// We can only have up to dataSize components.
final int realComponentCount = Math.min(this.componentCount, dataSize);
// Perform an eigendecomposition on the kernel matrix.
final EigenDecomposition decomposition =
EigenDecompositionRightMTJ.create(k);
// Now we need to take the result from the eigen-decomposition and
// transform it into a form we can use to transform data.
final Matrix components = MatrixFactory.getDenseDefault().createMatrix(
realComponentCount, dataSize);
for (int i = 0; i < realComponentCount; i++)
{
// Get the i-th eigenvector and eigenvalue.
final Vector eigenVector =
decomposition.getEigenVectorsRealPart().getColumn(i);
final double eigenValue = decomposition.getEigenValue(i).getRealPart();
// Create the component by scaling the eigenvector by one over
// the square root of the eigenvalue.
final Vector component = eigenVector.scale(
1.0 / Math.sqrt(Math.abs(eigenValue)));
components.setRow(i, component);
}
// Return the result.
return new Function(this.kernel, dataList, components,
this.centerData, kernelMatrix);
}
/**
* Gets the number of components the analysis attempts to find. If there
* are less data points than the number of components, then the number of
* data points is used instead.
*
* @return
* The number of components for the analysis. Must be positive.
*/
public int getComponentCount()
{
return this.componentCount;
}
/**
* Gets the number of components the analysis attempts to find. If there
* are less data points than the number of components, then the number of
* data points is used instead.
*
* @param componentCount
* The number of components for the analysis. Must be positive.
*/
public void setComponentCount(
final int componentCount)
{
ArgumentChecker.assertIsPositive("componentCount", componentCount);
this.componentCount = componentCount;
}
/**
* Gets whether or not the data needs to be centered in the kernel space
* before applying the algorithm. Only set this to false if the data has
* been pre-centered. If in doubt, set it to true.
*
* @return
* True if the algorithm will apply to the centered version of the
* input data. False if it will just apply directly to the given
* data.
*/
public boolean isCenterData()
{
return this.centerData;
}
/**
* Sets whether or not the data needs to be centered in the kernel space
* before applying the algorithm. Only set this to false if the data has
* been pre-centered. If in doubt, set it to true.
*
* @param centerData
* True if the algorithm will apply to the centered version of the
* input data. False if it will just apply directly to the given
* data.
*/
public void setCenterData(
final boolean centerData)
{
this.centerData = centerData;
}
/**
* The resulting transformation function learned by Kernel Principal
* Components Analysis. It can take any data item from the input space and
* return a vector version of that item transformed into the principal
* components space.
*
* @param
* The type of data used with the kernel that the function can be
* applied to.
*/
public static class Function
extends DefaultKernelContainer
implements VectorOutputEvaluator
{
/** The data that the KPCA was performed over. Each one corresponds to
* a column in the components matrix. */
protected List data;
/** The matrix of components for the function. The number of rows is
* the dimensionality of the reduction. The number of columns is equal
* to the number of data points that the KPCA was done over. */
protected Matrix components;
/** A flag indicating if the incoming data needs to be centered or not.
* Unless the data is being pre-centered, this should be true. */
protected boolean centerData;
/** The kernel matrix for all the data the KPCA was done over. It is
* a square matrix whose size is equal to the data. */
protected Matrix kernelMatrix;
/**
* Creates a new, empty Kernel Principal Components Analysis function.
*/
public Function()
{
this(null, null, null, DEFAULT_CENTER_DATA, null);
}
/**
* Creates a new Kernel Principal Components Analysis function.
*
* @param kernel
* The kernel to use.
* @param data
* The base data used in the KPCA analysis.
* @param components
* The matrix of components for the function.
* @param centerData
* True to center the data. Must correspond to whether or not
* the analysis was done with centered data.
* @param kernelMatrix
* The kernel matrix. Must be provided if data centering is
* true.
*/
public Function(
final Kernel kernel,
final List data,
final Matrix components,
final boolean centerData,
final Matrix kernelMatrix)
{
super(kernel);
this.setData(data);
this.setComponents(components);
this.setCenterData(centerData);
this.setKernelMatrix(kernelMatrix);
}
@Override
public Vector evaluate(
final DataType input)
{
final int dataSize = this.data.size();
// Create the kernel vector.
final Vector kernelVector =
VectorFactory.getDenseDefault().createVector(dataSize);
int index = 0;
for (DataType other : this.data)
{
final double value = this.kernel.evaluate(input, other);
kernelVector.setElement(index, value);
index++;
}
// Transform the kernel vector, if needed.
final Vector kInput;
if (!this.centerData || this.kernelMatrix == null)
{
// In the case we don't need to center the data just use the
// kernel vector directly.
kInput = kernelVector;
}
else
{
// Center the input before applying the transform.
// Kt2 = Kt - 1'_m K - Kt 1_m + 1'_m K 1_m
// Where 1_m is a m x m with 1/m on the diagonal
// and 1'_m is an m-dimensional vector filled with 1/m
// and m is the data size
// and K is the original kernel m x m matrix
// and Kt is the m-dimensional vector of K(t, x_i)
final Matrix centeringMatrix =
MatrixFactory.getDiagonalDefault().createIdentity(
dataSize, dataSize);
centeringMatrix.scaleEquals(1.0 / dataSize);
final Vector centeringVector =
VectorFactory.getDenseDefault().createVector(
dataSize, 1.0 / dataSize);
kInput = kernelVector.clone();
kInput.minusEquals(centeringVector.times(this.kernelMatrix));
kInput.minusEquals(kernelVector.times(centeringMatrix));
kInput.plusEquals(centeringVector.times(
this.kernelMatrix.times(centeringMatrix)));
}
return this.components.times(kInput);
}
@Override
public int getOutputDimensionality()
{
return this.components.getNumRows();
}
/**
* Gets the number of components in the analysis result.
*
* @return
* The number of components in the analysis result.
*/
public int getComponentCount()
{
return this.components.getNumRows();
}
/**
* Gets the data that was used in the analysis. It is still required for
* applying the transform.
*
* @return
* The data that was used in the analysis.
*/
public List getData()
{
return this.data;
}
/**
* Sets the data that was used in the analysis. It is still required for
* applying the transform.
*
* @param data
* The data that was used in the analysis.
*/
public void setData(
final List data)
{
this.data = data;
}
/**
* Gets the matrix of components from the analysis. The number of rows
* equals the number of components. The number of columns must equal
* the size of the data.
*
* @return
* The component matrix.
*/
public Matrix getComponents()
{
return this.components;
}
/**
* Sets the matrix of components from the analysis. The number of rows
* equals the number of components. The number of columns must equal
* the size of the data.
*
* @param components
* The component matrix.
*/
public void setComponents(
final Matrix components)
{
this.components = components;
}
/**
* Gets whether or not the data needs to be centered in the kernel space
* before applying the function. Only set this to false if the data has
* been pre-centered. If in doubt, set it to true. If this is set to
* true, the kernel matrix must be provided for the centering to be
* performed.
*
* @return
* True if the function will apply to the centered version of the
* input data. False if it will just apply directly to the given
* data.
*/
public boolean isCenterData()
{
return this.centerData;
}
/**
* Sets whether or not the data needs to be centered in the kernel space
* before applying the function. Only set this to false if the data has
* been pre-centered. If in doubt, set it to true. If this is set to
* true, the kernel matrix must be provided for the centering to be
* performed.
*
* @param centerData
* True if the function will apply to the centered version of the
* input data. False if it will just apply directly to the given
* data.
*/
public void setCenterData(
final boolean centerData)
{
this.centerData = centerData;
}
/**
* Gets the kernel matrix for the data that the analysis was done over.
* If the data needs to be centered, then this matrix must be provided.
* Each dimension corresponds to an element in the data list.
*
* @return
* The data kernel matrix.
*/
public Matrix getKernelMatrix()
{
return this.kernelMatrix;
}
/**
* Sets the kernel matrix for the data that the analysis was done over.
* If the data needs to be centered, then this matrix must be provided.
* Each dimension corresponds to an element in the data list.
*
* @param kernelMatrix
* The data kernel matrix.
*/
public void setKernelMatrix(
final Matrix kernelMatrix)
{
this.kernelMatrix = kernelMatrix;
}
}
}