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/* */
/* Copyright 2008-2011 by Michael Hanselmann and Ullrich Koethe */
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#ifndef VIGRA_UNSUPERVISED_DECOMPOSITION_HXX
#define VIGRA_UNSUPERVISED_DECOMPOSITION_HXX
#include <numeric>
#include "mathutil.hxx"
#include "matrix.hxx"
#include "singular_value_decomposition.hxx"
#include "random.hxx"
namespace vigra
{
/** \addtogroup Unsupervised_Decomposition Unsupervised Decomposition
Unsupervised matrix decomposition methods.
**/
//@{
/*****************************************************************/
/* */
/* principal component analysis (PCA) */
/* */
/*****************************************************************/
/** \brief Decompose a matrix according to the PCA algorithm.
This function implements the PCA algorithm (principal component analysis).
\arg features must be a matrix with shape <tt>(numFeatures * numSamples)</tt>, which is
decomposed into the matrices
\arg fz with shape <tt>(numFeatures * numComponents)</tt> and
\arg zv with shape <tt>(numComponents * numSamples)</tt>
such that
\f[
\mathrm{features} \approx \mathrm{fz} * \mathrm{zv}
\f]
(this formula requires that the features have been centered around the mean by
<tt>\ref linalg::prepareRows (features, features, ZeroMean)</tt>).
The shape parameter <tt>numComponents</tt> determines the complexity of
the decomposition model and therefore the approximation quality (if
<tt>numComponents == numFeatures</tt>, the representation becomes exact).
Intuitively, <tt>fz</tt> is a projection matrix from the reduced space
into the original space, and <tt>zv</tt> is the reduced representation
of the data, using just <tt>numComponents</tt> features.
<b>Declaration:</b>
<b>\#include</b> \<vigra/unsupervised_decomposition.hxx\>
\code
namespace vigra {
template <class U, class C1, class C2, class C3>
void
principalComponents(MultiArrayView<2, U, C1> const & features,
MultiArrayView<2, U, C2> fz,
MultiArrayView<2, U, C3> zv);
}
\endcode
<b>Usage:</b>
\code
Matrix<double> data(numFeatures, numSamples);
... // fill the input matrix
int numComponents = 3;
Matrix<double> fz(numFeatures, numComponents),
zv(numComponents, numSamples);
// center the data
prepareRows(data, data, ZeroMean);
// compute the reduced representation
principalComponents(data, fz, zv);
Matrix<double> model = fz*zv;
double meanSquaredError = squaredNorm(data - model) / numSamples;
\endcode
*/
template <class T, class C1, class C2, class C3>
void
principalComponents(MultiArrayView<2, T, C1> const & features,
MultiArrayView<2, T, C2> fz,
MultiArrayView<2, T, C3> zv)
{
using namespace linalg; // activate matrix multiplication and arithmetic functions
int numFeatures = rowCount(features);
int numSamples = columnCount(features);
int numComponents = columnCount(fz);
vigra_precondition(numSamples >= numFeatures,
"principalComponents(): The number of samples has to be larger than the number of features.");
vigra_precondition(numFeatures >= numComponents && numComponents >= 1,
"principalComponents(): The number of features has to be larger or equal to the number of components in which the feature matrix is decomposed.");
vigra_precondition(rowCount(fz) == numFeatures,
"principalComponents(): The output matrix fz has to be of dimension numFeatures*numComponents.");
vigra_precondition(columnCount(zv) == numSamples && rowCount(zv) == numComponents,
"principalComponents(): The output matrix zv has to be of dimension numComponents*numSamples.");
Matrix<T> U(numSamples, numFeatures), S(numFeatures, 1), V(numFeatures, numFeatures);
singularValueDecomposition(features.transpose(), U, S, V);
for(int k=0; k<numComponents; ++k)
{
rowVector(zv, k) = columnVector(U, k).transpose() * S(k, 0);
columnVector(fz, k) = columnVector(V, k);
}
}
/*****************************************************************/
/* */
/* probabilistic latent semantic analysis (pLSA) */
/* see T Hofmann, Probabilistic Latent Semantic */
/* Indexing for details */
/* */
/*****************************************************************/
/** \brief Option object for the \ref pLSA algorithm.
*/
class PLSAOptions
{
public:
/** Initialize all options with default values.
*/
PLSAOptions()
: min_rel_gain(1e-4),
max_iterations(50),
normalized_component_weights(true)
{}
/** Maximum number of iterations which is performed by the pLSA algorithm.
default: 50
*/
PLSAOptions & maximumNumberOfIterations(unsigned int n)
{
vigra_precondition(n >= 1,
"PLSAOptions::maximumNumberOfIterations(): number must be a positive integer.");
max_iterations = n;
return *this;
}
/** Minimum relative gain which is required for the algorithm to continue the iterations.
default: 1e-4
*/
PLSAOptions & minimumRelativeGain(double g)
{
vigra_precondition(g >= 0.0,
"PLSAOptions::minimumRelativeGain(): number must be positive or zero.");
min_rel_gain = g;
return *this;
}
/** Normalize the entries of the zv result array.
If true, the columns of zv sum to one. Otherwise, they have the same
column sum as the original feature matrix.
default: true
*/
PLSAOptions & normalizedComponentWeights(bool v = true)
{
normalized_component_weights = v;
return *this;
}
double min_rel_gain;
int max_iterations;
bool normalized_component_weights;
};
/** \brief Decompose a matrix according to the pLSA algorithm.
This function implements the pLSA algorithm (probabilistic latent semantic analysis)
proposed in
T. Hofmann: <a href="http://www.cs.brown.edu/people/th/papers/Hofmann-UAI99.pdf">
<i>"Probabilistic Latent Semantic Analysis"</i></a>,
in: UAI'99, Proc. 15th Conf. on Uncertainty in Artificial Intelligence,
pp. 289-296, Morgan Kaufmann, 1999
\arg features must be a matrix with shape <tt>(numFeatures * numSamples)</tt> and
non-negative entries, which is decomposed into the matrices
\arg fz with shape <tt>(numFeatures * numComponents)</tt> and
\arg zv with shape <tt>(numComponents * numSamples)</tt>
such that
\f[
\mathrm{features} \approx \mathrm{fz} * \mathrm{zv}
\f]
(this formula applies when pLSA is called with
<tt>PLSAOptions.normalizedComponentWeights(false)</tt>. Otherwise, you must
normalize the features by calling <tt>\ref linalg::prepareColumns (features, features, UnitSum)</tt>
to make the formula hold).
The shape parameter <tt>numComponents</tt> determines the complexity of
the decomposition model and therefore the approximation quality.
Intuitively, features are a set of words, and the samples a set of
documents. The entries of the <tt>features</tt> matrix denote the relative
frequency of the words in each document. The components represents a
(presumably small) set of topics. The matrix <tt>fz</tt> encodes the
relative frequency of words in the different topics, and the matrix
<tt>zv</tt> encodes to what extend each topic explains the content of each
document.
The option object determines the iteration termination conditions and the output
normalization. In addition, you may pass a random number generator to pLSA()
which is used to create the initial solution.
<b>Declarations:</b>
<b>\#include</b> \<vigra/unsupervised_decomposition.hxx\>
\code
namespace vigra {
template <class U, class C1, class C2, class C3, class Random>
void
pLSA(MultiArrayView<2, U, C1> const & features,
MultiArrayView<2, U, C2> & fz,
MultiArrayView<2, U, C3> & zv,
Random const& random,
PLSAOptions const & options = PLSAOptions());
template <class U, class C1, class C2, class C3>
void
pLSA(MultiArrayView<2, U, C1> const & features,
MultiArrayView<2, U, C2> & fz,
MultiArrayView<2, U, C3> & zv,
PLSAOptions const & options = PLSAOptions());
}
\endcode
<b>Usage:</b>
\code
Matrix<double> words(numWords, numDocuments);
... // fill the input matrix
int numTopics = 3;
Matrix<double> fz(numWords, numTopics),
zv(numTopics, numDocuments);
pLSA(words, fz, zv, PLSAOptions().normalizedComponentWeights(false));
Matrix<double> model = fz*zv;
double meanSquaredError = (words - model).squaredNorm() / numDocuments;
\endcode
*/
doxygen_overloaded_function(template <...> void pLSA)
template <class U, class C1, class C2, class C3, class Random>
void
pLSA(MultiArrayView<2, U, C1> const & features,
MultiArrayView<2, U, C2> fz,
MultiArrayView<2, U, C3> zv,
Random const& random,
PLSAOptions const & options = PLSAOptions())
{
using namespace linalg; // activate matrix multiplication and arithmetic functions
int numFeatures = rowCount(features);
int numSamples = columnCount(features);
int numComponents = columnCount(fz);
vigra_precondition(numFeatures >= numComponents && numComponents >= 1,
"pLSA(): The number of features has to be larger or equal to the number of components in which the feature matrix is decomposed.");
vigra_precondition(rowCount(fz) == numFeatures,
"pLSA(): The output matrix fz has to be of dimension numFeatures*numComponents.");
vigra_precondition(columnCount(zv) == numSamples && rowCount(zv) == numComponents,
"pLSA(): The output matrix zv has to be of dimension numComponents*numSamples.");
// random initialization of result matrices, subsequent normalization
UniformRandomFunctor<Random> randf(random);
initMultiArray(destMultiArrayRange(fz), randf);
initMultiArray(destMultiArrayRange(zv), randf);
prepareColumns(fz, fz, UnitSum);
prepareColumns(zv, zv, UnitSum);
// init vars
double eps = 1.0/NumericTraits<U>::max(); // epsilon > 0
double lastChange = NumericTraits<U>::max(); // infinity
double err = 0;
double err_old;
int iteration = 0;
// expectation maximization (EM) algorithm
Matrix<U> columnSums(1, numSamples);
features.sum(columnSums);
Matrix<U> expandedSums = ones<U>(numFeatures, 1) * columnSums;
while(iteration < options.max_iterations && (lastChange > options.min_rel_gain))
{
Matrix<U> fzv = fz*zv;
//if(iteration%25 == 0)
//{
//std::cout << "iteration: " << iteration << std::endl;
//std::cout << "last relative change: " << lastChange << std::endl;
//}
Matrix<U> factor = features / pointWise(fzv + (U)eps);
zv *= (fz.transpose() * factor);
fz *= (factor * zv.transpose());
prepareColumns(fz, fz, UnitSum);
prepareColumns(zv, zv, UnitSum);
// check relative change in least squares model fit
Matrix<U> model = expandedSums * pointWise(fzv);
err_old = err;
err = (features - model).squaredNorm();
//std::cout << "error: " << err << std::endl;
lastChange = abs((err-err_old) / (U)(err + eps));
//std::cout << "lastChange: " << lastChange << std::endl;
iteration += 1;
}
//std::cout << "Terminated after " << iteration << " iterations." << std::endl;
//std::cout << "Last relative change was " << lastChange << "." << std::endl;
if(!options.normalized_component_weights)
{
// undo the normalization
for(int k=0; k<numSamples; ++k)
columnVector(zv, k) *= columnSums(0, k);
}
}
template <class U, class C1, class C2, class C3>
inline void
pLSA(MultiArrayView<2, U, C1> const & features,
MultiArrayView<2, U, C2> & fz,
MultiArrayView<2, U, C3> & zv,
PLSAOptions const & options = PLSAOptions())
{
RandomNumberGenerator<> generator(RandomSeed);
pLSA(features, fz, zv, generator, options);
}
//@}
} // namespace vigra
#endif // VIGRA_UNSUPERVISED_DECOMPOSITION_HXX
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