svd_convolution.hpp

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#ifndef MLPACK_METHODS_ANN_CONVOLUTION_RULES_SVD_CONVOLUTION_HPP
#define MLPACK_METHODS_ANN_CONVOLUTION_RULES_SVD_CONVOLUTION_HPP

#include <mlpack/prereqs.hpp>
#include "border_modes.hpp"
#include "fft_convolution.hpp"
#include "naive_convolution.hpp"

namespace mlpack {
namespace ann  {

template<typename BorderMode = FullConvolution>
class SVDConvolution
{
 public:
  /*
   * Perform a convolution (valid or full mode) using singular value
   * decomposition. By using singular value decomposition of the filter matrix
   * the convolution can be expressed as a sum of outer products. Each product
   * can be computed efficiently as convolution with a row and a column vector.
   * The individual convolutions are computed with the naive implementation
   * which is fast if the filter is low-dimensional.
   *
   * @param input Input used to perform the convolution.
   * @param filter Filter used to perform the conolution.
   * @param output Output data that contains the results of the convolution.
   * @param dW Stride of filter application in the x direction.
   * @param dH Stride of filter application in the y direction.
   */
  template<typename eT>
  static void Convolution(const arma::Mat<eT>& input,
                          const arma::Mat<eT>& filter,
                          arma::Mat<eT>& output)
  {
    // Use the naive convolution in case the filter isn't two dimensional or the
    // filter is bigger than the input.
    if (filter.n_rows > input.n_rows || filter.n_cols > input.n_cols ||
        filter.n_rows == 1 || filter.n_cols == 1)
    {
      NaiveConvolution<BorderMode>::Convolution(input, filter, output);
    }
    else
    {
      arma::Mat<eT> U, V, subOutput;
      arma::Col<eT> s;

      arma::svd_econ(U, s, V, filter);

      // Rank approximation using the singular values calculated with singular
      // value decomposition of dense filter matrix.
      const size_t rank = arma::sum(s > (s.n_elem * arma::max(s) *
          arma::datum::eps));

      // Test for separability based on the rank of the kernel and take
      // advantage of the low rank.
      if (rank * (filter.n_rows + filter.n_cols) < filter.n_elem)
      {
        arma::Mat<eT> subFilter = V.unsafe_col(0) * s(0);
        NaiveConvolution<BorderMode>::Convolution(input, subFilter, subOutput);

        subOutput = subOutput.t();
        NaiveConvolution<BorderMode>::Convolution(subOutput, U.unsafe_col(0),
            output);

        for (size_t r = 1; r < rank; r++)
        {
          subFilter = V.unsafe_col(r) * s(r);
          NaiveConvolution<BorderMode>::Convolution(input, subFilter,
              subOutput);

          arma::Mat<eT> temp;
          subOutput = subOutput.t();
          NaiveConvolution<BorderMode>::Convolution(subOutput, U.unsafe_col(r),
              temp);
          output += temp;
        }

        output = output.t();
      }
      else
      {
        FFTConvolution<BorderMode>::Convolution(input, filter, output);
      }
    }
  }

  /*
   * Perform a convolution using 3rd order tensors.
   *
   * @param input Input used to perform the convolution.
   * @param filter Filter used to perform the conolution.
   * @param output Output data that contains the results of the convolution.
   * @param dW Stride of filter application in the x direction.
   * @param dH Stride of filter application in the y direction.
   */
  template<typename eT>
  static void Convolution(const arma::Cube<eT>& input,
                          const arma::Cube<eT>& filter,
                          arma::Cube<eT>& output)
  {
    arma::Mat<eT> convOutput;
    SVDConvolution<BorderMode>::Convolution(input.slice(0), filter.slice(0),
        convOutput);

    output = arma::Cube<eT>(convOutput.n_rows, convOutput.n_cols,
        input.n_slices);
    output.slice(0) = convOutput;

    for (size_t i = 1; i < input.n_slices; i++)
    {
      SVDConvolution<BorderMode>::Convolution(input.slice(i), filter.slice(i),
          convOutput);
      output.slice(i) = convOutput;
    }
  }

  /*
   * Perform a convolution using dense matrix as input and a 3rd order tensors
   * as filter and output.
   *
   * @param input Input used to perform the convolution.
   * @param filter Filter used to perform the conolution.
   * @param output Output data that contains the results of the convolution.
   * @param dW Stride of filter application in the x direction.
   * @param dH Stride of filter application in the y direction.
   */
  template<typename eT>
  static void Convolution(const arma::Mat<eT>& input,
                          const arma::Cube<eT>& filter,
                          arma::Cube<eT>& output)
  {
    arma::Mat<eT> convOutput;
    SVDConvolution<BorderMode>::Convolution(input, filter.slice(0), convOutput);

    output = arma::Cube<eT>(convOutput.n_rows, convOutput.n_cols,
        filter.n_slices);
    output.slice(0) = convOutput;

    for (size_t i = 1; i < filter.n_slices; i++)
    {
      SVDConvolution<BorderMode>::Convolution(input, filter.slice(i),
          convOutput);
      output.slice(i) = convOutput;
    }
  }

  /*
   * Perform a convolution using a 3rd order tensors as input and output and a
   * dense matrix as filter.
   *
   * @param input Input used to perform the convolution.
   * @param filter Filter used to perform the conolution.
   * @param output Output data that contains the results of the convolution.
   * @param dW Stride of filter application in the x direction.
   * @param dH Stride of filter application in the y direction.
   */
  template<typename eT>
  static void Convolution(const arma::Cube<eT>& input,
                          const arma::Mat<eT>& filter,
                          arma::Cube<eT>& output)
  {
    arma::Mat<eT> convOutput;
    SVDConvolution<BorderMode>::Convolution(input.slice(0), filter, convOutput);

    output = arma::Cube<eT>(convOutput.n_rows, convOutput.n_cols,
        input.n_slices);
    output.slice(0) = convOutput;

    for (size_t i = 1; i < input.n_slices; i++)
    {
      SVDConvolution<BorderMode>::Convolution(input.slice(i), filter,
          convOutput);
      output.slice(i) = convOutput;
    }
  }

};  // class SVDConvolution

} // namespace ann
} // namespace mlpack

#endif