FITCInferenceMethod.cpp

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/*
 * Copyright (c) The Shogun Machine Learning Toolbox
 * Written (W) 2015 Wu Lin
 * Written (W) 2013 Roman Votyakov
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions are met:
 *
 * 1. Redistributions of source code must retain the above copyright notice, this
 *    list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright notice,
 *    this list of conditions and the following disclaimer in the documentation
 *    and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
 * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
 * DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR
 * ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
 * (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
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 * ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
 * SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
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 */
#include <shogun/machine/gp/FITCInferenceMethod.h>

#ifdef HAVE_EIGEN3

#include <shogun/machine/gp/GaussianLikelihood.h>
#include <shogun/mathematics/Math.h>
#include <shogun/labels/RegressionLabels.h>
#include <shogun/mathematics/eigen3.h>

using namespace shogun;
using namespace Eigen;

CFITCInferenceMethod::CFITCInferenceMethod() : CSingleFITCLaplacianBase()
{
    init();
}

CFITCInferenceMethod::CFITCInferenceMethod(CKernel* kern, CFeatures* feat,
        CMeanFunction* m, CLabels* lab, CLikelihoodModel* mod, CFeatures* lat)
        : CSingleFITCLaplacianBase(kern, feat, m, lab, mod, lat)
{
    init();
}

void CFITCInferenceMethod::init()
{
}

CFITCInferenceMethod::~CFITCInferenceMethod()
{
}
void CFITCInferenceMethod::compute_gradient()
{
    CInferenceMethod::compute_gradient();

    if (!m_gradient_update)
    {
        update_deriv();
        m_gradient_update=true;
        update_parameter_hash();
    }
}

void CFITCInferenceMethod::update()
{
    SG_DEBUG("entering\n");

    CInferenceMethod::update();
    update_chol();
    update_alpha();
    m_gradient_update=false;
    update_parameter_hash();

    SG_DEBUG("leaving\n");
}

CFITCInferenceMethod* CFITCInferenceMethod::obtain_from_generic(
        CInferenceMethod* inference)
{
    if (inference==NULL)
        return NULL;

    if (inference->get_inference_type()!=INF_FITC_REGRESSION)
        SG_SERROR("Provided inference is not of type CFITCInferenceMethod!\n")

    SG_REF(inference);
    return (CFITCInferenceMethod*)inference;
}

void CFITCInferenceMethod::check_members() const
{
    CSingleFITCLaplacianBase::check_members();

    REQUIRE(m_model->get_model_type()==LT_GAUSSIAN,
            "FITC inference method can only use Gaussian likelihood function\n")
    REQUIRE(m_labels->get_label_type()==LT_REGRESSION, "Labels must be type "
            "of CRegressionLabels\n")
}

SGVector<float64_t> CFITCInferenceMethod::get_diagonal_vector()
{
    if (parameter_hash_changed())
        update();

    // get the sigma variable from the Gaussian likelihood model
    CGaussianLikelihood* lik=CGaussianLikelihood::obtain_from_generic(m_model);
    float64_t sigma=lik->get_sigma();
    SG_UNREF(lik);

    // compute diagonal vector: sW=1/sigma
    SGVector<float64_t> result(m_features->get_num_vectors());
    result.fill_vector(result.vector, m_features->get_num_vectors(), 1.0/sigma);

    return result;
}

float64_t CFITCInferenceMethod::get_negative_log_marginal_likelihood()
{
    if (parameter_hash_changed())
        update();

    //time complexity of the following operations is O(m*n)

    // create eigen representations of chol_utr, dg, r, be
    Map<MatrixXd> eigen_chol_utr(m_chol_utr.matrix, m_chol_utr.num_rows,
            m_chol_utr.num_cols);
    Map<VectorXd> eigen_t(m_t.vector, m_t.vlen);
    Map<VectorXd> eigen_r(m_r.vector, m_r.vlen);
    Map<VectorXd> eigen_be(m_be.vector, m_be.vlen);

    // compute negative log marginal likelihood:
    // nlZ=sum(log(diag(utr)))+(sum(log(dg))+r'*r-be'*be+n*log(2*pi))/2
    float64_t result=eigen_chol_utr.diagonal().array().log().sum()+
        (-eigen_t.array().log().sum()+eigen_r.dot(eigen_r)-eigen_be.dot(eigen_be)+
         m_ktrtr_diag.vlen*CMath::log(2*CMath::PI))/2.0;

    return result;
}

void CFITCInferenceMethod::update_chol()
{
    //time complexits O(m^2*n)

    // get the sigma variable from the Gaussian likelihood model
    CGaussianLikelihood* lik=CGaussianLikelihood::obtain_from_generic(m_model);
    float64_t sigma=lik->get_sigma();
    SG_UNREF(lik);

    // eigen3 representation of covariance matrix of inducing features (m_kuu)
    // and training features (m_ktru)
    Map<MatrixXd> eigen_kuu(m_kuu.matrix, m_kuu.num_rows, m_kuu.num_cols);
    Map<MatrixXd> eigen_ktru(m_ktru.matrix, m_ktru.num_rows, m_ktru.num_cols);

    Map<VectorXd> eigen_ktrtr_diag(m_ktrtr_diag.vector, m_ktrtr_diag.vlen);

    // solve Luu' * Luu = Kuu + m_ind_noise * I
    //Luu  = chol(Kuu+snu2*eye(nu));                         % Kuu + snu2*I = Luu'*Luu
    LLT<MatrixXd> Luu(eigen_kuu*CMath::exp(m_log_scale*2.0)+CMath::exp(m_log_ind_noise)*MatrixXd::Identity(
        m_kuu.num_rows, m_kuu.num_cols));

    // create shogun and eigen3 representation of cholesky of covariance of
    // inducing features Luu (m_chol_uu and eigen_chol_uu)
    m_chol_uu=SGMatrix<float64_t>(Luu.rows(), Luu.cols());
    Map<MatrixXd> eigen_chol_uu(m_chol_uu.matrix, m_chol_uu.num_rows,
        m_chol_uu.num_cols);
    eigen_chol_uu=Luu.matrixU();

    // solve Luu' * V = Ktru
    //V  = Luu'\Ku;                                     % V = inv(Luu')*Ku => V'*V = Q

    m_V=SGMatrix<float64_t>(m_chol_uu.num_cols, m_ktru.num_cols);
    Map<MatrixXd> V(m_V.matrix, m_V.num_rows, m_V.num_cols);
    V=eigen_chol_uu.triangularView<Upper>().adjoint().solve(eigen_ktru*
            CMath::exp(m_log_scale*2.0));

    // create shogun and eigen3 representation of
    // dg = diag(K) + sn2 - diag(Q)
    // t = 1/dg
    m_t=SGVector<float64_t>(m_ktrtr_diag.vlen);
    Map<VectorXd> eigen_t(m_t.vector, m_t.vlen);

    //g_sn2 = diagK + sn2 - sum(V.*V,1)';          % g + sn2 = diag(K) + sn2 - diag(Q)
    eigen_t=eigen_ktrtr_diag*CMath::exp(m_log_scale*2.0)+CMath::sq(sigma)*
        VectorXd::Ones(m_t.vlen)-(V.cwiseProduct(V)).colwise().sum().adjoint();
    eigen_t=MatrixXd::Ones(eigen_t.rows(),1).cwiseQuotient(eigen_t);

    // solve Lu' * Lu = V * diag(1/dg) * V' + I
    //Lu = chol(eye(nu) + (V./repmat(g_sn2',nu,1))*V');  % Lu'*Lu=I+V*diag(1/g_sn2)*V'
    LLT<MatrixXd> Lu(V*((VectorXd::Ones(m_t.vlen)).cwiseProduct(eigen_t)).asDiagonal()*
            V.adjoint()+MatrixXd::Identity(m_kuu.num_rows, m_kuu.num_cols));

    // create shogun and eigen3 representation of cholesky of covariance of
    // training features Luu (m_chol_utr and eigen_chol_utr)
    m_chol_utr=SGMatrix<float64_t>(Lu.rows(), Lu.cols());
    Map<MatrixXd> eigen_chol_utr(m_chol_utr.matrix, m_chol_utr.num_rows,
        m_chol_utr.num_cols);
    eigen_chol_utr=Lu.matrixU();

    // create eigen representation of labels and mean vectors
    SGVector<float64_t> y=((CRegressionLabels*) m_labels)->get_labels();
    Map<VectorXd> eigen_y(y.vector, y.vlen);
    SGVector<float64_t> m=m_mean->get_mean_vector(m_features);
    Map<VectorXd> eigen_m(m.vector, m.vlen);

    // compute sgrt_dg = sqrt(dg)
    VectorXd sqrt_t=eigen_t.array().sqrt();

    // create shogun and eigen3 representation of labels adjusted for
    // noise and means (m_r)
    //r  = (y-m)./sqrt(g_sn2);
    m_r=SGVector<float64_t>(y.vlen);
    Map<VectorXd> eigen_r(m_r.vector, m_r.vlen);
    eigen_r=(eigen_y-eigen_m).cwiseProduct(sqrt_t);

    // compute be
    //be = Lu'\(V*(r./sqrt(g_sn2)));
    m_be=SGVector<float64_t>(m_chol_utr.num_cols);
    Map<VectorXd> eigen_be(m_be.vector, m_be.vlen);
    eigen_be=eigen_chol_utr.triangularView<Upper>().adjoint().solve(
        V*eigen_r.cwiseProduct(sqrt_t));

    // compute iKuu
    //iKuu = solve_chol(Luu,eye(nu));
    MatrixXd iKuu=Luu.solve(MatrixXd::Identity(m_kuu.num_rows, m_kuu.num_cols));

    // create shogun and eigen3 representation of posterior cholesky
    MatrixXd eigen_prod=eigen_chol_utr*eigen_chol_uu;
    m_L=SGMatrix<float64_t>(m_kuu.num_rows, m_kuu.num_cols);
    Map<MatrixXd> eigen_chol(m_L.matrix, m_L.num_rows, m_L.num_cols);

    //post.L  = solve_chol(Lu*Luu,eye(nu)) - iKuu;
    eigen_chol=eigen_prod.triangularView<Upper>().adjoint().solve(
        MatrixXd::Identity(m_kuu.num_rows, m_kuu.num_cols));
    eigen_chol=eigen_prod.triangularView<Upper>().solve(eigen_chol)-iKuu;
}

void CFITCInferenceMethod::update_alpha()
{
    //time complexity O(m^2) since triangular.solve is O(m^2)
    Map<MatrixXd> eigen_chol_uu(m_chol_uu.matrix, m_chol_uu.num_rows,
        m_chol_uu.num_cols);
    Map<MatrixXd> eigen_chol_utr(m_chol_utr.matrix, m_chol_utr.num_rows,
        m_chol_utr.num_cols);
    Map<VectorXd> eigen_be(m_be.vector, m_be.vlen);

    // create shogun and eigen representations of alpha
    // and solve Luu * Lu * alpha = be
    m_alpha=SGVector<float64_t>(m_chol_uu.num_rows);
    Map<VectorXd> eigen_alpha(m_alpha.vector, m_alpha.vlen);

    //post.alpha = Luu\(Lu\be);
    eigen_alpha=eigen_chol_utr.triangularView<Upper>().solve(eigen_be);
    eigen_alpha=eigen_chol_uu.triangularView<Upper>().solve(eigen_alpha);
}

void CFITCInferenceMethod::update_deriv()
{
    //time complexits O(m^2*n)

    // create eigen representation of Ktru, Lu, Luu, dg, be
    Map<MatrixXd> eigen_Ktru(m_ktru.matrix, m_ktru.num_rows, m_ktru.num_cols);
    Map<MatrixXd> eigen_Lu(m_chol_utr.matrix, m_chol_utr.num_rows,
            m_chol_utr.num_cols);
    Map<MatrixXd> eigen_Luu(m_chol_uu.matrix, m_chol_uu.num_rows,
            m_chol_uu.num_cols);
    Map<VectorXd> eigen_t(m_t.vector, m_t.vlen);
    Map<VectorXd> eigen_be(m_be.vector, m_be.vlen);

    // get and create eigen representation of labels
    SGVector<float64_t> y=((CRegressionLabels*) m_labels)->get_labels();
    Map<VectorXd> eigen_y(y.vector, y.vlen);

    // get and create eigen representation of mean vector
    SGVector<float64_t> m=m_mean->get_mean_vector(m_features);
    Map<VectorXd> eigen_m(m.vector, m.vlen);

    // compute V=inv(Luu')*Ku
    Map<MatrixXd> V(m_V.matrix, m_V.num_rows, m_V.num_cols);

    // create shogun and eigen representation of al
    m_al=SGVector<float64_t>(m.vlen);
    Map<VectorXd> eigen_al(m_al.vector, m_al.vlen);

    // compute al=(Kt+sn2*eye(n))\y
    //r  = (y-m)./sqrt(g_sn2);
    //al = r./sqrt(g_sn2) - (V'*(Lu\be))./g_sn2;      % al = (Kt+sn2*eye(n))\(y-m)
    eigen_al=((eigen_y-eigen_m)-(V.adjoint()*
        eigen_Lu.triangularView<Upper>().solve(eigen_be))).cwiseProduct(eigen_t);

    // compute inv(Kuu+snu2*I)=iKuu
    MatrixXd iKuu=eigen_Luu.triangularView<Upper>().adjoint().solve(
            MatrixXd::Identity(m_kuu.num_rows, m_kuu.num_cols));
    iKuu=eigen_Luu.triangularView<Upper>().solve(iKuu);

    // create shogun and eigen representation of B
    m_B=SGMatrix<float64_t>(iKuu.rows(), eigen_Ktru.cols());
    Map<MatrixXd> eigen_B(m_B.matrix, m_B.num_rows, m_B.num_cols);

    //B = iKuu*Ku; w = B*al;
    eigen_B=iKuu*eigen_Ktru*CMath::exp(m_log_scale*2.0);

    // create shogun and eigen representation of w
    m_w=SGVector<float64_t>(m_B.num_rows);
    Map<VectorXd> eigen_w(m_w.vector, m_w.vlen);

    eigen_w=eigen_B*eigen_al;

    // create shogun and eigen representation of W
    m_Rvdd=SGMatrix<float64_t>(m_chol_utr.num_cols, m_t.vlen);
    Map<MatrixXd> eigen_W(m_Rvdd.matrix, m_Rvdd.num_rows, m_Rvdd.num_cols);

    // compute W=Lu'\(V./repmat(g_sn2',nu,1))
    //W = Lu'\(V./repmat(g_sn2',nu,1));
    eigen_W=eigen_Lu.triangularView<Upper>().adjoint().solve(V*VectorXd::Ones(
        m_t.vlen).cwiseProduct(eigen_t).asDiagonal());
}


SGVector<float64_t> CFITCInferenceMethod::get_posterior_mean()
{
    compute_gradient();

    m_mu=SGVector<float64_t>(m_al.vlen);
    Map<VectorXd> eigen_mu(m_mu.vector, m_mu.vlen);

    /*
    //true posterior mean with equivalent FITC prior
    //time complexity of the following operations is O(n)
    Map<VectorXd> eigen_al(m_al.vector, m_al.vlen);
    SGVector<float64_t> y=((CRegressionLabels*) m_labels)->get_labels();
    Map<VectorXd> eigen_y(y.vector, y.vlen);
    SGVector<float64_t> m=m_mean->get_mean_vector(m_features);
    Map<VectorXd> eigen_m(m.vector, m.vlen);
    CGaussianLikelihood* lik=CGaussianLikelihood::obtain_from_generic(m_model);
    float64_t sigma=lik->get_sigma();
    SG_UNREF(lik);
    eigen_mu=(eigen_y-eigen_m)-eigen_al*CMath::sq(sigma);
    */

    //FITC approximated posterior mean
    Map<VectorXd> eigen_alpha(m_alpha.vector, m_alpha.vlen);
    Map<MatrixXd> eigen_Ktru(m_ktru.matrix, m_ktru.num_rows, m_ktru.num_cols);
    //time complexity of the following operation is O(m*n)
    eigen_mu=CMath::exp(m_log_scale*2.0)*eigen_Ktru.adjoint()*eigen_alpha;

    return SGVector<float64_t>(m_mu);
}

SGMatrix<float64_t> CFITCInferenceMethod::get_posterior_covariance()
{
    compute_gradient();

    //time complexity of the following operations is O(m*n^2)
    //Warning: the the time complexity increases from O(m^2*n) to O(n^2*m) if this method is called
    m_Sigma=SGMatrix<float64_t>(m_ktrtr_diag.vlen, m_ktrtr_diag.vlen);
    Map<MatrixXd> eigen_Sigma(m_Sigma.matrix, m_Sigma.num_rows,
            m_Sigma.num_cols);
    Map<MatrixXd> eigen_V(m_V.matrix, m_V.num_rows, m_V.num_cols);
    Map<MatrixXd> eigen_Lu(m_chol_utr.matrix, m_chol_utr.num_rows,
            m_chol_utr.num_cols);

    /*
    //true posterior mean with equivalent FITC prior
    CGaussianLikelihood* lik=CGaussianLikelihood::obtain_from_generic(m_model);
    float64_t sigma=lik->get_sigma();
    SG_UNREF(lik);
    Map<VectorXd> eigen_t(m_t.vector, m_t.vlen);
    VectorXd diag_part=CMath::sq(sigma)*eigen_t;
    // diag(sigma2/dg)*V'*(Lu\eye(n))
    MatrixXd part1=diag_part.asDiagonal()*eigen_V.adjoint()*
        (eigen_Lu.triangularView<Upper>().solve(MatrixXd::Identity(
        m_kuu.num_rows, m_kuu.num_cols)));
    eigen_Sigma=part1*part1.adjoint();
    VectorXd part2=(VectorXd::Ones(m_t.vlen)-diag_part)*CMath::sq(sigma);
    eigen_Sigma+=part2.asDiagonal();
    */

    //FITC approximated posterior covariance
    //TO DO we only need tha diagonal elements of m_ktrtr
    Map<VectorXd> eigen_ktrtr_diag(m_ktrtr_diag.vector, m_ktrtr_diag.vlen);
    MatrixXd part1=eigen_V.adjoint()*(eigen_Lu.triangularView<Upper>().solve(MatrixXd::Identity(
        m_kuu.num_rows, m_kuu.num_cols)));
    eigen_Sigma=part1*part1.adjoint();
    VectorXd part2=eigen_ktrtr_diag*CMath::exp(m_log_scale*2.0)-(
        eigen_V.cwiseProduct(eigen_V)).colwise().sum().adjoint();
    eigen_Sigma+=part2.asDiagonal();

    return SGMatrix<float64_t>(m_Sigma);
}

SGVector<float64_t> CFITCInferenceMethod::get_derivative_wrt_likelihood_model(
        const TParameter* param)
{
    //time complexity O(m*n)
    REQUIRE(!strcmp(param->m_name, "log_sigma"), "Can't compute derivative of "
            "the nagative log marginal likelihood wrt %s.%s parameter\n",
            m_model->get_name(), param->m_name)

    // create eigen representation of dg, al, w, W and B
    Map<VectorXd> eigen_t(m_t.vector, m_t.vlen);
    Map<VectorXd> eigen_al(m_al.vector, m_al.vlen);
    Map<VectorXd> eigen_w(m_w.vector, m_w.vlen);
    Map<MatrixXd> eigen_W(m_Rvdd.matrix, m_Rvdd.num_rows, m_Rvdd.num_cols);
    Map<MatrixXd> eigen_B(m_B.matrix, m_B.num_rows, m_B.num_cols);

    // get the sigma variable from the Gaussian likelihood model
    CGaussianLikelihood* lik=CGaussianLikelihood::obtain_from_generic(m_model);
    float64_t sigma=lik->get_sigma();
    SG_UNREF(lik);

    SGVector<float64_t> result(1);

    //diag_dK = 1./g_sn2 - sum(W.*W,1)' - al.*al;                  % diag(dnlZ/dK)
    //dnlZ.lik = sn2*sum(diag_dK) without noise term
    result[0]=CMath::sq(sigma)*(VectorXd::Ones(m_t.vlen).cwiseProduct(
        eigen_t).sum()-eigen_W.cwiseProduct(eigen_W).sum()-eigen_al.dot(eigen_al));

    return result;
}

#endif /* HAVE_EIGEN3 */