LeastSquares.cc

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// -*- LSST-C++ -*-
 
/*
 * LSST Data Management System
 * Copyright 2008, 2009, 2010, 2011 LSST Corporation.
 *
 * This product includes software developed by the
 * LSST Project (http://www.lsst.org/).
 *
 * This program is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
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 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
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 */
 
#include "Eigen/Eigenvalues"
#include "Eigen/SVD"
#include "Eigen/Cholesky"
#include "boost/format.hpp"
#include <memory>
 
#include "lsst/afw/math/LeastSquares.h"
#include "lsst/pex/exceptions.h"
#include "lsst/log/Log.h"
 
namespace {
LOG_LOGGER _log = LOG_GET("lsst.afw.math.LeastSquares");
}
 
namespace lsst {
namespace afw {
namespace math {
 
class LeastSquares::Impl {
public:
    enum StateFlags {
        LOWER_FISHER_MATRIX = 0x001,
        FULL_FISHER_MATRIX = 0x002,
        RHS_VECTOR = 0x004,
        SOLUTION_ARRAY = 0x008,
        COVARIANCE_ARRAY = 0x010,
        DIAGNOSTIC_ARRAY = 0x020,
        DESIGN_AND_DATA = 0x040
    };
 
    int state;
    int dimension;
    int rank;
    Factorization factorization;
    Factorization whichDiagnostic;
    double threshold;
 
    Eigen::MatrixXd design;
    Eigen::VectorXd data;
    Eigen::MatrixXd fisher;
    Eigen::VectorXd rhs;
 
    ndarray::Array<double, 1, 1> solution;
    ndarray::Array<double, 2, 2> covariance;
    ndarray::Array<double, 1, 1> diagnostic;
 
    template <typename D>
    void setRank(Eigen::MatrixBase<D> const& values) {
        double cond = threshold * values[0];
        if (cond <= 0.0) {
            rank = 0;
        } else {
            for (rank = dimension; (rank > 1) && (values[rank - 1] < cond); --rank)
                ;
        }
    }
 
    void ensure(int desired) {
        if (state & FULL_FISHER_MATRIX) state |= LOWER_FISHER_MATRIX;
        if (desired & FULL_FISHER_MATRIX) desired |= LOWER_FISHER_MATRIX;
        int toAdd = ~state & desired;
        if (toAdd & LOWER_FISHER_MATRIX) {
            assert(state & DESIGN_AND_DATA);
            fisher = Eigen::MatrixXd::Zero(design.cols(), design.cols());
            fisher.selfadjointView<Eigen::Lower>().rankUpdate(design.adjoint());
        }
        if (toAdd & FULL_FISHER_MATRIX) {
            fisher.triangularView<Eigen::StrictlyUpper>() = fisher.adjoint();
        }
        if (toAdd & RHS_VECTOR) {
            assert(state & DESIGN_AND_DATA);
            rhs = design.adjoint() * data;
        }
        if (toAdd & SOLUTION_ARRAY) {
            if (solution.isEmpty()) solution = ndarray::allocate(dimension);
            updateSolution();
        }
        if (toAdd & COVARIANCE_ARRAY) {
            if (covariance.isEmpty()) covariance = ndarray::allocate(dimension, dimension);
            updateCovariance();
        }
        if (toAdd & DIAGNOSTIC_ARRAY) {
            if (diagnostic.isEmpty()) diagnostic = ndarray::allocate(dimension);
            updateDiagnostic();
        }
        state |= toAdd;
    }
 
    virtual void factor() = 0;
 
    virtual void updateRank() = 0;
 
    virtual void updateSolution() = 0;
    virtual void updateCovariance() = 0;
 
    virtual void updateDiagnostic() = 0;
 
    explicit Impl(
        int dimension_,
        Factorization factorization_,
        double threshold_ = std::numeric_limits<double>::epsilon()
    ) :
    state(0),
    dimension(dimension_),
    rank(dimension_),
    factorization(factorization_),
    whichDiagnostic(factorization_),
    threshold(threshold_)
    {}
 
    virtual ~Impl() = default;
};
 
namespace {
 
class EigensystemSolver : public LeastSquares::Impl {
public:
    explicit EigensystemSolver(int dimension)
        : Impl(dimension, LeastSquares::NORMAL_EIGENSYSTEM), _eig(dimension), _svd(), _tmp(dimension) {}
 
    void factor() override {
        ensure(LOWER_FISHER_MATRIX | RHS_VECTOR);
        _eig.compute(fisher);
        if (_eig.info() == Eigen::Success) {
            setRank(_eig.eigenvalues().reverse());
            LOGL_DEBUG(_log, "SelfAdjointEigenSolver succeeded: dimension=%d, rank=%d", dimension, rank);
        } else {
            // Note that the fallback is using SVD of the Fisher to compute the Eigensystem, because those
            // are the same for a symmetric matrix; this is very different from doing a direct SVD of
            // the design matrix.
            ensure(FULL_FISHER_MATRIX);
            _svd.compute(fisher, Eigen::ComputeFullU);  // Matrix is symmetric, so V == U == eigenvectors
            setRank(_svd.singularValues());
            LOGL_DEBUG(_log,
                       "SelfAdjointEigenSolver failed; falling back to equivalent SVD: dimension=%d, rank=%d",
                       dimension, rank);
        }
    }
 
    void updateRank() override {
        if (_eig.info() == Eigen::Success) {
            setRank(_eig.eigenvalues().reverse());
        } else {
            setRank(_svd.singularValues());
        }
    }
 
    void updateDiagnostic() override {
        if (whichDiagnostic == LeastSquares::NORMAL_CHOLESKY) {
            throw LSST_EXCEPT(
                    pex::exceptions::LogicError,
                    "Cannot compute NORMAL_CHOLESKY diagnostic from NORMAL_EIGENSYSTEM factorization.");
        }
        if (_eig.info() == Eigen::Success) {
            ndarray::asEigenMatrix(diagnostic) = _eig.eigenvalues().reverse();
        } else {
            ndarray::asEigenMatrix(diagnostic) = _svd.singularValues();
        }
        if (whichDiagnostic == LeastSquares::DIRECT_SVD) {
            ndarray::asEigenArray(diagnostic) = ndarray::asEigenArray(diagnostic).sqrt();
        }
    }
 
    void updateSolution() override {
        if (rank == 0) {
            ndarray::asEigenMatrix(solution).setZero();
            return;
        }
        if (_eig.info() == Eigen::Success) {
            _tmp.head(rank) = _eig.eigenvectors().rightCols(rank).adjoint() * rhs;
            _tmp.head(rank).array() /= _eig.eigenvalues().tail(rank).array();
            ndarray::asEigenMatrix(solution) = _eig.eigenvectors().rightCols(rank) * _tmp.head(rank);
        } else {
            _tmp.head(rank) = _svd.matrixU().leftCols(rank).adjoint() * rhs;
            _tmp.head(rank).array() /= _svd.singularValues().head(rank).array();
            ndarray::asEigenMatrix(solution) = _svd.matrixU().leftCols(rank) * _tmp.head(rank);
        }
    }
 
    void updateCovariance() override {
        if (rank == 0) {
            ndarray::asEigenMatrix(covariance).setZero();
            return;
        }
        if (_eig.info() == Eigen::Success) {
            ndarray::asEigenMatrix(covariance) =
                    _eig.eigenvectors().rightCols(rank) *
                    _eig.eigenvalues().tail(rank).array().inverse().matrix().asDiagonal() *
                    _eig.eigenvectors().rightCols(rank).adjoint();
        } else {
            ndarray::asEigenMatrix(covariance) =
                    _svd.matrixU().leftCols(rank) *
                    _svd.singularValues().head(rank).array().inverse().matrix().asDiagonal() *
                    _svd.matrixU().leftCols(rank).adjoint();
        }
    }
 
private:
    Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> _eig;
    Eigen::JacobiSVD<Eigen::MatrixXd> _svd;  // only used if Eigendecomposition fails, should be very rare
    Eigen::VectorXd _tmp;
};
 
class CholeskySolver : public LeastSquares::Impl {
public:
    explicit CholeskySolver(int dimension)
        : Impl(dimension, LeastSquares::NORMAL_CHOLESKY, 0.0), _ldlt(dimension) {}
 
    void factor() override {
        ensure(LOWER_FISHER_MATRIX | RHS_VECTOR);
        _ldlt.compute(fisher);
    }
 
    void updateRank() override {}
 
    void updateDiagnostic() override {
        if (whichDiagnostic != LeastSquares::NORMAL_CHOLESKY) {
            throw LSST_EXCEPT(
                    pex::exceptions::LogicError,
                    "Can only compute NORMAL_CHOLESKY diagnostic from NORMAL_CHOLESKY factorization.");
        }
        ndarray::asEigenMatrix(diagnostic) = _ldlt.vectorD();
    }
 
    void updateSolution() override { ndarray::asEigenMatrix(solution) = _ldlt.solve(rhs); }
 
    void updateCovariance() override {
        auto cov = ndarray::asEigenMatrix(covariance);
        cov.setIdentity();
        cov = _ldlt.solve(cov);
    }
 
private:
    Eigen::LDLT<Eigen::MatrixXd> _ldlt;
};
 
class SvdSolver : public LeastSquares::Impl {
public:
    explicit SvdSolver(int dimension)
        : Impl(dimension, LeastSquares::DIRECT_SVD), _svd(), _tmp(dimension) {}
 
    void factor() override {
        if (!(state & DESIGN_AND_DATA)) {
            throw LSST_EXCEPT(pex::exceptions::InvalidParameterError,
                              "Cannot initialize DIRECT_SVD solver with normal equations.");
        }
        _svd.compute(design, Eigen::ComputeThinU | Eigen::ComputeThinV);
        setRank(_svd.singularValues());
        LOGL_DEBUG(_log, "Using direct SVD method; dimension=%d, rank=%d", dimension, rank);
    }
 
    void updateRank() override { setRank(_svd.singularValues()); }
 
    void updateDiagnostic() override {
        switch (whichDiagnostic) {
            case LeastSquares::NORMAL_EIGENSYSTEM:
                ndarray::asEigenArray(diagnostic) = _svd.singularValues().array().square();
                break;
            case LeastSquares::NORMAL_CHOLESKY:
                throw LSST_EXCEPT(
                        pex::exceptions::LogicError,
                        "Can only compute NORMAL_CHOLESKY diagnostic from DIRECT_SVD factorization.");
            case LeastSquares::DIRECT_SVD:
                ndarray::asEigenMatrix(diagnostic) = _svd.singularValues();
                break;
        }
    }
 
    void updateSolution() override {
        if (rank == 0) {
            ndarray::asEigenMatrix(solution).setZero();
            return;
        }
        _tmp.head(rank) = _svd.matrixU().leftCols(rank).adjoint() * data;
        _tmp.head(rank).array() /= _svd.singularValues().head(rank).array();
        ndarray::asEigenMatrix(solution) = _svd.matrixV().leftCols(rank) * _tmp.head(rank);
    }
 
    void updateCovariance() override {
        if (rank == 0) {
            ndarray::asEigenMatrix(covariance).setZero();
            return;
        }
        ndarray::asEigenMatrix(covariance) =
                _svd.matrixV().leftCols(rank) *
                _svd.singularValues().head(rank).array().inverse().square().matrix().asDiagonal() *
                _svd.matrixV().leftCols(rank).adjoint();
    }
 
private:
    Eigen::JacobiSVD<Eigen::MatrixXd> _svd;
    Eigen::VectorXd _tmp;
};
 
}  // namespace
 
void LeastSquares::setThreshold(double threshold) {
    _impl->threshold = threshold;
    _impl->state &= ~Impl::SOLUTION_ARRAY;
    _impl->state &= ~Impl::COVARIANCE_ARRAY;
    _impl->updateRank();
}
 
double LeastSquares::getThreshold() const { return _impl->threshold; }
 
ndarray::Array<double const, 1, 1> LeastSquares::getSolution() {
    _impl->ensure(Impl::SOLUTION_ARRAY);
    return _impl->solution;
}
 
ndarray::Array<double const, 2, 2> LeastSquares::getCovariance() {
    _impl->ensure(Impl::COVARIANCE_ARRAY);
    return _impl->covariance;
}
 
ndarray::Array<double const, 2, 2> LeastSquares::getFisherMatrix() {
    _impl->ensure(Impl::FULL_FISHER_MATRIX);
    // Wrap the Eigen::MatrixXd in an ndarray::Array, using _impl as the reference-counted owner.
    // Doesn't matter if we swap strides, because it's symmetric.
    return ndarray::external(_impl->fisher.data(), ndarray::makeVector(_impl->dimension, _impl->dimension),
                             ndarray::makeVector(_impl->dimension, 1), _impl);
}
 
ndarray::Array<double const, 1, 1> LeastSquares::getDiagnostic(Factorization factorization) {
    if (_impl->whichDiagnostic != factorization) {
        _impl->state &= ~Impl::DIAGNOSTIC_ARRAY;
        _impl->whichDiagnostic = factorization;
    }
    _impl->ensure(Impl::DIAGNOSTIC_ARRAY);
    return _impl->diagnostic;
}
 
int LeastSquares::getDimension() const { return _impl->dimension; }
 
int LeastSquares::getRank() const { return _impl->rank; }
 
LeastSquares::Factorization LeastSquares::getFactorization() const { return _impl->factorization; }
 
LeastSquares::LeastSquares(Factorization factorization, int dimension) {
    switch (factorization) {
        case NORMAL_EIGENSYSTEM:
            _impl = std::make_shared<EigensystemSolver>(dimension);
            break;
        case NORMAL_CHOLESKY:
            _impl = std::make_shared<CholeskySolver>(dimension);
            break;
        case DIRECT_SVD:
            _impl = std::make_shared<SvdSolver>(dimension);
            break;
    }
}
 
LeastSquares::LeastSquares(LeastSquares const&) = default;
LeastSquares::LeastSquares(LeastSquares&&) = default;
LeastSquares& LeastSquares::operator=(LeastSquares const&) = default;
LeastSquares& LeastSquares::operator=(LeastSquares&&) = default;
 
LeastSquares::~LeastSquares() = default;
 
Eigen::MatrixXd& LeastSquares::_getDesignMatrix() { return _impl->design; }
Eigen::VectorXd& LeastSquares::_getDataVector() { return _impl->data; }
 
Eigen::MatrixXd& LeastSquares::_getFisherMatrix() { return _impl->fisher; }
Eigen::VectorXd& LeastSquares::_getRhsVector() { return _impl->rhs; }
 
void LeastSquares::_factor(bool haveNormalEquations) {
    if (haveNormalEquations) {
        if (_getFisherMatrix().rows() != _impl->dimension) {
            throw LSST_EXCEPT(pex::exceptions::InvalidParameterError,
                              (boost::format("Number of rows of Fisher matrix (%d) does not match"
                                             " dimension of LeastSquares solver.") %
                               _getFisherMatrix().rows() % _impl->dimension)
                                      .str());
        }
        if (_getFisherMatrix().cols() != _impl->dimension) {
            throw LSST_EXCEPT(pex::exceptions::InvalidParameterError,
                              (boost::format("Number of columns of Fisher matrix (%d) does not match"
                                             " dimension of LeastSquares solver.") %
                               _getFisherMatrix().cols() % _impl->dimension)
                                      .str());
        }
        if (_getRhsVector().size() != _impl->dimension) {
            throw LSST_EXCEPT(pex::exceptions::InvalidParameterError,
                              (boost::format("Number of elements in RHS vector (%d) does not match"
                                             " dimension of LeastSquares solver.") %
                               _getRhsVector().size() % _impl->dimension)
                                      .str());
        }
        _impl->state = Impl::RHS_VECTOR | Impl::FULL_FISHER_MATRIX | Impl::LOWER_FISHER_MATRIX;
    } else {
        if (_getDesignMatrix().cols() != _impl->dimension) {
            throw LSST_EXCEPT(
                    pex::exceptions::InvalidParameterError,
                    "Number of columns of design matrix does not match dimension of LeastSquares solver.");
        }
        if (_getDesignMatrix().rows() != _getDataVector().size()) {
            throw LSST_EXCEPT(pex::exceptions::InvalidParameterError,
                              (boost::format("Number of rows of design matrix (%d) does not match number of "
                                             "data points (%d)") %
                               _getDesignMatrix().rows() % _getDataVector().size())
                                      .str());
        }
        if (_getDesignMatrix().cols() > _getDataVector().size()) {
            throw LSST_EXCEPT(
                    pex::exceptions::InvalidParameterError,
                    (boost::format("Number of columns of design matrix (%d) must be smaller than number of "
                                   "data points (%d)") %
                     _getDesignMatrix().cols() % _getDataVector().size())
                            .str());
        }
        _impl->state = Impl::DESIGN_AND_DATA;
    }
    _impl->factor();
}
}  // namespace math
}  // namespace afw
}  // namespace lsst