devtools/eigen
1
0
Fork 0
mirror of https://gitlab.com/libeigen/eigen.git synced 2026-04-10 11:34:33 +08:00
Code Issues Packages Projects Releases Wiki Activity

Add a NNLS solver to unsupported - issue #655

Browse Source
This commit is contained in:
Essex Edwards
2022-03-23 20:20:44 +00:00
committed by Rasmus Munk Larsen
parent 0699fa06fe
commit cd3c81c3bc
6 changed files with 988 additions and 0 deletions
Download Patch File Download Diff File Expand all files Collapse all files

1
unsupported/Eigen/CMakeLists.txt
View File

@@ -12,6 +12,7 @@ set(Eigen_HEADERS
MatrixFunctions
MoreVectorization
MPRealSupport
NNLS
NonLinearOptimization
NumericalDiff
OpenGLSupport

388
unsupported/Eigen/NNLS Normal file
View File

@@ -0,0 +1,388 @@
/* Non-Negagive Least Squares Algorithm for Eigen.
*
* Copyright (C) 2021 Essex Edwards, <essex.edwards@gmail.com>
* Copyright (C) 2013 Hannes Matuschek, hannes.matuschek at uni-potsdam.de
*
* This Source Code Form is subject to the terms of the Mozilla
* Public License v. 2.0. If a copy of the MPL was not distributed
* with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
*/
/** \defgroup nnls Non-Negative Least Squares (NNLS) Module
* This module provides a single class @c Eigen::NNLS implementing the NNLS algorithm.
* The algorithm is described in "SOLVING LEAST SQUARES PROBLEMS", by Charles L. Lawson and
* Richard J. Hanson, Prentice-Hall, 1974 and solves optimization problems of the form
*
* \f[ \min \left\Vert Ax-b\right\Vert_2^2\quad s.t.\, x\ge 0\,.\f]
*
* The algorithm solves the constrained least-squares problem above by iteratively improving
* an estimate of which constraints are active (elements of \f$x\f$ equal to zero)
* and which constraints are inactive (elements of \f$x\f$ greater than zero).
* Each iteration, an unconstrained linear least-squares problem solves for the
* components of \f$x\f$ in the (estimated) inactive set and the sets are updated.
* The unconstrained problem minimizes \f$\left\Vert A^Nx^N-b\right\Vert_2^2\f$,
* where \f$A^N\f$ is a matrix formed by selecting all columns of A which are
* in the inactive set \f$N\f$.
*
*/
#ifndef EIGEN_NNLS_H
#define EIGEN_NNLS_H
#include "../../Eigen/Core"
#include "../../Eigen/QR"
#include <limits>
namespace Eigen {
/** \ingroup nnls
* \class NNLS
* \brief Implementation of the Non-Negative Least Squares (NNLS) algorithm.
* \tparam MatrixType The type of the system matrix \f$A\f$.
*
* This class implements the NNLS algorithm as described in "SOLVING LEAST SQUARES PROBLEMS",
* Charles L. Lawson and Richard J. Hanson, Prentice-Hall, 1974. This algorithm solves a least
* squares problem iteratively and ensures that the solution is non-negative. I.e.
*
* \f[ \min \left\Vert Ax-b\right\Vert_2^2\quad s.t.\, x\ge 0 \f]
*
* The algorithm solves the constrained least-squares problem above by iteratively improving
* an estimate of which constraints are active (elements of \f$x\f$ equal to zero)
* and which constraints are inactive (elements of \f$x\f$ greater than zero).
* Each iteration, an unconstrained linear least-squares problem solves for the
* components of \f$x\f$ in the (estimated) inactive set and the sets are updated.
* The unconstrained problem minimizes \f$\left\Vert A^Nx^N-b\right\Vert_2^2\f$,
* where \f$A^N\f$ is a matrix formed by selecting all columns of A which are
* in the inactive set \f$N\f$.
*
* See <a href="https://en.wikipedia.org/wiki/Non-negative_least_squares">the
* wikipedia page on non-negative least squares</a> for more background information.
*
* \note Please note that it is possible to construct an NNLS problem for which the
* algorithm does not converge. In practice these cases are extremely rare.
*/
template <class MatrixType_>
class NNLS {
public:
typedef MatrixType_ MatrixType;
enum {
RowsAtCompileTime = MatrixType::RowsAtCompileTime,
ColsAtCompileTime = MatrixType::ColsAtCompileTime,
Options = MatrixType::Options,
MaxRowsAtCompileTime = MatrixType::MaxRowsAtCompileTime,
MaxColsAtCompileTime = MatrixType::MaxColsAtCompileTime
};
typedef typename MatrixType::Scalar Scalar;
typedef typename MatrixType::RealScalar RealScalar;
typedef typename MatrixType::Index Index;
/** Type of a row vector of the system matrix \f$A\f$. */
typedef Matrix<Scalar, ColsAtCompileTime, 1> SolutionVectorType;
/** Type of a column vector of the system matrix \f$A\f$. */
typedef Matrix<Scalar, RowsAtCompileTime, 1> RhsVectorType;
typedef Matrix<Index, ColsAtCompileTime, 1> IndicesType;
/** */
NNLS();
/** \brief Constructs a NNLS sovler and initializes it with the given system matrix @c A.
* \param A Specifies the system matrix.
* \param max_iter Specifies the maximum number of iterations to solve the system.
* \param tol Specifies the precision of the optimum.
* This is an absolute tolerance on the gradient of the Lagrangian, \f$A^T(Ax-b)-\lambda\f$
* (with Lagrange multipliers \f$\lambda\f$).
*/
NNLS(const MatrixType &A, Index max_iter = -1, Scalar tol = NumTraits<Scalar>::dummy_precision());
/** Initializes the solver with the matrix \a A for further solving NNLS problems.
*
* This function mostly initializes/computes the preconditioner. In the future
* we might, for instance, implement column reordering for faster matrix vector products.
*/
template <typename MatrixDerived>
NNLS<MatrixType> &compute(const EigenBase<MatrixDerived> &A);
/** \brief Solves the NNLS problem.
*
* The dimension of @c b must be equal to the number of rows of @c A, given to the constructor.
*
* \returns The approximate solution vector \f$ x \f$. Use info() to determine if the solve was a success or not.
* \sa info()
*/
const SolutionVectorType &solve(const RhsVectorType &b);
/** \brief Returns the solution if a problem was solved.
* If not, an uninitialized vector may be returned. */
const SolutionVectorType &x() const { return x_; }
/** \returns the tolerance threshold used by the stopping criteria.
* \sa setTolerance()
*/
Scalar tolerance() const { return tolerance_; }
/** Sets the tolerance threshold used by the stopping criteria.
*
* This is an absolute tolerance on the gradient of the Lagrangian, \f$A^T(Ax-b)-\lambda\f$
* (with Lagrange multipliers \f$\lambda\f$).
*/
NNLS<MatrixType> &setTolerance(const Scalar &tolerance) {
tolerance_ = tolerance;
return *this;
}
/** \returns the max number of iterations.
* It is either the value set by setMaxIterations or, by default, twice the number of columns of the matrix.
*/
Index maxIterations() const { return max_iter_ < 0 ? 2 * A_.cols() : max_iter_; }
/** Sets the max number of iterations.
* Default is twice the number of columns of the matrix.
* The algorithm requires at least k iterations to produce a solution vector with k non-zero entries.
*/
NNLS<MatrixType> &setMaxIterations(Index maxIters) {
max_iter_ = maxIters;
return *this;
}
/** \returns the number of iterations (least-squares solves) performed during the last solve */
Index iterations() const { return iterations_; }
/** \returns Success if the iterations converged, and an error values otherwise. */
ComputationInfo info() const { return info_; }
private:
/** \internal Adds the given index @c idx to the inactive set N and updates the QR decomposition of \f$A^N\f$. */
void moveToInactiveSet_(Index idx);
/** \internal Removes the given index idx from the inactive set N and updates the QR decomposition of \f$A^N\f$. */
void moveToActiveSet_(Index idx);
/** \internal Solves the least-squares problem \f$\left\Vert y-A^Nx\right\Vert_2^2\f$. */
void solveInactiveSet_(const RhsVectorType &b);
private:
typedef Matrix<Scalar, ColsAtCompileTime, ColsAtCompileTime> MatrixAtAType;
/** \internal Holds the maximum number of iterations for the NNLS algorithm.
* @c -1 means to use the default value. */
Index max_iter_;
/** \internal Holds the number of iterations. */
Index iterations_;
/** \internal Holds success/fail of the last solve. */
ComputationInfo info_;
/** \internal Size of the inactive set. */
Index numInactive_;
/** \internal Accuracy of the algorithm w.r.t the optimality of the solution (gradient). */
Scalar tolerance_;
/** \internal The system matrix, a copy of the one given to the constructor. */
MatrixType A_;
/** \internal Precomputed product \f$A^TA\f$. */
MatrixAtAType AtA_;
/** \internal Will hold the solution. */
SolutionVectorType x_;
/** \internal Will hold the current gradient.\f$A^Tb - A^TAx\f$ */
SolutionVectorType gradient_;
/** \internal Will hold the partial solution. */
SolutionVectorType y_;
/** \internal Precomputed product \f$A^Tb\f$. */
SolutionVectorType Atb_;
/** \internal Holds the current permutation partitioning the active and inactive sets.
* The first @c numInactive_ elements form the inactive set and the rest the active set. */
IndicesType index_sets_;
/** \internal QR decomposition to solve the (inactive) sub system (together with @c qrCoeffs_). */
MatrixType QR_;
/** \internal QR decomposition to solve the (inactive) sub system (together with @c QR_). */
SolutionVectorType qrCoeffs_;
/** \internal Some workspace for QR decomposition. */
SolutionVectorType tempSolutionVector_;
RhsVectorType tempRhsVector_;
};
/* ********************************************************************************************
* Implementation
* ******************************************************************************************** */
template <typename MatrixType>
NNLS<MatrixType>::NNLS()
: max_iter_(-1),
iterations_(0),
info_(ComputationInfo::InvalidInput),
numInactive_(0),
tolerance_(NumTraits<Scalar>::dummy_precision()) {}
template <typename MatrixType>
NNLS<MatrixType>::NNLS(const MatrixType &A, Index max_iter, Scalar tol) : max_iter_(max_iter), tolerance_(tol) {
compute(A);
}
template <typename MatrixType>
template <typename MatrixDerived>
NNLS<MatrixType> &NNLS<MatrixType>::compute(const EigenBase<MatrixDerived> &A) {
// Ensure Scalar type is real. The non-negativity constraint doesn't obviously extend to complex numbers.
EIGEN_STATIC_ASSERT(!NumTraits<Scalar>::IsComplex, NUMERIC_TYPE_MUST_BE_REAL);
// max_iter_: unchanged
iterations_ = 0;
info_ = ComputationInfo::Success;
numInactive_ = 0;
// tolerance: unchanged
A_ = A.derived();
AtA_.noalias() = A_.transpose() * A_;
x_.resize(A_.cols());
gradient_.resize(A_.cols());
y_.resize(A_.cols());
Atb_.resize(A_.cols());
index_sets_.resize(A_.cols());
QR_.resize(A_.rows(), A_.cols());
qrCoeffs_.resize(A_.cols());
tempSolutionVector_.resize(A_.cols());
tempRhsVector_.resize(A_.rows());
return *this;
}
template <typename MatrixType>
const typename NNLS<MatrixType>::SolutionVectorType &NNLS<MatrixType>::solve(const RhsVectorType &b) {
// Initialize solver
iterations_ = 0;
info_ = ComputationInfo::NumericalIssue;
x_.setZero();
index_sets_ = IndicesType::LinSpaced(A_.cols(), 0, A_.cols() - 1); // Identity permutation.
numInactive_ = 0;
// Precompute A^T*b
Atb_.noalias() = A_.transpose() * b;
const Index maxIterations = this->maxIterations();
// OUTER LOOP
while (true) {
// Early exit if all variables are inactive, which breaks 'maxCoeff' below.
if (A_.cols() == numInactive_) {
info_ = ComputationInfo::Success;
return x_;
}
// Find the maximum element of the gradient in the active set.
// If it is small or negative, then we have converged.
// Else, we move that variable to the inactive set.
gradient_.noalias() = Atb_ - AtA_ * x_;
const Index numActive = A_.cols() - numInactive_;
Index argmaxGradient = -1;
const Scalar maxGradient = gradient_(index_sets_.tail(numActive)).maxCoeff(&argmaxGradient);
argmaxGradient += numInactive_; // beacause tail() skipped the first numInactive_ elements
if (maxGradient < tolerance_) {
info_ = ComputationInfo::Success;
return x_;
}
moveToInactiveSet_(argmaxGradient);
// INNER LOOP
while (true) {
// Check if max. number of iterations is reached
if (iterations_ >= maxIterations) {
info_ = ComputationInfo::NoConvergence;
return x_;
}
// Solve least-squares problem in inactive set only,
// this step is rather trivial as moveToInactiveSet_ & moveToActiveSet_
// updates the QR decomposition of inactive columns A^N.
// solveInactiveSet_ puts the solution in y_
solveInactiveSet_(b);
++iterations_; // The solve is expensive, so that is what we count as an iteration.
// Check feasability...
bool feasible = true;
Scalar alpha = NumTraits<Scalar>::highest();
Index infeasibleIdx = -1; // Which variable became infeasible first.
for (Index i = 0; i < numInactive_; i++) {
Index idx = index_sets_[i];
if (y_(idx) < 0) {
// t should always be in [0,1].
Scalar t = -x_(idx) / (y_(idx) - x_(idx));
if (alpha > t) {
alpha = t;
infeasibleIdx = i;
feasible = false;
}
}
}
eigen_assert(feasible || 0 <= infeasibleIdx);
// If solution is feasible, exit to outer loop
if (feasible) {
x_ = y_;
break;
}
// Infeasible solution -> interpolate to feasible one
for (Index i = 0; i < numInactive_; i++) {
Index idx = index_sets_[i];
x_(idx) += alpha * (y_(idx) - x_(idx));
}
// Remove these indices from the inactive set and update QR decomposition
moveToActiveSet_(infeasibleIdx);
}
}
}
template <typename MatrixType>
void NNLS<MatrixType>::moveToInactiveSet_(Index idx) {
// Update permutation matrix:
std::swap(index_sets_(idx), index_sets_(numInactive_));
numInactive_++;
// Perform rank-1 update of the QR decomposition stored in QR_ & qrCoeff_
internal::householder_qr_inplace_update(QR_, qrCoeffs_, A_.col(index_sets_(numInactive_ - 1)), numInactive_ - 1,
tempSolutionVector_.data());
}
template <typename MatrixType>
void NNLS<MatrixType>::moveToActiveSet_(Index idx) {
// swap index with last inactive one & reduce number of inactive columns
std::swap(index_sets_(idx), index_sets_(numInactive_ - 1));
numInactive_--;
// Update QR decomposition starting from the removed index up to the end [idx, ..., numInactive_]
for (Index i = idx; i < numInactive_; i++) {
Index col = index_sets_(i);
internal::householder_qr_inplace_update(QR_, qrCoeffs_, A_.col(col), i, tempSolutionVector_.data());
}
}
template <typename MatrixType>
void NNLS<MatrixType>::solveInactiveSet_(const RhsVectorType &b) {
eigen_assert(numInactive_ > 0);
tempRhsVector_ = b;
// tmpRHS(0:numInactive_-1) := Q'*b
// tmpRHS(numInactive_:end) := useless stuff we would rather not compute at all.
tempRhsVector_.applyOnTheLeft(
householderSequence(QR_.leftCols(numInactive_), qrCoeffs_.head(numInactive_)).transpose());
// tempSol(0:numInactive_-1) := inv(R) * Q' * b
// = the least-squares solution for the inactive variables.
tempSolutionVector_.head(numInactive_) = //
QR_.topLeftCorner(numInactive_, numInactive_) //
.template triangularView<Upper>() //
.solve(tempRhsVector_.head(numInactive_)); //
// tempSol(numInactive_:end) := 0 = the value for the constrained variables.
tempSolutionVector_.tail(y_.size() - numInactive_).setZero();
// Back permute into original column order of A
y_.noalias() = index_sets_.asPermutation() * tempSolutionVector_.head(y_.size());
}
} // namespace Eigen
#endif // EIGEN_NNLS_H

2
unsupported/test/CMakeLists.txt
View File

@@ -60,6 +60,8 @@ else()
ei_add_property(EIGEN_MISSING_BACKENDS "MPFR C++, ")
endif()
ei_add_test(NNLS)
ei_add_test(sparse_extra "" "")
find_package(FFTW)

472
unsupported/test/NNLS.cpp Normal file
View File

@@ -0,0 +1,472 @@
// This file is part of Eigen, a lightweight C++ template library
// for linear algebra.
//
// Copyright (C) Essex Edwards <essex.edwards@gmail.com>
//
// This Source Code Form is subject to the terms of the Mozilla
// Public License v. 2.0. If a copy of the MPL was not distributed
// with this file, You can obtain one at http://mozilla.org/MPL/2.0/.
#define EIGEN_RUNTIME_NO_MALLOC
#include "main.h"
#include <unsupported/Eigen/NNLS>
/// Check that 'x' solves the NNLS optimization problem `min ||A*x-b|| s.t. 0 <= x`.
/// The \p tolerance parameter is the absolute tolerance on the gradient, A'*(A*x-b).
template <typename MatrixType, typename VectorB, typename VectorX, typename Scalar>
static void verify_nnls_optimality(const MatrixType &A, const VectorB &b, const VectorX &x, const Scalar tolerance) {
// The NNLS optimality conditions are:
//
// * 0 = A'*A*x - A'*b - lambda
// * 0 <= x[i] \forall i
// * 0 <= lambda[i] \forall i
// * 0 = x[i]*lambda[i] \forall i
//
// we don't know lambda, but by assuming the first optimality condition is true,
// we can derive it and then check the others conditions.
const VectorX lambda = A.transpose() * (A * x - b);
// NNLS solutions are EXACTLY not negative.
VERIFY_LE(0, x.minCoeff());
// Exact lambda would be non-negative, but computed lambda might leak a little
VERIFY_LE(-tolerance, lambda.minCoeff());
// x[i]*lambda[i] == 0 <~~> (x[i]==0) || (lambda[i] is small)
VERIFY(((x.array() == Scalar(0)) || (lambda.array() <= tolerance)).all());
}
template <typename MatrixType, typename VectorB, typename VectorX>
static void test_nnls_known_solution(const MatrixType &A, const VectorB &b, const VectorX &x_expected) {
using Scalar = typename MatrixType::Scalar;
using std::sqrt;
const Scalar tolerance = sqrt(Eigen::GenericNumTraits<Scalar>::epsilon());
Index max_iter = 5 * A.cols(); // A heuristic guess.
NNLS<MatrixType> nnls(A, max_iter, tolerance);
const VectorX x = nnls.solve(b);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
VERIFY_IS_APPROX(x, x_expected);
verify_nnls_optimality(A, b, x, tolerance);
}
template <typename MatrixType>
static void test_nnls_random_problem() {
//
// SETUP
//
Index cols = MatrixType::ColsAtCompileTime;
if (cols == Dynamic) cols = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
Index rows = MatrixType::RowsAtCompileTime;
if (rows == Dynamic) rows = internal::random<Index>(cols, EIGEN_TEST_MAX_SIZE);
VERIFY_LE(cols, rows); // To have a unique LS solution: cols <= rows.
// Make some sort of random test problem from a wide range of scales and condition numbers.
using std::pow;
using Scalar = typename MatrixType::Scalar;
const Scalar sqrtConditionNumber = pow(Scalar(10), internal::random<Scalar>(Scalar(0), Scalar(2)));
const Scalar scaleA = pow(Scalar(10), internal::random<Scalar>(Scalar(-3), Scalar(3)));
const Scalar minSingularValue = scaleA / sqrtConditionNumber;
const Scalar maxSingularValue = scaleA * sqrtConditionNumber;
MatrixType A(rows, cols);
generateRandomMatrixSvs(setupRangeSvs<Matrix<Scalar, Dynamic, 1>>(cols, minSingularValue, maxSingularValue), rows,
cols, A);
// Make a random RHS also with a random scaling.
using VectorB = decltype(A.col(0).eval());
const Scalar scaleB = pow(Scalar(10), internal::random<Scalar>(Scalar(-3), Scalar(3)));
const VectorB b = scaleB * VectorB::Random(A.rows());
//
// ACT
//
using Scalar = typename MatrixType::Scalar;
using std::sqrt;
const Scalar tolerance =
sqrt(Eigen::GenericNumTraits<Scalar>::epsilon()) * b.cwiseAbs().maxCoeff() * A.cwiseAbs().maxCoeff();
Index max_iter = 5 * A.cols(); // A heuristic guess.
NNLS<MatrixType> nnls(A, max_iter, tolerance);
const NNLS<MatrixType>::SolutionVectorType &x = nnls.solve(b);
//
// VERIFY
//
// In fact, NNLS can fail on some problems, but they are rare in practice.
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
verify_nnls_optimality(A, b, x, tolerance);
}
static void test_nnls_handles_zero_rhs() {
//
// SETUP
//
const Index cols = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
const Index rows = internal::random<Index>(cols, EIGEN_TEST_MAX_SIZE);
const MatrixXd A = MatrixXd::Random(rows, cols);
const VectorXd b = VectorXd::Zero(rows);
//
// ACT
//
NNLS<MatrixXd> nnls(A);
const VectorXd x = nnls.solve(b);
//
// VERIFY
//
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
VERIFY_LE(nnls.iterations(), 1); // 0 or 1 would be be fine for an edge case like this.
VERIFY_IS_EQUAL(x, VectorXd::Zero(cols));
}
static void test_nnls_handles_Mx0_matrix() {
//
// SETUP
//
const Index rows = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
const MatrixXd A(rows, 0);
const VectorXd b = VectorXd::Random(rows);
//
// ACT
//
NNLS<MatrixXd> nnls(A);
const VectorXd x = nnls.solve(b);
//
// VERIFY
//
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
VERIFY_LE(nnls.iterations(), 0);
VERIFY_IS_EQUAL(x.size(), 0);
}
static void test_nnls_handles_0x0_matrix() {
//
// SETUP
//
const MatrixXd A(0, 0);
const VectorXd b(0);
//
// ACT
//
NNLS<MatrixXd> nnls(A);
const VectorXd x = nnls.solve(b);
//
// VERIFY
//
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
VERIFY_LE(nnls.iterations(), 0);
VERIFY_IS_EQUAL(x.size(), 0);
}
static void test_nnls_handles_dependent_columns() {
//
// SETUP
//
const Index rank = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE / 2);
const Index cols = 2 * rank;
const Index rows = internal::random<Index>(cols, EIGEN_TEST_MAX_SIZE);
const MatrixXd A = MatrixXd::Random(rows, rank) * MatrixXd::Random(rank, cols);
const VectorXd b = VectorXd::Random(rows);
//
// ACT
//
const double tolerance = 1e-8;
NNLS<MatrixXd> nnls(A);
const VectorXd &x = nnls.solve(b);
//
// VERIFY
//
// What should happen when the input 'A' has dependent columns?
// We might still succeed. Or we might not converge.
// Either outcome is fine. If Success is indicated,
// then 'x' must actually be a solution vector.
if (nnls.info() == ComputationInfo::Success) {
verify_nnls_optimality(A, b, x, tolerance);
}
}
static void test_nnls_handles_wide_matrix() {
//
// SETUP
//
const Index cols = internal::random<Index>(2, EIGEN_TEST_MAX_SIZE);
const Index rows = internal::random<Index>(2, cols - 1);
const MatrixXd A = MatrixXd::Random(rows, cols);
const VectorXd b = VectorXd::Random(rows);
//
// ACT
//
const double tolerance = 1e-8;
NNLS<MatrixXd> nnls(A);
const VectorXd &x = nnls.solve(b);
//
// VERIFY
//
// What should happen when the input 'A' is wide?
// The unconstrained least-squares problem has infinitely many solutions.
// Subject the the non-negativity constraints,
// the solution might actually be unique (e.g. it is [0,0,..,0]).
// So, NNLS might succeed or it might fail.
// Either outcome is fine. If Success is indicated,
// then 'x' must actually be a solution vector.
if (nnls.info() == ComputationInfo::Success) {
verify_nnls_optimality(A, b, x, tolerance);
}
}
// 4x2 problem, unconstrained solution positive
static void test_nnls_known_1() {
Matrix<double, 4, 2> A(4, 2);
Matrix<double, 4, 1> b(4);
Matrix<double, 2, 1> x(2);
A << 1, 1, 2, 4, 3, 9, 4, 16;
b << 0.6, 2.2, 4.8, 8.4;
x << 0.1, 0.5;
return test_nnls_known_solution(A, b, x);
}
// 4x3 problem, unconstrained solution positive
static void test_nnls_known_2() {
Matrix<double, 4, 3> A(4, 3);
Matrix<double, 4, 1> b(4);
Matrix<double, 3, 1> x(3);
A << 1, 1, 1, 2, 4, 8, 3, 9, 27, 4, 16, 64;
b << 0.73, 3.24, 8.31, 16.72;
x << 0.1, 0.5, 0.13;
test_nnls_known_solution(A, b, x);
}
// Simple 4x4 problem, unconstrained solution non-negative
static void test_nnls_known_3() {
Matrix<double, 4, 4> A(4, 4);
Matrix<double, 4, 1> b(4);
Matrix<double, 4, 1> x(4);
A << 1, 1, 1, 1, 2, 4, 8, 16, 3, 9, 27, 81, 4, 16, 64, 256;
b << 0.73, 3.24, 8.31, 16.72;
x << 0.1, 0.5, 0.13, 0;
test_nnls_known_solution(A, b, x);
}
// Simple 4x3 problem, unconstrained solution non-negative
static void test_nnls_known_4() {
Matrix<double, 4, 3> A(4, 3);
Matrix<double, 4, 1> b(4);
Matrix<double, 3, 1> x(3);
A << 1, 1, 1, 2, 4, 8, 3, 9, 27, 4, 16, 64;
b << 0.23, 1.24, 3.81, 8.72;
x << 0.1, 0, 0.13;
test_nnls_known_solution(A, b, x);
}
// Simple 4x3 problem, unconstrained solution indefinite
static void test_nnls_known_5() {
Matrix<double, 4, 3> A(4, 3);
Matrix<double, 4, 1> b(4);
Matrix<double, 3, 1> x(3);
A << 1, 1, 1, 2, 4, 8, 3, 9, 27, 4, 16, 64;
b << 0.13, 0.84, 2.91, 7.12;
// Solution obtained by original nnls() implementation in Fortran
x << 0.0, 0.0, 0.1106544;
test_nnls_known_solution(A, b, x);
}
static void test_nnls_small_reference_problems() {
test_nnls_known_1();
test_nnls_known_2();
test_nnls_known_3();
test_nnls_known_4();
test_nnls_known_5();
}
static void test_nnls_with_half_precision() {
// The random matrix generation tools don't work with `half`,
// so here's a simpler setup mostly just to check that NNLS compiles & runs with custom scalar types.
using Mat = Matrix<half, 8, 2>;
using VecB = Matrix<half, 8, 1>;
using VecX = Matrix<half, 2, 1>;
Mat A = Mat::Random(); // full-column rank with high probability.
VecB b = VecB::Random();
NNLS<Mat> nnls(A, 20, half(1e-2f));
const VecX x = nnls.solve(b);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
verify_nnls_optimality(A, b, x, half(1e-1));
}
static void test_nnls_special_case_solves_in_zero_iterations() {
// The particular NNLS algorithm that is implemented starts with all variables
// in the active set.
// This test builds a system where all constraints are active at the solution,
// so that initial guess is already correct.
//
// If the implementation changes to another algorithm that does not have this property,
// then this test will need to change (e.g. starting from all constraints inactive,
// or using ADMM, or an interior point solver).
const Index n = 10;
const Index m = 3 * n;
const VectorXd b = VectorXd::Random(m);
// With high probability, this is full column rank, which we need for uniqueness.
MatrixXd A = MatrixXd::Random(m, n);
// Make every column of `A` such that adding it to the active set only /increases/ the objective,
// this ensuring the NNLS solution is all zeros.
const VectorXd alignment = -(A.transpose() * b).cwiseSign();
A = A * alignment.asDiagonal();
NNLS<MatrixXd> nnls(A);
nnls.solve(b);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
VERIFY(nnls.iterations() == 0);
}
static void test_nnls_special_case_solves_in_n_iterations() {
// The particular NNLS algorithm that is implemented starts with all variables
// in the active set and then adds one variable to the inactive set each iteration.
// This test builds a system where all variables are inactive at the solution,
// so it should take 'n' iterations to get there.
//
// If the implementation changes to another algorithm that does not have this property,
// then this test will need to change (e.g. starting from all constraints inactive,
// or using ADMM, or an interior point solver).
const Index n = 10;
const Index m = 3 * n;
// With high probability, this is full column rank, which we need for uniqueness.
const MatrixXd A = MatrixXd::Random(m, n);
const VectorXd x = VectorXd::Random(n).cwiseAbs().array() + 1; // all positive.
const VectorXd b = A * x;
NNLS<MatrixXd> nnls(A);
nnls.solve(b);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
VERIFY(nnls.iterations() == n);
}
static void test_nnls_returns_NoConvergence_when_maxIterations_is_too_low() {
// Using the special case that takes `n` iterations,
// from `test_nnls_special_case_solves_in_n_iterations`,
// we can set max iterations too low and that should cause the solve to fail.
const Index n = 10;
const Index m = 3 * n;
// With high probability, this is full column rank, which we need for uniqueness.
const MatrixXd A = MatrixXd::Random(m, n);
const VectorXd x = VectorXd::Random(n).cwiseAbs().array() + 1; // all positive.
const VectorXd b = A * x;
NNLS<MatrixXd> nnls(A);
const Index max_iters = n - 1;
nnls.setMaxIterations(max_iters);
nnls.solve(b);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::NoConvergence);
VERIFY(nnls.iterations() == max_iters);
}
static void test_nnls_default_maxIterations_is_twice_column_count() {
const Index cols = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
const Index rows = internal::random<Index>(cols, EIGEN_TEST_MAX_SIZE);
const MatrixXd A = MatrixXd::Random(rows, cols);
NNLS<MatrixXd> nnls(A);
VERIFY_IS_EQUAL(nnls.maxIterations(), 2 * cols);
}
static void test_nnls_does_not_allocate_during_solve() {
const Index cols = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
const Index rows = internal::random<Index>(cols, EIGEN_TEST_MAX_SIZE);
const MatrixXd A = MatrixXd::Random(rows, cols);
const VectorXd b = VectorXd::Random(rows);
NNLS<MatrixXd> nnls(A);
internal::set_is_malloc_allowed(false);
nnls.solve(b);
internal::set_is_malloc_allowed(true);
}
static void test_nnls_repeated_calls_to_compute_and_solve() {
const Index cols2 = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
const Index rows2 = internal::random<Index>(cols2, EIGEN_TEST_MAX_SIZE);
const MatrixXd A2 = MatrixXd::Random(rows2, cols2);
const VectorXd b2 = VectorXd::Random(rows2);
NNLS<MatrixXd> nnls;
for (int i = 0; i < 4; ++i) {
const Index cols = internal::random<Index>(1, EIGEN_TEST_MAX_SIZE);
const Index rows = internal::random<Index>(cols, EIGEN_TEST_MAX_SIZE);
const MatrixXd A = MatrixXd::Random(rows, cols);
nnls.compute(A);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
for (int j = 0; j < 3; ++j) {
const VectorXd b = VectorXd::Random(rows);
const VectorXd x = nnls.solve(b);
VERIFY_IS_EQUAL(nnls.info(), ComputationInfo::Success);
verify_nnls_optimality(A, b, x, 1e-4);
}
}
}
EIGEN_DECLARE_TEST(NNLS) {
// Small matrices with known solutions:
CALL_SUBTEST_1(test_nnls_small_reference_problems());
CALL_SUBTEST_1(test_nnls_handles_Mx0_matrix());
CALL_SUBTEST_1(test_nnls_handles_0x0_matrix());
for (int i = 0; i < g_repeat; i++) {
// Essential NNLS properties, across different types.
CALL_SUBTEST_2(test_nnls_random_problem<MatrixXf>());
CALL_SUBTEST_3(test_nnls_random_problem<MatrixXd>());
using MatFixed = Matrix<double, 12, 5>;
CALL_SUBTEST_4(test_nnls_random_problem<MatFixed>());
CALL_SUBTEST_5(test_nnls_with_half_precision());
// Robustness tests:
CALL_SUBTEST_6(test_nnls_handles_zero_rhs());
CALL_SUBTEST_6(test_nnls_handles_dependent_columns());
CALL_SUBTEST_6(test_nnls_handles_wide_matrix());
// Properties specific to the implementation,
// not NNLS in general.
CALL_SUBTEST_7(test_nnls_special_case_solves_in_zero_iterations());
CALL_SUBTEST_7(test_nnls_special_case_solves_in_n_iterations());
CALL_SUBTEST_7(test_nnls_returns_NoConvergence_when_maxIterations_is_too_low());
CALL_SUBTEST_7(test_nnls_default_maxIterations_is_twice_column_count());
CALL_SUBTEST_8(test_nnls_repeated_calls_to_compute_and_solve());
// This test fails. It hits allocations in HouseholderSequence.h
// test_nnls_does_not_allocate_during_solve();
}
}