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Addresses comments on Eigen pull request PR-174.

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* Get rid of code-duplication for real vs. complex matrices.
* Fix flipped arguments to select.
* Make the condition estimation functions free functions.
* Use Vector::Unit() to generate canonical unit vectors.
* Misc. cleanup.
This commit is contained in:
Rasmus Munk Larsen
2016-04-04 14:20:01 -07:00
parent 30242b7565
commit 86e0ed81f8
7 changed files with 177 additions and 267 deletions
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401
Eigen/src/Core/ConditionEstimator.h
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@@ -13,19 +13,35 @@
namespace Eigen {
namespace internal {
template <typename Decomposition, bool IsSelfAdjoint, bool IsComplex>
struct EstimateInverseMatrixL1NormImpl {};
template <typename MatrixType>
inline typename MatrixType::RealScalar MatrixL1Norm(const MatrixType& matrix) {
return matrix.cwiseAbs().colwise().sum().maxCoeff();
}
template <typename Vector>
inline typename Vector::RealScalar VectorL1Norm(const Vector& v) {
return v.template lpNorm<1>();
}
template <typename Vector, typename RealVector, bool IsComplex>
struct SignOrUnity {
static inline Vector run(const Vector& v) {
const RealVector v_abs = v.cwiseAbs();
return (v_abs.array() == 0).select(Vector::Ones(v.size()), v.cwiseQuotient(v_abs));
}
};
// Partial specialization to avoid elementwise division for real vectors.
template <typename Vector>
struct SignOrUnity<Vector, Vector, false> {
static inline Vector run(const Vector& v) {
return (v.array() < 0).select(-Vector::Ones(v.size()), Vector::Ones(v.size()));
}
};
} // namespace internal
template <typename Decomposition, bool IsSelfAdjoint = false>
class ConditionEstimator {
public:
typedef typename Decomposition::MatrixType MatrixType;
typedef typename internal::traits<MatrixType>::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef typename internal::plain_col_type<MatrixType>::type Vector;
/** \class ConditionEstimator
/** \class ConditionEstimator
* \ingroup Core_Module
*
* \brief Condition number estimator.
@@ -41,265 +57,148 @@ class ConditionEstimator {
*
* \sa FullPivLU, PartialPivLU.
*/
static RealScalar rcond(const MatrixType& matrix, const Decomposition& dec) {
eigen_assert(matrix.rows() == dec.rows());
eigen_assert(matrix.cols() == dec.cols());
eigen_assert(matrix.rows() == matrix.cols());
if (dec.rows() == 0) {
return RealScalar(1);
}
return rcond(MatrixL1Norm(matrix), dec);
template <typename Decomposition>
typename Decomposition::RealScalar ReciprocalConditionNumberEstimate(
const typename Decomposition::MatrixType& matrix,
const Decomposition& dec) {
eigen_assert(matrix.rows() == dec.rows());
eigen_assert(matrix.cols() == dec.cols());
eigen_assert(matrix.rows() == matrix.cols());
if (dec.rows() == 0) {
return Decomposition::RealScalar(1);
}
return ReciprocalConditionNumberEstimate(MatrixL1Norm(matrix), dec);
}
/** \class ConditionEstimator
* \ingroup Core_Module
*
* \brief Condition number estimator.
*
* Computing a decomposition of a dense matrix takes O(n^3) operations, while
* this method estimates the condition number quickly and reliably in O(n^2)
* operations.
*
* \returns an estimate of the reciprocal condition number
* (1 / (||matrix||_1 * ||inv(matrix)||_1)) of matrix, given ||matrix||_1 and
* its decomposition. Supports the following decompositions: FullPivLU,
* PartialPivLU.
*
* \sa FullPivLU, PartialPivLU.
*/
static RealScalar rcond(RealScalar matrix_norm, const Decomposition& dec) {
eigen_assert(dec.rows() == dec.cols());
if (dec.rows() == 0) {
return 1;
}
if (matrix_norm == 0) {
return 0;
}
const RealScalar inverse_matrix_norm = EstimateInverseMatrixL1Norm(dec);
return inverse_matrix_norm == 0 ? 0
: (1 / inverse_matrix_norm) / matrix_norm;
/** \class ConditionEstimator
* \ingroup Core_Module
*
* \brief Condition number estimator.
*
* Computing a decomposition of a dense matrix takes O(n^3) operations, while
* this method estimates the condition number quickly and reliably in O(n^2)
* operations.
*
* \returns an estimate of the reciprocal condition number
* (1 / (||matrix||_1 * ||inv(matrix)||_1)) of matrix, given ||matrix||_1 and
* its decomposition. Supports the following decompositions: FullPivLU,
* PartialPivLU.
*
* \sa FullPivLU, PartialPivLU.
*/
template <typename Decomposition>
typename Decomposition::RealScalar ReciprocalConditionNumberEstimate(
typename Decomposition::RealScalar matrix_norm, const Decomposition& dec) {
eigen_assert(dec.rows() == dec.cols());
if (dec.rows() == 0) {
return 1;
}
/**
* \returns an estimate of ||inv(matrix)||_1 given a decomposition of
* matrix that implements .solve() and .adjoint().solve() methods.
*
* The method implements Algorithms 4.1 and 5.1 from
* http://www.maths.manchester.ac.uk/~higham/narep/narep135.pdf
* which also forms the basis for the condition number estimators in
* LAPACK. Since at most 10 calls to the solve method of dec are
* performed, the total cost is O(dims^2), as opposed to O(dims^3)
* needed to compute the inverse matrix explicitly.
*
* The most common usage is in estimating the condition number
* ||matrix||_1 * ||inv(matrix)||_1. The first term ||matrix||_1 can be
* computed directly in O(n^2) operations.
*/
static RealScalar EstimateInverseMatrixL1Norm(const Decomposition& dec) {
eigen_assert(dec.rows() == dec.cols());
if (dec.rows() == 0) {
return 0;
}
return internal::EstimateInverseMatrixL1NormImpl<
Decomposition, IsSelfAdjoint,
NumTraits<Scalar>::IsComplex != 0>::compute(dec);
if (matrix_norm == 0) {
return 0;
}
const typename Decomposition::RealScalar inverse_matrix_norm = InverseMatrixL1NormEstimate(dec);
return inverse_matrix_norm == 0 ? 0 : (1 / inverse_matrix_norm) / matrix_norm;
}
/**
* \returns the induced matrix l1-norm
* ||matrix||_1 = sup ||matrix * v||_1 / ||v||_1, which is equal to
* the greatest absolute column sum.
*/
static inline Scalar MatrixL1Norm(const MatrixType& matrix) {
return matrix.cwiseAbs().colwise().sum().maxCoeff();
}
};
namespace internal {
template <typename Decomposition, typename Vector, bool IsSelfAdjoint = false>
struct solve_helper {
static inline Vector solve_adjoint(const Decomposition& dec,
const Vector& v) {
return dec.adjoint().solve(v);
}
};
// Partial specialization for self_adjoint matrices.
template <typename Decomposition, typename Vector>
struct solve_helper<Decomposition, Vector, true> {
static inline Vector solve_adjoint(const Decomposition& dec,
const Vector& v) {
return dec.solve(v);
}
};
// Partial specialization for real matrices.
template <typename Decomposition, bool IsSelfAdjoint>
struct EstimateInverseMatrixL1NormImpl<Decomposition, IsSelfAdjoint, false> {
/**
* \returns an estimate of ||inv(matrix)||_1 given a decomposition of
* matrix that implements .solve() and .adjoint().solve() methods.
*
* The method implements Algorithms 4.1 and 5.1 from
* http://www.maths.manchester.ac.uk/~higham/narep/narep135.pdf
* which also forms the basis for the condition number estimators in
* LAPACK. Since at most 10 calls to the solve method of dec are
* performed, the total cost is O(dims^2), as opposed to O(dims^3)
* needed to compute the inverse matrix explicitly.
*
* The most common usage is in estimating the condition number
* ||matrix||_1 * ||inv(matrix)||_1. The first term ||matrix||_1 can be
* computed directly in O(n^2) operations.
*/
template <typename Decomposition>
typename Decomposition::RealScalar InverseMatrixL1NormEstimate(
const Decomposition& dec) {
typedef typename Decomposition::MatrixType MatrixType;
typedef typename internal::traits<MatrixType>::Scalar Scalar;
typedef typename Decomposition::Scalar Scalar;
typedef typename Decomposition::RealScalar RealScalar;
typedef typename internal::plain_col_type<MatrixType>::type Vector;
typedef typename internal::plain_col_type<MatrixType, RealScalar>::type RealVector;
const bool is_complex = (NumTraits<Scalar>::IsComplex != 0);
// Shorthand for vector L1 norm in Eigen.
static inline Scalar VectorL1Norm(const Vector& v) {
return v.template lpNorm<1>();
eigen_assert(dec.rows() == dec.cols());
const int n = dec.rows();
if (n == 0) {
return 0;
}
Vector v = Vector::Ones(n) / n;
v = dec.solve(v);
static inline Scalar compute(const Decomposition& dec) {
const int n = dec.rows();
const Vector plus = Vector::Ones(n);
Vector v = plus / n;
v = dec.solve(v);
Scalar lower_bound = VectorL1Norm(v);
if (n == 1) {
return lower_bound;
}
// lower_bound is a lower bound on
// ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
// and is the objective maximized by the ("super-") gradient ascent
// algorithm.
// Basic idea: We know that the optimum is achieved at one of the simplices
// v = e_i, so in each iteration we follow a super-gradient to move towards
// the optimal one.
Scalar old_lower_bound = lower_bound;
const Vector minus = -Vector::Ones(n);
Vector sign_vector = (v.cwiseAbs().array() == 0).select(plus, minus);
Vector old_sign_vector = sign_vector;
int v_max_abs_index = -1;
int old_v_max_abs_index = v_max_abs_index;
for (int k = 0; k < 4; ++k) {
// argmax |inv(matrix)^T * sign_vector|
v = solve_helper<Decomposition, Vector, IsSelfAdjoint>::solve_adjoint(dec, sign_vector);
v.cwiseAbs().maxCoeff(&v_max_abs_index);
if (v_max_abs_index == old_v_max_abs_index) {
// Break if the solution stagnated.
break;
}
// Move to the new simplex e_j, where j = v_max_abs_index.
v.setZero();
v[v_max_abs_index] = 1;
v = dec.solve(v); // v = inv(matrix) * e_j.
lower_bound = VectorL1Norm(v);
if (lower_bound <= old_lower_bound) {
// Break if the gradient step did not increase the lower_bound.
break;
}
sign_vector = (v.array() < 0).select(plus, minus);
// lower_bound is a lower bound on
// ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
// and is the objective maximized by the ("super-") gradient ascent
// algorithm below.
RealScalar lower_bound = internal::VectorL1Norm(v);
if (n == 1) {
return lower_bound;
}
// Gradient ascent algorithm follows: We know that the optimum is achieved at
// one of the simplices v = e_i, so in each iteration we follow a
// super-gradient to move towards the optimal one.
RealScalar old_lower_bound = lower_bound;
Vector sign_vector(n);
Vector old_sign_vector;
int v_max_abs_index = -1;
int old_v_max_abs_index = v_max_abs_index;
for (int k = 0; k < 4; ++k) {
sign_vector = internal::SignOrUnity<Vector, RealVector, is_complex>::run(v);
if (k > 0 && !is_complex) {
if (sign_vector == old_sign_vector) {
// Break if the solution stagnated.
break;
}
}
// v_max_abs_index = argmax |real( inv(matrix)^T * sign_vector )|
v = dec.adjoint().solve(sign_vector);
v.real().cwiseAbs().maxCoeff(&v_max_abs_index);
if (v_max_abs_index == old_v_max_abs_index) {
// Break if the solution stagnated.
break;
}
// Move to the new simplex e_j, where j = v_max_abs_index.
v = dec.solve(Vector::Unit(n, v_max_abs_index)); // v = inv(matrix) * e_j.
lower_bound = internal::VectorL1Norm(v);
if (lower_bound <= old_lower_bound) {
// Break if the gradient step did not increase the lower_bound.
break;
}
if (!is_complex) {
old_sign_vector = sign_vector;
old_v_max_abs_index = v_max_abs_index;
old_lower_bound = lower_bound;
}
// The following calculates an independent estimate of ||matrix||_1 by
// multiplying matrix by a vector with entries of slowly increasing
// magnitude and alternating sign:
// v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
// This improvement to Hager's algorithm above is due to Higham. It was
// added to make the algorithm more robust in certain corner cases where
// large elements in the matrix might otherwise escape detection due to
// exact cancellation (especially when op and op_adjoint correspond to a
// sequence of backsubstitutions and permutations), which could cause
// Hager's algorithm to vastly underestimate ||matrix||_1.
Scalar alternating_sign = 1;
for (int i = 0; i < n; ++i) {
v[i] = alternating_sign * static_cast<Scalar>(1) +
(static_cast<Scalar>(i) / (static_cast<Scalar>(n - 1)));
alternating_sign = -alternating_sign;
}
v = dec.solve(v);
const Scalar alternate_lower_bound =
(2 * VectorL1Norm(v)) / (3 * static_cast<Scalar>(n));
return numext::maxi(lower_bound, alternate_lower_bound);
old_v_max_abs_index = v_max_abs_index;
old_lower_bound = lower_bound;
}
};
// Partial specialization for complex matrices.
template <typename Decomposition, bool IsSelfAdjoint>
struct EstimateInverseMatrixL1NormImpl<Decomposition, IsSelfAdjoint, true> {
typedef typename Decomposition::MatrixType MatrixType;
typedef typename internal::traits<MatrixType>::Scalar Scalar;
typedef typename NumTraits<Scalar>::Real RealScalar;
typedef typename internal::plain_col_type<MatrixType>::type Vector;
typedef typename internal::plain_col_type<MatrixType, RealScalar>::type
RealVector;
// Shorthand for vector L1 norm in Eigen.
static inline RealScalar VectorL1Norm(const Vector& v) {
return v.template lpNorm<1>();
// The following calculates an independent estimate of ||matrix||_1 by
// multiplying matrix by a vector with entries of slowly increasing
// magnitude and alternating sign:
// v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
// This improvement to Hager's algorithm above is due to Higham. It was
// added to make the algorithm more robust in certain corner cases where
// large elements in the matrix might otherwise escape detection due to
// exact cancellation (especially when op and op_adjoint correspond to a
// sequence of backsubstitutions and permutations), which could cause
// Hager's algorithm to vastly underestimate ||matrix||_1.
Scalar alternating_sign = 1;
for (int i = 0; i < n; ++i) {
v[i] = alternating_sign * static_cast<RealScalar>(1) +
(static_cast<RealScalar>(i) / (static_cast<RealScalar>(n - 1)));
alternating_sign = -alternating_sign;
}
v = dec.solve(v);
const RealScalar alternate_lower_bound =
(2 * internal::VectorL1Norm(v)) / (3 * static_cast<RealScalar>(n));
return numext::maxi(lower_bound, alternate_lower_bound);
}
static inline RealScalar compute(const Decomposition& dec) {
const int n = dec.rows();
const Vector ones = Vector::Ones(n);
Vector v = ones / n;
v = dec.solve(v);
RealScalar lower_bound = VectorL1Norm(v);
if (n == 1) {
return lower_bound;
}
// lower_bound is a lower bound on
// ||inv(matrix)||_1 = sup_v ||inv(matrix) v||_1 / ||v||_1
// and is the objective maximized by the ("super-") gradient ascent
// algorithm.
// Basic idea: We know that the optimum is achieved at one of the simplices
// v = e_i, so in each iteration we follow a super-gradient to move towards
// the optimal one.
RealScalar old_lower_bound = lower_bound;
int v_max_abs_index = -1;
int old_v_max_abs_index = v_max_abs_index;
for (int k = 0; k < 4; ++k) {
// argmax |inv(matrix)^* * sign_vector|
RealVector abs_v = v.cwiseAbs();
const Vector psi =
(abs_v.array() == 0).select(v.cwiseQuotient(abs_v), ones);
v = solve_helper<Decomposition, Vector, IsSelfAdjoint>::solve_adjoint(dec, psi);
const RealVector z = v.real();
z.cwiseAbs().maxCoeff(&v_max_abs_index);
if (v_max_abs_index == old_v_max_abs_index) {
// Break if the solution stagnated.
break;
}
// Move to the new simplex e_j, where j = v_max_abs_index.
v.setZero();
v[v_max_abs_index] = 1;
v = dec.solve(v); // v = inv(matrix) * e_j.
lower_bound = VectorL1Norm(v);
if (lower_bound <= old_lower_bound) {
// Break if the gradient step did not increase the lower_bound.
break;
}
old_v_max_abs_index = v_max_abs_index;
old_lower_bound = lower_bound;
}
// The following calculates an independent estimate of ||matrix||_1 by
// multiplying matrix by a vector with entries of slowly increasing
// magnitude and alternating sign:
// v_i = (-1)^{i} (1 + (i / (dim-1))), i = 0,...,dim-1.
// This improvement to Hager's algorithm above is due to Higham. It was
// added to make the algorithm more robust in certain corner cases where
// large elements in the matrix might otherwise escape detection due to
// exact cancellation (especially when op and op_adjoint correspond to a
// sequence of backsubstitutions and permutations), which could cause
// Hager's algorithm to vastly underestimate ||matrix||_1.
RealScalar alternating_sign = 1;
for (int i = 0; i < n; ++i) {
v[i] = alternating_sign * static_cast<RealScalar>(1) +
(static_cast<RealScalar>(i) / (static_cast<RealScalar>(n - 1)));
alternating_sign = -alternating_sign;
}
v = dec.solve(v);
const RealScalar alternate_lower_bound =
(2 * VectorL1Norm(v)) / (3 * static_cast<RealScalar>(n));
return numext::maxi(lower_bound, alternate_lower_bound);
}
};
} // namespace internal
} // namespace Eigen
#endif