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| /* Copyright 2015 Google Inc. All Rights Reserved. | ||
| Licensed under the Apache License, Version 2.0 (the "License"); | ||
| you may not use this file except in compliance with the License. | ||
| You may obtain a copy of the License at | ||
| http://www.apache.org/licenses/LICENSE-2.0 | ||
| Unless required by applicable law or agreed to in writing, software | ||
| distributed under the License is distributed on an "AS IS" BASIS, | ||
| WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. | ||
| See the License for the specific language governing permissions and | ||
| limitations under the License. | ||
| ==============================================================================*/ | ||
|
|
||
| // See docs in ../ops/linalg_ops.cc. | ||
| #include <cmath> | ||
|
|
||
| #include "third_party/eigen3/Eigen/Cholesky" | ||
| #include "third_party/eigen3/Eigen/Core" | ||
| #include "third_party/eigen3/Eigen/QR" | ||
| #include "tensorflow/core/framework/kernel_def_builder.h" | ||
| #include "tensorflow/core/framework/op_kernel.h" | ||
| #include "tensorflow/core/kernels/binary_linalg_ops_common.h" | ||
| #include "tensorflow/core/lib/core/errors.h" | ||
| #include "tensorflow/core/platform/logging.h" | ||
| #include "tensorflow/core/platform/port.h" | ||
| #include "tensorflow/core/public/tensor_shape.h" | ||
|
|
||
| namespace tensorflow { | ||
|
|
||
| template <class Scalar, bool SupportsBatchOperationT> | ||
| class MatrixSolveLsOp | ||
| : public BinaryLinearAlgebraOp<Scalar, SupportsBatchOperationT> { | ||
| public: | ||
| explicit MatrixSolveLsOp(OpKernelConstruction* context) | ||
| : BinaryLinearAlgebraOp<Scalar, SupportsBatchOperationT>(context) { | ||
| OP_REQUIRES_OK(context, context->GetAttr("fast", &fast_)); | ||
| } | ||
|
|
||
| ~MatrixSolveLsOp() override {} | ||
|
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| TensorShape GetOutputMatrixShape( | ||
| const TensorShape& input_matrix_shape, | ||
| const TensorShape& rhs_matrix_shape) override { | ||
| CHECK_EQ(input_matrix_shape.dims(), rhs_matrix_shape.dims()); | ||
| TensorShape output_matrix_shape = rhs_matrix_shape; | ||
| output_matrix_shape.set_dim( | ||
| output_matrix_shape.dims() - 2, | ||
| input_matrix_shape.dim_size(output_matrix_shape.dims() - 1)); | ||
| return output_matrix_shape; | ||
| } | ||
|
|
||
| int64 GetCostPerUnit(const TensorShape& input_matrix_shape, | ||
| const TensorShape& rhs_matrix_shape) override { | ||
| const int64 rows = input_matrix_shape.dim_size(0); | ||
| const int64 rhss = rhs_matrix_shape.dim_size(1); | ||
| if (rows > (1LL << 20)) { | ||
| // A big number to cap the cost in case overflow. | ||
| return kint32max; | ||
| } else { | ||
| return 2 * rows * rows * (rows + rhss); | ||
| } | ||
| } | ||
|
|
||
| using typename BinaryLinearAlgebraOp<Scalar, SupportsBatchOperationT>::Matrix; | ||
| using typename BinaryLinearAlgebraOp<Scalar, | ||
| SupportsBatchOperationT>::MatrixMap; | ||
| using typename BinaryLinearAlgebraOp<Scalar, | ||
| SupportsBatchOperationT>::ConstMatrixMap; | ||
|
|
||
| void ComputeMatrix(OpKernelContext* context, const ConstMatrixMap& matrix, | ||
| const ConstMatrixMap& rhs, MatrixMap* output) override { | ||
| const int64 rows = matrix.rows(); | ||
| const int64 cols = matrix.cols(); | ||
| OP_REQUIRES( | ||
| context, rows == rhs.rows(), | ||
| errors::InvalidArgument("Input matrix and rhs are incompatible.")); | ||
| const auto& l2_regularizer_in = context->input(2); | ||
| OP_REQUIRES( | ||
| context, TensorShapeUtils::IsScalar(l2_regularizer_in.shape()), | ||
| errors::InvalidArgument("l2_regularizer must be scalar, got shape ", | ||
| l2_regularizer_in.shape().DebugString())); | ||
| const double l2_regularizer = l2_regularizer_in.scalar<double>()(); | ||
|
|
||
| OP_REQUIRES(context, l2_regularizer >= 0, | ||
| errors::InvalidArgument("l2_regularizer must be >= 0.")); | ||
| if (rows == 0 || cols == 0) { | ||
| // The result is the empty matrix. | ||
| return; | ||
| } | ||
| if (fast_) { | ||
| // The fast branch assumes that matrix is not rank deficient and | ||
| // not too ill-conditioned. Specifically, the reciprobal condition number | ||
| // should be greater than the square root of the machine precision, i.e. | ||
| // 1 / cond(matrix) > sqrt(std::numeric_limits<Scalar>::epsilon()). | ||
| // This branch solves over- or underdetermined least-squares problems | ||
| // via the normal equations and Cholesky decomposition. | ||
| if (matrix.rows() >= matrix.cols()) { | ||
| // Overdetermined case (rows >= cols): Solves the ordinary (possibly | ||
| // regularized) least-squares problem | ||
| // min || A * X - RHS ||_F^2 + l2_regularizer ||X||_F^2 | ||
| // by solving the normal equations | ||
| // (A^T * A + l2_regularizer * I) X = A^T RHS | ||
| // using Cholesky decomposition. | ||
| Matrix gramian(cols, cols); | ||
| gramian.template triangularView<Eigen::Lower>() = | ||
| matrix.transpose() * matrix; | ||
| if (l2_regularizer > 0) { | ||
| gramian += | ||
| (Scalar(l2_regularizer) * Matrix::Ones(cols, 1)).asDiagonal(); | ||
| } | ||
| const Eigen::LLT<Matrix, Eigen::Lower> llt(gramian); | ||
| OP_REQUIRES( | ||
| context, llt.info() == Eigen::Success, | ||
| errors::InvalidArgument("Input matrix was rank deficient or " | ||
| "ill-conditioned. Try setting fast=False " | ||
| "or provide a larger l2_regularizer > 0.")); | ||
| *output = llt.solve(matrix.transpose() * rhs); | ||
| } else { | ||
| // Underdetermined case (rows < cols): Solves the minimum-norm problem | ||
| // min ||X||_F^2 s.t. A*X = RHS | ||
| // by solving the normal equations of the second kind | ||
| // (A * A^T + l2_regularizer * I) Z = RHS, X = A^T * Z | ||
| // using Cholesky decomposition. | ||
| Matrix gramian(rows, rows); | ||
| gramian.template triangularView<Eigen::Lower>() = | ||
| matrix * matrix.transpose(); | ||
| if (l2_regularizer > 0) { | ||
| gramian += | ||
| (Scalar(l2_regularizer) * Matrix::Ones(rows, 1)).asDiagonal(); | ||
| } | ||
| const Eigen::LLT<Matrix, Eigen::Lower> llt(gramian); | ||
| OP_REQUIRES( | ||
| context, llt.info() == Eigen::Success, | ||
| errors::InvalidArgument("Input matrix was rank deficient or " | ||
| "ill-conditioned. Try setting fast=False " | ||
| "or provide an l2_regularizer > 0.")); | ||
| *output = matrix.transpose() * llt.solve(rhs); | ||
| } | ||
| } else { | ||
| // Use a rank revealing factorization (QR with column pivoting). | ||
| // | ||
| // NOTICE: Currently, Eigen's implementation of column pivoted Householder | ||
| // QR has a few deficiencies: | ||
| // 1. It does not implement the post-processing step to compute a | ||
| // complete orthogonal factorization. This means that it does not | ||
| // return a minimum-norm solution for underdetermined and | ||
| // rank-deficient matrices. We could use the Eigen SVD instead, but | ||
| // the currently available JacobiSVD is so slow that is it is | ||
| // essentially useless (~100x slower than QR). | ||
| // 2. The implementation is not blocked, so for matrics that do not fit | ||
| // in cache, it is significantly slower than the equivalent blocked | ||
| // LAPACK routine xGEQP3 (e.g. Eigen is ~3x slower for 4k x 4k | ||
| // matrices). See http://www.netlib.org/lapack/lawnspdf/lawn114.pdf | ||
| // 3. The implementation uses the numerically unstable norm downdating | ||
| // formula from the original 1965 Businger & Golub paper. This can | ||
| // lead to incorrect rank determination for graded matrices. I | ||
| // (rmlarsen@) have a patch to bring this up to date by implementing | ||
| // the robust formula from | ||
| // http://www.netlib.org/lapack/lawnspdf/lawn176.pdf | ||
| // | ||
| // TODO(rmlarsen): a) Contribute 1. and 2. to Eigen. | ||
| // b) Evaluate new divide-and-conquer SVD in Eigen when | ||
| // it becomes available & robust. | ||
| *output = matrix.colPivHouseholderQr().solve(rhs); | ||
| } | ||
| } | ||
|
|
||
| private: | ||
| bool fast_; | ||
| }; | ||
|
|
||
| REGISTER_BINARY_LINALG_OP("MatrixSolveLs", (MatrixSolveLsOp<float, false>), | ||
| float); | ||
| REGISTER_BINARY_LINALG_OP("MatrixSolveLs", (MatrixSolveLsOp<double, false>), | ||
| double); | ||
| REGISTER_BINARY_LINALG_OP("BatchMatrixSolveLs", (MatrixSolveLsOp<float, true>), | ||
| float); | ||
| REGISTER_BINARY_LINALG_OP("BatchMatrixSolveLs", (MatrixSolveLsOp<double, true>), | ||
| double); | ||
|
|
||
| } // namespace tensorflow |
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