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LinearAlgebra.cpp
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#include <ATen/ATen.h>
#include <ATen/core/grad_mode.h>
#include <ATen/Dispatch.h>
#include <ATen/ExpandUtils.h>
#include <ATen/NamedTensorUtils.h>
#include <ATen/native/CPUBlas.h>
#include <ATen/native/IndexingUtils.h>
#include <ATen/native/LinearAlgebra.h>
#include <ATen/native/LinearAlgebraUtils.h>
#include <ATen/native/ReduceOps.h>
#include <ATen/native/ReduceOpsUtils.h>
#include <ATen/native/Resize.h>
#include <ATen/native/TensorIterator.h>
#include <ATen/NativeFunctions.h>
#include <ATen/Parallel.h>
#include <ATen/TensorUtils.h>
#include <ATen/Utils.h>
#include <c10/util/accumulate.h>
#include <c10/util/irange.h>
#include <c10/util/variant.h>
#include <functional>
#include <limits>
#include <numeric>
namespace at {
namespace meta {
TORCH_META_FUNC(addmm)(const Tensor& self, const Tensor& mat1, const Tensor& mat2, const Scalar& beta, const Scalar& alpha) {
TORCH_CHECK(mat1.dim() == 2, "mat1 must be a matrix, got ", mat1.dim(), "-D tensor");
TORCH_CHECK(mat2.dim() == 2, "mat2 must be a matrix, got ", mat2.dim(), "-D tensor");
TORCH_CHECK(
mat1.sizes()[1] == mat2.sizes()[0], "mat1 and mat2 shapes cannot be multiplied (",
mat1.sizes()[0], "x", mat1.sizes()[1], " and ", mat2.sizes()[0], "x", mat2.sizes()[1], ")");
auto names = at::namedinference::propagate_names_for_addmm(mat1, mat2, self);
set_output(0, {mat1.sizes()[0], mat2.sizes()[1]}, {}, self.options(), names);
const auto& result = maybe_get_output(0);
//this check can fire for inplace op only, for all other versions result is guaranteed to be correct size
TORCH_CHECK(((result.dim() == 2) && (result.sizes()[0] == mat1.sizes()[0]) && (result.sizes()[1] == mat2.sizes()[1])),
"The input tensor must be a matrix with size ", mat1.sizes()[0], "x", mat2.sizes()[1], ", but got a ", result.dim(),
"-D tensor with size ", result.sizes()[0], "x", result.sizes()[1]);
}
TORCH_META_FUNC(mm)(const Tensor & self, const Tensor & mat2) {
TORCH_CHECK(self.dim() == 2, "self must be a matrix");
TORCH_CHECK(mat2.dim() == 2, "mat2 must be a matrix");
TORCH_CHECK(
self.sizes()[1] == mat2.sizes()[0], "mat1 and mat2 shapes cannot be multiplied (",
self.sizes()[0], "x", self.sizes()[1], " and ", mat2.sizes()[0], "x", mat2.sizes()[1], ")");
auto names = at::namedinference::compute_matmul_outnames(self, mat2);
set_output(0, {self.sizes()[0], mat2.sizes()[1]}, {}, self.options(), names);
const auto& result = maybe_get_output(0);
//this check can fire for inplace op only, for all other versions result is guaranteed to be correct size
TORCH_CHECK(((result.dim() == 2) && (result.sizes()[0] == self.sizes()[0]) && (result.sizes()[1] == mat2.sizes()[1])),
"The input tensor must be a matrix with size ", self.sizes()[0], "x", mat2.sizes()[1], ", but got a ", result.dim(),
"-D tensor with size ", result.sizes()[0], "x", result.sizes()[1]);
}
} // namespace meta
namespace native {
DEFINE_DISPATCH(addr_stub);
DEFINE_DISPATCH(linalg_vector_norm_stub);
// Helper function for det methods.
// For pivoted LU factorization A = P * L * U. Since we always have det(L) = 1,
// det(P) = \pm 1, this method returns a 3-tuple:
// (det(P), diag(U), info),
// where info helps us identify singular matrices.
static inline std::tuple<c10::ExclusivelyOwned<Tensor>, c10::ExclusivelyOwned<Tensor>> _lu_det_P_diag_U(const Tensor& self) {
Tensor pivs, lu, infos;
std::tie(lu, pivs, infos) = at::_lu_with_info(self, /*pivot=*/true, /*check_errors=*/false);
TORCH_CHECK(infos.ge(0).all().item<uint8_t>(), "Invalid argument passed to lu");
auto n = self.size(-1);
auto num_exchanges = (at::arange(1, n + 1, pivs.options()) != pivs)
.sum(-1, /*keepdim=*/false, /*dtype=*/at::kLong).fmod_(2);
auto u_diagonal = lu.diagonal(/*offset=*/0, /*dim1=*/-2, /*dim2=*/-1);
num_exchanges.mul_(-2).add_(1);
return std::make_tuple(c10::ExclusivelyOwned<Tensor>(std::move(num_exchanges)), c10::ExclusivelyOwned<Tensor>(std::move(u_diagonal)));
}
// Given a pivoted LU factorization A = P L U,
// det(A) = det(P) * det(L) * det(U).
// Since det(P) = +- 1 (even or odd permutation), and diag(L) = I, we get that
// det(A) = ([is P odd] * -2 + 1) * prod(diag(U))
std::tuple<Tensor, Tensor, Tensor> _det_lu_based_helper(const Tensor& self) {
Tensor lu, pivs, infos;
std::tie(lu, pivs, infos) = at::_lu_with_info(self, /*pivot=*/true, /*check_errors*/false);
TORCH_CHECK(infos.ge(0).all().item<uint8_t>(), "at::_det_lu_based_helper(): Invalid argument passed to LU");
// find det(P)
auto n = self.size(-1);
auto n_transpositions_mod_2 = (at::arange(1, n + 1, pivs.options()) != pivs)
.sum(-1, /*keepdim=*/false, /*dtype=*/at::kLong).fmod_(2);
auto p_det = n_transpositions_mod_2.mul_(-2).add_(1);
auto det = p_det * at::prod(lu.diagonal(0, -2, -1), /*dim=*/-1);
return std::make_tuple(det, lu, pivs);
}
// torch.det, alias for torch.linalg.det
Tensor det(const Tensor& self) {
return at::linalg_det(self);
}
Tensor& linalg_det_out(const Tensor& self, Tensor& out) {
checkSameDevice("torch.linalg.det", out, self, "out");
checkLinalgCompatibleDtype("torch.linalg.det", out, self, "out");
squareCheckInputs(self);
TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())),
"Expected a floating point or complex tensor as input");
IntArrayRef out_sizes(self.sizes().data(), self.dim() - 2);
at::native::resize_output(out, out_sizes);
auto det = std::get<0>(at::native::_det_lu_based_helper(self));
out.copy_(det);
return out;
}
Tensor linalg_det(const Tensor& self) {
squareCheckInputs(self);
TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())),
"Expected a floating point or complex tensor as input");
return std::get<0>(at::_det_lu_based_helper(self));
}
Tensor logdet(const Tensor& self) {
squareCheckInputs(self);
TORCH_CHECK((at::isFloatingType(self.scalar_type()) || at::isComplexType(self.scalar_type())),
"Expected a floating point tensor as input");
c10::ExclusivelyOwned<Tensor> det_P, diag_U;
std::tie(det_P, diag_U) = _lu_det_P_diag_U(self);
Tensor det_sign = diag_U->sign().prod(-1).mul_(*det_P);
// If det_sign > 0, diag_U.abs_().log_().sum(-1) gives logdet (this means U is not singular).
// If det_sign <= 0, then we get proper nan (when det < 0, i.e., det_sign) or -inf (when det = 0, i.e., U is singular).
// U is singular when U(i, i) = 0 for some i in [1, self.size(-1)].
Tensor logdet_vals = diag_U->abs_().log_().sum(-1);
if (self.dim() > 2) {
auto indices = toListOfOptionalTensors((det_sign < 0).nonzero_numpy());
// NOLINTNEXTLINE(performance-move-const-arg)
logdet_vals.index_put_(std::move(indices), at::full({}, NAN, self.options()));
} else if (det_sign.item<double>() < 0) {
logdet_vals.fill_(NAN);
}
return logdet_vals;
}
std::tuple<Tensor, Tensor> linalg_slogdet(const Tensor& self) {
squareCheckInputs(self);
ScalarType t = self.scalar_type();
TORCH_CHECK(t == ScalarType::Double || t == ScalarType::Float || t == ScalarType::ComplexFloat || t == ScalarType::ComplexDouble,
"linalg_slogdet: expected a tensor of float, double, cfloat or cdouble types but got ", t);
c10::ExclusivelyOwned<Tensor> det_P, diag_U;
std::tie(det_P, diag_U) = _lu_det_P_diag_U(self);
auto det_sign = diag_U->sgn().prod(-1).mul_(*det_P);
// abslogdet_val is -inf if U is singular, in which case diag_U.abs_().log_().sum(-1) will return -inf.
// U is singular when U(i, i) = 0 for some i in [1, self.size(-1)].
// Since abslogdet_val cannot take nan, no special case handling is required.
// in-place abs is not supported for complex tensors
auto abslogdet_val = isComplexType(t) ? diag_U->abs().log_().sum(-1) : diag_U->abs_().log_().sum(-1);
return std::make_tuple(det_sign, abslogdet_val);
}
// TODO: implement _out variant avoiding copy and using already allocated storage directly
std::tuple<Tensor&, Tensor&> linalg_slogdet_out(const Tensor& input, Tensor& sign, Tensor& logabsdet) {
checkSameDevice("linalg_slogdet", sign, input, "sign");
checkSameDevice("linalg_slogdet", logabsdet, input, "logabsdet");
checkLinalgCompatibleDtype("linalg_slogdet", sign, input, "sign");
ScalarType real_dtype = toValueType(input.scalar_type());
// logabsdet is always real-valued here
checkLinalgCompatibleDtype("linalg_slogdet", logabsdet.scalar_type(), real_dtype, "logabsdet");
Tensor sign_tmp, logabsdet_tmp;
std::tie(sign_tmp, logabsdet_tmp) = at::linalg_slogdet(input);
at::native::resize_output(sign, sign_tmp.sizes());
sign.copy_(sign_tmp);
at::native::resize_output(logabsdet, logabsdet_tmp.sizes());
logabsdet.copy_(logabsdet_tmp);
return std::tuple<Tensor&, Tensor&>(sign, logabsdet);
}
std::tuple<Tensor, Tensor> slogdet(const Tensor& self) {
return at::linalg_slogdet(self);
}
Tensor linalg_pinv(const Tensor& input, const Tensor& rcond, bool hermitian) {
NoTF32Guard disable_tf32;
ScalarType t = input.scalar_type();
TORCH_CHECK((t == ScalarType::Double || t == ScalarType::Float || t == ScalarType::ComplexFloat || t == ScalarType::ComplexDouble)
&& input.dim() >= 2,
"linalg_pinv(", t, "{", input.sizes(), "}): expected a tensor with 2 or more dimensions "
"of float, double, cfloat or cdouble types");
TORCH_CHECK(rcond.device() == input.device(),
"Expected rcond and input to be on the same device, but found rcond on ",
rcond.device(), " and input on ", input.device(), " instead.");
TORCH_CHECK(!at::isComplexType(rcond.scalar_type()),
"linalg_pinv: rcond tensor of complex type is not supported.");
if (input.numel() == 0) {
// The implementation below uses operations that do not work for zero numel tensors
// therefore we need this early return for 'input.numel() == 0' case
Tensor U, S, V;
// TODO: replace input.svd with linalg_svd when torch/xla can work with at::linalg_svd
std::tie(U, S, V) = input.svd();
return at::matmul(V * S.reciprocal().unsqueeze(-2), U.conj().transpose(-2, -1));
}
// If not Hermitian use singular value decomposition, else use eigenvalue decomposition
if (!hermitian) {
Tensor U, S, V;
// TODO: replace input.svd with linalg_svd
// using linalg_svd breaks pytorch/xla, see https://github.com/pytorch/xla/issues/2755
std::tie(U, S, V) = input.svd();
Tensor max_val = at::narrow(S, /*dim=*/-1, /*start=*/0, /*length=*/1); // singular values are sorted in descending order
Tensor S_pseudoinv = at::where(S > (rcond.unsqueeze(-1) * max_val), S.reciprocal(), at::zeros({}, S.options())).to(input.dtype());
// computes V @ diag(S_pseudoinv) @ U.conj().T
return at::matmul(V * S_pseudoinv.unsqueeze(-2), U.conj().transpose(-2, -1));
} else {
Tensor S, U;
std::tie(S, U) = at::linalg_eigh(input);
// For Hermitian matrices, singular values equal to abs(eigenvalues)
Tensor S_abs = S.abs();
// eigenvalues are sorted in ascending order starting with negative values, we need a maximum value of abs(eigenvalues)
Tensor max_val = S_abs.amax(/*dim=*/-1, /*keepdim=*/true);
Tensor S_pseudoinv = at::where(S_abs > (rcond.unsqueeze(-1) * max_val), S.reciprocal(), at::zeros({}, S.options())).to(input.dtype());
// computes U @ diag(S_pseudoinv) @ U.conj().T
return at::matmul(U * S_pseudoinv.unsqueeze(-2), U.conj().transpose(-2, -1));
}
}
Tensor linalg_pinv(const Tensor& input, double rcond, bool hermitian) {
Tensor rcond_tensor = at::full({}, rcond, input.options().dtype(ScalarType::Double));
return at::linalg_pinv(input, rcond_tensor, hermitian);
}
// TODO: implement _out variant avoiding copy and using already allocated storage directly
Tensor& linalg_pinv_out(const Tensor& input, const Tensor& rcond, bool hermitian, Tensor& result) {
checkSameDevice("linalg_pinv", result, input);
checkLinalgCompatibleDtype("linalg_pinv", result, input);
Tensor result_tmp = at::linalg_pinv(input, rcond, hermitian);
at::native::resize_output(result, result_tmp.sizes());
result.copy_(result_tmp);
return result;
}
Tensor& linalg_pinv_out(const Tensor& input, double rcond, bool hermitian, Tensor& result) {
Tensor rcond_tensor = at::full({}, rcond, input.options().dtype(ScalarType::Double));
return at::linalg_pinv_out(result, input, rcond_tensor, hermitian);
}
Tensor pinverse(const Tensor& self, double rcond) {
return at::linalg_pinv(self, rcond, /*hermitian=*/false);
}
// matrix_power implementation
namespace {
/**
* @brief Raises the input matrix to the given power n
*
* If the exponent n is negative, the inverse of the input
* matrix will be raised to power abs(n).
*
* @param self (batched) square matrix to raise to power n
* @param n exponent to raise matrix (or matrices in batch) to
* @param _out optional tensor to write the output to
* @return Tensor input matrix raised to power n
*/
Tensor linalg_matrix_power_impl(
const Tensor& self,
int64_t n,
c10::optional<Tensor> _out) {
auto out = _out.value_or(Tensor());
squareCheckInputs(self);
if (_out.has_value()) {
checkSameDevice("matrix_power", out, self);
checkLinalgCompatibleDtype("matrix_power", out, self);
at::native::resize_output(out, self.sizes());
}
// For n=0 we return the identity matrix of the same shape as input.
if (n == 0) {
if (!_out.has_value()) {
// Clone input to include result in the autograd graph
out = self.clone(at::MemoryFormat::Contiguous);
}
return out.copy_(at::eye(self.size(-2), self.options()));
}
if (n == 1) {
return _out.has_value() ? out.copy_(self)
: self.clone(at::MemoryFormat::Contiguous);
}
if (n == -1) {
return _out.has_value() ? at::linalg_inv_out(out, self)
: at::linalg_inv(self);
}
// For negative n we inverte the input matrix before raising to power abs(n)
auto a = n < 0 ? at::linalg_inv(self) : self;
n = std::abs(n);
// Fast paths for small powers
if (n == 2) {
return _out.has_value() ? at::matmul_out(out, a, a) : at::matmul(a, a);
}
if (n == 3) {
return _out.has_value() ? at::matmul_out(out, at::matmul(a, a), a)
: at::matmul(at::matmul(a, a), a);
}
// This is a binary decomposition of n.
// Moving from the least significant bit to the most significant bit
// This is done to reduce the number of matrix multiplications
// by raising the input matrix in powers of 2
// The total number of matrix multiplications are
// number of bits + number of bits that equal 1 ~ O(log n)
// instead of O(n)
Tensor z, result;
while (n > 0) {
const auto bit = n % 2;
n = n / 2;
z = z.defined() ? at::matmul(z, z) : a;
if (bit == 1) {
if (_out.has_value() && n <= 0) {
// Last multiplication can use the out version
return result.defined() ? at::matmul_out(out, result, z) : out.copy_(z);
}
result = result.defined() ? at::matmul(result, z) : z;
}
}
return result;
}
} // namespace
Tensor& linalg_matrix_power_out(const Tensor& self, int64_t n, Tensor& result) {
linalg_matrix_power_impl(self, n, result);
return result;
}
Tensor linalg_matrix_power(const Tensor& self, int64_t n) {
return linalg_matrix_power_impl(self, n, c10::nullopt);
}
Tensor& matrix_power_out(const Tensor& self, int64_t n, Tensor& result) {
return at::native::linalg_matrix_power_out(self, n, result);
}
Tensor matrix_power(const Tensor& self, int64_t n) {
return at::native::linalg_matrix_power(self, n);
}
// Computes the rank of 'input' and saves the result in-place in 'result'
// 'hermitian' controls whether SVD or eigendecomposition is used for computing the singular values
// 'atol' and 'rtol' are the absolute and relative tolerances, respectively.
// TODO: this function can be made public, see: https://github.com/pytorch/pytorch/issues/54151
static Tensor& linalg_matrix_rank_out_helper(const Tensor& input, const Tensor& atol, const Tensor& rtol, bool hermitian, Tensor& result) {
checkSameDevice("torch.linalg.matrix_rank", result, input);
checkSameDevice("torch.linalg.matrix_rank", atol, input, "atol");
checkSameDevice("torch.linalg.matrix_rank", rtol, input, "rtol");
ScalarType output_type = ScalarType::Long;
checkLinalgCompatibleDtype("torch.linalg.matrix_rank", result.scalar_type(), output_type);
// Matrices or batch of matrices are allowed
TORCH_CHECK(input.dim() >= 2, "torch.linalg.matrix_rank: Expected as input a matrix or a batch of matrices, but got a tensor of size: ", input.sizes());
TORCH_CHECK(!at::isComplexType(atol.scalar_type()),
"torch.linalg.matrix_rank: atol tensor of complex type is not supported.");
TORCH_CHECK(!at::isComplexType(rtol.scalar_type()),
"torch.linalg.matrix_rank: rtol tensor of complex type is not supported.");
// matrix_rank assigns a scalar value for each matrix in the batch so
// result's shape is equal to input.shape[0:input.ndim-2]
// for single matrix result_shape = {}
auto result_shape = IntArrayRef(input.sizes().cbegin(), input.sizes().cend() - 2);
at::native::resize_output(result, result_shape);
// NumPy doesn't take into account possible input with no elements and it errors on max not defined for this case
// Let's output 0 for this case, since that kind of matrices have zero number of non-zero rows, hence rank is 0.
if (input.numel() == 0) {
result.fill_(0);
return result;
}
// We compute matrix rank as the number of singular or absolute eigen values
// that are above max(atol, rtol * max(S)) threshold
Tensor S, max_S;
if (!hermitian) {
S = at::linalg_svdvals(input);
// singular values are sorted in descending order
max_S = at::narrow(S, /*dim=*/-1, /*start=*/0, /*length=*/1);
} else {
S = at::linalg_eigvalsh(input);
S = S.abs();
// eigenvalues are sorted in ascending order starting with negative values, we need a maximum value of abs(eigenvalues)
max_S = S.amax(/*dim=*/-1, /*keepdim=*/true);
}
Tensor tol = at::max(atol.unsqueeze(-1), rtol * max_S);
result = at::sum_out(result, S > tol, /*dim=*/-1);
return result;
}
Tensor& linalg_matrix_rank_out(const Tensor& input, const Tensor& tol, bool hermitian, Tensor& result) {
// For NumPy compatibility tol is not scaled with max(singular_value) if the value for tol is provided
// It is assumed that the provided value is the absolute tolerance
Tensor rtol = at::zeros({}, tol.options());
result = linalg_matrix_rank_out_helper(input, tol, rtol, hermitian, result);
return result;
}
Tensor& linalg_matrix_rank_out(const Tensor& input, optional<double> tol, bool hermitian, Tensor& result) {
double tol_value;
Tensor atol, rtol;
if (tol.has_value()) {
tol_value = tol.value();
// For NumPy compatibility tol is not scaled with max(singular_value) if the value for tol is provided
// It is assumed that the provided value is the absolute tolerance
atol = at::full({}, tol_value, input.options().dtype(ScalarType::Double));
rtol = at::zeros({}, input.options().dtype(ScalarType::Double));
} else {
ScalarType real_dtype = toValueType(input.scalar_type());
// This is NumPy compatible default value
tol_value = _get_epsilon(real_dtype) * std::max(input.size(-1), input.size(-2));
// It is assumed that the default tolerance is the relative tolerance
atol = at::zeros({}, input.options().dtype(ScalarType::Double));
rtol = at::full({}, tol_value, input.options().dtype(ScalarType::Double));
}
result = linalg_matrix_rank_out_helper(input, atol, rtol, hermitian, result);
return result;
}
Tensor linalg_matrix_rank(const Tensor& input, const Tensor& tol, bool hermitian) {
Tensor result = at::empty({0}, input.options().dtype(ScalarType::Long));
result = at::linalg_matrix_rank_outf(input, tol, hermitian, result);
return result;
}
Tensor linalg_matrix_rank(const Tensor& input, optional<double> tol, bool hermitian) {
Tensor result = at::empty({0}, input.options().dtype(ScalarType::Long));
result = at::linalg_matrix_rank_outf(input, tol, hermitian, result);
return result;
}
Tensor matrix_rank(const Tensor& self, double tol, bool symmetric) {
TORCH_WARN_ONCE(
"torch.matrix_rank is deprecated in favor of torch.linalg.matrix_rank",
"and will be removed in a future PyTorch release. The parameter 'symmetric' was ",
"renamed in torch.linalg.matrix_rank to 'hermitian'."
);
return at::linalg_matrix_rank(self, optional<double>(tol), symmetric);
}
Tensor matrix_rank(const Tensor& self, bool symmetric) {
TORCH_WARN_ONCE(
"torch.matrix_rank is deprecated in favor of torch.linalg.matrix_rank",
"and will be removed in a future PyTorch release. The parameter 'symmetric' was ",
"renamed in torch.linalg.matrix_rank to 'hermitian'."
);
return at::linalg_matrix_rank(self, c10::nullopt, symmetric);
}
// multi_dot helper functions
namespace {
/**
* @brief Computes the optimal matrix chain multiplication order
*
* Follows the dynamic programming algorithm from Cormen et al,
* "Introduction to Algorithms, Third Edition", Chapter 15.2,
* p. 370-378. Note that the book uses 1-based indexing.
*
* The cost of multiplying two matrices with sizes p x q and q x r
* is defined here as p * q * r. The optimal multiplication order
* is the one that minimizes the total cost.
*
* @param tensors list of 2D tensors
* @return a 2D vector s used by #matrix_chain_multiplication to construct
* the optimal matrix multiplication order. The optimal multiplication
* order for multiplying tensors i...j is to multiply tensors i...s[i, j]
* and tensors (s[i, j] + 1)...j first and then the result of that.
*/
std::vector<std::vector<int64_t>> matrix_chain_order(TensorList tensors) {
const size_t n = tensors.size();
// Tensor i has dimensions p[i] x p[i + 1]
std::vector<int64_t> p(n + 1);
for (const auto i : c10::irange(n)) {
p[i] = tensors[i].size(0);
}
p[n] = tensors[n - 1].size(1);
// m[i, j] = k where k is the minimum cost for multiplying tensors i...j
std::vector<std::vector<int64_t>> m(n, std::vector<int64_t>(n, 0));
// s[i, j] = k where k is the index at which to split the list such that
// optimally multiplying matrices i...k and k...j first and then the resulting
// matrices is the optimal order for multiplying matrices i...j.
std::vector<std::vector<int64_t>> s(n, std::vector<int64_t>(n));
// Compute the optimal multiplication order
for (const auto l : c10::irange(1, n)) {
for (const auto i : c10::irange(n - l)) {
const auto j = i + l;
m[i][j] = std::numeric_limits<int64_t>::max();
for (const auto k : c10::irange(i, j)) {
const auto q = m[i][k] + m[k + 1][j] + p[i] * p[k + 1] * p[j + 1];
if (q < m[i][j]) {
m[i][j] = q;
s[i][j] = k;
}
}
}
}
return s;
}
/**
* @brief Recursively multiplies the tensors i...j using the given order
*
* @param tensors matrices to multiply togther
* @param order optimal chain multiplication order from #matrix_chain_order
* @param i index of first tensor to be multiplied
* @param j index of last tensor to be multiplied
* @return Tensor result of multiplying tensors[i...j] together.
*/
Tensor matrix_chain_multiplication(
TensorList tensors,
const std::vector<std::vector<int64_t>>& order,
int64_t i,
int64_t j) {
if (i == j) {
return tensors[i];
}
return at::mm(
matrix_chain_multiplication(tensors, order, i, order[i][j]),
matrix_chain_multiplication(tensors, order, order[i][j] + 1, j));
}
// Implements torch.linalg.multi_dot
Tensor multi_dot_impl(TensorList _tensors, c10::optional<Tensor> _out) {
const size_t n = _tensors.size();
TORCH_CHECK(n >= 2, "multi_dot(): expected at least 2 tensors but got ", n);
std::vector<int64_t> out_shape;
std::vector<Tensor> tensors(n);
// If the first tensor is 1D of size n view it as a row vector (1, n)
if (_tensors[0].dim() == 1) {
tensors[0] = _tensors[0].unsqueeze(0);
} else if (_tensors[0].dim() == 2) {
tensors[0] = _tensors[0];
out_shape.emplace_back(tensors[0].size(0));
} else {
TORCH_CHECK(
false,
"multi_dot(): the first tensor must be 1D or 2D but got ",
_tensors[0].dim(),
"D");
}
// If the last tensor is 1D of size n view it as a column vector (n, 1)
if (_tensors[n - 1].dim() == 1) {
tensors[n - 1] = _tensors[n - 1].unsqueeze(-1);
} else if (_tensors[n - 1].dim() == 2) {
tensors[n - 1] = _tensors[n - 1];
out_shape.emplace_back(tensors[n - 1].size(1));
} else {
TORCH_CHECK(
false,
"multi_dot(): the last tensor must be 1D or 2D but got ",
_tensors[n - 1].dim(),
"D");
}
// Ensure middle tensors are 2D
for (const auto i : c10::irange(1, n - 1)) {
TORCH_CHECK(
_tensors[i].dim() == 2,
"multi_dot(): tensor ",
i,
" must be 2D but got ",
_tensors[i].dim(),
"D");
tensors[i] = _tensors[i];
}
// Ensure all tensors have the same device and dtype and check
// that the shapes can be multiplied
const auto dtype = tensors[0].dtype();
const auto device = tensors[0].device();
for (const auto i : c10::irange(1, n)) {
TORCH_CHECK(
tensors[i].dtype() == dtype,
"multi_dot(): all tensors must have be the same dtype but tensor 0 is ",
dtype,
" and tensor ",
i,
" ",
tensors[i].dtype());
TORCH_CHECK(
tensors[i].device() == device,
"multi_dot(): all tensors must be on the same device but tensor 0 is on ",
device,
" and tensor ",
i,
" on ",
tensors[i].device());
TORCH_CHECK(
tensors[i - 1].size(-1) == tensors[i].size(0),
"multi_dot(): tensors ",
i - 1,
" and ",
i,
" with shapes ",
_tensors[i - 1].sizes(),
" and ",
_tensors[i].sizes(),
" cannot be multiplied")
}
Tensor result;
if (_out.has_value()) {
auto out = *_out;
TORCH_CHECK(
dtype == out.dtype(),
"multi_dot(): expected out tensor to have dtype ",
dtype,
" but got ",
out.dtype());
TORCH_CHECK(
device == out.device(),
"multi_dot(): expected out tensor to be on device ",
device,
" but got ",
out.device());
// If the last and last tensors have shapes (a, b) and (b, c) the
// output has shape (a, c). If either the first or last tensor is 1D
// a and/or c dimensions will be implicitely size 1 and will be ommited
// from the output. e.g. for inputs (a, b) x (b) the output has shape (a,).
at::native::resize_output(out, out_shape);
// View output as 2D for simplicity of computation.
result = out.view({tensors[0].size(0), tensors.back().size(-1)});
}
// The resize_ and view calls below are to ensure the
// output shape respects the original dimensionality of
// the first and last tensors which we are now viewed as 2D
if (tensors.size() == 2) {
return _out.has_value() ? at::mm_out(result, tensors[0], tensors[1])
: at::mm(tensors[0], tensors[1]).view(out_shape);
}
// Why the separate implementation for 3 matrices?
// The logic for three matrices is much faster when done directly
// Requires 1 comparison to 4 comparisons and fewer arithmetic operations
if (tensors.size() == 3) {
const auto a = tensors[0].size(0);
const auto b = tensors[1].size(0);
const auto c = tensors[2].size(0);
const auto d = tensors[2].size(1);
// The matrices are of size (a x b), (b x c), (c x d)
// cost_1 is the cost of parenthesizing (a x b) and (b x c) and then
// combining (c x d) cost_2 is the cost of parenthesizing (b x c) and (c x
// d) and then combining (a x b)
const auto cost_1 = (a * c) * (b + d);
const auto cost_2 = (b * d) * (a + c);
if (cost_1 > cost_2) {
return _out.has_value()
? at::mm_out(result, tensors[0], at::mm(tensors[1], tensors[2]))
: at::mm(tensors[0], at::mm(tensors[1], tensors[2])).view(out_shape);
} else {
return _out.has_value()
? at::mm_out(result, at::mm(tensors[0], tensors[1]), tensors[2])
: at::mm(at::mm(tensors[0], tensors[1]), tensors[2]).view(out_shape);
}
}
// Algorithm for multiplying 4 or more matrices
const auto order = matrix_chain_order(tensors);
const int64_t i = 0;
const int64_t j = n - 1;
if (_out.has_value()) {
// We manually implement the first recursive layer here so we can use mm_out
// for the final multiplication
return at::mm_out(
result,
matrix_chain_multiplication(tensors, order, i, order[i][j]),
matrix_chain_multiplication(tensors, order, order[i][j] + 1, j));
}
return matrix_chain_multiplication(tensors, order, i, j).view(out_shape);
}
} // namespace
Tensor linalg_multi_dot(TensorList tensors) {
return multi_dot_impl(tensors, c10::nullopt);
}
Tensor& linalg_multi_dot_out(TensorList tensors, Tensor& result) {
multi_dot_impl(tensors, result);
return result;
}
Tensor chain_matmul(TensorList matrices) {
TORCH_WARN_ONCE(
"torch.chain_matmul is deprecated and will be removed in a future PyTorch release. ",
"Use torch.linalg.multi_dot instead, which accepts a list of two or more tensors rather than ",
"multiple parameters."
);
checkAllSameDim(matrices, 2);
TORCH_CHECK(
matrices.size() > 0, "chain_matmul(): Expected one or more matrices");
if (matrices.size() == 1) {
return matrices[0].clone();
}
return at::native::linalg_multi_dot(matrices);
}
Tensor& chain_matmul_out(TensorList matrices, Tensor& result) {
TORCH_WARN_ONCE(
"torch.chain_matmul is deprecated and will be removed in a future PyTorch release. ",
"Use torch.linalg.multi_dot instead, which accepts a list of two or more tensors rather than ",
"multiple parameters."
);
checkAllSameDim(matrices, 2);
TORCH_CHECK(
matrices.size() > 0, "chain_matmul(): Expected one or more matrices");
if (matrices.size() == 1) {
at::native::resize_output(result, matrices[0].sizes());
return result.copy_(matrices[0]);
}
return at::native::linalg_multi_dot_out(matrices, result);
}
static void check_1d(const Tensor& t, const char* arg, const char* fn) {
TORCH_CHECK(t.dim() == 1, fn, ": Expected 1-D argument ", arg, ", but got ", t.dim(), "-D");
}
static void check_addr_scalar(const ScalarType dtype,
const Scalar& scalar,
const std::string& scalar_name) {
TORCH_CHECK(
!scalar.isBoolean() || dtype == ScalarType::Bool,
"Boolean ", scalar_name, " only supported for Boolean results.");
TORCH_CHECK(
isFloatingType(dtype) || isComplexType(dtype) || scalar.isIntegral(true),
"For integral input tensors, "
"argument ", scalar_name ," must not be a floating point number.");
}
static TensorIterator build_addr_iter(Tensor& result,
const Tensor& self,
const Tensor& vec1,
const Tensor& vec2) {
check_1d(vec1, "vec1", "addr");
check_1d(vec2, "vec2", "addr");
const auto vec1_size0 = vec1.sizes()[0];
const auto vec2_size0 = vec2.sizes()[0];
auto self_ = &result == &self
? c10::MaybeOwned<Tensor>::borrowed(self)
: expand_size(self, {vec1_size0, vec2_size0}, "addr");
TORCH_CHECK(
self_->dim() == 2,
"2D tensor expected, got ", self_->dim(), "D tensor for input"
);
TORCH_CHECK(
self_->sizes()[0] == vec1_size0 && self_->sizes()[1] == vec2_size0,
"size mismatch, input: ", self_->sizes(),
", v1: ", vec1.sizes(),
", v2: ", vec2.sizes()
);
auto iter = TensorIteratorConfig()
.set_check_mem_overlap(true)
.add_output(result)
.add_owned_input(*self_)
.add_owned_input(vec1.reshape({vec1_size0, 1}))
.add_input(vec2)
.allow_cpu_scalars(true)
.promote_inputs_to_common_dtype(true)
.cast_common_dtype_to_outputs(true)
.enforce_safe_casting_to_output(true)
.build();
return iter;
}
Tensor addr(const Tensor& self,
const Tensor& vec1, const Tensor& vec2,
const Scalar& beta, const Scalar& alpha) {
Tensor result;
auto iter = build_addr_iter(result, self, vec1, vec2);
check_addr_scalar(iter.dtype(), beta, "beta");
check_addr_scalar(iter.dtype(), alpha, "alpha");
addr_stub(iter.device_type(), iter, beta, alpha);
return iter.output();
}
Tensor& addr_(Tensor& self,
const Tensor& vec1, const Tensor& vec2,
const Scalar& beta, const Scalar& alpha) {
return at::addr_out(self, self, vec1, vec2, beta, alpha);
}
Tensor& addr_out(const Tensor& self,
const Tensor& vec1, const Tensor& vec2,
const Scalar& beta, const Scalar& alpha, Tensor &result) {
auto iter = build_addr_iter(result, self, vec1, vec2);
check_addr_scalar(iter.dtype(), beta, "beta");
check_addr_scalar(iter.dtype(), alpha, "alpha");
addr_stub(iter.device_type(), iter, beta, alpha);
return result;
}
// The math_addr and math_addr_out functions support backends
// other than CPU and CUDA, such as XLA.
// They are implemented using the composition of existing ops
Tensor math_addr(const Tensor& self,
const Tensor& vec1, const Tensor& vec2,
const Scalar& beta, const Scalar& alpha) {
// when beta==0, values in self should be ignored,
// nans and infs in self should not propagate.
if (beta.toComplexDouble() == 0.0) {
if (alpha.toComplexDouble() == 1.0) {
return at::outer(vec1, vec2);
}
return alpha * at::outer(vec1, vec2);
}
if (beta.toComplexDouble() == 1.0) {
if (alpha.toComplexDouble() == 1.0) {
return self + at::outer(vec1, vec2);
}
return self + alpha * at::outer(vec1, vec2);
}
if (alpha.toComplexDouble() == 1.0) {
return beta * self + at::outer(vec1, vec2);
}
return beta * self + alpha * at::outer(vec1, vec2);
}
Tensor& math_addr_out(const Tensor& self,
const Tensor& vec1, const Tensor& vec2,
const Scalar& beta, const Scalar& alpha, Tensor &result) {
auto addr_result = at::addr(self, vec1, vec2, beta, alpha);
// Validates safe casting
const auto result_dtype = addr_result.scalar_type();
TORCH_CHECK(canCast(result_dtype, result.scalar_type()),
"result type ", result_dtype,
" can't be cast to the desired output type ", result.scalar_type());
at::native::resize_output(result, addr_result.sizes().vec());
result.copy_(addr_result);
return result;
}
// torch.ger, alias for torch.outer
Tensor& ger_out(const Tensor& self, const Tensor& vec2, Tensor &result) {
TORCH_WARN("torch.ger is deprecated and will be removed in a future PyTorch release. "
"Use torch.outer instead.");
return at::outer_out(result, self, vec2);
}
Tensor ger(const Tensor& self, const Tensor& vec2) {
return self.outer(vec2);
}
Tensor& inner_out(const Tensor& self, const Tensor& other, Tensor& out) {
checkDeviceType("inner()", {out, self, other}, self.device().type());
// If either self or other is a scalar just multiply them
if (self.dim() == 0 || other.dim() == 0) {
at::mul_out(out, self, other);
return out;
}
// Last dimension should match (tensordot does not enforce this)
TORCH_CHECK(
self.size(-1) == other.size(-1),
"inner() the last dimension must match on both input tensors but got shapes ",
self.sizes(),
" and ",
other.sizes());
at::tensordot_out(out, self, other, -1, -1);
return out;
}
Tensor inner(const Tensor& self, const Tensor& other) {
checkDeviceType("inner()", {self, other}, self.device().type());
// If either self or other is a scalar just multiply them
if (self.dim() == 0 || other.dim() == 0) {
return self * other;
}
// Last dimension should match (tensordot does not enforce this)
TORCH_CHECK(
self.size(-1) == other.size(-1),
"inner() the last dimension must match on both input tensors but got shapes ",
self.sizes(),
" and ",
other.sizes());
return at::tensordot(self, other, -1, -1);
}
Tensor& outer_out(const Tensor& self, const Tensor& vec2, Tensor &result) {
check_1d(self, "self", "outer");
check_1d(vec2, "vec2", "outer");
// torch.outer is implemented as a composite op using reshape and mul
at::mul_out(result, self.reshape({self.size(0), 1}), vec2);
return result;
}
Tensor outer(const Tensor& self, const Tensor& vec2) {
check_1d(self, "self", "outer");
check_1d(vec2, "vec2", "outer");
return self.reshape({self.size(0), 1}) * vec2;
}
static void addmm_impl_cpu_(
Tensor &result, const Tensor &self, Tensor m1, Tensor m2, const Scalar& beta, const Scalar& alpha) {
TORCH_INTERNAL_ASSERT(self.dim() == 2 && m1.dim() == 2 && m2.dim() == 2);
// Array access is faster than .size(n) and .stride(n)
const auto self_sizes = self.sizes();
auto m1_strides = m1.strides();
auto m1_sizes = m1.sizes();
auto m2_strides = m2.strides();
auto m2_sizes = m2.sizes();
TORCH_CHECK(
self_sizes[0] == m1_sizes[0] && self_sizes[1] == m2_sizes[1],
"input shape is incompatible with matrix multiplication (",
m1_sizes[0], "x", m1_sizes[1], " @ ", m2_sizes[0], "x", m2_sizes[1], " != ",
self_sizes[0], "x", self_sizes[1], ")");
at::native::resize_output(result, self_sizes);
const auto result_strides = result.strides();
const auto result_sizes = result.sizes();
if (result.numel() == 0) {
return;
}
if (beta.toComplexDouble() != 0.0 && !self.is_same(result)) {
result.copy_(self);
}
bool transpose_c = false;
Tensor c;
// Cast result as matrix a
if (result_strides[0] == 1 &&
(result_sizes[1] == 1 || result_strides[1] >= std::max(int64_t{1}, result_sizes[0]))) {
transpose_c = false;
c = result;
} else if (result_strides[1] == 1 &&
(result_sizes[0] == 1 || result_strides[0] >= std::max(int64_t{1}, result_sizes[1]))) {
std::swap(m1, m2);
std::swap(m1_sizes, m2_sizes);
std::swap(m1_strides, m2_strides);