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LinearAlgebra.cpp
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#define TORCH_ASSERT_ONLY_METHOD_OPERATORS
#include <ATen/Context.h>
#include <ATen/Dispatch.h>
#include <ATen/ExpandUtils.h>
#include <ATen/NamedTensorUtils.h>
#include <ATen/OpMathType.h>
#include <ATen/Parallel.h>
#include <ATen/TensorIndexing.h>
#include <ATen/TensorIterator.h>
#include <ATen/TensorOperators.h>
#include <ATen/TensorSubclassLikeUtils.h>
#include <ATen/TensorUtils.h>
#include <ATen/core/Tensor.h>
#include <ATen/native/CPUBlas.h>
#include <ATen/native/cpu/int_mm_kernel.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/mkldnn/Matmul.h>
#include <ATen/native/mkldnn/Utils.h>
#include <ATen/cpu/Utils.h>
#include <c10/core/GradMode.h>
#include <c10/util/accumulate.h>
#include <c10/util/irange.h>
#include <variant>
#ifndef AT_PER_OPERATOR_HEADERS
#include <ATen/Functions.h>
#include <ATen/NativeFunctions.h>
#else
#include <ATen/ops/_addmm_activation_native.h>
#include <ATen/ops/_compute_linear_combination_native.h>
#include <ATen/ops/_convert_weight_to_int4pack_for_cpu_native.h>
#include <ATen/ops/_int_mm_native.h>
#include <ATen/ops/_linalg_check_errors.h>
#include <ATen/ops/_linalg_det.h>
#include <ATen/ops/_linalg_det_native.h>
#include <ATen/ops/_linalg_slogdet.h>
#include <ATen/ops/_linalg_slogdet_native.h>
#include <ATen/ops/_unsafe_view.h>
#include <ATen/ops/_weight_int4pack_mm_for_cpu_native.h>
#include <ATen/ops/_weight_int8pack_mm_native.h>
#include <ATen/ops/abs.h>
#include <ATen/ops/addbmm_native.h>
#include <ATen/ops/addmm_native.h>
#include <ATen/ops/addr.h>
#include <ATen/ops/addr_native.h>
#include <ATen/ops/arange.h>
#include <ATen/ops/argsort.h>
#include <ATen/ops/baddbmm_native.h>
#include <ATen/ops/bmm.h>
#include <ATen/ops/bmm_native.h>
#include <ATen/ops/cat.h>
#include <ATen/ops/ceil.h>
#include <ATen/ops/chain_matmul_native.h>
#include <ATen/ops/cumsum.h>
#include <ATen/ops/det_native.h>
#include <ATen/ops/diag_embed.h>
#include <ATen/ops/diff.h>
#include <ATen/ops/dot.h>
#include <ATen/ops/dot_native.h>
#include <ATen/ops/empty.h>
#include <ATen/ops/empty_like.h>
#include <ATen/ops/eye.h>
#include <ATen/ops/floor.h>
#include <ATen/ops/frobenius_norm_native.h>
#include <ATen/ops/from_blob.h>
#include <ATen/ops/full.h>
#include <ATen/ops/full_like.h>
#include <ATen/ops/gelu.h>
#include <ATen/ops/ger_native.h>
#include <ATen/ops/index_select.h>
#include <ATen/ops/inner_native.h>
#include <ATen/ops/is_complex_native.h>
#include <ATen/ops/is_floating_point_native.h>
#include <ATen/ops/kron_native.h>
#include <ATen/ops/linalg_cond.h>
#include <ATen/ops/linalg_cond_native.h>
#include <ATen/ops/linalg_det.h>
#include <ATen/ops/linalg_det_native.h>
#include <ATen/ops/linalg_diagonal_native.h>
#include <ATen/ops/linalg_eigh.h>
#include <ATen/ops/linalg_eigvalsh.h>
#include <ATen/ops/linalg_inv.h>
#include <ATen/ops/linalg_inv_ex.h>
#include <ATen/ops/linalg_lu_factor_ex.h>
#include <ATen/ops/linalg_matmul_native.h>
#include <ATen/ops/linalg_matrix_exp.h>
#include <ATen/ops/linalg_matrix_exp_native.h>
#include <ATen/ops/linalg_matrix_norm.h>
#include <ATen/ops/linalg_matrix_norm_native.h>
#include <ATen/ops/linalg_matrix_power_native.h>
#include <ATen/ops/linalg_matrix_rank.h>
#include <ATen/ops/linalg_matrix_rank_native.h>
#include <ATen/ops/linalg_multi_dot_native.h>
#include <ATen/ops/linalg_norm.h>
#include <ATen/ops/linalg_norm_native.h>
#include <ATen/ops/linalg_pinv.h>
#include <ATen/ops/linalg_pinv_native.h>
#include <ATen/ops/linalg_slogdet.h>
#include <ATen/ops/linalg_slogdet_native.h>
#include <ATen/ops/linalg_solve.h>
#include <ATen/ops/linalg_svdvals.h>
#include <ATen/ops/linalg_tensorinv.h>
#include <ATen/ops/linalg_tensorinv_native.h>
#include <ATen/ops/linalg_tensorsolve.h>
#include <ATen/ops/linalg_tensorsolve_native.h>
#include <ATen/ops/linalg_vector_norm.h>
#include <ATen/ops/linalg_vector_norm_native.h>
#include <ATen/ops/log2.h>
#include <ATen/ops/logdet_native.h>
#include <ATen/ops/matmul.h>
#include <ATen/ops/matmul_native.h>
#include <ATen/ops/matrix_exp_backward_native.h>
#include <ATen/ops/matrix_exp_native.h>
#include <ATen/ops/matrix_power_native.h>
#include <ATen/ops/max.h>
#include <ATen/ops/mm.h>
#include <ATen/ops/mm_native.h>
#include <ATen/ops/movedim.h>
#include <ATen/ops/mul.h>
#include <ATen/ops/mv.h>
#include <ATen/ops/narrow.h>
#include <ATen/ops/ne.h>
#include <ATen/ops/norm.h>
#include <ATen/ops/nuclear_norm_native.h>
#include <ATen/ops/ones.h>
#include <ATen/ops/outer.h>
#include <ATen/ops/outer_native.h>
#include <ATen/ops/pinverse_native.h>
#include <ATen/ops/pow.h>
#include <ATen/ops/prod.h>
#include <ATen/ops/real.h>
#include <ATen/ops/relu.h>
#include <ATen/ops/slogdet_native.h>
#include <ATen/ops/sort.h>
#include <ATen/ops/sqrt.h>
#include <ATen/ops/sum.h>
#include <ATen/ops/tensordot.h>
#include <ATen/ops/unique_consecutive.h>
#include <ATen/ops/vdot_native.h>
#include <ATen/ops/where.h>
#include <ATen/ops/zeros.h>
#include <ATen/ops/zeros_like.h>
#endif
#include <limits>
#include <numeric>
#include <string>
#include <tuple>
#include <utility>
#if !defined(__s390x__) && !defined(__powerpc__)
#include <cpuinfo.h>
#endif
namespace at {
namespace detail {
static void check_linalg_norm_dtype(std::optional<ScalarType> opt_dtype, ScalarType self_dtype, const char* const name) {
if (opt_dtype.has_value()) {
auto dtype = opt_dtype.value();
TORCH_CHECK(isFloatingType(dtype) || isComplexType(dtype), name, ": dtype should"
" be floating point or complex, but got ", dtype);
TORCH_CHECK(isComplexType(self_dtype) == isComplexType(dtype),
name, ": dtype should be ", isComplexType(self_dtype) ? "complex" : "real",
" for ", isComplexType(self_dtype) ? "complex" : "real", " inputs, but got ", dtype);
TORCH_CHECK(promoteTypes(self_dtype, dtype) == dtype,
name, ": the dtype of the input ", "(", self_dtype, ") should be convertible ",
"without narrowing to the specified dtype (", dtype, ")");
}
}
}
namespace meta {
#define ADDMM_META() \
TORCH_CHECK(self.scalar_type() == mat2.scalar_type(), "self and mat2 must have the same dtype, but got ", self.scalar_type(), " and ", mat2.scalar_type()); \
TORCH_CHECK(mat1.scalar_type() == mat2.scalar_type(), "mat1 and mat2 must have the same dtype, but got ", mat1.scalar_type(), " and ", mat2.scalar_type()); \
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_raw_strided(0, {mat1.sizes()[0], mat2.sizes()[1]}, {}, mat1.options(), names);
TORCH_META_FUNC(addmm)(const Tensor& self, const Tensor& mat1, const Tensor& mat2, const Scalar& beta, const Scalar& alpha) {
ADDMM_META();
}
TORCH_META_FUNC(_addmm_activation)(const Tensor& self, const Tensor& mat1, const Tensor& mat2, const Scalar& beta, const Scalar& alpha, bool use_gelu) {
ADDMM_META();
}
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_raw_strided(0, {self.sizes()[0], mat2.sizes()[1]}, {}, self.options(), names);
}
TORCH_META_FUNC(linalg_vector_norm)(const Tensor& self, const Scalar& scalar_ord, OptionalIntArrayRef opt_dim, bool keepdim, std::optional<ScalarType> opt_dtype) {
at::native::checkFloatingOrComplex(self, "linalg.vector_norm");
TORCH_CHECK(!at::isComplexType(scalar_ord.type()), "linalg.vector_norm: Expected a non-complex scalar as the order of norm.");
auto dim = opt_dim.value_or(IntArrayRef{});
// Casting a large integer to a double will just introduce an error for
// values larger than 10^53 (same for negative numbers), so that's fine.
auto ord = scalar_ord.toDouble();
// For more context, see issue 52783
// If the tensor is empty and norm < 0 || norm == infty
// - We cannot reduce the whole tensor
// - We cannot reduce over an empty dimension
if (self.numel() == 0 && (ord < 0. || ord == INFINITY)) {
// dim=None or dim=() reduces the whole tensor
TORCH_CHECK(opt_dim.has_value() && !opt_dim->empty(),
"linalg.vector_norm cannot compute the ", scalar_ord, " norm on an empty ",
"tensor because the operation does not have an identity");
for (auto dim_num : dim) {
TORCH_CHECK(self.size(dim_num) != 0,
"linalg.vector_norm cannot compute the ", scalar_ord, " norm on the dimension ", dim_num ,
"because this dimension is empty and the operation does not have an identity");
}
}
at::detail::check_linalg_norm_dtype(opt_dtype, self.scalar_type(), "linalg.vector_norm");
auto mask = at::native::make_dim_mask(dim, self.dim());
auto shape = at::native::shape_from_dim_mask(self, std::move(mask), keepdim);
auto options = self.options()
.dtype(toRealValueType(opt_dtype.value_or(self.scalar_type())));
set_output_raw_strided(0, shape, {}, options);
}
TORCH_META_FUNC(_linalg_det)(const Tensor& A) {
at::native::squareCheckInputs(A, "linalg.det");
at::native::checkFloatingOrComplex(A, "linalg.det");
auto shape = A.sizes();
auto ndim = shape.size();
// det
set_output_contiguous(0, shape.slice(0, ndim - 2), A.options());
// LU
auto LU_strides = at::native::batched_matrix_contiguous_strides(shape, /*f-contig*=*/true);
set_output_strided(1, shape, LU_strides, A.options());
// pivots
set_output_contiguous(2, shape.slice(0, ndim - 1), A.options().dtype(kInt));
}
TORCH_META_FUNC(_linalg_slogdet)(const Tensor& A) {
at::native::squareCheckInputs(A, "linalg.slogdet");
at::native::checkFloatingOrComplex(A, "linalg.slogdet", /*low_precision*/false);
auto shape= A.sizes();
auto ndim = shape.size();
auto shape_outputs = shape.slice(0, ndim - 2);
// sign
set_output_contiguous(0, shape_outputs, A.options());
// logabsdet
set_output_contiguous(1, shape_outputs, A.options().dtype(toRealValueType(A.scalar_type())));
// LU
auto LU_strides = at::native::batched_matrix_contiguous_strides(shape, /*f-contig*=*/true);
set_output_strided(2, shape, LU_strides, A.options());
// pivots
set_output_contiguous(3, shape.slice(0, ndim - 1), A.options().dtype(kInt));
}
template <typename Meta>
void common_checks_baddbmm_bmm(Meta& meta, const Tensor& batch1, const Tensor& batch2, const Scalar& beta, const Scalar& alpha, bool is_bmm, const std::optional<Tensor>& self_baddbmm = std::nullopt) {
TORCH_CHECK(batch1.dim() == 3, "batch1 must be a 3D tensor");
TORCH_CHECK(batch2.dim() == 3, "batch2 must be a 3D tensor");
const auto batch1_sizes = batch1.sizes();
const auto batch2_sizes = batch2.sizes();
int64_t bs = batch1_sizes[0];
int64_t contraction_size = batch1_sizes[2];
int64_t res_rows = batch1_sizes[1];
int64_t res_cols = batch2_sizes[2];
std::vector<int64_t> output_size {bs, res_rows, res_cols};
TORCH_CHECK(batch2_sizes[0] == bs && batch2_sizes[1] == contraction_size,
"Expected size for first two dimensions of batch2 tensor to be: [",
bs, ", ", contraction_size, "] but got: [", batch2_sizes[0], ", ", batch2_sizes[1], "].");
auto& result = meta.maybe_get_output(0);
// 'set_output' does not resize for in-place calls
meta.set_output_raw_strided(0, output_size, {}, batch2.options());
const auto result_sizes = result.sizes();
// Error is raised if called from in-place overload with incorrect shape
TORCH_CHECK(result_sizes == output_size,
"Expected an output tensor with shape [", output_size, "] but got shape ", result_sizes);
std::vector<Dimname> outnames = {};
if (!is_bmm) {
if (self_baddbmm.has_value()) {
const auto& self = self_baddbmm.value();
if (beta.toComplexDouble() != 0.0) result.copy_(self);
TORCH_CHECK(self.dim() == 3, "self must be a 3D tensor");
const auto self_sizes = self.sizes();
TORCH_CHECK(self_sizes == output_size,
"Expected an input tensor shape with shape ", output_size, " but got shape: ", self_sizes);
outnames = namedinference::compute_baddbmm_outnames(result, batch1, batch2, self);
}
} else {
outnames = namedinference::compute_bmm_outnames(result, batch1, batch2);
}
namedinference::propagate_names_if_nonempty(
result,
outnames
);
}
TORCH_META_FUNC(bmm)(const Tensor& self, const Tensor& mat2) {
common_checks_baddbmm_bmm(*this, self, mat2, Scalar(0.0), Scalar(1.0), true);
}
TORCH_META_FUNC(baddbmm)(const Tensor& self, const Tensor& batch1, const Tensor& batch2, const Scalar& beta, const Scalar& alpha) {
auto self_ = expand_size(self, {batch1.size(0), batch1.size(1), batch2.size(2)}, "baddbmm");
TORCH_CHECK(self.dtype() == batch1.dtype(), "Input dtypes must be the same, got: input ", self.dtype(), ", batch1: ", batch1.dtype(), ", batch2: ", batch2.dtype());
common_checks_baddbmm_bmm(*this, batch1, batch2, beta, alpha, false, *self_);
}
} // namespace meta
namespace native {
DEFINE_DISPATCH(addr_stub);
// ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ linalg.det ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// As P is a permutation matrix
// det(P) = 1 if it's an even permutation and det(P) = -1 if it's an odd permutation
static Tensor lu_det_P(const Tensor& pivots) {
return (at::arange(1, pivots.size(-1) + 1, pivots.options()) != pivots)
.sum(-1, /*keepdim=*/false, /*dtype=*/at::kLong)
.fmod_(2)
// take 0 to 1 and 1 to -1
.mul_(-2)
.add_(1);
}
// Auxiliary function that returns the LU decomposition to use it in the backward
TORCH_IMPL_FUNC(_linalg_det_out)(const Tensor& A, const Tensor& result, const Tensor& LU, const Tensor& pivots) {
// info is an aux tensor
auto info = at::empty({0}, A.options().dtype(kInt));
// Optimisation: lu_factor_ex requires the input to be F-contig, otherwise it copies
// Use the transpose of if A is contiguous since det(A^T) = det(A)
// We limit this to real matrices, but it could also be implemented for complex matrices
at::linalg_lu_factor_ex_out(const_cast<Tensor&>(LU), const_cast<Tensor&>(pivots), const_cast<Tensor&>(info), A.is_contiguous() && !A.is_complex() ? A.mH() : A);
// det = det_P * prod(diag(LU))
at::mul_out(const_cast<Tensor&>(result), lu_det_P(pivots), at::prod(LU.diagonal(0, -2 ,-1), /*dim=*/-1));
}
Tensor linalg_det(const Tensor& A) {
return std::get<0>(at::_linalg_det(A));
}
Tensor& linalg_det_out(const Tensor& A, Tensor& result) {
auto LU = at::empty({0}, A.options());
auto pivots = at::empty({0}, A.options().dtype(kInt));
at::_linalg_det_out(result, LU, pivots, A);
return result;
}
// torch.det, alias for torch.linalg.det
Tensor det(const Tensor& self) {
return at::linalg_det(self);
}
//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ linalg.slogdet ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
// Auxiliary function that returns the LU decomposition to use it in the backward
TORCH_IMPL_FUNC(_linalg_slogdet_out)(const Tensor& A, const Tensor& sign, const Tensor& logabsdet, const Tensor& LU, const Tensor& pivots) {
// info is an aux tensor
auto info = at::empty({0}, A.options().dtype(kInt));
// Optimisation: lu_factor_ex requires the input to be F-contig, otherwise it copies
// Use the transpose of if A is contiguous since det(A^T) = det(A)
// We limit this to real matrices, but it could also be implemented for complex matrices
at::linalg_lu_factor_ex_out(const_cast<Tensor&>(LU), const_cast<Tensor&>(pivots), const_cast<Tensor&>(info), A.is_contiguous() && !A.is_complex() ? A.mH() : A);
auto diag_U = LU.diagonal(0, -2, -1);
// sign
at::mul_out(const_cast<Tensor&>(sign), diag_U.sgn().prod(-1), lu_det_P(pivots));
// logabsdet
at::sum_out(const_cast<Tensor&>(logabsdet), diag_U.abs().log_(), -1);
}
std::tuple<Tensor, Tensor> linalg_slogdet(const Tensor& A) {
auto out = at::_linalg_slogdet(A);
return std::make_tuple(std::move(std::get<0>(out)), std::move(std::get<1>(out)));
}
std::tuple<Tensor&, Tensor&> linalg_slogdet_out(const Tensor& A, Tensor& sign, Tensor& logabsdet) {
auto LU = at::empty({0}, A.options());
auto pivots = at::empty({0}, A.options().dtype(kInt));
at::_linalg_slogdet_out(sign, logabsdet, LU, pivots, A);
return std::tie(sign, logabsdet);
}
// Alias
std::tuple<Tensor, Tensor> slogdet(const Tensor& A) {
return at::linalg_slogdet(A);
}
std::tuple<Tensor&, Tensor&> slogdet_out(const Tensor& A, Tensor& sign, Tensor& logabsdet) {
return at::linalg_slogdet_out(sign, logabsdet, A);
}
//~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ logdet ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Tensor logdet(const Tensor& A) {
squareCheckInputs(A, "logdet");
checkFloatingOrComplex(A, "logdet", /*low_precision*/false);
auto [sign, logabsdet] = at::linalg_slogdet(A);
if (A.is_complex()) {
return sign.log() + logabsdet;
} else {
return at::where(sign == -1., NAN, logabsdet);
}
}
namespace {
// This function extracts the optional Tensors for atol and rtol
// Default value for atol is zero
// Default value for rtol is eps*max(rows, cols)
// If atol is specified and rtol is not specified then default value for rtol is zero
// It is used for matrix_rank and pinv
std::tuple<Tensor, Tensor> get_atol_rtol(
const Tensor& input,
const std::optional<Tensor>& atol_opt,
const std::optional<Tensor>& rtol_opt,
const c10::string_view function_name) {
auto options = input.options();
if (input.device().type() == kMetal || input.device().type() == kMPS) {
options = options.dtype(ScalarType::Float);
} else {
options = options.dtype(ScalarType::Double);
}
auto atol = atol_opt.has_value() ? atol_opt.value() : at::zeros({}, options);
checkNotComplexTolerance(atol, function_name, "atol");
Tensor rtol;
if (rtol_opt.has_value()) {
rtol = rtol_opt.value();
checkNotComplexTolerance(rtol, function_name, "rtol");
} else {
ScalarType real_dtype = toRealValueType(input.scalar_type());
auto default_rtol = at::full({}, _get_epsilon(real_dtype) * std::max(input.sym_size(-1), input.sym_size(-2)), options);
rtol = atol_opt.has_value()
? at::where(atol_opt.value() > 0, at::zeros({}, options), default_rtol)
: std::move(default_rtol);
}
return std::make_tuple(atol, rtol);
}
std::tuple<Tensor, Tensor> get_atol_rtol(
const Tensor& input,
std::optional<double> atol_opt,
std::optional<double> rtol_opt) {
auto atol = atol_opt.has_value() ? atol_opt.value() : 0.0;
c10::SymFloat rtol;
if (rtol_opt.has_value()) {
rtol = rtol_opt.value();
} else {
ScalarType real_dtype = toRealValueType(input.scalar_type());
auto default_rtol = _get_epsilon(real_dtype) * std::max(input.sym_size(-1), input.sym_size(-2));
rtol = (atol_opt.has_value() && atol_opt.value() > 0.0)
? 0.0
: default_rtol;
}
auto options = input.options();
if (input.device().type() == kMetal || input.device().type() == kMPS) {
options = options.dtype(ScalarType::Float);
} else {
options = options.dtype(ScalarType::Double);
}
auto atol_tensor = at::full({}, atol, options);
auto rtol_tensor = at::full({}, rtol, options);
return std::make_tuple(atol_tensor, rtol_tensor);
}
} // anonymous namespace
Tensor linalg_pinv(
const Tensor& input,
const std::optional<Tensor>& atol_opt,
const std::optional<Tensor>& rtol_opt,
bool hermitian) {
// FIXME: Whenever we have a nice lstsq, we should dispatch this function to simply be
// `torch.lstsq(A, torch.eye(A.shape[-1]), atol=atol, rtol=rtol)`
// with a driver that supports singular inputs
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");
auto [atol, rtol] = get_atol_rtol(input, atol_opt, rtol_opt, "torch.linalg.pinv");
if (input.sym_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
// TODO: replace input.svd with linalg_svd when torch/xla can work with at::linalg_svd
auto [U, S, V] = input.svd();
return at::matmul(V * S.reciprocal().unsqueeze(-2), U.mH());
}
// If not Hermitian use singular value decomposition, else use eigenvalue decomposition
if (!hermitian) {
// TODO: replace input.svd with linalg_svd
// using linalg_svd breaks pytorch/xla, see https://github.com/pytorch/xla/issues/2755
auto [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 tol = at::max(atol.unsqueeze(-1), rtol.unsqueeze(-1) * max_val);
Tensor S_pseudoinv = at::where(S > tol, 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.mH());
} else {
auto [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 tol = at::max(atol.unsqueeze(-1), rtol.unsqueeze(-1) * max_val);
Tensor S_pseudoinv = at::where(S_abs > tol, 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.mH());
}
}
Tensor linalg_pinv(const Tensor& input, std::optional<double> atol, std::optional<double> rtol, bool hermitian) {
auto [atol_tensor, rtol_tensor] = get_atol_rtol(input, atol, rtol);
return at::linalg_pinv(input, atol_tensor, rtol_tensor, hermitian);
}
Tensor linalg_pinv(const Tensor& input, const Tensor& rcond, bool hermitian) {
// For NumPy compatibility the rcond argument is used as relative tolerance
checkNotComplexTolerance(rcond, "torch.linalg.pinv", "rcond");
auto options = input.options();
if (input.device().type() == kMetal || input.device().type() == kMPS) {
options = options.dtype(ScalarType::Float);
} else {
options = options.dtype(ScalarType::Double);
}
return at::linalg_pinv(input, at::zeros({}, options), rcond, hermitian);
}
Tensor linalg_pinv(const Tensor& input, double rcond, bool hermitian) {
// For NumPy compatibility the rcond argument is used as relative tolerance
return at::linalg_pinv(input, 0.0, rcond, hermitian);
}
// TODO: implement _out variant avoiding copy and using already allocated storage directly
Tensor& linalg_pinv_out(
const Tensor& input,
const std::optional<Tensor>& atol,
const std::optional<Tensor>& rtol,
bool hermitian,
Tensor& result) {
checkSameDevice("linalg.pinv", result, input);
checkLinalgCompatibleDtype("linalg.pinv", result, input);
Tensor result_tmp = at::linalg_pinv(input, atol, rtol, hermitian);
at::native::resize_output(result, result_tmp.sizes());
result.copy_(result_tmp);
return result;
}
Tensor& linalg_pinv_out(
const Tensor& input,
std::optional<double> atol,
std::optional<double> rtol,
bool hermitian,
Tensor& result) {
checkSameDevice("linalg.pinv", result, input);
checkLinalgCompatibleDtype("linalg.pinv", result, input);
Tensor result_tmp = at::linalg_pinv(input, atol, rtol, hermitian);
at::native::resize_output(result, result_tmp.sizes());
result.copy_(result_tmp);
return result;
}
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,
std::optional<Tensor> _out) {
NoTF32Guard disable_tf32;
auto out = _out.value_or(Tensor());
squareCheckInputs(self, "linalg.matrix_power");
if (_out.has_value()) {
checkSameDevice("matrix_power", out, self);
checkLinalgCompatibleDtype("matrix_power", out, self);
at::native::resize_output_symint(out, self.sym_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_symint(self.sym_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, std::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);
}
namespace {
// 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.
Tensor& matrix_rank_impl(
const Tensor& input,
const std::optional<Tensor>& atol_opt,
const std::optional<Tensor>& rtol_opt,
bool hermitian,
Tensor& result) {
auto [atol, rtol] = get_atol_rtol(input, atol_opt, rtol_opt, "torch.linalg.matrix_rank");
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);
checkNotComplexTolerance(atol, "torch.linalg.matrix_rank", "atol");
checkNotComplexTolerance(rtol, "torch.linalg.matrix_rank", "rtol");
// 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.sym_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.unsqueeze(-1) * max_S);
if (isTensorSubclassLike(input)) {
result = at::sum(S > tol, /*dim=*/-1);
return result;
}
result = at::sum_out(result, S > tol, /*dim=*/-1);
return result;
}
Tensor get_matrix_rank_result_tensor(const Tensor& input) {
// Matrices or batch of matrices are allowed
checkIsMatrix(input, "torch.linalg.matrix_rank", "input");
// For Composite Compliance, allocate `result` of correct shape to
// avoid resizing in `out` variant.
// See also `NOTE [matrix rank output shape]`
auto result_shape =
SymIntArrayRef(input.sym_sizes().cbegin(), input.sym_sizes().cend() - 2);
Tensor result =
at::empty_symint(result_shape, input.options().dtype(ScalarType::Long));
return result;
}
} // anonymous namespace
Tensor& linalg_matrix_rank_out(
const Tensor& input,
const std::optional<Tensor>& atol_opt,
const std::optional<Tensor>& rtol_opt,
bool hermitian,
Tensor& result) {
// Matrices or batch of matrices are allowed
checkIsMatrix(input, "torch.linalg.matrix_rank", "input");
auto result_shape =
IntArrayRef(input.sizes().cbegin(), input.sizes().cend() - 2);
at::native::resize_output(result, result_shape);
return matrix_rank_impl(input, atol_opt, rtol_opt, hermitian, result);
}
Tensor& linalg_matrix_rank_out(const Tensor& input, std::optional<double> atol, std::optional<double> rtol, bool hermitian, Tensor& result) {
auto [atol_tensor, rtol_tensor] = get_atol_rtol(input, atol, rtol);
result = linalg_matrix_rank_out(input, atol_tensor, rtol_tensor, hermitian, result);
return result;
}
Tensor linalg_matrix_rank(const Tensor& input, const std::optional<Tensor>& atol, const std::optional<Tensor>& rtol, bool hermitian) {
auto result = get_matrix_rank_result_tensor(input);
return matrix_rank_impl(input, atol, rtol, hermitian, result);
}
Tensor linalg_matrix_rank(const Tensor& input, std::optional<double> atol, std::optional<double> rtol, bool hermitian) {
auto result = get_matrix_rank_result_tensor(input);
auto [atol_tensor, rtol_tensor] = get_atol_rtol(input, atol, rtol);
return matrix_rank_impl(input, atol_tensor, rtol_tensor, hermitian, 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 = at::linalg_matrix_rank_outf(input, tol, rtol, hermitian, result);
return result;
}
Tensor& linalg_matrix_rank_out(const Tensor& input, double 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
result = at::linalg_matrix_rank_outf(input, tol, 0.0, hermitian, result);
return result;
}
Tensor linalg_matrix_rank(const Tensor& input, const Tensor& tol, bool hermitian) {
auto result = get_matrix_rank_result_tensor(input);
return matrix_rank_impl(input, tol, at::zeros({}, tol.options()), hermitian, result);
}
Tensor linalg_matrix_rank(const Tensor& input, double tol, bool hermitian) {
auto result = get_matrix_rank_result_tensor(input);
auto [atol_tensor, rtol_tensor] = get_atol_rtol(input, tol, 0.0);
return matrix_rank_impl(input, atol_tensor, rtol_tensor, hermitian, result);
}
// 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 together
* @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, std::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,