svd(input, some=True, compute_uv=True, *, out=None) -> (Tensor, Tensor, Tensor)¶
Computes the singular value decomposition of either a matrix or batch of matrices
input. The singular value decomposition is represented as a namedtuple (U, S, V), such that
input= U diag(S) Vᴴ. where Vᴴ is the transpose of V for real inputs, and the conjugate transpose of V for complex inputs. If
inputis a batch of matrices, then U, S, and V are also batched with the same batch dimensions as
someis True (default), the method returns the reduced singular value decomposition. In this case, if the last two dimensions of
inputare m and n, then the returned U and V matrices will contain only min(n, m) orthonormal columns.
compute_uvis False, the returned U and V will be zero-filled matrices of shape (m, m) and (n, n) respectively, and the same device as
input. The argument
somehas no effect when
inputof float, double, cfloat and cdouble data types. The dtypes of U and V are the same as
input’s. S will always be real-valued, even if
U, S, V = torch.svd(A, some=some, compute_uv=True)(default) should be replaced with
U, S, Vh = torch.linalg.svd(A, full_matrices=not some) V = Vh.transpose(-2, -1).conj()
_, S, _ = torch.svd(A, some=some, compute_uv=False)should be replaced with
S = torch.svdvals(A)
someis the opposite of
full_matrices. Note that default value for both is True, so the default behavior is effectively the opposite.
The singular values are returned in descending order. If
inputis a batch of matrices, then the singular values of each matrix in the batch are returned in descending order.
The S tensor can only be used to compute gradients if
someis False, the gradients on U[…, :, min(m, n):] and V[…, :, min(m, n):] will be ignored in the backward pass, as those vectors can be arbitrary bases of the corresponding subspaces.
The implementation of
torch.linalg.svd()on CPU uses LAPACK’s routine ?gesdd (a divide-and-conquer algorithm) instead of ?gesvd for speed. Analogously, on GPU, it uses cuSOLVER’s routines gesvdj and gesvdjBatched on CUDA 10.1.243 and later, and MAGMA’s routine gesdd on earlier versions of CUDA.
The returned U will not be contiguous. The matrix (or batch of matrices) will be represented as a column-major matrix (i.e. Fortran-contiguous).
The gradients with respect to U and V will only be finite when the input does not have zero nor repeated singular values.
If the distance between any two singular values is close to zero, the gradients with respect to U and V will be numerically unstable, as they depends on . The same happens when the matrix has small singular values, as these gradients also depend on S⁻¹.
inputthe singular value decomposition is not unique, as U and V may be multiplied by an arbitrary phase factor on every column. The same happens when
inputhas repeated singular values, where one may multiply the columns of the spanning subspace in U and V by a rotation matrix and the resulting vectors will span the same subspace. Different platforms, like NumPy, or inputs on different device types, may produce different U and V tensors.
input (Tensor) – the input tensor of size (*, m, n) where * is zero or more batch dimensions consisting of (m, n) matrices.
some (bool, optional) – controls whether to compute the reduced or full decomposition, and consequently, the shape of returned U and V. Default: True.
compute_uv (bool, optional) – controls whether to compute U and V. Default: True.
- Keyword Arguments
out (tuple, optional) – the output tuple of tensors
>>> a = torch.randn(5, 3) >>> a tensor([[ 0.2364, -0.7752, 0.6372], [ 1.7201, 0.7394, -0.0504], [-0.3371, -1.0584, 0.5296], [ 0.3550, -0.4022, 1.5569], [ 0.2445, -0.0158, 1.1414]]) >>> u, s, v = torch.svd(a) >>> u tensor([[ 0.4027, 0.0287, 0.5434], [-0.1946, 0.8833, 0.3679], [ 0.4296, -0.2890, 0.5261], [ 0.6604, 0.2717, -0.2618], [ 0.4234, 0.2481, -0.4733]]) >>> s tensor([2.3289, 2.0315, 0.7806]) >>> v tensor([[-0.0199, 0.8766, 0.4809], [-0.5080, 0.4054, -0.7600], [ 0.8611, 0.2594, -0.4373]]) >>> torch.dist(a, torch.mm(torch.mm(u, torch.diag(s)), v.t())) tensor(8.6531e-07) >>> a_big = torch.randn(7, 5, 3) >>> u, s, v = torch.svd(a_big) >>> torch.dist(a_big, torch.matmul(torch.matmul(u, torch.diag_embed(s)), v.transpose(-2, -1))) tensor(2.6503e-06)