torch.svd

torch.svd(input, some=True, compute_uv=True, *, out=None)

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 $= U \text{diag}(S) V^{\text{H}}$. where $V^{\text{H}}$ is the transpose of V for real inputs, and the conjugate transpose of V for complex inputs. If input is a batch of matrices, then U, S, and V are also batched with the same batch dimensions as input.

If some is True (default), the method returns the reduced singular value decomposition. In this case, if the last two dimensions of input are m and n, then the returned U and V matrices will contain only min(n, m) orthonormal columns.

If compute_uv is 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 some has no effect when compute_uv is False.

Supports input of 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 input is complex.

Warning

torch.svd() is deprecated in favor of torch.linalg.svd() and will be removed in a future PyTorch release.

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.mH

_, S, _ = torch.svd(A, some=some, compute_uv=False) should be replaced with

S = torch.linalg.svdvals(A)

Note

Differences with torch.linalg.svd():

• some is the opposite of torch.linalg.svd()’s full_matrices. Note that default value for both is True, so the default behavior is effectively the opposite.

• torch.svd() returns V, whereas torch.linalg.svd() returns Vh, that is, $V^{\text{H}}$.

• If compute_uv is False, torch.svd() returns zero-filled tensors for U and Vh, whereas torch.linalg.svd() returns empty tensors.

Note

The singular values are returned in descending order. If input is a batch of matrices, then the singular values of each matrix in the batch are returned in descending order.

Note

The S tensor can only be used to compute gradients if compute_uv is True.

Note

When some is 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.

Note

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.

Note

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).

Warning

The gradients with respect to U and V will only be finite when the input does not have zero nor repeated singular values.

Warning

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 $\frac{1}{\min_{i \neq j} \sigma_i^2 - \sigma_j^2}$. The same happens when the matrix has small singular values, as these gradients also depend on S⁻¹.

Warning

For complex-valued input the singular value decomposition is not unique, as U and V may be multiplied by an arbitrary phase factor $e^{i \phi}$ on every column. The same happens when input has 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.

Parameters:

• 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

Example:

>>> 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.mT))
tensor(2.6503e-06)