Svd

## Arguments

- self
(Tensor) the input tensor of size \((*, m, n)\) where

`*`

is zero or more batch dimensions consisting of \(m \times n\) matrices.- some
(bool, optional) controls the shape of returned

`U`

and`V`

- compute_uv
(bool, optional) option whether to compute

`U`

and`V`

or not

## 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 is returned in descending order.

The implementation of SVD on CPU uses the LAPACK routine `?gesdd`

(a divide-and-conquer
algorithm) instead of `?gesvd`

for speed. Analogously, the SVD on GPU uses the MAGMA routine
`gesdd`

as well.

Irrespective of the original strides, the returned matrix `U`

will be transposed, i.e. with strides `U.contiguous().transpose(-2, -1).stride()`

Extra care needs to be taken when backward through `U`

and `V`

outputs. Such operation is really only stable when `input`

is
full rank with all distinct singular values. Otherwise, `NaN`

can
appear as the gradients are not properly defined. Also, notice that
double backward will usually do an additional backward through `U`

and
`V`

even if the original backward is only on `S`

.

When `some`

= `FALSE`

, the gradients on `U[..., :, min(m, n):]`

and `V[..., :, min(m, n):]`

will be ignored in backward as those vectors
can be arbitrary bases of the subspaces.

When `compute_uv`

= `FALSE`

, backward cannot be performed since `U`

and `V`

from the forward pass is required for the backward operation.

## svd(input, some=TRUE, compute_uv=TRUE) -> (Tensor, Tensor, Tensor)

This function returns a namedtuple `(U, S, V)`

which is the singular value
decomposition of a input real matrix or batches of real matrices `input`

such that
\(input = U \times diag(S) \times V^T\).

If `some`

is `TRUE`

(default), the method returns the reduced singular value decomposition
i.e., 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`

matrices will be zero matrices
of shape \((m \times m)\) and \((n \times n)\) respectively. `some`

will be ignored here.

## Examples

```
if (torch_is_installed()) {
a = torch_randn(c(5, 3))
a
out = torch_svd(a)
u = out[[1]]
s = out[[2]]
v = out[[3]]
torch_dist(a, torch_mm(torch_mm(u, torch_diag(s)), v$t()))
a_big = torch_randn(c(7, 5, 3))
out = torch_svd(a_big)
u = out[[1]]
s = out[[2]]
v = out[[3]]
torch_dist(a_big, torch_matmul(torch_matmul(u, torch_diag_embed(s)), v$transpose(-2, -1)))
}
#> torch_tensor
#> 2.64772e-06
#> [ CPUFloatType{} ]
```