# Derivative of covariance matrix

**URL:** <https://discuss.pytorch.org/t/derivative-of-covariance-matrix/109798>\
**Category:** autograd\
**Created:** [January 23, 2021, 6:58pm UTC](https://discuss.pytorch.org/t/derivative-of-covariance-matrix/109798 "2021-01-23T18:58:36Z")\
**Posts on this page:** 2\
**Page:** 1

<div class="post-metadata">

**Author:** ![Mr\_Cool](https://discuss.pytorch.org/letter_avatar_proxy/v4/letter/m/e99b99/32.png) [@Mr\_Cool](https://discuss.pytorch.org/u/Mr_Cool)\
**Post date:** [January 23, 2021, 6:58pm UTC](https://discuss.pytorch.org/t/derivative-of-covariance-matrix/109798/1 "2021-01-23T18:58:36Z")

</div>

Suppose I have a bivariate covariance function `c(x,y)` and at input `[x1,...,xn]` and `[y1,...,ym]` and I would like to compute the covariance matrix with `i,j`th element `d/dx c(xi,yj)`. Here I have assumed that each `xi` and `yj` are scalar valued, however this could easily be extended to finding the `k`th partial derivative of the covariance matrix. In general, I would like this to be extended to higher derivatives too (e.g. I would like to compute the matrix with `i,j`th element `d/dx d/dy c(xi,yj)`), but this may be an easy generalisation of the first issue.

I feel that PyTorch’s automatic differentiation system should be able to handle this, but I haven’t been able to come up with any nice solution. I think the issue is to do with broadcasting derivatives. For instance, to compute the non-differentiated covariance matrix, I would do something like `c(x.unsqueeze(1),y)`, but the following code doesn’t work as I would like (using `autograd.grad`):

```auto
X = torch.Tensor([1,2,3]).requires_grad_(True).unsqueeze(1)
Y = torch.Tensor([4,5]).requires_grad_(True)
cov_matrix = c(X,Y)
dx_cov_matrix = grad(cov_matrix, [X], grad_outputs=[torch.ones([X.size(0),Y.size(0)])], create_graph=True)

```

Any help would be appreciated! Thanks

---

<div class="post-metadata">

**Author:** ![pascal\_notsawo](https://discuss.pytorch.org/user_avatar/discuss.pytorch.org/pascal_notsawo/32/22829_2.png) [@pascal\_notsawo](https://discuss.pytorch.org/u/pascal_notsawo)\
**Post date:** [January 24, 2021, 7:53am UTC](https://discuss.pytorch.org/t/derivative-of-covariance-matrix/109798/2 "2021-01-24T07:53:02Z")

</div>

Here is an implementation of the covariance with pytorch from [covariance-and-gradient-support](https://discuss.pytorch.org/t/covariance-and-gradient-support/16217/2)

```auto
import torch

def cov(m, rowvar=False, inplace=False):
    '''Estimate a covariance matrix given data.

    Thanks :
    - https://discuss.pytorch.org/t/covariance-and-gradient-support/16217/2
    - https://github.com/numpy/numpy/blob/master/numpy/lib/function_base.py#L2276-L2494

    Covariance indicates the level to which two variables vary together.
    If we examine N-dimensional samples, `X = [x_1, x_2, ... x_N]^T`,
    then the covariance matrix element `C_{ij}` is the covariance of
    `x_i` and `x_j`. The element `C_{ii}` is the variance of `x_i`.

    Args:
        m: A 1-D or 2-D array containing multiple variables and observations.
            Each row of `m` represents a variable, and each column a single
            observation of all those variables.
        rowvar: If `rowvar` is True, then each row represents a
            variable, with observations in the columns. Otherwise, the
            relationship is transposed: each column represents a variable,
            while the rows contain observations.

    Returns:
        The covariance matrix of the variables.
    '''
    if m.dim() > 2:
        raise ValueError('m has more than 2 dimensions')
    if m.dim() < 2:
        m = m.view(1, -1)
    if not rowvar and m.size(0) != 1:
        m = m.t()
    m = m.type(torch.double)  
    fact = 1.0 / (m.size(1) - 1)
    if inplace:
        m -= torch.mean(m, dim=1, keepdim=True)
    else :
        m = m - torch.mean(m, dim=1, keepdim=True)
    mt = m.t() # if complex: mt = m.t().conj()
    return fact * m.matmul(mt).squeeze()

```

You can also calculate the gradient as follows.

```auto
def gradient(y, x, grad_outputs=None):
    """Compute dy/dx @ grad_outputs"""
    if grad_outputs is None:
        grad_outputs = torch.ones_like(y)
    grad = torch.autograd.grad(y, [x], grad_outputs=grad_outputs, create_graph=True)[0]
    return grad

```

Now let’s take an example.

```auto
x, y = [90, 60, 90], [90, 90, 30]
A = torch.tensor([x, y], dtype = torch.double, requires_grad=True)
c = cov(m = A, rowvar = True)

```

> tensor([[300., -300.], [-300., 1200.]], dtype=torch.float64, grad\_fn=\<MulBackward0\>)

You can now calculate the gradient of `c` w.r.t to `A`, like this.

```auto
gradient(y = c, x = A)

```

> tensor([[30., 0., -30.], [30., 0., -30.]], dtype=torch.float64, grad\_fn=\<AddBackward0\>)

But I guess this is not the behaviour you want, because `torch.autograd.grad` makes dimension reductions for multi-dimensional parameters.

So you can calculate `dc(xi, yj)/dx or dc(xi, yj)/dy` like this :

```auto
grad_cij = gradient(y = c[i][j], x = A)

dc(xi, yj)/dx = grad_cij[0]

dc(xi, yj)/dy = grad_cij[1]

```

```auto
i = 0
j = 0
dcij_dA = gradient(y = c[i][j], x = A)

```

> tensor([[10., -20., 10.], [0., 0., 0.]], dtype=torch.float64, grad\_fn=)

Since `c[0][0]` is independent of `y`, `dcij_dA[1]` is null. Similarly, you can see that the derivative of `c[1][1]` with respect to `x` is also null.

```auto
i = 1
j = 1
dcij_dA = gradient(y = c[i][j], x = A)

```

> tensor([[0., 0., 0.], [20., 20., -40.]], dtype=torch.float64, grad\_fn=\<AddBackward0\>)

You can also notice that `c[1][0] = c[0][1]`, all having non-zero derivatives with respect to `x` and `y`.

```auto
i = 0
j = 1
dcij_dA = gradient(y = c[i][j], x = A)

```

> tensor([[10., 10., -20.], [5., -10., 5.]], dtype=torch.float64, grad\_fn=\<AddBackward0\>)

```auto
i = 1
j = 0
dcij_dA = gradient(y = c[i][j], x = A)
dcij_dA

```

> tensor([[10., 10., -20.], [5., -10., 5.]], dtype=torch.float64, grad\_fn=\<AddBackward0\>)

You can also calculate everything like this :

```auto
n, m = c.shape
dc_dA = torch.stack([gradient(y = c[i][j], x = A) for j in range(m) for i in range(n)])

```