# Quick example¶

With NNGeometry, you can easily manipulate \(d \times d\) matrices and \(d\) vectors where \(d\) is the number of parameter of your neural network, for modern neural networks where \(d\) can be as big as \(10^8\). These matrices include for instance:

- The
Fisher Information Matrix(FIM) used in statistics, in the natural gradient algorithm, or as an approximate of the Hessian matrix in some applications.Posterior covariancesin Bayesian Deep Learning.

You can also compute finite *tangent kernels*.

A naive computation of the FIM would require storing \(d \times d\) scalars in memory. This is prohibitively large for modern neural network architectures, and a line of research has focused at finding lower memory intensive approximations specific to neural networks, such as KFAC, EKFAC, low-rank approximations, etc. This library proposes a common interface for manipulating these different approximations, called *representations*.

Let us now illustrate this by computing the FIM using the KFAC representation.

```
>>> F_kfac = FIM(model=model,
loader=loader,
representation=PMatKFAC,
n_output=10,
variant='classif_logits',
device='cuda')
>>> print(F_kfac.trace())
```

Computing the FIM requires the following arguments:

- The
`torch.nn.Module`

`model`

object is the PyTorch model used as our neural network.- The
`torch.utils.data.DataLoader`

`loader`

object is the dataloader that contains examples used for computing the FIM.- The
`object.PMatKFAC`

`PMatKFAC`

argument specifies which representation to use in order to store the FIM.We will next define a vector in parameter space, by using the current value given by our model:

>>> v = PVector.from_model(model)We can now compute the matrix-vector product \(F v\) by simply calling:

>>> Fv = F_kfac.mv(v)Note that switching from the

`object.PMatKFAC`

representation to any other representation such as`object.PMatDense`

is as simple as passing`representation=PMatDense`

when building the`F_kfac`

object.More notebook examples can be found at https://github.com/tfjgeorge/nngeometry/tree/master/examples