Fix logsignatures to divide by window length about torchcde HOT 7 CLOSED

Provided the integral is integrated between [0, 1] (which it always should be), then where does this become an issue? The log-signature never needs to be normalised by the interval length.

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The integral isn't always integrated over [0, 1]. (Why should it always be?)

I think all that's necessary is removing the multiplication on this line:

from torchcde.

The log-ODE method is defined as:

Z_b = Z_a + int_0^1 f(Z)logsig_[a, b](X) ds

There is never any need to evaluate an integral with limits that are not [0, 1].

Perhaps I am misunderstanding, but if it is done in this way then you do not need to worry about dividing by b-a.

from torchcde.

What you've written is correct if a=0, b=1.
We can certainly reparameterise for that to be true each time... by dividing by b - a.

from torchcde.

Check definition A.8. in https://arxiv.org/pdf/2009.08295.pdf

The integral is over [0, 1] for any interval [a, b]. We divide by (b-a) if we integrate from a to b.

from torchcde.

Okay, flipping this around: integrating over [0, 1] (or rather [n, n+1] later on) is indeed what torchcde will be doing by default later on.

Over that region it should be just the logsignature, as you say. Therefore the multiplication I highlighted above should be removed?

To emphasise this: the tests for this check that the derivative on each piece is equal to the logsignature -- whilst using the optional t argument. (They predate the t argument being optional.) So the tests are verifying that over an interval determined by t (i.e. not of unit length), the vector field is the logsignature (i.e. without a b-a normalisation). Which is a mismatch.

Do you agree?

from torchcde.

Closed in version 0.2.0 with the introduction of logsig_windows.

from torchcde.