Higher Derivatives

[altaetran]

Hey,

I've been trying to input higher order derivatives for a while using GaussianProcess.add_data(), but I am really confused as to how one should input gradients or hessians this way since the documentation says that y should have 1 non trivial dimension, which is not possible for say, a list of gradients. Thanks!

Best,

Han

[altaetran]

Specifically, I am curious about Hessians. Perhaps we can write a wrapper function that lets the user pass in a list of higher order derivatives (e.g. list of numpy vectors or matrices for gradients and hessians), because this is more intuitive to me than the 'n' input parameter. So for instance, I would expect n=1 to allow me to pass in a gradient, rather than to have me pass each component of the gradient to different calls of add_data. I guess I am still kind of confused. Thanks!

[markchil]

Han:

Thank you for the input. Yes, a wrapper like that would be convenient, I’ll have to think about the best way to implement that.

In the meantime, the “n” input is designed to let you make as general of a specification of inputs as possible. Essentially, it is asking you to unravel the Hessian. So, say you are working with a 2d process and you have the Hessian H=[[1, 2], [2, 3]] at point x = [0, 1]. You then unravel H and you probably want to drop the redundant element(s). So, your inputs would be:

y = [1, 2, 3]
X = [[0, 1], [0, 1], [0, 1]]
n = [[2, 0], [1, 1], [0, 2]]

You can then add these all at once with gp.add_data(X, y, n=n)

-Mark

On Aug 15, 2014, at 7:05, altaetran notifications@github.com wrote:

Specifically, I am curious about Hessians. Perhaps we can write a wrapper function that lets the user pass in a list of higher order derivatives (e.g. list of numpy vectors or matrices for gradients and hessians), because this is more intuitive to me than the 'n' input parameter. So for instance, I would expect n=1 to allow me to pass in a gradient, rather than to have me pass each component of the gradient to different calls of add_data. I guess I am still kind of confused. Thanks!

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[altaetran]

Ahh, that makes more sense. I foresee that it will be really helpful to have this wrapper function. I would it to take inputs of the following form:
X : (num_points x num_dims array, like in the case without derivatives),
grad_list : (list of gradient vectors if n = 1, or list of hessians if n = 2)

The way to keep the original functionality and add this functionality is to check whether or not n is an array or a positive integer because if we're in multiple dimensions, then n cannot be an positive integer in the original implementation. That will keep backward compatibility, and also allow for a very natural input of gradients and hessians (nothing above that though, which are rarely used in multiple dimensions anyway). I can write the code for this, since I will be doing it for my work anyway. Thanks!

[altaetran]

Ok, I've finished with the wrapper function. I can submit a pull request in a little bit.

[nxtruong]

@markchil You should include the above example in the documentation of your code. When I read the docs, it's really confusing how to add derivatives for multi-dimensional inputs. Your demo code doesn't help as the input is scalar. The above example is very helpful for users of your code.