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If F_x, F_y, and F_z have significant shared computation, but also significant independent computation, you will have to disentangle the shared computation by hand if you want to calculate your divergence with maximum efficiency.

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How to calculate the divergence efficiently? … Assume f(x): R^d-> R^d. I could use autograd to get the derivative matrix (of size d x d) and then simply take the sum of the diagonals. But this is seems terribly inefficient and wasteful. There has to be a better way!

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If F_u, F_v, and F_w have almost all of their calculational work shared, very little efficiency will be lost by using one run of autograd’s backward() to calculate all three components (x, y, and z) of the gradient at the same time and discarding the off-diagonal elements. That is, you already had to do almost all of the calculational work needed for the off-diagonal elements, so the incremental cost of calculating them (and then not using them) is small.

The short answer is to use requires_grad = True on each of your d input variables one at at time, and calculate one derivative at a time for each of your d output variables. Then you sum the derivatives together.

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If you calculate the three needed partial derivatives independently (using requires_grad = True one at a time on each variable), you will be needlessly repeating the shared computation. But if you calculate the full gradient all at once (performing one backward() run with requires_grad = True set on all the input variables at the same time), you will be needlessly performing the off-diagonal pieces of the “independent” computation.

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Hope I’m not late. I think there are two ways, 1) Divergence = Trace(Jacobian). This is a trivial extension of the definition, however, computation of Jacobian using torch.autograd.functional.jacobian might be memory intensive. 2) You can use the torch.autograd.grad module for this.

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Note, in a typical multi-layer neural network, most of the computation that leads to the value of your output values will be shared, so, as a practical matter, I would use autograd to calculate the full gradient all at once, sum the diagonal elements to get the divergence, and discard the off-diagonal elements. I would be wasting a little bit of effort to calculate the off-diagonal elements, but, because most of the calculational cost was shared in the earlier, upstream layers of the neural network, the wasted work would likely be small.

I’m not aware of any easy way to do precisely what you want. The reason is that there is no automatic way to disentangle calculation that is shared by your d output variables.

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How to calculate the divergence efficiently? I’m not talking about a GAN divergence, but the actual divergence which is the sum of the partial derivative of all elements of a vector (Divergence - Wikipedia).

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In three-dimensional language, if your three output variables as functions of your three input variables are F_u (x, y, z), F_v (x, y, z), and F_w (x, y, z), and those three functions are completely unrelated to one another, then no efficiency is is lost by calculating them separately, and using three separate runs of autograd’s backward() to calculate d F_u / d x, d F_v / d u, and d F_w_ / d z separately.

Assume f(x): R^d-> R^d. I could use autograd to get the derivative matrix (of size d x d) and then simply take the sum of the diagonals. But this is seems terribly inefficient and wasteful. There has to be a better way!

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