KL-divergence between two multivariate gaussian

Rojin · August 15, 2019, 3:48pm

I have two multivariate Gaussian distributions that I would like to calculate the kl divergence between them. each is defined with a vector of mu and a vector of variance (similar to VAE mu and sigma layer). What is the best way to calculate the KL between the two? Is this even doable? because I do not have the covariance matrix. If that is not doable, what if I take samples from both distributions and calculate the KL between the samplings? is that the right way of doing this? is there any build-in function for any of these calculations?

Nikronic · August 9, 2019, 8:19pm

Hi,

Yes, this is the correct approach.
Just be aware that the input a must should contain log-probabilities and the target b should contain probability.

github.com

pytorch/pytorch/blob/35fed93b1ef05175143f883c6f89f06c6dd9429b/aten/src/ATen/native/Loss.cpp#L71


  return apply_loss_reduction(output, reduction);
}


Tensor margin_ranking_loss(const Tensor& input1, const Tensor& input2, const Tensor& target, double margin, int64_t reduction) {
  auto output =  (-target * (input1 - input2) + margin).clamp_min_(0);
  return apply_loss_reduction(output, reduction);
}


Tensor kl_div(const Tensor& input, const Tensor& target, int64_t reduction) {
  auto zeros = at::zeros_like(target);
  auto output_pos = target * (at::log(target) - input);
  auto output = at::where(target > 0, output_pos, zeros);
  return apply_loss_reduction(output, reduction);
}


Tensor kl_div_backward_cpu(const Tensor& grad, const Tensor& input, const Tensor& target, int64_t reduction) {
  auto grad_input = at::zeros_like(input);
  auto grad_expand = grad.expand_as(input);
  AT_DISPATCH_FLOATING_TYPES(input.scalar_type(), "kl_div_backward_cpu", [&]() {
    at::CPU_tensor_apply3<scalar_t, scalar_t, scalar_t>(
        grad_input,

https://pytorch.org/docs/stable/nn.functional.html?highlight=kl_div#kl-div

By the way, PyTorch use this approach:

https://pytorch.org/docs/stable/distributions.html?highlight=kl_div#torch.distributions.kl.kl_divergence

Good luck
Nik

Rojin · August 13, 2019, 11:52pm

Thanks Nick for your input. I should restate my question. I have two multi-variate distributions each defined with “n” mu and sigma. Now I would like to do a KL between these two. Any idea how that can be done?

zhl515 · August 14, 2019, 6:42am

You can sample x1 and x2 from 𝑝1(𝑥|𝜇1,Σ1) and 𝑝2(𝑥|𝜇2,Σ2) respectively, then compute KL divergence using torch.nn.functional.kl_div(x1, x2).

Rojin · August 14, 2019, 4:33pm

As @Nikronic mentioned the kl_div requires a to be log-probabilities and the target b to be probability. correct me if I’m wrong, but I think sampling from the two distributions is not going to give log-probabilities and probabilities

Rojin · August 16, 2019, 12:02am

any suggestion? I don’t think that what you said can be used for sampling layer

Nikronic · August 16, 2019, 12:14am

@Rojin
Actually, I have never used distribution class but based on docs, in this situation you just need to pass them directly to kl_div from distribution if the aforementioned distributions are obtained from Distribution subclasses or you have registered your own.

Please refer to this links for more information:

well, you know, my name is Nik, somehow abbreviated of Nikan (a Persian name)

Bests

Rojin · August 16, 2019, 12:47am

Hey Nikan!

Yeah I realized that from your last name!

Sorry if I’m restating my question ( I modified it a couple of times at the top lol) and probably repeating your answer, I’m kinda new to the KL implementation in pytorch, so I really appreciate your help.

My question is :
I have two VAEs, each with a mu and standard deviation layers. Now I would like to calculate the KL between these two VAEs using either the mu and sds, or the sampling layers. This sounds to me like a multivariate gaussian KL divergence problem, so I looked at the formula and I noticed that I actually need the covariance matrix of q (if we assume that KL(p||q) ). And as far as I know there is no way to calculate the covariance in this case, am I correct?

@zhl515 suggested to directly calculate the KL between the two sampling layers as you said before using the bellow function, but as you said this requires probabilities which I do not have, so I think the following function cannot be used on the sampling layers, is that right? :

 torch.nn.functional.kl_div(p, q).

And based on your last comment, you are suggesting to register the distribution, and then use

torch.distributions.kl.kl_divergence(p, q)

The only problem is that in order to register the distribution I need to have the covariance matrix, and I can’t obtain that because I only have mu and std. It makes me think whether that is a possible thing to do at all in neural network. In the VAE paper they don’t have that problem because the are assuming that q has mean 0 and covariance I

Thank you!

zhl515 · August 16, 2019, 7:07am

You said you can’t obtain covariance matrix. In VAE paper, the author assume the true (but intractable) posterior takes on a approximate Gaussian form with an approximately diagonal covariance. So just place the std on diagonal of convariance matrix, and other elements of matrix are zeros.

zhl515 · August 16, 2019, 7:16am

Now you can compute KL-divargence of two multivariate Gaussians directly from the below formula:
1565939371(1)

Rojin · August 16, 2019, 7:31am

Thank you that solves the problem