# L_inf norm equivalent of L_2 norm. in higher dimensions

Hi all,

I am sorry, my question is not related with Pytorch at all, but I would be glad if somebody could help on my problem.

I am working on a research subject where I need to implement different adversarial attack types on MNIST data.

I use max perturbation limit “epsilon” under L_infinity norm. This L_inf norm is limiting the max amount of allowed perturbation to each pixel of 28x28 image using a chosen epsilon parameter.

For example: I use epsilon value of 0.15 with L_inf norm metric, use PGD attack type and get an attack success rate around %60 for all MNIST test data.

But I am not sure which epsilon I need to chose when I need to implement the same attack under L2_norm metric. ( to simulate the same volume of perturbation)

I know that in higher dimensions the volume of L2 norm ball is smaller than the L_inf norm cube. And for the same amount of perturbation ( to make the volumes of L_2 ball equal to L_inf cube), I need to use a bigger epsilon value for L_2 norm.

For example, if I chose an epsilon of 0.1 for L_inf norm, then may be I need to chose an epsilon value of around 1.3 for L_2 norm.

But I don’t know the exact conversion formula for the correct epsilon from L_inf norm to L2_norm.

I would be very happy if somebody could help me on this.

I still couldn’t figure out how to calculate necessary l2 norm perturbation epsilon_1 value which will result in the same volume of perturbation for l’inf norm under epsilon_2 value.

I need to calculate epsilon_1 using epsilon_2 and the image dimensions. Can somebody help me to find out the conversion formula?