# Find inputs that give highest variance in output space

I have a relatively simple problem that I think should have a known solution, but that I can't figure out. Any help would be greatly appreciated.

Basically, I have a function $$f : \mathbb{R}^d \rightarrow \mathbb{R}^p$$ which has vector input of features and vector output of images. Given a training set of combinations of the input variables and their corresponding outputs $$\{x_i, y_i\}_{i=1}^N$$, I would like to find the input feature (i.e. element of the inputs) that leads to the highest variability in the outputs, i.e. highest pixel-wise variance. Broadly speaking, I'm hoping to quantify the relationship between each feature and its effect on the variance of the output, to get an idea of how "sensitive" that feature is.

Methods like PCA wouldn't work because I am working in a supervised setting. Whereas PCA tries to find the directions of maximum variance of an unsupervised dataset, I am trying, in a sense, to find the directions of maximum variance in the label (output) space as it relates to the input space. Does an answer to this exist? Thanks in advance.

I propose you directly learn the relation between input and the variance of output images. For simplicity, I assume you target the pixle-wise variance within an output image. Then your training data would become $$\{x_i, Var(y_i)\}_{i=1}^N$$ instead of $$\{x_i, y_i\}_{i=1}^N$$!