I want to find a smooth function, say $f$ that condenses partitions and I want to find it by optimization process. For example, let's say we want $f: D \times \mathbb{R}^k \to \mathbb{R}^k$ to condense $D \times p_j(\mathbf{1})$ ($j=1,\dots,k$) where $D \subseteq \mathbb{R}^d$ is a compact set, $p_j$ is the $j$-th element projection, and $\mathbf{1} \in \mathbb{R}^k$ is a vector with all ones. Then one optimization criterion able to do it is to minimize the sample variances on each $D \times p_j(\mathbf{1})$; i.e., $$ \min_f\sum_j \text{Var}_{x \sim \mu_j} (f(x)) $$ where $\mu_j$ is a uniform distribution on $D \times p_j(\mathbf{1})$. ($f$ might be modeled as a multilayer perceptron or something. Of cousre we need some constraints not to get a trivial solution.)
I am sorry but I do not know a proper term for this type of 'condensation'. Basically, what I want to find is a function that maps near things to nearer points.
As you see in the above, such a variance minimization criterion is highly prone to overfitting; gets a very trivial solution such as a constant function or something. So would there be any better idea??