I’m interested in characterizing functions which are “insensitive” to subsets of their input variables, especially in high-dimensional spaces.
There’s a field called “Analysis of boolean functions” (essentially Fourier analysis of functions f:{0,1}n→{0,1}) that seems relevant to this question and perhaps to your specific problem statement. In particular, the notion of “total influence” of a boolean function is meant to capture its sensitivity (e.g. the XOR function on all inputs has maximal total influence). This is the standard reference, see section 2.3 for total influence. Boolean functions with low influence (i.e. “insensitive” functions) are an important topic in this field, so I expect there are some relevant results (see e.g. tribes functions and the KKL theorem, though those specifically address a somewhat different question than your problem statement).
There’s a field called “Analysis of boolean functions” (essentially Fourier analysis of functions f:{0,1}n→{0,1}) that seems relevant to this question and perhaps to your specific problem statement. In particular, the notion of “total influence” of a boolean function is meant to capture its sensitivity (e.g. the XOR function on all inputs has maximal total influence). This is the standard reference, see section 2.3 for total influence. Boolean functions with low influence (i.e. “insensitive” functions) are an important topic in this field, so I expect there are some relevant results (see e.g. tribes functions and the KKL theorem, though those specifically address a somewhat different question than your problem statement).