This question is related to research on Tishby's information bottleneck principle as seen here, the problem at hand is inherently an optimization problem as seen directly on page 7 section 3.2, my question concerns an added equality constraint where I am also looking at the mentioned optimization problem subject to the equation $ p(Y|X)=p(Y|T(X)) $ where T is a well defined well behaved function, and my question is if one cas say something nice about the problem statement with added constraint $ p(Y|X)=p(Y|T(X)) $ if T is nice enough or is of a nice enough family of transformations, maybe if it has a structure (topological or algebraic or any other) which reduces the problem at hand to something more approachable. I thought the problem might be easier to approach if it is restricted to discrete random variables on a Euclidean space, but maybe this is not necessary. I appreciate all help on this.
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