Topological representations of data are inherently coarse summaries which endows them with certain desirable properties like stability but also potentially inhibits their discriminatory power relative to fine-scale learned features. In this work we present a novel framework for enriching the discriminatory power of topological representations based on random filters and capturing “interference topology” rather than direct topology. We show that our random filters outperform previously explored structured image filters while requiring orders of magnitudeless computational time. The approach is demonstrated on the MNIST dataset but is broadly applicable across data sets and modalities. This work is concluded with a discussion of the mathematical intuition underlying the approach and identification of future directions to enable deeper understanding and theoretical results.