Biological fitness arises from interactions between molecules, genes, and organisms. To discover the causative mechanisms of this complexity, we must differentiate the significant interactions from a large number of possibilities. Epistasis is the standard way to identify interactions in fitness landscapes. However, this intuitive approach breaks down in higher dimensions for example because the sign of epistasis takes on an arbitrary meaning, and the false discovery rate becomes high. These limitations make it difficult to evaluate the role of epistasis in higher dimensions. Here we develop epistatic filtrations, a dimensionally-normalized approach to define fitness landscape topography for higher dimensional spaces. We apply the method to higher-dimensional datasets from genetics and the gut microbiome. This reveals a sparse higher-order structure that often arises from lower-order. Despite sparsity, these higher-order effects carry significant effects on biological fitness and are consequential for ecology and evolution.
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