There are some ∼1,100 known antimicrobial peptides (AMPs), which permeabilize microbial membranes but have diverse sequences. Here, we develop a support vector machine (SVM)-based classifier to investigate ⍺-helical AMPs and the interrelated nature of their functional commonality and sequence homology. SVM is used to search the undiscovered peptide sequence space and identify Pareto-optimal candidates that simultaneously maximize the distance σ from the SVM hyperplane (thus maximize its “antimicrobialness”) and its ⍺-helicity, but minimize mutational distance to known AMPs. By calibrating SVM machine learning results with killing assays and small-angle X-ray scattering (SAXS), we find that the SVM metric σ correlates not with a peptide’s minimum inhibitory concentration (MIC), but rather its ability to generate negative Gaussian membrane curvature. This surprising result provides a topological basis for membrane activity common to AMPs. Moreover, we highlight an important distinction between the maximal recognizability of a sequence to a trained AMP classifier (its ability to generate membrane curvature) and its maximal antimicrobial efficacy. As mutational distances are increased from known AMPs, we find AMP-like sequences that are increasingly difficult for nature to discover via simple mutation. Using the sequence map as a discovery tool, we find a unexpectedly diverse taxonomy of sequences that are just as membrane-active as known AMPs, but with a broad range of primary functions distinct from AMP functions, including endogenous neuropeptides, viral fusion proteins, topogenic peptides, and amyloids. The SVM classifier is useful as a general detector of membrane activity in peptide sequences.