Accurate binding affinity prediction is crucial to structure-based drug design. Recent work used computational topology to obtain an effective representation of protein-ligand interactions. While algorithms using algebraic topology have proven useful in predicting properties of biomolecules, previous algorithms employed uninterpretable machine learning models which failed to explain the underlying geometric and topological features that drive accurate binding affinity prediction. Moreover, they had high computational complexity which made them intractable for large proteins. We present the fastest known algorithm to compute persistent homology features for protein-ligand complexes using opposition distance, with a runtime that is independent of the protein size. Then, we exploit these features in a novel, interpretable algorithm to predict protein-ligand binding affinity. Our algorithm achieves interpretability through an effective embedding of distances across bipartite matchings of the protein and ligand atoms into real-valued functions by summing Gaussians centered at features constructed by persistent homology. We name these functions internuclear persistent contours (IPCs) . Next, we introduce persistence fingerprints , a vector with 10 components that sketches the distances of different bipartite matching between protein and ligand atoms, refined from IPCs. Let the number of protein atoms in the protein-ligand complex be n , number of ligand atoms be m , and ω ≈ 2.4 be the matrix multiplication exponent. We show that for any 0 < ε < 1, after an 𝒪 ( mn log( mn )) preprocessing procedure, we can compute an ε -accurate approximation to the persistence fingerprint in 𝒪 ( m log 6 ω ( m/ε )) time, independent of protein size. This is an improvement in time complexity by a factor of 𝒪 (( m + n ) 3 ) over any previous binding affinity prediction that uses persistent homology. We show that the representational power of persistence fingerprint generalizes to protein-ligand binding datasets beyond the training dataset. Then, we introduce PATH , Predicting Affinity Through Homology, a two-part algorithm consisting of PATH + and PATH - . PATH + is an interpretable, small ensemble of shallow regression trees for binding affinity prediction from persistence fingerprints. We show that despite using 1,400-fold fewer features, PATH + has comparable performance to a previous state-of-the-art binding affinity prediction algorithm that uses persistent homology. Moreover, PATH + has the advantage of being interpretable. We visualize the features captured by persistence fingerprint for variant HIV-1 protease complexes and show that persistence fingerprint captures binding-relevant structural mutations. PATH - , in turn, uses regression trees over IPCs to differentiate between binding and decoy complexes. Finally, we benchmarked PATH versus established binding affinity prediction algorithms spanning physics-based, knowledge-based, and deep learning methods, revealing that PATH has comparable or better performance with less overfitting, compared to these state-of-the-art methods. The source code for PATH is released open-source as part of the osprey protein design software package.