Topological Alternatives for Precision and Recall in Generative Models

We introduce the Normalized Topological Divergence (NTD), a fully differentiable metric that simultaneously quantifies fidelity and diversity of generative models directly in raw pixel or spectrogram space, eliminating reliance on pretrained feature extractors. For two empirical distributions P (model) and Q (reference), NTD builds a Vietoris–Rips filtration over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$P \cup Q$ </tex-math></inline-formula> where distance matrix within Q is equal to 0. Extensive experiments on six vision and audio benchmarks: ImageNet-1k, CIFAR-10, MNIST, AFHQv2, AFHQ-Cat, LJSpeech-1, and Gaussian mixtures covering ten generator families show that NTD exposes blur, mode collapse, variance inflation and other generation artifacts. As a result, our metrics are domain-agnostic, provide a precision-recall trade-off, not offered by FID. It represent difference in variance better than density-coverage, TopP&R and P-Precision (Recall) while indicating problems of VAE-like generation more effectively. Moreover, it can be directly optimized by gradient descent algorithms.