Researchers have modified principal component analysis to account for the curvature of data, not just its spread.
The method, called SHOPCA (Shape Operator-based Principal Component Analysis), folds geometric information into the covariance matrix that ordinary PCA relies on. It works by averaging the absolute local shape operators - a measure of how a data manifold curves - across the dataset, then using that average to steer principal components toward directions with both high variance and meaningful curvature. A single tuning parameter, alpha, controls how much curvature factors in: set it to zero and the method reduces to plain PCA, push it toward infinity and it leans almost entirely on curvature. The team also built an unsupervised way to pick alpha automatically, based on the gap between top and lower eigenvalues, so the method calibrates itself without labeled data.
Tested against PCA, ISOMAP, and UMAP on more than 50 real-world benchmark datasets, SHOPCA improved clustering scores over standard PCA consistently and beat UMAP specifically in small-sample settings. That distinction matters because UMAP's neighborhood-based manifold estimation tends to degrade when data is scarce - exactly the situation many real-world scientific and business datasets are stuck in.
It's not a wholesale replacement for UMAP or ISOMAP, just a tunable middle ground - and a single parameter beats wrestling with UMAP's fussier hyperparameters, even if that's a modest win.