Predictive Affinity Learning for Laplacian Eigenmaps

In many scientific studies, reducing high-dimensional measurements to a few coordinates simplifies prediction and visualization. Laplacian Eigenmaps construct coordinates from a neighborhood graph, but nuisance features can distort its distances and the standard eigenmap loss can favor almost disconnected graphs. In this article, we study how to learn a feature-weighted distance for Laplacian Eigenmaps using a predictive criterion. We propose predictive affinity learning, which predicts excluded features from neighbor averages and constructs spectral coordinates from the selected features. Theoretical results show that, under the stated identification and regularity conditions, the method removes independent nuisance features and its spectral representation converges to that of the population predictive distance, even when the same observations estimate the distance and construct the graph. The selected features persist under small departures from independence when the population criterion strictly separates the selected and excluded coordinates. Through simulations and multiview data analyses, we identify settings where the learned distance improves neighborhood recovery and prediction over isotropic eigenmaps.