HyperShadow: A Benchmark for Detecting 3D Projections of Higher-Dimensional Spatial Objects
This benchmark provides a controlled instrument for studying observable statistics that can certify incompatibility with a purely 3D explanation, relevant for researchers in geometry and machine learning.
HyperShadow introduces the first benchmark for detecting whether a 3D point cloud is a native 3D shape or the projection of a higher-dimensional (4-6D) rigid object. A 190k-parameter point network achieves 96.6% accuracy across corruption tiers, and a zero-parameter rigidity witness reaches AUROC 0.982 on a temporal track.
Machine-learning datasets labelled "4D" universally denote three spatial dimensions plus time. We introduce HyperShadow, the first public benchmark in which the fourth, fifth, and sixth dimensions are spatial: the task is to decide whether a 3D point cloud is a native three-dimensional shape or the projection, the "shadow", of a rigid object living in R^N (N = 4-6). We show this task is fundamentally distinct from intrinsic-dimension estimation: a shadow is still at-most-3-dimensional data, and standard estimators (TwoNN, Levina-Bickel MLE) reach only 71-73% accuracy. Detection instead requires projection signatures, density folds, filled volumes with characteristic radial profiles, and topology changes, which a 190k-parameter point network recovers at 96.6% accuracy across four corruption tiers, generalizing at 79-91% to object families never seen in training. On a temporal track of rigidly rotating objects we introduce a zero-parameter rigidity witness: the residual of the optimal rigid 3D alignment (Kabsch) between consecutive frames, which must vanish for any rigid 3D motion but cannot vanish for the shadow of a rigid rotation in R^N. This single interpretable statistic separates the classes at AUROC 0.982. All data are generated reproducibly from seeds; the dataset, models, and code are released publicly. HyperShadow makes no claim about physical reality; it is a controlled instrument for studying which observable statistics can certify incompatibility with a purely three-dimensional explanation.