HyperShadow

A benchmark for one question: given a 3D point cloud, can you tell whether it is an ordinary 3D object or the 3D projection (the "shadow") of an object from a higher spatial dimension?

Most datasets that say "4D" mean 3D plus time. Here the extra dimensions are spatial: label 1 clouds are projections of objects living in R^4, R^5 or R^6 (hyperspheres, tesseracts, Clifford tori, duocylinders, hypertori, random smooth manifolds), rotated randomly in their ambient space before projection. Label 0 clouds are native 3D shapes given the same rotation and corruption treatment, so nothing in the preprocessing separates the classes.

Why this is an interesting task

A shadow is still at most 3-dimensional data, so intrinsic-dimension estimation cannot solve this. On this benchmark, TwoNN and the Levina-Bickel MLE reach about 71-73% accuracy. What actually separates the classes are the traces the projection map leaves behind: density folds, volumes filled with a particular radial profile, changed topology. A small PointNet (190k parameters) picks these up at 96.6%, and still detects shape families it was never trained on (79-91%).

The temporal track has the strongest result, and it needs no learning at all. A rigid 3D object in rigid motion can be aligned frame to frame by a rigid 3D transform with near-zero residual. The shadow of a rigidly rotating 4D object cannot: no rigid 3D motion explains it. The mean Kabsch alignment residual, one number per sequence, separates the classes at AUROC 0.982.

Files

file	contents
static.npz	points: float32 (10800, 1024, 3); labels: int64 (10800,)
static_meta.json	per-sample shape name, ambient dimension, corruption tier, projection type
temporal.npz	points: float32 (1800, 16, 512, 3), rigid-rotation sequences; labels
temporal_meta.json	per-sequence metadata

Labels: 0 = native 3D, 1 = projection of a higher-dimensional object. Classes are balanced across corruption tiers. All clouds are centred, scaled to unit mean radius, and resampled to a fixed point count.

Corruption tiers (cumulative): 0 clean; 1 Gaussian jitter (2% of scale); 2 plus random half-space occlusion; 3 plus heavier jitter and distance-dependent dropout.

Loading

import numpy as np

d = np.load("static.npz")
x, y = d["points"], d["labels"]     # (10800, 1024, 3), (10800,)

t = np.load("temporal.npz")
seq = t["points"]                    # (1800, 16, 512, 3)

Baseline numbers to beat

Method	Accuracy (static)
intrinsic dimension, best threshold	0.737 +/- 0.004
persistent homology + GBT	0.904
geometric features + GBT	0.956
PointNet-lite, 190k params	0.962 +/- 0.003

Uncertainty is the standard deviation over five random splits or training seeds. Temporal track: Kabsch rigidity residual threshold, AUROC 0.982, held-out accuracy 0.978 +/- 0.004.

Reproducing or extending

The generator is plain seeded NumPy, no GPU needed. Code, tests, and all baseline implementations are at https://github.com/AkshaySasi/hypershadow. Trained checkpoints are at https://huggingface.co/AkshaySasi/hypershadow-models. Regenerating with the published seeds reproduces these exact files. New shape families can be added with a single sampler function.

Limitations

Everything here is simulated, under specific choices of sampling measure, projection model and noise. Results on this data say nothing about physical reality. The intended uses are: benchmarking point-cloud models on a task where the ground truth is known and the shortcut routes have been closed off, studying out-of-distribution detection with a mathematically defined "out", and comparing machine performance against the human 4D-rigidity perception results reported by He, Bi and Zaidi (2023).

Citation

@misc{hypershadow2026,
  title  = {HyperShadow: A Benchmark for Detecting 3D Projections of
            Higher-Dimensional Spatial Objects},
  author = {Sasi, Akshay},
  year   = {2026},
}