Beyond Feature Subspaces: Query-Conditioned Projection Learning for Outlier Interpretation

Most outlier interpretation methods answer why a point is anomalous by returning a feature subspace: an axis-aligned subset $S$ of the $d$ observed coordinates in which the query stands out. We show that this format can be expressively inadequate, but in a way that depends on dimensionality and on the search budget in a manner no method can infer a priori. Enumerating every non-empty coordinate subspace of all $47$ ADBench Classical benchmarks, we find that $94.9\%$ of outliers admit no subspace that isolates them from every inlier when $d\le 6$, where only $31$–$63$ candidates exist, yet $0\%$ when $13\le d\le 20$, where exhaustive search succeeds. Beyond $d\approx 20$ exhaustive search is impossible (MNIST: $2^{512}-1$ subsets), and under the cardinality budget $|S|\le 3$ that any practical search must adopt, $56.0\%$ of outliers remain unseparated for $21\le d\le 40$. Pooled over all $47$ benchmarks ($33,170$ outliers) the rate is $47.7\%$ under the strict criterion and $16.7\%$ under a $99$th-percentile reference. Where the search is exhaustive the failures cannot be attributed to pruning or heuristics. We therefore study TMQ, whose parameters are re-estimated for that query alone, so the contrast may be a weighted feature combination rather than a coordinate subset. Sampled quadruplets (query, other outlier, near inlier, far inlier) are embedded; a global attention branch and four leave-one-position-out local branches estimate latent contributions; and three margin constraints organize inlier–inlier, query–outlier and query–inlier distances. We characterize the exact zero-loss distance interval and the subtype condition under which its inter-outlier lower bound is useful. Extensive experiments across twelve benchmarks, nine baselines and seven detector references show that TMQ attains the best pooled mean on six of seven metrics.