ERank in Latent Space as an Image-Complexity and Richness Measure

Deep networks trained with label noise often learn clean structure before memorizing corrupted labels. We show that this transition leaves a spectral signature in the centered scatter of per-example last-layer gradients. Its effective rank transiently expands during memorization and contracts after corrupted labels are fit. We call this phenomenon Fisher Rank Inflation. Corrupted labels increase effective rank by injecting spectral mass into low-energy or previously unused eigendirections, increasing the entropy of the gradient spectrum. We derive a first-order leave-one-out attribution formula, identify conditions under which corrupted examples contribute more strongly than clean examples, and explain why attribution signals weaken once the normalized Fisher-gradient spectrum stabilizes. We test these predictions on CIFAR-10, CIFAR-100, and CIFAR-10N using SmallCNN, ResNet18, and Vision Transformers. Across settings, Fisher effective rank exhibits a consistent inflation--collapse trajectory aligned with memorization. At peak-rank checkpoints, corrupted examples are enriched among the highest rank-contributing samples, with top-100 noisy fractions from $69.2\%$ to $96.2\%$ across five-seed synthetic-corruption experiments and $94.4\%\pm1.9\%$ on CIFAR-10N. First-order spectral attribution closely matches exact leave-one-out contributions in convolutional models and remains enriched in the Vision Transformer. Peak effective rank increases monotonically with corruption severity, from $28.88\pm1.95$ under clean training to $97.09\pm1.78$ at $60\%$ corruption. In several settings, the retrospectively identified onset of rank inflation precedes observable test degradation. These results establish Fisher Rank Inflation as a spectral signature connecting corrupted-example enrichment, corruption severity, and the transition from structure learning to memorization.

Image quality in modern imaging systems emerges from the coupled effects of the sensor, optics, and computational reconstruction. Ultra-thin metalenses offer a path toward substantial miniaturization of optical modules, but practical designs often exhibit pronounced chromatic and field-dependent aberrations that necessitate computational reconstruction. In current metalens pipelines, reconstruction models are commonly trained and selected using distortion-based fidelity objectives, such as PSNR, yet these proxies can be weakly correlated with human preference and downstream utility, reflecting the well-known perception--distortion trade-off. We introduce MetaRanker, a human-in-the-loop active ranking framework that formalizes metalens image quality in terms of semantic interpretability, defined as the degree to which humans can reliably recognize objects and structures in the presence of optical artifacts. MetaRanker combines a probabilistic preference model with uncertainty-aware query selection, and leverages vision--language models to provide lightweight semantic priors. Importantly, these priors are used only to guide the sampling of informative comparisons; human judgments remain the primary supervision signal throughout. Across real-world and synthetic metalens datasets with distinct degradation profiles, MetaRanker produces rankings that align most closely with human assessments, while reducing the number of pairwise annotations required by approximately 80% relative to exhaustive pairwise evaluation. Finally, we show that standard image quality assessment metrics exhibit limited alignment with human interpretability in the metalens domain, positioning MetaRanker as a practical step toward perceptually grounded metalens evaluation and co-design.

How many directions does a neural representation use to encode a concept? A common answer repeatedly erases probe directions and reports the stopping count or cumulative removed rank. We show that both quantities can change under an information-preserving invertible reparameterization, so neither is intrinsically a concept dimension. We distinguish model-defined population quantities (generating dimension, sufficient linear dimension, and minimum guarding rank) from procedure-defined quantities such as stopping count and cumulative edit rank. In a population Gaussian construction, an invertible shear preserves the prediction problem and all three quantities, yet changes the cumulative Euclidean erasure count from one to two. The separation holds for Moore--Penrose ordinary least squares and every finite nonnegative ridge weight. For a two-output full-QR procedure matching our motivating video analysis, cumulative edit rank similarly changes from two to the ambient dimension four. Conversely, the complete cumulative metric-QR trajectory is affine-equivariant when its positive-definite metric, probe, regularizer, and tie-breaking are transported consistently; exact covariance is one corollary, not a canonical semantic metric. In a known-rank finite-sample Adam/QR calibration, identity mixing stops after one accepted update in all 20 large-sample runs, whereas each tested shear $a\in\{.5,.75,1,1.25,2\}$ accepts at least two updates in all 20 runs. Controlled reparameterizations of frozen V-JEPA2 features preserve rank-zero predictions yet alter later Euclidean trajectories under practical optimization. These visual contact experiments are stress tests, not estimates of contact dimension. Iterative erasure therefore returns a procedure-relative estimand jointly determined by representation geometry and the full measurement procedure, not a semantic dimension by itself.