Studies the geometric relationships — angles, isomorphisms, reflections, geodesic distances — between feature directions that have already been found, rather than predicting a shape in advance.

Estimates the local dimensionality of the manifold activations lie on directly from the ratio of each point's distances to its first and second nearest neighbors, without needing an explicit low-dimensional embedding.

Derives the expected representation geometry from first principles — proofs, self-consistent quantization conditions — ahead of and independent from empirical observation.

Computes Betti numbers (counts of connected components, independent loops, and enclosed voids) directly from a point cloud of activations via topological data analysis, to empirically distinguish a disc from a circle from a torus without assuming a parametric shape in advance.

Numerically locates approximate fixed points of a recurrent network's zero-input update rule, then linearizes the dynamics around each one via its Jacobian and studies the Jacobian's eigenvalues/eigenvectors — converting an opaque nonlinear recurrent system into a locally-linear, interpretable dynamical description.

Estimates the local intrinsic dimensionality around each activation vector from the rate of growth of its neighbor-distance distribution, tracked per-layer and compared between clean, benign-perturbed, and adversarially-perturbed inputs — a diagnostic use of local dimensionality as an anomaly signature rather than a static quality measure.

Replica mean-field theory from statistical physics that predicts a class's linear separability (manifold capacity) directly from three measured geometric properties of its activation cloud — radius, dimension, and inter-class center correlation.

Treats each class or category's set of activations as a geometric 'object manifold' and computes its capacity (how many such manifolds can be linearly separated per feature dimension), effective dimension (participation ratio), and radius directly from the manifold's covariance structure, plus inter-manifold correlation/axis-alignment statistics — a mean-field-theoretic generalization of linear separability from points to whole manifolds.

A covariance-whitened inner product on the unembedding space, ⟨γ,γ'⟩_C = γᵀΣγ⁻¹γ', constructed so that causally separable concepts become approximately orthogonal under it — turning 'is concept A independent of concept B' into an angle you can measure directly, rather than a raw cosine similarity that correlated training data can distort.

Recovers a piecewise-linear network's induced polyhedral partition of input space directly, either by exact combinatorial traversal (BFS over the region-adjacency graph, following shared facets between activation patterns) or by grid/Monte-Carlo sampling of forward passes to approximate region boundaries — as opposed to bounding the number of regions purely theoretically without measuring a trained network.

Measures how much a point's set of k-nearest-neighbors changes between two representations — between successive layers (tracking how fast a layer rewrites the data's local structure) or between a representation and ground-truth class labels (tracking how much semantic information the representation carries).

Computes the participation ratio (an effective-rank measure derived from the entropy of the squared singular-value distribution) of each attention head's per-token output activation matrix, integrated over training, to identify which heads are doing specialized computation without any behavioral labels.

Decomposes the asymptotic contrastive loss into an alignment term (closeness of positive pairs) and a uniformity term (a Gaussian-potential measure of how close the induced feature distribution is to the uniform measure on the hypersphere), then causally tests the decomposition by training directly on the two decomposed losses in place of the original contrastive objective.

Tests whether a fitted simplex's barycentric coordinates carry genuine mixture information by comparing held-out predictive R^2 of a barycentric combination against the single best individual feature, separately on near-vertex and simplex-interior sample splits, to rule out a tiling artifact.

Recovers the Bayes-optimal class posterior at each diffusion noise level from a trained diffusion model's own conditional and unconditional noise predictions (a classifier-free-guidance-style Bayesian update), then differentiates the resulting conditional entropy with respect to diffusion time to obtain a semantic-information-production rate -- a purely information-theoretic diagnostic requiring no auxiliary classifier.

Proves matching upper and lower bounds on how many k-sparse features m a d-dimensional linear representation can store such that a second linear map can recover them, using incoherent-matrix constructions for the upper bound and a rank/Turán-theorem argument for the lower bound.

Estimates the local surface volume of a classifier's decision boundary by Monte Carlo sampling of an epsilon-tubular neighborhood, using an adversarial-example-finding method (e.g. FGSM) as a cheap distance-to-boundary oracle, justified by Weyl's classical tube formula relating small-epsilon neighborhood volume to the hypersurface's own volume.

Identifies cluster centers as points that are both high-density and far from any other higher-density point, then assigns every other point to the nearest higher-density neighbor's cluster — a nonparametric way to count how many distinct semantic modes a representation manifold organizes into, and how sharply separated they are, without pre-specifying a cluster count.

Projects a class's within-class covariance onto the class-mean-separating (decision) axis before computing the class-distance-normalized-variance ratio, isolating collapse specifically along task-relevant directions from variance left standing in orthogonal, task-irrelevant (nuisance) directions.

Estimates local decision-boundary curvature during a black-box adversarial attack's own query trajectory, tracking a 'curvature dynamic trajectory' as the attack converges, rather than a one-shot Monte Carlo volume estimate.

Builds a provably complete concept lattice from a binary object-attribute incidence relation via the Galois connection, then softens every step (incidence, inclusion, meet, join) into graded scores so the construction runs directly on continuous embedding projections instead of a hand-annotated discrete table.

Fits a low-rank activation-derived tangent-subspace proxy once from pooled early-checkpoint activations, then tests whether true backpropagated gradients concentrate in it more than in a matched-rank random subspace, via a gradient-energy ratio (with a Monte Carlo null) and an IsoScore*-removal test (does ablating the proxy recover more isotropy than ablating a random subspace).

Detects context-modifying words by comparing the input Jacobian evaluated after that word (h_mod) to the input Jacobian at the nearest line-attractor fixed point (h*), then fits a low-rank bilinear correction model to how the input Jacobian itself changes as a function of displacement off the attractor.

A k-NN neighborhood-overlap statistic computed across languages rather than across layers or against ground-truth labels — measures how much a translation-equivalent point's local neighborhood in one language's representations is shared with its neighborhood in another language's representations, isolating a shared interlingual region from language-specific structure.

A Fisher-information-based metric that decomposes a diffusion model's noise-level-conditioned features into an 'invariant' component (signal that should be constant across augmentations/noise realizations of the same underlying image) and a 'residual' component, then quantifies how much residual variation contaminates the invariant signal — unifying diagnosis of classification-optimal noise level, generative quality, and memorization onset in one measurement.

Replaces a sub-network's raw hidden-state activation with the row-sum of its local input-output Jacobian (mu = J^T * 1, evaluated at the affine region containing the input) as the feature-representation space, then applies standard representation-analysis tools (PCA, sparse dictionary learning, probing) to these centroids instead of to activations.

Runs k-means over a PCA-reduced activation population for a sweep of cluster counts k, reporting the maximum silhouette score achieved as a scalar 'how cleanly does this population split into discrete modes' statistic — used to compare whether one domain's representations organize into sharply-separated discrete clusters while another remains a single smooth blob.

A four-metric suite — token/state entropy, effective rank of the activation covariance, anisotropy, and intrinsic dimension — applied jointly to a sequence of latent states (e.g. a recurrent model's iterated latent 'thoughts') to characterize how tightly organized versus diffuse a trajectory through representation space is, rather than any single scalar.

Estimates the local density of a piecewise-linear network's region-boundary ('nonlinear locus') near a data distribution, without enumerating the full polyhedral complex, by perturbing each unit's bias with small noise and measuring how often the perturbation flips the unit's activation pattern near a given input.

A formal criterion for when a sparse dictionary (e.g. an SAE) 'captures' a known manifold — a small, fixed group of atoms whose span contains it and whose encoder consistently selects that group — plus a taxonomy of three regimes (compact capture, shattering/tiling, dilution) describing how real dictionaries fall short of that ideal.

Estimates a representation manifold's intrinsic dimension from the growth rate of the total length of a minimum spanning tree over sampled points, as a function of sample size — an alternative to nearest-neighbor-ratio estimators like TwoNN that uses global tree structure rather than only first/second nearest-neighbor distances.

Classifies each activation by which class-mean centroid it is closest to (no trained classifier), giving a parameter-free measure of how tightly representations already cluster by a given label — applied at both the individual-sample level and the semantic-class level to separate augmentation-invariance clustering from genuine semantic clustering.

Performs SVD on a context/next-token co-occurrence matrix implicit in next-token-prediction training, then reads off semantic structure from the sign pattern (orthant) of each embedding's coordinates in the singular-vector basis, showing increasingly fine-grained real semantic categories emerge as more singular components are included.

Defines 'word transformation' via an information-theoretic paraphrase condition (small KL divergence between the context distributions two word sets induce), then algebraically decomposes any embedding-sum residual into named, exactly-quantified error terms (paraphrase error, plus two conditional/unconditional dependence error terms) rather than assuming or spectrally deriving a generative structure for co-occurrence.

Projects hidden states into a fixed low-rank subspace, then measures whether relation-bound token tuples share a more consistent determinant-sign (orientation) pattern at the arity-matched rank than scrambled or random controls — a rank-indexed diagnostic for relation frames.

Extends Neural Collapse-style within-class variance collapse to visual-control regression pipelines with no explicit label set, by defining implicit classes from the sign pattern (orthant) of a relative pose vector — e.g. object-to-target displacement — and measuring whether visual representations cluster by that orthant just as classifier representations cluster by label.

A normalized Frobenius-norm distance between two entity-token pairwise-distance matrices — one from an edited model, one from the unedited baseline — quantifying how much a knowledge-editing intervention distorts the relative geometric positions of entities that were never themselves targeted by the edit.

Trains a Recursive Feature Machine (RFM) as a target-vs-rest classifier directly on transformer residual-stream activations, then takes the ranked top-k eigenvectors of the resulting Average Gradient Outer Product (AGOP) matrix as a multi-dimensional concept subspace — a cheaper kernel-machine alternative to iterative diff-in-means/PCA subspace search that yields an explicit ranking of how many dimensions a concept needs.

A Fisher-discriminant-style score S(l,k) = ||mu_A - mu_B|| / (intra-cluster variances) computed per layer/attention-head on a frozen model's real activations, quantifying how entangled or separable two behaviorally-defined classes' representation manifolds are at that specific site — used to rank sites for a targeted geometric intervention rather than intervening uniformly.

Proves classification accuracy is upper-bounded by the maximum linear separability of hidden states, with equality when the separating direction also aligns with the label unembedding-difference direction — then measures both quantities layer-by-layer to explain when and why interventions work.

Treats a whole protein (or its per-residue PLM representation) as a curve rather than a single pooled vector, maps it to the square-root-velocity (SRV) representation to quotient out rotations/translations, and then measures the resulting Riemannian shape space's Fréchet radius (spread) and tangent-PCA effective dimension (number of shape-variation degrees of freedom) — a curvature-aware generalization of [[intrinsic-dimension-estimation|TwoNN/effective-rank dimensionality profiling]] to whole-sequence shape rather than pointwise activation geometry.

Fits a linear probe per task and compares the resulting probe subspaces directly to each other (rather than only reporting each probe's own accuracy), letting a single procedure track how much information distinct tasks' subspaces share, and how that sharing changes across training checkpoints.

Splits a learned square weight matrix (e.g. an attention query-key bilinear form) into its unique symmetric and skew-symmetric parts via the Toeplitz decomposition, then scores which part dominates by a Frobenius-norm ratio — a purely linear-algebraic diagnostic of whether a trained operator behaves as an order-independent or a genuinely directional interaction.