Principal Component Analysis — projects high-dimensional activations onto the low-dimensional linear basis that captures the most variance, revealing geometric structure such as manifolds.

Decomposes weights or activations into a fixed harmonic (discrete Fourier) basis, rather than a data-fitted one, revealing sparse support on a handful of frequencies when the underlying computation is periodic — and enabling ablation directly in the frequency domain.

Singular Value Decomposition — factors an activation matrix directly as $X = U\Sigma V^\top$ and inspects the singular-value spectrum or a low-rank ($U\Sigma$) projection, without necessarily centering the data first — the same linear-algebra machinery PCA relies on, but reported directly in terms of singular values/vectors rather than an eigendecomposition of a centered covariance matrix.

Uniform Manifold Approximation and Projection — a nonlinear dimensionality-reduction technique that preserves local neighborhood structure, used when linear projection (PCA) doesn't reveal the shape.

Fits a linear projection whose Euclidean distances match a user-specified target distance function of a numeric label, turning manifold discovery into a model-selection problem across a menu of candidate geometries.

Supervised dimensionality reduction: projects activations onto the low-dimensional basis that maximizes covariance with a known target value (e.g. the number, or a date), rather than PCA's target-blind maximum-variance basis.

Projects data onto the eigenmodes of (between-cluster covariance) / (within-cluster covariance), maximizing separation between labeled clusters while suppressing variation within them — used here to strip distractor directions (e.g. word length) out of concept-difference vectors before checking for parallelogram/trapezoid structure.

Fits a K-vertex simplex to a point cloud by learning extreme points (archetypes) whose convex hull contains the data and re-expresses each point by its barycentric coordinates with respect to those archetypes, via a neural archetypal-analysis autoencoder (AANet).

Nonlinear dimensionality reduction that approximates geodesic (manifold) distance via shortest paths on a neighbor graph, then applies classical MDS to embed those geodesic distances in a low-dimensional space.

Decomposes the cross-covariance between two modalities' embedding spaces (e.g. text and audio) via SVD into paired, covariance-ranked orthogonal axes, splitting each embedding into a shared/aligned 'head' and a modality-private 'tail'.

A supervised variant of PCA that finds a subspace maximally explaining one labeled variable's variance (e.g. an answer identity) while separating it from other variables (e.g. a query's string length), imported from neuroscience population-coding analysis; its null space can then be used to selectively remove one variable's information via projection.

Fits an implicit polynomial surface p(y)=0 of increasing degree (affine, quadric, cubic) to a reduced-dimension point cloud via total-least-squares (SVD on the monomial design matrix), then compares held-out residual error across degrees and classifies the fitted quadric's type (ellipsoid, hyperboloid, paraboloid, cylinder, cone) to characterize local curvature.