behavior manifold

One-dimensional curved surface in output probability space; the paper shows this mirrors representation manifold structure.

Neighborhood — ranked by edge-count

framework

Geometry-Aware Steering Framework
uses
The overarching theoretical framework proposed in the paper, asserting that steering interventions should be aligned with the geometric structure of the model's representation manifold.

method

Pullback Steering
uses
The method of optimizing steering interventions in activation space to produce outputs that follow the behavior manifold, independent of the representation manifold.
Hellinger distance
implements
Metric used to define geometric space of output token probability distributions in behavior manifold analysis.

concept

behavior manifold M_y
related_tosame_as
Manifold fitted to output probability distributions (behavior).
Manifold Steering
implements
Central framework: steering neural networks by intervening along the curved manifold where a concept lives, rather than in straight lines through activation space.
representation manifold
associated_with
One-dimensional curved surface in internal activation space; the paper demonstrates alignment with behavior manifold.
Activation Manifold
associated_with
The low-dimensional geometric structure discovered in neural activation space; contrasted with linear/Euclidean geometry.
days of the week
implements
Primary case study demonstrating circular manifold structure in both behavior and representation space of Llama-3.1-8B.
Behavior Space
associated_with
A geometric space of all output token probability distributions, equipped with Hellinger distance, used to visualize model behavior.
Behavior-based Path
associated_with
The path in activation space derived by optimizing steering interventions to produce outputs along the behavior manifold, independent of representation geometry.
One-dimensional Manifold
associated_with
The type of manifold fitted to the cyclic concept structure in both activation and behavior space — a path along which steering moves the model.
concept geometry
associated_with
The spatial/geometric organization of conceptual structure within neural network representations; central to the paper's thesis.

cosine ≥ 0.65 · no typed edge

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

Manifold Fitting to Representation/Behavior Spacemethod0.828
The procedure of fitting a one-dimensional manifold (path) to clusters in activation or behavior space to capture the geometric structure of a concept.
behavior manifold fitting (M_y)method0.806
Method to fit a manifold M_y to output probability distributions.
Activation-Behavior Manifold Isometryconcept0.803
Activation manifolds and behavior manifolds are approximately isometric across cyclic and sequential concepts.claim0.802
optimization of interventions to follow behavior manifold M_ymethod0.800
Method that optimizes activation interventions so that resulting behaviors trace M_y, recovering activation paths that follow M_h curvature.
manifoldconcept0.799
A smooth, potentially curved surface in activation space along which activations vary according to a coherent semantic dimension.
Manifold geometry provides a practical blueprint for steering model behavior across diverse tasks and modalities.hypothesis0.786
The generalizing predictive claim that manifold steering is a broadly applicable framework beyond the days-of-week case study.
Analogous alignment between representation manifold and behavior manifold is found across months, letters, ages, and synthetic in-context learning tasks in language models.finding0.786
Generalization finding from the full paper extending beyond days-of-week to other structured concepts.