Concept Purity

Measures how tightly the activations for a single concept are grouped. High purity requires that all exemplars of a concept—such as 'stripes'—map to a compact, dense region in the activation space. Silhouette Score and Davies-Bouldin Index are standard metrics for quantifying this compactness. Poor cohesion suggests the model is encoding multiple distinct sub-features under one label, a phenomenon known as polysemanticity.

Quantifies the distinctness of the boundary between different concept clusters. A pure concept is linearly separable from unrelated concepts with a wide margin. This is typically measured by training a linear probe and evaluating its classification accuracy. High separability confirms that the concept direction is not an artifact of noise but a distinct, orthogonal axis in the activation space.

Assesses whether the concept's representation is uniformly distributed in its local subspace or if it collapses to a narrow, degenerate cone. High isotropy indicates a robust representation that doesn't rely on a few outlier dimensions. This is often measured by the average cosine similarity of activations within the concept cluster; a value approaching 1.0 indicates a collapse, while a controlled, moderate value suggests a healthy, high-capacity representation.

Tests purity by adversarially removing the concept's information. A pure concept can be cleanly erased via Concept Subspace Projection without degrading the model's performance on unrelated tasks. If erasing 'color' significantly harms 'texture' classification, the two concepts were entangled, indicating low purity. This interventionist test validates causal monosemanticity.

Evaluates the variance explained by the concept vector relative to total variance. High purity implies a high signal-to-noise ratio (SNR), where the concept direction captures the dominant source of variation for its exemplars. This is computed by projecting activations onto the Concept Activation Vector (CAV) and comparing the variance along that axis to the variance in orthogonal directions.

A hallmark of a truly pure, universal concept is its stability across different architectures and random seeds. If a concept vector learned from one model can be used to accurately classify the same concept in another independently trained model, it exhibits high cross-model purity. This transferability validates that the concept captures a fundamental statistical regularity of the data rather than a model-specific idiosyncrasy.

The degree to which a linear or non-linear classifier can distinguish between the activation patterns of two different concepts. High separability is a prerequisite for high purity. It is often measured using the Area Under the Receiver Operating Characteristic Curve (AUC-ROC) or the silhouette score on the activations of concept exemplars. A low separability score indicates that the network's internal representations for the two concepts are entangled, leading to ambiguous explanations.

A measure of whether a specific neuron, channel, or direction in the activation space responds exclusively to a single concept and not to others. A highly selective unit is a pure encoder. This is distinct from purity, which measures the clustering of the concept as a whole. A concept can have high purity (a tight cluster) but be encoded by neurons with low individual selectivity if the representation is distributed across many polysemantic neurons.

A technique for removing a specific, often sensitive, concept's information from a model's latent representation. This is achieved by projecting the activations onto a subspace orthogonal to the concept vector. The effectiveness of erasure is a direct test of concept purity: a perfectly pure, linearly separable concept can be completely removed with a single projection. Residual information after erasure indicates a non-pure, entangled representation.

A module that replaces a standard batch normalization layer, aligning the latent space axes with predefined concepts to produce a disentangled and interpretable representation. By enforcing orthogonality between concept axes during training, Concept Whitening directly optimizes for concept purity. The resulting latent space has the property that moving along a single axis changes exactly one known concept, maximizing both purity and separability.

An inherently interpretable architecture that first predicts a set of predefined human-understandable concepts from the input and then uses only those concept scores to make the final prediction. The purity of these intermediate concept predictions is paramount. An impure concept predictor will leak information from other concepts, undermining the bottleneck's promise of a fully transparent and auditable reasoning chain.

A metric that evaluates how sufficient a set of discovered or defined concepts is for explaining a model's full behavior on a given task. A high completeness score with low purity suggests the model uses a few highly entangled, polysemantic concepts. In contrast, a high completeness score with high purity indicates the model's reasoning is based on a rich vocabulary of distinct, well-separated, and internally consistent concepts, which is the ideal state for interpretability.