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Feature discovery from activation geometry

ICA Lens

Non-Gaussianity Reveals Interpretable Directions in Language Models

We introduce ICA Lens, a practical workflow for efficient discovery, reliable annotation, and faithful evaluation of non-Gaussian directions in LLM residual-stream activations.

Read the paperOpen quickstartDocumentationPyPI

The central result

Interpretable structure leaves a statistical signature.

Along a typical random direction, natural-text activations look close to Gaussian. Along many interpretable directions, most contexts form a quiet background while a recurring pattern occupies a selective tail. ICA Lens searches for that structure directly.

01 / A toy example

A selective tail can reveal a coherent concept.

At GPT-2 layer 0, a random projection of all 50,256 non-special token activations is nearly Gaussian. An ICA direction instead places a related family—neither, either, both, and between—in a sparse, distant tail while most tokens remain near zero.

Whitening equalizes variance, then ICA finds a rotation whose standardized projection is unusually non-Gaussian.

Run the GPT-2 toy example
A random GPT-2 token projection produces a near-Gaussian distribution while an ICA direction isolates related tokens in one tail.
Figure 1 intuition: a generic direction mixes the vocabulary; a recovered non-Gaussian direction separates recurring structure into a selective tail.

02 / Evidence across models

SAE directions expose the signal. ICA searches for it directly.

Across GPT-2 Small, Gemma 2 2B, and Qwen 3.5 9B Base, random directions remain near the Gaussian reference and PCA captures little heavy-tailed structure. Public SAE feature directions are already substantially non-Gaussian. ICA optimizes this property directly and finds the strongest heavy-tailed projections throughout the residual stream.

GPT-2 Small 12 layers · d=768Gemma 2 2B 26 layers · d=2304Qwen 3.5 9B Base 32 layers · d=4096
Mean excess kurtosis of ICA, SAE, PCA, and random directions across three models.
Figure 3: mean excess kurtosis by layer. ICA consistently recovers stronger departures from Gaussianity than the comparison directions.

03 / Practical workflow

Collect, fit, profile, apply.

ICA Lens connects statistical discovery to inspectable evidence and downstream use in four concrete stages.

1

Collect

Capture residual-stream activations from representative text or conversations.

2

Fit

Center, whiten, and run GPU-parallel symmetric FastICA.

3

Profile

Store score statistics, the skewness-selected semantic tail, corpus occurrences, Logit Lens tokens, and optional R-Lens readouts.

4

Apply

Inspect new inputs, reconstruct activations, probe concepts, or intervene on generation.

A complete mini target

Fit GPT-2 layer 8 from 100,000 Pile-10k activations for 50 iterations, profile the components, then apply the new Lens to held-out text. Every stage reports its runtime and artifact path.

Run the full workflow

04 / Interpretability

Two complementary routes from direction to meaning.

Human annotation tests whether convergent evidence supports a usable label. Automated annotation scales an explanation–prediction test across thousands of components.

Manual annotation

Inspect, hypothesize, then test.

Annotators combine top activation contexts with Logit Lens and R-Lens vocabulary readouts, then use controlled counterfactual prompts when the evidence is ambiguous.

123/150high-confidence labels (scores 4–5)
GPT-2 · L8/C396Programming/API specification language
Gemma 2 · L20/C11541960s–1970s American culture and history
Qwen 3.5 · L20/C352Both / paired alternatives
GPT-2 Small
41/50 · 82%
Gemma 2 2B
42/50 · 84%
Qwen 3.5 9B
40/50 · 80%
Browse all 150 componentsTest GPT-2 L8/C396

Automated interpretability

Explanation–prediction, with ICA’s sign handled fairly.

Following the Cunningham-style protocol, an explainer proposes a description from high-scoring examples and a simulator predicts token scores on held-out fragments. Agreement is measured by Pearson correlation.

Paper Figure 4a reproducing the sign-sensitive Pythia 70M comparison.
Paper Figure 4(a): arbitrary-sign evaluation reproduces the reported ICA–SAE gap on Pythia 70M; stronger-tail orientation recovers much of it.

05 / The statistical link

Interpretability rises as fitting finds more non-Gaussian directions.

The experiment follows the same 400 Qwen components—100 each from layers 7, 15, 23, and 31—throughout FastICA. Mean sign-corrected automated interpretability rises with non-Gaussianity and then saturates.

ρs = 0.45At iteration 50, excess kurtosis is positively associated with automated interpretability across the 400 components.
Automated interpretability, logcosh deviation, and excess kurtosis across FastICA iterations for 400 Qwen components.
Figure 5(a): fitting dynamics averaged across four Qwen 3.5 9B Base layers. Error bars show the 95% confidence interval for auto-interpretability.

06 / ICA versus SAE

Evaluate the coordinates on the tasks SAEs are built to serve.

The comparison covers information preservation, sparse concept readout, and selective causal removal using matched public SAEs.

Sparse Reconstruction

How much of an activation survives a small coordinate budget?

Keeping only the largest coordinates, ICA is competitive with matched public SAEs and consistently stronger than PCA and random-direction baselines.

Sparse reconstruction cosine similarity across three models.

07 / Intervention

One Qwen coordinate can steer language in both directions.

In the paper’s Qwen 3.5 9B Base case study, layer 26 component C9 is causally linked to French-language generation. Adding its writing direction moves an English prompt toward French; clamping the coordinate moves a French prompt toward English.

Paper-exact demo: requires an A100-class/high-memory CUDA GPU. A free T4 is insufficient. The notebook can also be downloaded and run locally.
Run in ColabDownload notebookSteering documentation
English promptAdd C9 → French

Deep learning is a machine learning technique that…

Baseline

uses artificial neural networks to learn from data.

With L26/C9 steering

utilise des réseaux de neurones artificiels pour apprendre à partir de données.

French promptRemove C9 → English

L’apprentissage profond est une technique d’apprentissage automatique qui…

Baseline

utilise des réseaux de neurones artificiels pour apprendre à partir de données.

With L26/C9 clamped to zero

uses neural networks to learn from data.

08 / Practical efficiency

Full-width ICA fits in minutes per layer.

Reported fitting times use one million activations, 50 fixed FastICA iterations, and one NVIDIA GB10.

<1 minGPT-2 Small / layer
5.5 minGemma 2 2B / layer
12–13 minQwen 3.5 9B / layer
7–43×less parameter memory than the matched SAEs

The practical advantage is fitting: ICA recovers one coordinate per residual-stream dimension without training a large overcomplete dictionary.

09 / Use ICA Lens

Install, load a Lens, inspect text.

The Python package supports text, conversations, profiles, reconstruction, interventions, fitting, and publication of new lenses.

Quick start
pip install icalens

from icalens import ICALens

lens = ICALens.from_pretrained(
    "sida/icalens-gpt2-small-pile10k"
)
result = lens.analyze(
    "The API accepts a JSON object and returns a response.",
    layer=8,
)
result
Open quickstartDocumentationModel collection
Interactive ICA Lens token-level analysis in a notebook.
Interactive signed scores and component evidence display directly in Jupyter and Colab.

Paper and resources

Non-Gaussianity Reveals Interpretable Directions in Language Models

Sida Liu and Feijiang Han

arXivCodeDocumentationPyPIModels
BibTeX
@article{liu2026nonGaussianity,
  title={Non-Gaussianity Reveals Interpretable
         Directions in Language Models},
  author={Liu, Sida and Han, Feijiang},
  journal={arXiv preprint arXiv:2606.11722},
  year={2026}
}