# Function Lives Where Variance Doesn't: Task-Weighted Charts of a Language Model's Computation

arXiv: https://arxiv.org/abs/2609.18989 · HTML: https://mlsanity.com/paper/2609.18989/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 21 references that are in the map.
- First posted on arXiv 12 Aug 2026.

## Builds on (most cited first, 15 of 15)

- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Intrinsic Dimensionality Explains the Effectiveness of Language Model Fine-Tuning](https://mlsanity.com/paper/2012.13255/) (2020)
- [Eliciting Latent Predictions from Transformers with the Tuned Lens](https://mlsanity.com/paper/2303.08112/) (2023)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [SVD-LLM: Truncation-aware Singular Value Decomposition for Large Language Model Compression](https://mlsanity.com/paper/2403.07378/) (2024)
- [Language model compression with weighted low-rank factorization](https://mlsanity.com/paper/2207.00112/) (2022)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [All Bark and No Bite: Rogue Dimensions in Transformer Language Models Obscure Representational Quality](https://mlsanity.com/paper/2109.04404/) (2021)
- [The Truth is in There: Improving Reasoning in Language Models with Layer-Selective Rank Reduction](https://mlsanity.com/paper/2312.13558/) (2023)
- [Future Lens: Anticipating Subsequent Tokens from a Single Hidden State](https://mlsanity.com/paper/2311.04897/) (2023)
- [The geometry of hidden representations of large transformer models](https://mlsanity.com/paper/2302.00294/) (2023)
- [Verbalizable Representations Form a Global Workspace in Language Models](https://mlsanity.com/paper/2607.15495/) (2026)
- [Identifying Functionally Important Features with End-to-End Sparse Dictionary Learning](https://mlsanity.com/paper/2405.12241/) (2024)
- [Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDE](https://mlsanity.com/paper/2607.07034/) (2026)

## Abstract (arXiv)

How many dimensions does a language model's computation actually use? The question is ill-posed until one names a functional. Task-weighted charts make it well-posed: low-dimensional coordinate systems fit against a chosen functional of the representation, under the functional's own metric, turning distillation into plain least squares. Across six models from three families, spanning 70m to 7B parameters, next-token prediction needs 70--90% of the residual stream's width to stay within 5% of intact perplexity, a width consumed by the rare tail of language, and the variance profile predicts none of it: two directions carry 90% of GPT-2's activation variance and almost none of its function. Dimension is per-functional: the model's own uncertainty reads from six coordinates where the full predictive distribution needs hundreds; and it grows with depth. The dissociation is exploitable: when only a few dimensions can be kept, charts trained under the functional's metric preserve the model's predictions better than variance-based or optimal linear compression.