Disentangling Representation Evolution in Transformers through Directional Decomposition
Shwai He, Haichao Zhang, Shen Yan
University of Maryland, College Park · Northeastern University · ByteDance
Published September 14, 2026 · on the feed September 16, 2026 · ▲ 9 · arXiv 2609.15975
The abstract
Transformer representations evolve through learned additive transformations that either preserve their current direction or redirect it. We study this evolution as a functional geometry, decomposing learned updates into parallel and perpendicular components. Across pretrained models, we find substantial parallel components beyond the residual identity path. We then apply the decomposition in two spaces: to attention and MLP updates relative to the hidden state, and to attention value aggregation relative to the current token's value. Targeted edits reveal a strongly space-dependent asymmetry: exclude-self value-space parallel manipulation is markedly more robust than residual-space and perpendicular counterparts, preserving the direct self message while scaling only the non-self aggregate. The same decomposition gives a component-resolved description of compression-induced update error: perpendicular error separates compression methods more clearly than parallel error. Extensive experiments further demonstrate that full-aggregate parallel suppression during from-scratch pretraining lowers validation-loss trajectories and improves downstream averages, with the value-space variant strongest. Together, these results connect representation geometry to editing robustness, compression diagnosis, and training-time intervention. Code is available in the https://github.com/Shwai-He/Transformer-Geometry{project repository}.
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Transcript
Disentangling Representation Evolution in Transformers through Directional Decomposition. By Shwai He and two others, from the University of Maryland, College Park, Northeastern University, and one other institution. Published on the fourteenth of September. Its code repository is worked out: about one thousand source files, with a readme.
Transformer representations change through learned additive updates, and each update either keeps the current direction of a representation or redirects it. The authors study that evolution as a functional geometry. Every learned update is split into two components: one parallel to the current direction, which lengthens or shortens it, and one perpendicular, which turns it. Across pretrained models, they find substantial parallel components, beyond what the residual identity path alone would contribute.
The decomposition is then applied in two spaces. The first is attention and MLP updates, taken relative to the hidden state. The second is attention's value aggregation, taken relative to the current token's own value. Targeted edits reveal a strong asymmetry that depends on the space. Manipulating the parallel component in value space, while excluding the token's own contribution, is markedly more robust than the residual-space and perpendicular versions. It preserves the direct self message and scales only the aggregate arriving from other tokens.
The same lens gives a component-by-component description of the error introduced by compression. Perpendicular error separates compression methods more clearly than parallel error does, so it is the more useful signal for telling methods apart. Finally, in from-scratch pretraining, suppressing the full-aggregate parallel component lowers validation-loss trajectories and improves downstream averages, with the value-space variant strongest.
The observations concern pretrained transformers, and the specific edits and interventions the authors tested. The robustness asymmetry is a statement about those targeted edits, and the compression result is a description of error, not a new compression method. The training result is about pretraining from scratch with this suppression applied.
One decomposition, put to three uses: explaining which edits a model tolerates, diagnosing how compression damages it, and changing how it trains. The common thread is that separating how much a representation grows from where it turns reveals structure that the whole update hides. Code is available in the project repository.
Paper Quest · 7 chapters · about 40 min
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Learn it by doing: make the calls the authors made, guess the result before you see it, and defend it in a viva graded on your own words. Chapter 1 is right here.
Chapter 1 🧩 · The problem
Every layer nudges every token. Some nudges stretch; some turn. Which ones matter?
You are studying a transformer. Each layer adds an update to every token's running representation. Part of each update points along the representation, just stretching it; part points sideways, turning it.
The road ahead
- 1 · The problem ~7 min Every transformer layer adds an update that partly stretches a token's representation and partly turns it, and it is an open question whether the stretching matters.
- 2 · What came before ~3 min
- 3 · The key idea ~8 min Split each update into a stretching part and a turning part; the model tolerates rescaled stretching, but rescaling the turning breaks it.
- 4 · Choose your own research ~3 min
- 5 · Build the method ~8 min One split, into stretching and turning parts, is applied in two spaces, tested by edits, used on compression error, and finally built into pretraining.
- 6 · The experiment ~6 min You’ll guess: When researchers rescaled parts of transformer updates at inference, which manipulation proved markedly the most robust?
- 7 · The viva ~5 min Three questions from the examiners, answered in your own words. Win it for the paper’s trophy.
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