[Proposal] SVD Circuits: singular-vector decomposition of a head's QK/ OV into causally-validated subfunctions

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Proposal

Component-level circuit analysis (direct_logit_attribution, path patching, head detectors) treats an attention head as an indivisible unit. But a single canonical head — e.g. a GPT-2-small IOI name-mover — empirically superposes multiple subfunctions on distinct low-rank directions of its QK and OV maps. This proposes svd_circuits, a self-contained tool in transformer_lens/tools/analysis/ that takes the SVD of a head's QK (W_Q W_Kᵀ) and OV (W_V W_O) matrices, exposes each orthogonal singular direction as an interpretable sub-component, and — crucially — causally gates every claimed subfunction by patching activations along that individual singular direction. This is a finer-than-component decomposition that resolves within-head superposition component-level tools miss.

Capabilities:

1. Per-head QK/OV SVD via FactoredMatrix — get singular values/vectors of W_Q W_Kᵀ and W_V W_O without materializing the d_model × d_model matrix.
2. Direction → vocab readout — project each OV singular direction into the unembedding basis (reusing SVDInterpreter) and each singular direction into logit space via direct_logit_attribution, so a direction gets a human-readable "what tokens does this subfunction move" signature.
3. Activation projection / attribution — project cached per-head activations onto the singular directions to measure how much each subfunction fires on a given prompt (per token/position).
4. Causal validation (patch-along-a-direction) — reconstruct the head output using only a chosen singular subspace (or ablate a single direction) via generic_activation_patch, and report the behavior change (e.g. IOI logit-diff), so a subfunction claim is accepted only if its direction is causally load-bearing.
5. Degeneracy guard — detect near-equal singular values (rotation/degeneracy ambiguity) and refuse to attribute individual directions inside a degenerate block, reporting the block as a subspace instead.

Status: PR1 (#1768) and PR2 (#1775) merged; PR3 in progress
Suggested labels: enhancement, tooling, TransformerBridge, complexity: moderate

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Motivation

The existing TL baseline for "what does this head do" is direct_logit_attribution (transformer_lens/tools/analysis/direct_logit_attribution.py) and head_detector — both operate at head granularity. The IOI work established name-mover / S-inhibition / duplicate-token heads as the atomic units of the circuit. The claim in Beyond Components is that this granularity is too coarse: an IOI name-mover head is not one function but several, packed onto near-orthogonal low-rank directions of its QK/OV weights, and only an SVD-based decomposition surfaces them. This tool lets a TL user go inside a canonical head and attribute/patch its subfunctions — the natural next rung below component-level circuit analysis.

Why no maintained TL implementation exists. TL already ships the two hardest primitives — FactoredMatrix (low-rank OV/QK SVD/eigendecomposition without materializing the full matrix) and SVDInterpreter (projects OV/w_in/w_out singular vectors into vocab) — but neither closes the loop from "singular direction" to "causally-validated subfunction." SVDInterpreter is a static weight-inspection utility (its own docstring warns SVD directions are not reliably interpretable); it never touches activations or patching. The missing piece is the activation-projection + causal-patch layer, which is exactly what makes the decomposition trustworthy.

Implementation status elsewhere. The authors' research repo exists — github.com/Exploration-Lab/Beyond-Components (~7 commits; IOI / GP / GT examples) — but it is not maintained or TL-integrated. This would be the first TL-native implementation. This is not SAE/dictionary-learning territory (no learned features, no overcomplete basis) — it is a closed-form weight decomposition, so it sits inside TL rather than at the SAELens boundary.

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Pitch

Add svd_circuits as a self-contained analysis tool under transformer_lens/tools/analysis/. It builds on FactoredMatrix for the weight-space SVD, SVDInterpreter for vocab readout, ActivationCache/direct_logit_attribution for activation projection and logit signatures, and generic_activation_patch for the mandatory causal gate.

Proposed API

Names adjustable to maintainer preference.

from transformer_lens.model_bridge import TransformerBridge
from transformer_lens.tools.analysis import svd_circuits

# TransformerBridge is the supported model system
model = TransformerBridge.boot_transformers("gpt2-small")
model.enable_compatibility_mode()  # folds final LN into W_U — required for the OV vocab/logit readout

# 1. Decompose a head's QK and OV into singular directions (FactoredMatrix under the hood).
decomp = svd_circuits.decompose_head(model, layer=9, head=9, which=("QK", "OV"))
decomp.OV.S            # singular values, sorted desc
decomp.OV.rank_report  # (idx, sigma, sigma/sigma_max, is_degenerate) per direction

# 2. Readout: each OV singular direction -> top tokens in the unembedding basis.
decomp.OV.vocab_readout(k=10)     # wraps SVDInterpreter.get_singular_vectors
decomp.OV.logit_signature(model, prompt=ioi_prompt)  # via direct_logit_attribution

# 3. Attribution: how much does each direction fire on this prompt?
proj = svd_circuits.project_activations(model, decomp, prompt=ioi_prompt)  # [pos, direction]

# 4. Causal gate: keep only directions {0,3}, measure IOI logit-diff change.
result = svd_circuits.patch_along_directions(
    model, decomp.OV, keep=[0, 3], prompt=ioi_prompt,
    metric=ioi_logit_diff,      # any callable(logits) -> scalar
)
result.delta_metric   # behavior change attributable to that subspace
result.gated          # True only if |delta| exceeds the causal threshold

Design (algorithm)

For a head (layer ℓ, head h):

1. Build factored maps. OV = FactoredMatrix(W_V[h], W_O[h]) (shape d_model × d_model, rank ≤ d_head); QK = FactoredMatrix(W_Q[h], W_K[h].T). Both stay factored — never materialize d_model².
2. SVD. U, S, V = M.svd() (M == U @ S.diag() @ V.transpose(-2, -1); U and V are both [d_model, rank], with the i-th singular direction in column i — U[:, i], V[:, i]). Right singular vectors V[:, i] are the input directions (which residual-stream directions the subfunction reads); left singular vectors U[:, i] are the output directions. Access them via the .V property — .Vh is a deprecated alias that returns the same tensor as .V (it emits a DeprecationWarning), not its Hermitian transpose; do not introduce a new call site that relies on it.
3. Degeneracy guard. Flag any run of singular values with σᵢ/σᵢ₊₁ − 1 < ε (default ε = 1e-2) as a degenerate block: individual directions inside it are rotation-ambiguous, so attribute the block as a subspace, not per-direction.
4. Vocab readout (OV). For each output direction U[:, i], project through the (LN-folded) unembedding — reuse SVDInterpreter.get_singular_vectors("OV", ℓ, head_index=h) — to get its top-token signature.
5. Activation projection. Cache the head's per-position input (residual stream into the head; for OV attribution use stack_head_results / the head's value stream). Project onto V[:, i] to get a [pos] firing coefficient per direction i.
6. Logit signature. Feed the rank-1 reconstruction σᵢ · U[:, i] V[:, i]ᵀ of the head output through direct_logit_attribution to get each direction's signed logit effect on the task metric.
7. Causal patch-along-direction. Using generic_activation_patch, replace the head's output (at blocks.ℓ.attn.hook_z / hook_result) with its projection onto the chosen singular subspace span(U[:, keep]) (or zero out a single direction for ablation), rerun, and record Δmetric. A subfunction is accepted only if its direction/subspace produces a causal Δmetric above threshold — the decomposition alone never suffices.

Reuse map:

Sub-step	TL primitive	file:symbol
Factored QK/OV, low-rank SVD/eigen	FactoredMatrix (.svd(), .U/.S/.V, .eigenvalues)	transformer_lens/FactoredMatrix.py:FactoredMatrix
OV direction → vocab readout	SVDInterpreter	transformer_lens/SVDInterpreter.py:SVDInterpreter.get_singular_vectors
Per-head activations for projection	ActivationCache	transformer_lens/ActivationCache.py:stack_head_results (:957)
Direction → logit effect	direct_logit_attribution	transformer_lens/tools/analysis/direct_logit_attribution.py:direct_logit_attribution
Bridge LN-folding guard (reuse pattern)	_validate_bridge_compatibility	transformer_lens/tools/analysis/direct_logit_attribution.py:_validate_bridge_compatibility
Characterize resulting subfunction	head detector	transformer_lens/head_detector.py
Causal patch-along-direction	generic_activation_patch + setter	transformer_lens/patching.py:generic_activation_patch
Weights / unembed / final LN	W_Q/W_K/W_V/W_O, model.W_U	Bridge tl_parameters()

Main correctness risk (named): interpreting individual singular directions under rotation/degeneracy ambiguity

SVD directions are unique only when singular values are distinct; equal (or near-equal) singular values leave the corresponding singular subspace defined only up to an arbitrary rotation, so any "direction 3 = the surname subfunction" claim inside a degenerate block is meaningless. Compounding this, "the SVD direction is interpretable" is not guaranteed even for well-separated directions (SVDInterpreter's own docstring flags numerical instability and cross-device inconsistency). Mitigation, not a warning: (a) the degeneracy guard (step 3) raises/refuses per-direction attribution inside a near-degenerate block and downgrades it to a subspace; (b) every claimed subfunction must pass the causal patch-along-direction gate (step 7) — a direction that is not causally load-bearing is never reported as a subfunction, regardless of how clean its vocab readout looks. The tool's contract is "causal evidence gates interpretive claims," not "SVD is interpretable."

Validation plan (falsifiable)

1. Analytic SVD oracle. On a tiny synthetic head with hand-set W_V/W_O (and W_Q/W_K), the tool's S, U, V match torch.linalg.svd of the explicitly materialized matrix to atol=1e-5 (sign/degeneracy handled).
2. Reconstruction fidelity. Summing all rank-1 direction reconstructions reproduces the full head output (via the cache) to atol=1e-4; keeping top-k directions drives abs(Δmetric) toward zero (non-increasing)slow
3. Degeneracy guard fires. A synthetic head with two exactly-equal singular values → the guard flags the block and per-direction attribution raises; subspace attribution still works.
4. Causal gate discriminates. Ablating a high-σ, causally-relevant direction changes the IOI logit-diff materially; ablating a null-space / random direction does not (baseline reported alongside).
5. Paper sanity check (slow). On gpt2-small head L9H9 (name-mover), the tool surfaces ≥2 causally-gated OV subfunctions whose vocab readouts qualitatively match the paper's name-mover split; reported as a qualitative sanity check, not a pinned numeric threshold.

Scope of the vertical slice

Per-head QK/OV SVD + direction→vocab readout + direction→logit signature + one causal patch-along-direction validation, on gpt2-small IOI. Defer full multi-head circuit assembly and automated subfunction labeling. Ships as 3 sequential PRs, not one — base off dev:

• feat(svd_circuits): per-head QK/OV SVD with degeneracy guard — decompose_head, HeadSVD, degeneracy guard. Pure weight-space, no public export yet.
• feat(svd_circuits): singular-direction readout, projection, and causal patch gate — vocab/logit readout, project_activations, patch_along_directions, public exports, Bridge compatibility-mode validation, gpt2-small IOI integration test.
• docs(svd_circuits): add demo notebook, slow name-mover parity, and tool docs — Reflect the combined demo+docs commit, SVD_Circuits_Demo.ipynb, docs section.

Files:

• New: transformer_lens/tools/analysis/svd_circuits.py
• Export: transformer_lens/tools/analysis/__init__.py
• Unit: tests/unit/tools/test_svd_circuits.py (synthetic head; checks 1–4, no HF download)
• Integration: tests/integration/test_svd_circuits.py (gpt2-small, IOI patch-along-direction end-to-end)
• Oracle-parity (slow): tests/integration/test_svd_circuits_oracle_parity.py (check 5, @pytest.mark.slow)
• Demo: demos/SVD_Circuits_Demo.ipynb (nbval)

Follow-up work

Deferred to a separate tiered issue, filed after PR3 merges: multi-head circuit assembly, automated subfunction labeling, QK-side pattern attribution, more models.

Model coverage

• CI reference / oracle: tiny synthetic head (no download) for unit correctness; gpt2-small for the integration + slow paper sanity check (CI-cacheable, canonical IOI).
• Demo model: gpt2-small.
• Honesty caveat: the paper's headline subfunction taxonomy is scale- and model-dependent; gpt2-small reproduces the mechanism (a name-mover splits into causally-distinct directions) but not necessarily the exact subfunction inventory of larger models. Baselines (random-direction ablation) are reported next to every headline Δmetric.

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Alternatives

• Stay at head granularity (direct_logit_attribution / head detectors). Rejected: cannot resolve within-head superposition — the whole point.
• Use SVDInterpreter alone. Rejected: static weight readout with no activation projection and no causal gate; its own docstring disclaims interpretability of the directions.
• Sparse autoencoders on head outputs (SAELens). Rejected here: needs training, an overcomplete learned basis, and lives outside TL; svd_circuits is closed-form, training-free, and reuses TL primitives.
• Full path-patching over a rank-1 direction grid. Rejected as the first slice: combinatorially expensive and premature before the single-direction gate is validated; folded into Tier 2 of the follow-up issue.

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Correctness oracle

Brute-force / analytic reference (no external frozen impl exists). Primary oracle: torch.linalg.svd on an explicitly materialized synthetic head, compared to the FactoredMatrix-based path within threshold (atol=1e-5), plus a reconstruction-fidelity check. Secondary (slow, @pytest.mark.slow): the gpt2-small name-mover paper sanity check, treated as qualitative because the paper ships no numeric reference to pin against.

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Additional context

• Paper: Areeb Ahmad, Abhinav Joshi, Ashutosh Modi (IIT Kanpur), "Beyond Components: Singular Vector-Based Interpretability of Transformer Circuits," arXiv 2511.20273, NeurIPS 2025 poster (poster 119702, OpenReview 7UbXEQNny7).
• External implementation status: authors' research repo Exploration-Lab/Beyond-Components exists (research-grade, ~7 commits, 4 stars, no transformer_lens dependency); this would be the first TL-native implementation. Paper nuance: it augments weight matrices with biases folded in and applies SVD uniformly to attention and MLP — this slice covers attention QK/OV (OV logit-facing) only.
• Artifact sources: none required — the decomposition is computed from model weights at call time; no downloaded lens/SAE artifacts, so no registry entry is needed for the vertical slice.
• Algorithm note for reviewers: OV singular directions are attributed through the unembedding (output side); QK singular directions describe which residual directions the head compares (attention-pattern side) and are attributed via the pattern, not the unembedding — the vertical slice focuses on OV for logit-facing claims and includes QK SVD as a readout only. The causal gate is what separates this from a pretty-picture SVD tool; keep it mandatory.
• Substrate history: SVD convention fix #341 (closed) → #1300 (merged) (torch.linalg.svd; .Vh deprecated in favour of .V); #328 (closed, 2023, "SVD tests fail on GPU") is direct evidence for the cross-device instability the degeneracy guard + causal gate backstop.
• Provisional-citation flags: arXiv id 2511.20273 and NeurIPS-2025 venue are confirmed (no open citation flags).

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Checklist

• I have checked that there is no similar issue in the repo (required).
• Checked no similar issue / tool exists (SVDInterpreter is static weight readout only; no causal singular-direction tool in tools/analysis/).
• Scope limited to a vertical slice (single head, OV logit-facing, one causal gate); follow-up tiered separately.
• Correctness oracle identified (analytic torch.linalg.svd + reconstruction; slow qualitative paper check).
• Targets TransformerBridge (the supported TL 3.x path) via shared ActivationCache
• Claim-in-comments before starting a tier item.

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