B01 — SVD / Spectral Analysis

This framework asks: what are the principal directions of computation in a circuit’s weight matrices, and how concentrated is their energy?

The singular value decomposition (SVD) of a weight matrix reveals which linear subspaces carry most of the matrix’s computational energy. In mechanistic interpretability, applying SVD to attention weight matrices (W_QK, W_OV) decomposes multi-dimensional transformations into rank-one contributions ordered by importance. A circuit claim that identifies specific heads as load-bearing should predict that those heads have spectrally distinct weight structure — concentrated singular values, interpretable singular vectors, or spectral gaps separating signal from noise.

SVD is the workhorse of structural circuit analysis because it provides a canonical, rotation-invariant decomposition. Unlike probing (which requires labeled data) or activation patching (which requires paired inputs), SVD operates directly on weights and reveals capacity-level structure: what the circuit could compute, not merely what it does compute on a given distribution.

Theoretical grounding

Source	Year	Key contribution
Elhage et al., “A Mathematical Framework for Transformer Circuits”	2021	SVD of W_OV and W_QK as interpretive decomposition
Millidge & Black, arXiv 2202.05924	2022	SVD-based analysis of singular value spectra in GPT-2 heads
Eckart & Young, “The approximation of one matrix by another”	1936	Optimal low-rank approximation theorem
Henighan et al., arXiv 2309.14322	2023	Superposition and spectral structure of MLP weight matrices

Core concept

Given a weight matrix ( W \in \mathbb{R}^{m \times n} ), the SVD factorizes it as:

[ W = U \Sigma V^T = \sum_{i=1}^{r} \sigma_i , u_i , v_i^T ]

where ( \sigma_1 \geq \sigma_2 \geq \ldots \geq \sigma_r > 0 ) are the singular values, ( u_i ) are left singular vectors (output directions), and ( v_i ) are right singular vectors (input directions). The effective rank measures how many of these directions carry meaningful energy.

For attention heads, the composed matrices ( W_Q^T W_K \in \mathbb{R}^{d_h \times d_h} ) (QK circuit) and ( W_O W_V \in \mathbb{R}^{d_m \times d_m} ) (OV circuit) are the natural objects to decompose. A head performing a single interpretable operation (e.g., “copy the previous token”) will have a rapidly decaying spectrum dominated by one or two singular values. A head performing multiple superposed operations will have a flatter spectrum.

Instruments under B01

Effective Rank of W_QK (18_weight_extended.py)

Computes the SVD of the composed QK matrix for each attention head and reports the effective rank (see B02 for the entropy-based definition) alongside the top-k singular value ratios ( \sigma_1 / \sigma_k ).

What it establishes: Whether circuit heads have spectrally concentrated attention patterns — low effective rank indicates a head is implementing a small number of attention motifs.

What it does not establish: Whether those motifs are correct for the task, or whether the head is necessary (that requires causal validation from Pillar A).

Usage:

uv run python 18_weight_extended.py --tasks ioi sva

Reading the scores

Pattern	What it means
Low effective rank in circuit heads	Head implements a concentrated, potentially interpretable operation
High ( \sigma_1 / \sigma_2 ) ratio	Dominated by a single rank-one term (e.g., copy or induction)
Flat spectrum across all heads	No structural differentiation — circuit claim lacks weight-level support
Circuit heads spectrally distinct from non-circuit	Structural signature corroborates causal findings

Connection to other frameworks

SVD provides the substrate for B02 (effective rank as a scalar summary), B03 (OV/QK decomposition interprets the singular vectors), and B04 (weight alignment compares top singular directions across heads). Causal findings from A01 (activation patching) identify which heads matter; B01 then asks whether those heads have structurally distinctive spectra that explain why they were selected.