NOTE-659: Small Singular Values Matter

Contribution

Before this paper, random-matrix readings of trained weights looked for learning in the outliers above the Marchenko–Pastur bulk, or in the heavy upper tail, and low-rank compression discarded the bottom of the spectrum as the cheapest part to lose. This paper points out that a rectangular matrix’s null spectrum ends above zero, so “below the bulk” is a departure from the null too. It measures such departures in three trained transformers, shows that in the MLP up and gate projections their singular vectors line up with the principal directions of the matrix’s input, shows that removing them costs more than removing most larger singular values, and gives one solvable model in which a learned direction ends up below the bulk.

Key insight

What locates information in a learned matrix is distance from the null spectrum, not size. Under the i.i.d. null, the singular values of a rectangular matrix fill an interval [ν₋, ν₊] with ν₋ > 0; a learned direction can leave that interval at either end. Magnitude order, which Frobenius-optimal truncation follows, is the right order only when the null reaches zero, as it does for square matrices; and there the paper’s ablations do fall monotonically with magnitude.

Assumptions

• The null. Entries i.i.d., zero mean, finite variance; m, n → ∞ with q = n/m fixed (n ≥ m). Then the singular-value density is Eq. 4 on [ν₋, ν₊], ν± = σ̃(1 ± √(1/q)), σ̃ = σ√n. The variance of each empirical matrix is estimated by Azadkia’s method, not fitted to the bulk.
• Finite size. BERT and Pythia matrices have a smaller dimension of 768 and 1024, Llama 4096. Outlier counts (Table 3) depend on where the finite matrix’s edge is drawn; no finite-size correction or edge-fluctuation scale (Tracy–Widom) is used, so “outlier” means “outside the asymptotic support”.
• Overlap. O_k = max_j (v_k · f_j), v_k the k-th right singular vector, f_j the eigenvectors of the covariance of the activations entering the matrix, estimated on WikiText (BookCorpus in Appendix B). As printed it has no absolute value, so it depends on the arbitrary signs of v_k and f_j; an absolute value is presumably meant. The 3σ reference band is for a random unit vector.
• Ablation. Zero one decile of the rank-ordered singular values (Eq. 9) in every matrix of one type, or (RULER, BERT) of all types at once, and reconstruct from the original singular vectors. No re-fitting.
• The model (Section 6). Inputs ξ ~ N(0, 𝟙); teacher uᵀλξ, ‖u‖ = 1; student vᵀWξ with v fixed and normalised, W ∈ ℝ^{K×N}, q = K/N < 1. Weights are drawn from the Gibbs ensemble P(W) ∝ exp[−Nβε_g(W) − (N/2α) Tr WᵀW], β the inverse temperature (in an annealed approximation, examples per input dimension) and α a Gaussian prior or L2 strength. A linear network, one rank-one rule, one fixed second layer.

Key results

• Spectral shape (Figure 1, Table 3). Initialized matrices agree with Eq. 4. Trained ones show outliers above the bulk in all matrix types and below it in rectangular ones: Llama-3.1-8B, layer 9, Up-Projection 120 below and 253 above; Query 0 below, as it must be. Below-bulk counts are 2–6.5% of singular values in the up and down projections (Appendix H), with no clear trend across the three model sizes.
• Overlap (Figures 2–4, 8–9, 12). Above-bulk outliers overlap the activation-covariance eigenvectors beyond the 3σ band in Query, Key and Value. Below-bulk outliers do so in Pythia’s Up- and Down-Projection and Llama’s Up- and Gate-Projection. Not in Attention-Output (any model), not in Llama’s Down-Projection, and not for the below-bulk end of Llama’s Key and Value although those are rectangular. Over Pythia’s pretraining (Figure 10) above-bulk overlap appears within 1,000 steps; below-bulk overlap in the projections appears late.
• Perplexity (Figure 5; Figure 11 on BookCorpus). Removing the largest decile always hurts most. In square matrices damage decreases monotonically down the deciles; in rectangular ones the smallest decile outranks some larger ones, and for the Down-Projection it is second.
• Benchmarks (Llama-3 8B). GSM8K 3-shot, baseline 43.2% (Table 1): Down-Projection deciles 1 to 10 leave 2.0, 30.2, 28.0, 27.1, 26.2, 22.0, 11.0, 15.0, 5.0 and 0.0%; Attention-Output decile 1 leaves 40.0%; Query 40.3%; Gate-Projection 34.1%, the fourth most damaging. HumanEval, baseline 32.32% (Table 5): Down-Projection decile 1 leaves 0.0%, as do deciles 9 and 10. RULER at 8192 tokens, all matrix types at once (Table 2): decile 1 scores 0.0 on all five tasks, as does decile 10; decile 9 scores 0.0 on four and decile 8 on three.
• Layers (Table 4, BookCorpus perplexity, baseline 6.0045). Removing a decile from one layer barely matters except layer 0, where the top decile gives 62,355.6 and the bottom decile 6.3161, the second largest.
• Fine-tuning order (Figure 6, BERT, all matrices, 3σ from six unpruned runs). Prune decile 1 then fine-tune: BoolQ recovers, RTE and SST-2 lose a little but significantly. Fine-tune then prune: all three lose significantly.
• The model (Eqs. 11–13, Appendix F). Completing the square gives W = W₀ + X with W₀ = αβλ/(1 + αβ) vuᵀ and X Gaussian with columns uncorrelated and row covariance Σ = α𝟙 − α²β/(1 + αβ) vvᵀ (Eq. 12): the noise variance along v drops from α to α/(1 + αβ). For 0 < (1 + αβ)⁻¹ < 1, X has a singular value below its bulk, and for λ < √(α(1 − q)) this stabilises a below-bulk outlier of W at ⟨ν_min⟩ (Eq. 13), which as β → ∞ tends to λ√(1 − αq/(α − λ²)), the teacher’s λ shifted by level repulsion. The outlier merges with the bulk at β_bulk = α²(1 − √(α² − 4√q αλ²))/2λ². The expected generalisation error is α/2(1 + βα) + λ²/2(1 + βα)² (Eq. 23), smallest where the outlier is farthest from the bulk. Numerically (Figure 7; α = 1, λ = 0.2, N = 2048, K = 512, 100 draws) the overlaps of the outlier’s singular vectors with u and v fall from their maximum at perfect learning to chance at the merger.

Claims

id	claim	strength	support
C1	Initialized transformer matrices follow the Marchenko–Pastur law; trained ones depart from it	strong	Figure 1, all three models
C2	Rectangular trained matrices have outliers below the bulk; square ones cannot	strong (the second half is the null’s geometry; the first is counted)	Eqs. 4–5, Table 3
C3	Directions that depart from the null align with the input’s principal directions	moderate, and not uniform: holds for above-bulk outliers and for the up and gate projections’ lower outliers, fails for Attention-Output, Llama’s Down-Projection and the lower end of Key and Value	Figures 2–4, 8–9, 12
C4	In rectangular matrices the smallest decile is more important to keep than some larger deciles; in square ones importance falls with magnitude	strong for the perplexity and GSM8K ablations	Figure 5, Tables 1 and 5, Figure 11
C5	The singular values that depart from the null are what matter in that decile	weak	the decile is 10% of the spectrum while lower outliers are 2–6.5%; no ablation of the outliers alone
C6	Fine-tuning writes task information into the small directions	moderate, for one model	Figure 6, BERT, three datasets
C7	Noise suppressed along a learned direction can place that direction below the bulk, with information growing with distance from it	moderate within the model: an exact ensemble, with the outlier location quoted from an unpublished thesis and checked numerically	Section 6, Appendix F, Figure 7
C8	This is the mechanism in real networks	not supported	the authors say the noise covariance of real weights would need many training runs to estimate

Method

Two measurements and a model. The spectrum of each weight matrix is compared with the Marchenko–Pastur density at an estimated variance; each right singular vector is projected on the eigenbasis of the activation covariance at the matrix’s input and its largest coefficient recorded. Then deciles of the spectrum are zeroed across all matrices of a type and the damage measured by perplexity and benchmark accuracy. The model is a Gibbs (Gaussian) posterior over the first layer of a two-layer linear student; the loss and the prior combine into a Gaussian whose mean is the rule and whose covariance is the prior’s isotropic noise with one direction squeezed.

Concepts

• zero-information hypothesis: the i.i.d. random matrix whose spectrum is Marchenko–Pastur; agreement with it is read as “nothing learned here”.
• bulk and outlier: the asymptotic support [ν₋, ν₊], and a singular value outside it, at either end.
• activation covariance matrix: C = ⟨(h_i − h̄)(h_i − h̄)ᵀ⟩ over tokens i, for the activations h entering one matrix (Eq. 6).
• overlap O_k: the largest coefficient of the k-th right singular vector in that covariance’s eigenbasis (Eq. 7). M_k (Appendix C) is the same in the standard basis, a localisation measure.
• decile: a tenth of the rank-ordered singular values, decile 1 the smallest.
• lazy regime: training that leaves weights near initialisation, and so near the null; offered as the reason Attention-Output shows little overlap (Appendix D).

Connections

The method is the authors’ own earlier programme, random-matrix filtering of weight matrices (Thamm, Staats and Rosenow 2022 and 2023, Phys. Rev. E), which follows Martin and Mahoney’s reading of trained spectra and Papyan’s class structure in deep-learning spectra. The upper outlier is the BBP transition (Baik, Ben Arous and Péché; Benaych-Georges and Nadakuditi). The lower outlier of Section 6 is, in my reading, the same transition at the lower edge for a noise with one reduced covariance direction, the sample-covariance case of a spike below one; the paper does not name it so. The empirical target is Eckart and Young’s theorem, which the paper cites for the view that small singular values are the cheapest to lose. The fine-tuning experiment is aimed at Hsu et al. (weighted low-rank factorisation) against Sharma et al. (LASER).

Bearing on the record

• LIT-330 (Eckart and Young). The theorem is untouched: the best rank-r Frobenius approximation is still the magnitude truncation. What this reading adds is that Frobenius distance weights a direction by its singular value squared, while the function of a trained rectangular matrix does not; Table 1’s Down-Projection loses 41 points of GSM8K to the 10% of its spectrum that Eckart–Young treats as cheapest. The record’s reading of Eckart and Young (NOTE-301) already says the theorem fixes a truncation of one matrix and nothing more; this is a measured case of that limit.
• THEORY-078 (the Fisher’s top C eigenvalues stand clear of the bulk and a cut at the bulk edge keeps the span of the class means). A different matrix, the Fisher rather than a weight, so nothing here changes it. But the convention of cutting at the upper bulk edge is one-sided, and this paper shows that for a rectangular matrix, information can also sit below the lower edge. Whether the Fisher’s Gram form has such lower outliers is not asked anywhere in the record.
• LIT-672, LIT-369, LIT-617, LIT-619. The same line of random-matrix readings of trained networks: Martin and Mahoney’s statistical mechanics, the HTSR tail exponent in the grokking study, and Papyan’s bulk-and-outlier Hessians. All of them read the upper end; this paper is the record’s first reading of the lower end.
• The anthology’s reading (ANTH-LIT-517, ANTH-NOTE-263). It reads the paper for low-rank-plus-quantization practice, and its account matches this one on the numbers. One point it states more broadly than the paper supports: its key results put the lower-end overlap “in Up/Down/Gate projections of Pythia and Llama”, while the paper finds none in Llama’s Down-Projection (the anthology’s own limitations note this) and none for Llama’s rectangular Key and Value (not noted there). Its reading of RULER, decile 1 scoring the same as decile 10, is right, and it means Table 2 cannot rank decile 1 second, as the paper’s text does. Neither the anthology’s reading nor the paper notes the decile-versus-outlier gap (C5). Reported per ADR-013, not fixed from here.
• The manuscript’s argument. No claim on the pragmatic-transport line is about weight spectra. The only echo is with the task-relative view of preservation (CLAIM-088, QUESTION-016): a reduction optimal under a task-blind distortion here discards what the task needed. That is an analogy, not evidence for those claims.
• Practice. The paper’s guidance for SVD pruning is ML practice and belongs to the anthology, which holds it.
• No THEORY filed. The measurements are of three models and already read in the anthology. The model’s lower-outlier result rests on a formula quoted from an unpublished thesis, and the general random-matrix fact behind it, that a covariance spike below the bulk produces a lower outlier, is older than the paper and not established by it.

Limitations

• The ablation does not test the outliers. Lower outliers are 2–6.5% of the spectrum; the smallest decile is 10%, so it mixes them with the lower bulk. The claim that RMT-violating values matter more than bulk values is therefore inferred, not measured; ablating exactly the below-bulk values, and an equal number of lower-bulk values, would test it.
• Overlap and importance come apart. Llama’s Down-Projection has no lower-end overlap yet the most damaging smallest decile; Llama’s Key and Value have hundreds of lower outliers (Table 3) and no lower-end overlap. Attention-Output has no overlap anywhere; Appendix D reads its averaged spectrum as having far fewer outliers than Query, yet Table 3 still counts hundreds per layer (723 in Llama’s layer 0). The paper offers GLU placement, localisation (M_k) and lazy training as speculations.
• Single runs. The perplexity and benchmark ablations report no repetitions; RULER is visibly non-monotone (on the needle tasks decile 6 scores about 99 while deciles 4 and 7 score 0–13). Only the BERT fine-tuning has a 3σ band.
• Removed norm is not controlled. A higher decile removes more Frobenius norm than a lower one; damage per unit of removed norm is not reported, though it would make the paper’s case stronger, not weaker.
• The model is linear, two-layer, rank-one, Gaussian, with a fixed second layer. It shows a lower outlier can arise and carry information; the authors say themselves that showing it does so in an LLM needs the noise covariance of real weights, which needs many training runs.
• Three models, differing in size, data and architecture; Appendix H declines to read a scaling trend from them.

Open questions

• Ablate the below-bulk outliers alone, against an equal number of values from the lower bulk, with seeds: does the importance follow the departure from the null, or just the smallest decile?
• Does the information measured by overlap with the teacher in the model correspond to anything measurable in a real matrix, for example the overlap gap between below-bulk and lower-bulk vectors?
• Why do Key and Value (rectangular under grouped-query attention, by my reading of Llama’s shapes) have lower outliers without overlap? Is the lower edge populated there by finite-size effects rather than learning?
• Do Gram-type matrices in the record, the Fisher of THEORY-078 among them, have structure below their lower bulk edge when they are rectangular in the relevant sense?