Quantization Dominates Rank Reduction for KV-Cache Compression
arXiv:2604.11501v1 [cs.LG] 13 Apr 2026
Samuel Salfati fraQtl AI Research [email protected]
Abstract We compare two strategies for compressing the KV cache in transformer inference: rank reduction (discard dimensions) and quantization (keep all dimensions, reduce precision). At matched storage budgets across five models (124M–14B, MHA and GQA), we find that quantization consistently outperforms rank reduction—by 4–364 PPL depending on model and compression level. The gap persists even when rank reduction is combined with quantization in hybrid baselines, and it grows with GQA aggressiveness. On LAMBADA, INT4 matches FP16 accuracy (+0.23 PPL on Mistral 7B, +0.58 on GPT-2) while rank-32 at identical storage collapses to 0.4%. We trace this gap to a structural asymmetry: under softmax attention routing, removing a dimension can flip which token is attended (a discrete failure), while quantization noise is bounded and typically preserves score ordering. We formalize this via a perturbation result showing projection damage exceeds quantization damage by 3 × 22b per direction under the softmax Fisher metric. A basis ablation confirms the finding is basis-independent (spread <0.4 PPL), establishing that the advantage comes from preserving dimensions, not from a better coordinate system. Joint K+V INT4 quantization achieves 75% total KV reduction at only +0.18 PPL on Mistral 7B.
1
Introduction
The KV cache stores key and value activations for all previous tokens during transformer inference, consuming 2LHdT elements for L layers, H heads, dimension d, and sequence length T . Two strategies exist for reducing this cost: rank reduction, which projects activations to fewer dimensions, and quantization, which keeps all dimensions but represents them at lower precision. Both have active research literatures [Yang et al., 2024, Lesens et al., 2025, Hsu et al., 2022, Liu et al., 2024, Kang et al., 2024], but they are rarely compared at the same storage budget. We provide this comparison. The result is clear: in every setting we test, quantization outperforms rank reduction at matched storage. INT4 achieves near-lossless compression (+0.23 PPL on Mistral 7B at 75% reduction), while rank-32 at identical storage loses 34.77 PPL. Even hybrid approaches (rank reduction followed by quantization of the retained dimensions) cannot match full-rank quantization. This is not because rank reduction uses the wrong basis—our basis ablation shows quantization quality is independent of the coordinate system (spread <0.4 PPL). The issue is that deleting dimensions is qualitatively worse than reducing their precision in a system governed by softmax routing. We explain why through the geometry of softmax decision boundaries and formalize it with a perturbation asymmetry result. Contributions. Preprint.
1. An empirical comparison of quantization and rank reduction at matched storage across five models (124M–14B), two metrics (PPL, LAMBADA), two datasets (WikiText-2, C4), and hybrid baselines. No prior work provides this comparison. 2. A theoretical explanation: under the softmax Fisher metric, projection damage exceeds quantization damage by 3 × 22b per direction (Proposition 2), because deletion can flip attention routing while bounded noise typically preserves it. 3. A downstream-optimal compression framework (Theorem 1) that provides the best rank-reduction basis (52% gap recovery at zero cost) and unifies PCA, Fisher-SVD, GPTQ [Frantar et al., 2022], AWQ [Lin et al., 2023], and KQ-SVD as special cases.
2
Background and Related Work
KV-cache quantization. KIVI [Liu et al., 2024] quantizes the KV cache at INT2/INT4 and achieves equivalent PPL to our approach (44.21 on GPT-2). GEAR [Kang et al., 2024] adds low-rank residual correction. We do not claim a better quantization method; our contribution is the matched-budget comparison and the explanation of why quantization wins. Low-rank KV compression. MatryoshkaKV [Yang et al., 2024] learns nested subspaces. KQSVD [Lesens et al., 2025] derives closed-form projections using V W O ; our attention-output metric αV outperforms theirs by 31.6 PPL under joint compression (Appendix G). Sensitivity-weighted methods. FWSVD [Hsu et al., 2022] shows Fisher-weighted metrics beat PCA. Our framework recovers this as a special case and identifies an exact solution for attention values. Token eviction. H2O [Zhang et al., 2023] and Scissorhands [Liu et al., 2023] reduce sequence length rather than dimension; the paradigms are orthogonal. The gap we fill. KIVI evaluates quantization only. MatryoshkaKV and KQ-SVD evaluate rank reduction only. Neither compares the paradigms at matched bit budgets with perplexity and downstream validation. Our experiments provide this comparison for the first time.
3
Why Deletion Is Worse Than Noise Under Softmax
3.1
A General Framework for Compression
The right way to compress a linear component depends on what happens downstream. We formalize this with a general result. Theorem 1 (Downstream-Optimal Compression). For a linear component W ∈ Rm×n feeding into differentiable f , the rank-r projection P ∈ St(r, n) minimizing E∥f (W x) − f (W P P ⊤ x)∥2 (first-order) uses the top-r eigenvectors of M = E[W ⊤ Jf⊤ Jf W · xx⊤ ] ∈ Rn×n , where Jf = ∂f /∂z. Exact when f is linear. For attention values, f (·) = α · (·) is linear, giving an exact eigenbasis MV = V ⊤ α⊤ αV . This is PCA on the attention output αV , not on V itself, and recovers 52% of rank-reduction damage at zero cost vs PCA’s 0%. The framework unifies existing methods as special cases (Appendix F). But this framework also reveals why rank reduction has a fundamental problem that quantization avoids. 3.2
Softmax Creates Decision Boundaries
√ Attention routing is governed by α = softmax(QK ⊤ / d). The softmax Jacobian J = diag(αt ) − αt αt⊤ (where αt ∈ RS is row t) shows that sensitivity peaks when the score gap ∆ = si1 − si2 between the top-2 competing tokens is small. At these decision boundaries, a small perturbation can flip which token receives attention. 2
Table 1: Quantization vs rank reduction at matched storage (d × b vs r × 16 bits/token/head).
3.3
Model
Budget
GPT-2 (MHA) GPT-2 (MHA) Llama 3.2 (GQA-3) Llama 3.2 (GQA-3) Mistral 7B (GQA-4) Mistral 7B (GQA-4) Qwen 3B (GQA-8) Qwen 3B (GQA-8) Qwen 14B (GQA-5) Qwen 14B (GQA-5)
256b (75%) 512b (50%) 512b (75%) 1024b (50%) 512b (75%) 1024b (50%) 512b (75%) 1024b (50%) 512b (75%) 1024b (50%)
Rank (FP16)
Quant (INT)
Margin
96.71 56.30 69.10 21.50 44.19 13.07 377.00 33.99 161.54 13.90
44.57 44.00 13.61 13.22 9.42 9.19 12.93 12.69 7.23 6.99
+52.14 +12.30 +55.50 +8.29 +34.77 +3.88 +364.07 +21.30 +154.31 +6.91
The Asymmetry Between Deletion and Noise
Proposition 2 (Perturbation Asymmetry). Under the softmax Fisher metric G = diag(α) − αα⊤ , direction u with signal σu incurs: KLproj = 12 σu2 · u⊤ Gu
(1)
σ2 E[KLquant ] = 21 · 3·2u2b · u⊤ Gu
(2)
Ratio: 3 × 22b (768× at INT4). This is an upper bound; wider quantization ranges decrease the ratio. The sensitivity u⊤ Gu cancels—both methods see the same softmax geometry. The difference is entirely in perturbation magnitude. Rank reduction deletes a direction’s contribution to the score gap ∆; if |δi | > |∆|, attention flips—a discrete failure. Quantization perturbs by O(λi /2b ), crossing the boundary only when noise exceeds ∆, which is rare at b ≥ 4. Below 3 bits, quantization noise exceeds typical gaps, triggering the same instability (INT2: 73–138 PPL). Under GQA [Ainslie et al., 2023], a single deleted direction can flip routing for multiple query heads at different positions, amplifying the damage. This explains the growing margin: MHA (+12–52), GQA-3 (+8–55), GQA-4 (+4–35), GQA-8 (+21–364).
4
Experiments
Setup. Five models: GPT-2 (124M, MHA, d=64), Llama-3.2 3B (GQA-3, d=128), Mistral 7B (GQA-4, d=128), Qwen 3B (GQA-8, d=128), Qwen 14B (GQA-5, d=128). WikiText-2, strict three-way splits. All FP16. LAMBADA (500 examples) and C4 for validation. Rank reduction uses the V Theorem eigenbasis with K-PCA at matching rank. Quantization: per-channel symmetric, V only. 4.1
Core Result: Matched-Budget Comparison
Quantization outperforms rank reduction at every budget above 3 bits on all five models (Table 1). The margin grows with GQA aggressiveness.1 4.2
Hybrid Baselines
Can rank reduction recover by quantizing the retained directions? No. Even Rank-64 INT8 fails to match full INT4 at the same budget (Table 2). The information destroyed by deleting dimensions cannot be recovered by giving more bits to what remains. The Pareto frontier (Figure 1) makes the gap visual: quantization traces the FP16 baseline closely above 3 bits, while rank reduction remains orders of magnitude worse at every matched budget. 3
Table 2: More directions at lower precision beats fewer at higher precision, at matched storage. Model (budget)
Method
Dirs
PPL
Mistral (512b) Mistral (512b) Mistral (512b)
Full INT4 Rank-64 INT8 Rank-32 FP16
128 64 32
9.42 13.06 44.19
Qwen 14B (512b) Qwen 14B (512b) Qwen 14B (512b)
Full INT4 Rank-64 INT8 Rank-32 FP16
128 64 32
7.23 13.90 161.54
Figure 1: Perplexity vs storage budget on Mistral 7B. Quantization (blue) achieves near-FP16 quality above 3 bits per dimension. Rank reduction at matched storage (red) is catastrophically worse at every budget. The gap between curves is the paper’s central finding.
4.3
Basis Ablation
All bases give equivalent quantization quality (spread <0.4 PPL, Table 3). Simple INT4 in the original basis matches KIVI [Liu et al., 2024]. The advantage of quantization over rank reduction is not about the coordinate system—it is about preserving all dimensions.
4.4
Mechanism Validation
Rank-32 induces 13× larger attention KL than INT4 (Table 4). Attention KL is monotonically correlated with PPL (r = 0.99). At gap-conditioned thresholds (∆ > 0.05), INT4 flips 0.03% of routing decisions while rank-32 flips 4.6%. This confirms the mechanism: rank reduction causes discrete routing failures that quantization avoids.
4.5
Joint K+V Quantization
The results above compress V only. Joint K+V INT4 quantization achieves 75% total KV reduction: Joint INT4 costs only +0.18 PPL for 75% total KV memory reduction. K quantization adds negligible damage (+0.04 PPL alone), confirming that both K and V are safely quantizable. 4
Table 3: Quantization quality is basis-independent (GPT-2 INT4, FP16: 43.99). Basis
INT4 PPL
Original (no rotation) Theorem (V ⊤ α⊤ αV ) PCA (V ⊤ V ) Random orthogonal
44.21 44.57 44.48 44.39
Table 4: Attention KL at matched 512-bit budget (Mistral, per-sequence, 3 seeds).
4.6
Method
Attention KL
Flip rate
PPL
INT4 full Rank-32 FP16
0.013 ± 0.002 0.174 ± 0.023
2.4% 10.4%
9.42 44.19
Downstream and Cross-Dataset Validation
INT4 matches FP16 accuracy on LAMBADA; rank reduction collapses (57× gap on Mistral, Table 6). On C4 validation (Mistral), INT4 achieves 10.17 PPL (+0.13) vs Rank-32 at 69.08—a 58.91 PPL margin. 4.7
Calibration and Stability
Eigenbasis calibration: 1.87 s (GPT-2), 9.04 s (Mistral). One-time cost. Across 3 seeds on Mistral: INT4 σ = 0.03, Rank-32 σ = 0.86 (29× more variable). The INT4 vs Rank-32 gap is 626× the INT4 standard deviation.
5
Discussion and Conclusion
What we show. For KV-cache compression in the models and budgets we test, quantization consistently outperforms rank reduction. This holds across five models (124M–14B), MHA and multiple GQA configurations, perplexity and LAMBADA, WikiText-2 and C4, and even when rank reduction is combined with quantization. Why it happens. Attention is a routing system governed by softmax competition. Removing a dimension can change which token wins that competition—a discrete failure that bounded quantization noise typically avoids. The asymmetry is formalized by Proposition 2 and validated by the 13× attention KL gap. What we do not claim. We do not claim this extends to all compression settings, all models, or all tasks. Our theory provides an explanation for the observed results, not a universal law. Practical implication. Per-channel INT4 quantization of the KV cache requires no special basis, no optimization, and no retraining. At 75% KV reduction on Mistral 7B: +0.23 PPL. Limitations. Five models up to 14B; larger models untested. LAMBADA on two models. No wallclock latency (rank reduction saves attention compute; quantization saves bandwidth). Cache-level quantization validated at INT8/INT4 only. Broader lesson. At fixed storage budgets, the dominant source of damage in attention compression is dimensional deletion rather than low-precision noise. Compression of routing-sensitive systems should preserve decision geometry, not just representation energy.
References Ainslie, J., et al. GQA. arXiv:2305.13245, 2023. 1 Bit budgets count data bits only. Per-channel scale overhead: <2% at INT4 for T ≥ 256.
5
Figure 2: Softmax decision boundary. (a) Two tokens compete. (b) INT4 adds bounded noise—stays on correct side (flip rate 0.03% at gap >0.05). (c) Rank reduction removes a dimension—crosses the boundary (flip rate 4.6%).
Figure 3: Attention routing disruption (KL divergence) predicts compression damage. Monotonic across all three methods at matched 512-bit budget on Mistral 7B. Frantar, E., et al. GPTQ. arXiv:2210.17323, 2022. Hsu, Y.-C., et al. FWSVD. ICLR, 2022. Kang, H., et al. GEAR. arXiv:2403.05527, 2024. Lesens, D., et al. KQ-SVD. arXiv:2512.05916, 2025. Lin, J., et al. AWQ. arXiv:2306.00978, 2023. Liu, Z., et al. Scissorhands. NeurIPS, 2023. Liu, Z., et al. KIVI. arXiv:2402.02750, 2024. Merity, S., et al. Pointer Sentinel Mixture Models. arXiv:1609.07843, 2016. Paperno, D., et al. LAMBADA. ACL, 2016. Yang, L., et al. MatryoshkaKV. arXiv:2410.14731, 2024. Zhang, Z., et al. H2O: Heavy-Hitter Oracle for Efficient Generative Inference. NeurIPS, 2023.
6
Table 5: Joint K+V quantization (Mistral 7B, FP16: 9.19). Configuration
PPL
∆
K full + V full K INT8 + V INT8 K INT4 + V INT4
9.19 9.20 9.37
+0.00 +0.01 +0.18
Table 6: LAMBADA (500 examples) at matched storage. Model
Method
Bits
Accuracy
GPT-2 GPT-2 Mistral Mistral
INT4 Rank-16 FP16 INT4 Rank-32 FP16
256 256 512 512
7.20% 0.20% 23.00% 0.40%
A
Mechanism: Full Analysis
A.1
Influence vs Energy (Spice Ratio)
For direction ui with eigenvalue λi , the spice ratio Si = Influence(ui )/λi measures damage per unit energy. On GPT-2, Spearman ρ(energy rank, influence rank) = 0.38—62% of the damage ranking is invisible to energy-based methods. Block 0 exhibits 84× higher maximum spice than late blocks. A.2
Projection vs Quantization: Discrete vs Continuous
Rank reduction removes a direction’s contribution to ∆ = stop1 − stop2 . If |δi | > |∆|, the softmax flips—a discrete event. Quantization perturbs by O(λi /2b ); the boundary is crossed only when this exceeds ∆, rare at b ≥ 4. A.3
Gap-Sensitive Importance
Compression damage at position t scales as (∆t + ε)−1 · ∥perturbation∥. Near-boundary positions (∆t ≈ 0) are exclusively vulnerable to projection. Quantization distributes errors uniformly. A.4
Cross-Layer Propagation
A routing flip at layer ℓ produces a corrupted hidden state affecting all subsequent layers. On Mistral (32 layers), top-1 agreement is 0.966 per layer under rank-64 projection, but 3.4% misrouted tokens per layer compound through residual connections (assuming approximately independent errors across layers, which provides an upper bound on accumulation). Under quantization: no routing flips, per-layer noise adds linearly, producing +0.23 PPL at INT4. A.5
GQA Amplification
Under GQA, a single removed direction can flip routing for multiple query heads at different positions. Damage multiplies with the number of consumers: MHA (+12–52), GQA-3 (+8–55), GQA-4 (+5–55), GQA-8 (+21–364). A.6
Calibration Stability
Three seeds on Mistral (held-out split): INT4 10.04 ± 0.03, Rank-32 28.83 ± 0.86. Gap is 626× INT4 std. Rank reduction is 29× more sensitive to calibration seed.
B
Full Proof of the General Theorem
Step 1.
∆y = f (W P P ⊤ x) − f (W x) ≈ −Jf W (I − P P ⊤ )x. 7
Step 2.
∥∆y∥2 ≈ x⊤ (I − P P ⊤ )W ⊤ Jf⊤ Jf W (I − P P ⊤ )x.
Step 3. Taking expectations, applying idempotency (I − P P ⊤ )2 = I − P P ⊤ and cyclicity: E[∥∆y∥2 ] = Tr[(I − P P ⊤ )M ] = Tr[M ] − Tr[P ⊤ M P ]. Step 4. Maximizing Tr[P ⊤ M P ] over P P ∈ St(r, n): decompose M = U ΛU ⊤ . For any P : P ⊤ ⊤ 2 2 ∗ Tr[P M P ] = i λi ∥ui P ∥ . Subject to i ∥u⊤ i P ∥ = r, this is maximized by P = [u1 , . . . , ur ]. □ For the V Theorem: M = V ⊤ α⊤ αV exactly (no approximation since Jf = α is constant).
C
Loss-Theoretic Foundation
Under Fisher approximation HO ≈ FO at a trained model: E[∆L] ≈ 21 E[vec(∆O)⊤ FO vec(∆O)]. With FO ≈ cI: E[∆L] ≈ 2c Tr[(I − P P ⊤ )E[V ⊤ α⊤ αV ]]—exactly the V Theorem objective. Validation. Full loss Jacobian: +2.3 PPL over V Theorem on GPT-2, +0.02 on Mistral. W O orthonormality: blocks 3–9 CV(σ) < 0.27 (good); block 11 CV= 1.04 (poor). The isotropic assumption holds for most of the network.
D
Rayleigh–Schrödinger Perturbation Analysis
Expanding V Theorem eigenvectors around PCA via RS perturbation: Table 7: RS convergence (GPT-2, rank 32). Perturbation ∥∆∥/δmin ∼ 5000–10000. Method
PPL
Theorem gap rec.
∆/order
PCA RS Order 1 RS Order 2 RS Order 3 Padé [2/1] V Theorem
104.14 96.80 93.86 95.67 93.49 85.36
0% 39% 55% 45% 57% 100%
— +7.35 +2.94 −1.82 (diverges) +0.37 —
The series converges through order 2 but diverges at order 3 (∥c3 ∥/∥c2 ∥ up to 6675×). This is asymptotic series behavior: the first terms are useful despite formal divergence. The correction has exactly two perturbative layers: per-direction (RS1, 39%) and pairwise interactions (RS2, +16%). The remaining 45% is genuinely non-perturbative—requiring the full eigendecomposition. Subspace overlaps. RS2 moves PCA by only 1–2% in subspace overlap—but in the right direction. The PPL improvement comes from tiny corrections concentrated in high-impact directions.
E
The Geometric Dissection (Experiments 68–73)
E.1
Exp 68: The Gain Law
Per-direction gain prediction (GPT-2, block 6, rank 64): Table 8: Gain law predictors. Predictor
R2
λ alone θ√alone λ×θ Fisher
0.813 0.511 0.887 0.018
8
Direction 0 (λ = 34,218) alone: +26.25 PPL, closing 36% of the gap. The predicted ranking nearly matches the oracle. Fisher is useless (R2 = 0.018). E.2
Exp 69: The Angle Law
θi ≈ a − b log λi , R2 = 0.885, corr(θ, 1/λ) = 0.965. Best two-predictor model (log λ, Fisher): R2 = 0.925. Combined closed-form gain prediction: R2 = 0.820. E.3
Exp 70: Can We Construct the Correction?
Membership test. Swapping directions: n = 1: +1 PPL; n = 32: −65 PPL. PCA membership is correct. Heuristic rotation. Gradient-based: zero useful signal. Restricted learning: random beats spectrum at every budget (8 dirs: 36.6% vs 23.4%; 32 dirs: 83.6% vs 63.3%). Structure. Each corrected direction: 91% self-overlap, ∼10% from many complement directions. Small (∼6◦ ), distributed rotation. E.4
Exp 71: The Metric Hypothesis
Fisher: 5230 PPL (−6754%). Σ + 0.01F : 218 (7.6%). Σ0.8 F Σ0.8 : 206 (23.3%). Learned: 148 (100%). Fisher-learned cosine: ∼0.05. No second-order matrix explains the correction. E.5
Exp 72: Sequential Correction
Across all 12 GPT-2 blocks at k = 64 (8% retention): 84% gap recovery. At k = 256 (33%): 33%. Sequential correction propagates through residuals as coupled rotations. E.6
Exp 73: Isotropy
SVD energy of correction operator A: identical in PCA, Fisher, Jacobian, and sandwich bases (2.6%, 10.4%, 20.8%, 41.4% at ranks 1, 4, 8, 16). The correction is isotropic on the Grassmannian. Column norms: tail directions receive 6× larger corrections (0.496 vs 0.080), consistent with angle law. E.7
Summary 1. PCA membership correct; correction is ∼6◦ distributed rotation (Exp 70) √ 2. Gain law λθ: R2 = 0.887; angle law θ ∝ 1/λ: R2 = 0.885 (Exp 68-69) 3. No heuristic, metric, or perturbation series (beyond order 2) captures the correction (Exp 70-71, THEORY-01) 4. Correction operator is isotropic—no basis simplifies it (Exp 73) 5. 23% from second-order structure, 77% from joint interactions (Exp 71, 73)
F
The Rank-Reduction Objective Hierarchy: Full Data
F.1
GPT-2 Closed-Form Methods
Four independent methods converge to ∼82 PPL: the closed-form ceiling under rank reduction. This ceiling is an artifact of the rank-reduction paradigm, not of the metric. F.2
GPT-2 Learned Methods
F.3
GPT-2 E2E Convergence
On Mistral, theorem-init at 0 steps (13.50) beats PCA-init at 25 steps (14.58). The theorem starts ahead and PCA never catches up in the tested range. 9
Table 9: GPT-2 closed-form methods (rank 32, 50% KV dim reduction). Method
PPL
Gap rec.
What it adds
PCA V Theorem V Theorem + W O V Theorem + W O + LN Entropy-weighted Full loss Jacobian
130.95 85.36 84.22 82.42 82.27 83.02
0% 52% 54% 56% 56% 55%
— Attention weights α Output projection LayerNorm Jacobian Attention confidence Full downstream gradient
Table 10: GPT-2 learned methods (rank 32). Method
PPL
Gap rec.
Steps
Learned MSE KL from PCA init KL from theorem init KL from entropy init
102.17 85.26 77.74 76.73
33% 52% 61% 62%
57K 58K 58K 58K
F.4
Mistral Full Hierarchy
F.5
Mistral Diagnostics
Routing preservation. V Theorem top-1 agreement: 0.966 (vs PCA 0.956). Wins 31/32 layers. Gap is value distortion through 32 residual layers, not routing error. GQA aggregation. Arithmetic (15.94), geometric (15.92), harmonic (16.20). ∆ < 0.3 PPL. Not the bottleneck. GQA limitation. Simulated on GPT-2: V Theorem beneficial at group=1-2, harmful at group≥3. Real Mistral (co-trained heads, alignment 0.75–0.89) still benefits for both rank reduction and quantization.
G
KQ-SVD Head-to-Head
KQ-SVD degrades below PCA under joint compression: their static W O metric doesn’t adapt to changed attention patterns. Combined metric fails—negative eigenvalues in ∼90% of heads.
H
Entropy-Weighted Metric
MVent = V ⊤ α⊤ Dent αV where Dent = diag(exp(−H(αt ))). Middle layers have focused attention (entropy 1.6–2.3) → entropy weighting upweights. Early/late layers diffuse (3.7–4.4) → downweighted. W O orthonormality: blocks 3–9 deviation <5% (theorem ≈ loss-optimal). Block 11: 18% deviation, κ = 19.6 (poor).
I
Quantization Stacking Under Rank Reduction
Negative quant cost on PCA/learned: quantization noise regularizes suboptimal projections. Good projections (V Theorem, E2E) pay a small positive cost. Note: These rank-reduction+INT4 results (11.20 at 87.5%) are now superseded by basis-only INT4 (9.43 at 75%) from the main experiments. Basis quantization achieves better quality at more aggressive compression. 10
Table 11: Theorem-init → E2E convergence (GPT-2, K+V KL). Steps
PCA→E2E
Thm→E2E
Speedup
10 50 200
99.59 73.73 62.78
74.14 66.39 61.42
5× 2× Converging
Table 12: Theorem-init → KL convergence (Mistral 7B, V-only).
J
Steps
PCA→KL
Thm→KL
Advantage
0 5 10 25
15.74 15.44 15.19 14.58
13.50 13.49 13.46 —
+2.24 +1.95 +1.73 Thm@0 still ahead
Downstream Evaluation
Cross-dataset PPL (WikiText-2 train): E2E transfers cleanly (85–91% gap, ∆ < 0.5 PPL across seeds). Reconstruction-learned worsens LAMBADA vs PCA at all ratios—magnitude weighting biases toward common patterns. PPL-LAMBADA divergence: 92% PPL recovery → modest downstream gain. Perplexity alone insufficient.
K
Ablation: 10 Structural Refinements
Objective-based methods transfer; structure-based do not. GPT-2: bottleneck-dominated (block 0 = +32.69). Mistral: cumulative-error-dominated (flat sensitivity). All of these refinements are moot under the quantization-in-basis paradigm, which avoids rank reduction entirely.
L
Experimental Details
Data. WikiText-2: calibration (32 seq Mistral, 64 seq GPT-2), E2E training (64 seq), evaluation (held out). LAMBADA: 500 examples. V Theorem basis. Per head per layer: collect α, V from calibration forward pass, compute MV = V ⊤ α⊤ αV , eigendecompose (d × d). For GQA: average α across Q heads sharing the KV cache. Quantization. Symmetric per-channel: scalei = max |xi |/(2b−1 − 1), xq = round(x/scale) · scale. Applied in the eigenbasis (rotate, quantize, store). Cache quantization: per-channel scales computed once per sequence. Rank reduction (comparison). Projection V̂ = V P P ⊤ where P = top-r eigenvectors. Applied via weight absorption: Wv ← P P ⊤ Wv . KL learning. Adam (lr= 3×10−3 , wd= 0.01), 200 steps, QR orthogonalization. Hook: output += input × Wslice (P P ⊤ − I). E2E. All projections jointly, Adam (lr= 3 × 10−3 , batch= 4, 80–200 steps). Gradient checkpointing.
M
Variance ̸= Importance: The Foundational Observation
The observation that variance and importance are decoupled is the starting point of this work. We document it here as the first link in the chain from “PCA is wrong” to “quantize in the right basis.” 11
Table 13: Mistral 7B complete hierarchy (rank 64, 50% KV dim reduction). Method
PPL
Gap rec.
Cost
PCA V Theorem MSE Learned KL Learned MSE → E2E KL → E2E
21.04 16.33 16.93 15.96 11.92 11.37
0% 40% 35% 43% 77% 82%
Free Free 150 steps/head 40 steps/block +50 E2E +50 E2E
Table 14: V Theorem vs KQ-SVD (GPT-2, rank 32, 336 eval seq). Method PCA KQ-SVD (V W O ) V Theorem (αV ) Combined
M.1
V-only PPL
K+V PPL
65.44 64.22 62.46 63.09
131.09 139.86 108.22 115.66
The Residual Stream Experiment
On GPT-2, the residual stream between transformer blocks has extreme variance concentration: 90% of variance in 2–7 dimensions after middle blocks. Projecting to the high-variance subspace destroys perplexity: Interpretation. Low-variance directions carry decision-critical information invisible to PCA. The model’s predictions depend on subtle distinctions in directions that have almost no energy. This is the fundamental insight: PCA tells you where the model lives; it does not tell you how the model thinks. M.2
The Sensitivity Experiment
Average per-direction perturbation sensitivity (Exp 29): all 768 directions are equally sensitive (u⊤ Gu ≈ const), meaning the Jacobian Gram G = J ⊤ J is approximately isotropic. Since importance = variance × sensitivity, and sensitivity is uniform, importance ≈ variance on average. But this is a population average—per-token sensitivity varies massively (1–367 active directions). The divergence between variance and importance is token-conditional, not global. Ablation confirmation. Per-direction ablation importance correlates with eigenvalue at r = 0.69 (Exp 52b)—substantial but far from 1.0. The 31% unexplained variance is the signal that energy-blind methods (like our theorem) can exploit, and why the spice ratio (Si = Ti /mi ) is a better predictor than eigenvalue alone.
N
The Spice Ratio: Full Analysis
N.1
Definition and Measurement
The spice ratio Si = Ti /mi for direction i in a given head, where: • mi = λi (MV ): the eigenvalue (energy/mass in the attention output) • Ti : the toxicity—KL divergence between clean model and model with direction i perturbed by ε = 0.5 standard deviations High Si = low energy but high damage when perturbed. These are the “jalapeño” directions: small, hot, and devastating when removed. 12
Table 15: Closed-form metrics (GPT-2, K+V, 336 eval seq). Method
PPL
Gap rec.
PCA KQ-SVD V Theorem Entropy-weighted
130.95 99.39 85.38 82.30
0% 36% 52% 56%
Table 16: Rank reduction + INT4 (GPT-2). Method PCA V Theorem Learned E2E
N.2
50% FP16
+INT4
Quant cost
130.95 85.36 102.16 51.52
127.25 87.26 99.25 52.85
−3.70 (helps!) +1.90 −2.91 (helps!) +1.33
Results (GPT-2)
Global correlation. Spearman ρ(mass rank, toxicity rank) = 0.38. Energy explains only 38% of the damage ranking. The remaining 62% is invisible to any eigenvalue-based method—including PCA, the V Theorem, and KQ-SVD. Practical application. Exempting block 0 from rank-reduction compression: V Theorem PPL improves from 85.36 to 76.35 (+9.01 PPL), matching 200 steps of KL optimization at zero compute. The spice ratio correctly identifies block 0 as the dominant liability without running any compression experiment. N.3
Why Quantization Sidesteps the Spice Problem
Under rank reduction, high-spice (low-mass) directions are the first discarded—they have the smallest eigenvalues. But their influence is disproportionate: discarding a direction with Si = 0.84 costs 84× more per unit energy than a direction with Si = 0.01. Under quantization, all directions are preserved. High-spice directions receive their own per-channel scale, keeping the quantized value within the correct region of the softmax decision boundary. The spice ratio becomes irrelevant because no direction is eliminated. This is the mechanistic explanation for the 34.77 PPL gap at 512 bits: rank reduction kills the jalapeños; quantization preserves them.
O
The Road to Quantization: Every Attempt to Improve Rank Reduction
The quantization result (EXP-26) was not the starting hypothesis—it was discovered after systematically exhausting every approach to improve rank reduction. We document this journey because the negative results are informative: they establish that the barrier is the paradigm, not the implementation. O.1
Closed-Form Metric Improvements (All Converge to ∼82 PPL on GPT-2)
Four independent approaches, four paths to the same wall. O.2
Attempts to Break the Per-Head Ceiling on Mistral
Per-head corrections overfit. The gap is cross-layer error accumulation, not per-head metric quality. 13
Table 17: Mistral: rank reduction + INT4 at 87.5% total. Method
50% FP16
+INT4
Quant cost
PCA Learned E2E
18.78 14.48 10.80
19.31 14.67 11.20
+0.53 +0.19 +0.40
Table 18: LAMBADA across compression ratios (GPT-2, rank reduction). FP16: 24.2% acc. PCA
O.3
Stage 1
E2E
Reduction
Acc
LogP
Acc
LogP
Acc
LogP
25% 50% 75%
19.7 10.0 0.5
−5.66 −8.23 −13.41
18.3 7.2 0.2
−5.68 −8.37 −12.77
17.0 10.3 2.3
−6.14 −8.08 −12.53
Skip-Layer and Variable Rank on Mistral (EXP-18, EXP-19)
Compression is binary: full rank or skip. Variable rank redistribution destroys the robust layers without helping the fragile ones. This is consistent with the quantization finding: every direction matters, and reducing any layer below full rank incurs non-recoverable damage. O.4
GQA Fix Attempts (EXP-12, EXP-13, EXP-16)
Aggregation method is not the Mistral bottleneck (∆ < 0.3 PPL across all variants). Q-head alignment is 0.75–0.89—co-trained heads already share enough structure. O.5
Perturbation Theory Ceiling (THEORY-01)
Rayleigh–Schrödinger perturbation recovers 55% of the PCA→theorem gap but diverges at order 3. Padé resummation adds only +2% over RS2. The correction has exactly two perturbative layers (per-direction + pairwise); the rest is non-perturbative (see Appendix D). O.6
Random Matrix Theory (EXP-02)
The correction operator A has eigenvalues above the Marchenko–Pastur edge (not random) with Poisson spacing (weakly localized). But extracting the spikes gives only +0.1–0.6% subspace overlap improvement. Structure exists but is not actionable. O.7
The Lesson
Every approach to improve rank reduction—better metrics, deeper Jacobians, perturbation theory, random matrix theory, polynomial kernels, learned corrections, GQA fixes, variable rank, skip strategies—either hits the same ceiling or makes things worse. The breakthrough came from changing the compression operator (quantization instead of projection), not the compression basis. The theorem was right all along; we were just using it wrong.
14
Table 19: Cross-architecture transferability of structural methods. Strategy
GPT-2
Mistral
Transfers?
V Theorem Learned > PCA E2E > per-layer
52% 47% 80%
40% 35% 82%
✓ ✓ ✓
+11% +14.58 +32.69 Negative Negative — ∆=0
−8% −4.08 +1.02 — — Negative ∆=0
× × × × × × ×
K query-weighted K-heavy budget Skip layer 0 Head-wise alloc. Position-aware (9 configs) Non-uniform rank Sequential calib.
Table 20: Residual stream projection test (GPT-2, block 6). Dims kept
% variance
PPL
8 64 256 768
∼95% ∼99% ∼99.9% 100%
3,441 (destroyed) 524 (destroyed) 145 (degraded) 47 (baseline)
Table 21: Spice concentration by block (GPT-2). Max spice = max Si across all heads and directions. Block
Max spice
Mean spice
0 1 3 6 9 11
0.84 0.31 0.21 0.03 0.01 0.01
0.12 0.05 0.04 0.01 0.003 0.002
Character 84× spicier than late blocks Moderate Moderate Low Very low Very low
Table 22: Every closed-form metric converges to the same ceiling (GPT-2, rank 32). Method
PPL
Gap rec.
What it adds
PCA V Theorem V Theorem + W O V Theorem + W O + LN Entropy-weighted Full loss Jacobian
130.95 85.36 84.22 82.42 82.27 83.02
0% 52% 54% 56% 56% 55%
— Attention α Output projection LayerNorm Jacobian Attention confidence Full backward pass
Table 23: Every per-head improvement attempt on Mistral (rank 64). Baseline: V Theorem 16.33. Experiment
PPL
Result
Full loss Jacobian (Theory-10b) Deeper linearization L1 (+W O ) Deeper linearization L2 (+RMSNorm) Theorem → KL per-head (EXP-20) Polynomial kernel (EXP-24) Learned rank-4 correction (EXP-21) Learned rank-8 correction (EXP-21) Learned rank-16 correction (EXP-21)
16.35 16.01 16.03 15.74 16.30 14.24 14.47 15.27
+0.02 (nothing) +0.32 (marginal) +0.30 (marginal) +0.59 (ceiling holds) +0.03 (noise) −0.24 (worse) −0.47 (worse) −1.27 (worse)
15
Table 24: Layer-level strategies on Mistral (rank 64, baseline: 13.07). Strategy
PPL
Improvement
Uniform compression (all layers) Skip worst 8 layers (EXP-18) Variable rank proportional to drift (EXP-19) Fragile=96, robust=48 (EXP-19)
13.07 11.50 44.15 18.02
— +2.50 catastrophic worse than uniform
Table 25: Attempts to fix GQA metric aggregation. Method Arithmetic mean of per-Q metrics Geometric mean Harmonic mean Shared + residual (geometric) Norm-weighted (EXP-13) Norm + attention (EXP-13)
Best PPL 15.94 15.92 16.20 15.98 — —
16
Outcome Baseline ∆ < 0.1 Worse ∆ < 0.1 Marginal on MHA Catastrophic on simulated GQA