IO-SVD: Input-Output Whitened SVD for Adaptive-Rank LLM Compression

Ali Abbasi; Chayne Thrash; Hamed Pirsiavash; Haoran Qin; Soheil Kolouri

arxiv: 2605.15626 · v1 · pith:2CURW7NInew · submitted 2026-05-15 · 💻 cs.LG

IO-SVD: Input-Output Whitened SVD for Adaptive-Rank LLM Compression

Ali Abbasi , Chayne Thrash , Haoran Qin , Hamed Pirsiavash , Soheil Kolouri This is my paper

Pith reviewed 2026-05-20 21:10 UTC · model grok-4.3

classification 💻 cs.LG

keywords LLM compressionSVDlow-rank approximationpost-training compressionadaptive rank allocationKL divergencemodel quantization

0 comments

The pith

IO-SVD compresses LLMs by whitening both input activations and output prediction sensitivity to limit accuracy loss.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Large language models remain expensive to store and run, so post-training methods that shrink their weight matrices without retraining are valuable. IO-SVD creates a whitening space that incorporates both the statistics of typical inputs and a measure of how weight changes affect the model's final token predictions. The output measure comes from expanding the KL divergence to second order over the most probable tokens. A separate step then assigns different compression levels to different singular components by estimating their individual contribution to loss under a fixed total budget. If the approach works, models can be made substantially smaller while still producing nearly the same answers on downstream tasks and running faster at inference time.

Core claim

IO-SVD forms a KL-aware double-sided whitening space for model weights. Using a second-order expansion of the KL loss over the top-K token probabilities, it constructs an output-side metric that captures predictive sensitivity, while input whitening captures activation statistics. It further introduces an efficient heterogeneous rank-allocation strategy that scores whitened singular components using first-order calibration loss estimates and prunes the least sensitive components under a global budget. The same sensitivity estimates also guide loss-aware remapping when combining the low-rank factors with 8-bit quantization.

What carries the argument

The KL-aware double-sided whitening space that combines input activation statistics with an output metric derived from second-order KL expansion over top token probabilities.

If this is right

Models retain higher task performance at the same compression ratio compared with input-only whitening.
Inference speed increases because the resulting low-rank matrices require fewer operations during forward passes.
Hybrid low-rank plus quantization achieves better quality by using the loss estimates to decide which factors to quantize to 8 bits.
The same construction applies to both pure language models and vision-language models with only minor changes to the calibration data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The whitening construction could be tested with other divergence measures or with full-sequence losses instead of top-K tokens.
The global budget for rank allocation might be replaced by per-layer hardware constraints to target specific latency targets.
Sensitivity scores computed once could be reused to guide further compression steps such as pruning or distillation.

Load-bearing premise

The second-order expansion of the KL loss over the top-K token probabilities accurately captures predictive sensitivity for the output-side metric.

What would settle it

Apply the chosen ranks and factors to a held-out calibration set, then measure the actual change in KL divergence or downstream accuracy and check whether it matches the first-order and second-order estimates used to select the components.

Figures

Figures reproduced from arXiv: 2605.15626 by Ali Abbasi, Chayne Thrash, Hamed Pirsiavash, Haoran Qin, Soheil Kolouri.

**Figure 1.** Figure 1: Overview of IO-SVD. (a) Comparison of whitening strategies: standard SVD reconstructs the weight directly, one-sided whitening incorporates only input activation statistics, and double-sided whitening incorporates both input statistics and output-side sensitivity before SVD. (b) Heterogeneous rank allocation. For each whitened matrix B, singular components are sorted by singular-value magnitude, and the sm… view at source ↗

**Figure 2.** Figure 2: Loss-aware remapping: (a) SVD-truncate each weight to rank k; (b) score factor rows by first-order calibration-loss change under int8 quantization; (c) greedily keep low-score rows in int8 until meeting Crem; (d) assign the remaining rows to fp16. 3.2 Adaptive rank allocation The SVD solution above assumes fixed per-layer ranks. Under a global compression budget, we instead allocate ranks by estimating the… view at source ↗

**Figure 3.** Figure 3: Top-K ablation for output-side KL curvature. Normalized perplexity on wiki2, C4, PTB [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: Throughput vs. peak memory on LLaMA-2- 7B (batch 64, seq 1024+1024). Peak GPU memory. The dense baseline consumes 77.6 GB, dominated by a 64.0 GB KV cache and a 12.6 GB weight tensor. IO-SVD without cache optimization shrinks the weight footprint to 5.4 GB but leaves the KV cache untouched at 64.0 GB, giving a 70.4 GB peak. Adding V-cache compression reduces the cache to 38.8 GB and yields a 50.3 GB peak,… view at source ↗

read the original abstract

Large language models deliver strong performance across language and reasoning tasks, but their storage and compute costs remain major barriers to deployment in resource-constrained and latency-sensitive settings. SVD-based post-training compression offers a hardware-agnostic way to reduce model size and improve inference efficiency through low-rank factorization. However, existing methods often rely on input-only whitening spaces, homogeneous rank allocation, or loss-agnostic allocation heuristics, limiting their ability to preserve model quality under aggressive compression. We propose Input-Output Whitened SVD (IO-SVD), a post-training compression method that forms a KL-aware double-sided whitening space for model weights. Using a second-order expansion of the KL loss over the top-K token probabilities, IO-SVD constructs an output-side metric that captures predictive sensitivity, while input whitening captures activation statistics. We further introduce an efficient heterogeneous rank-allocation strategy that scores whitened singular components using first-order calibration loss estimates and prunes the least sensitive components under a global budget. Inspired by prior work that combines SVD truncation with quantization, we improve hybrid SVD-quantization compression through loss-aware remapping, which selects low-rank factor rows for 8-bit quantization based on the predicted loss change incurred by quantizing them. Extensive experiments across diverse LLM and VLM families, and inference-time analysis shows that IO-SVD compresses LLMs with minimal performance degradation while delivering practical inference speedups. Code is available at https://github.com/mint-vu/IO-SVD.git

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

IO-SVD adds output-side KL sensitivity to SVD compression and heterogeneous rank allocation, but the second-order approximation is the part that needs the most scrutiny under real compression ratios.

read the letter

IO-SVD builds a double-sided whitening space for SVD-based LLM compression: input side uses activation statistics as usual, while the output side uses a second-order Taylor expansion of KL divergence over top-K token probabilities to score predictive sensitivity. It then scores whitened components with first-order calibration loss estimates and prunes under a global rank budget, plus a loss-aware remapping step when mixing with 8-bit quantization. That combination of output metric and heterogeneous allocation is the concrete step beyond prior input-only or uniform-rank SVD work. The experiments run across several LLM and VLM families and include inference timing, which is useful for anyone who actually ships compressed models. Code release helps too. The soft spot is exactly the one the stress-test flags. The second-order KL expansion assumes the low-rank perturbation stays small enough that higher-order terms can be ignored. At the compression ratios the paper targets, singular-value truncation can shift output distributions more than that, so the sensitivity scores used for both whitening and pruning may not track actual loss as cleanly as claimed. Without seeing the full derivation, error bounds, or ablations that compare the approximation to measured loss change, it is hard to know how much this affects the final numbers. The central claim of minimal degradation therefore rests on the experiments holding up rather than on a proven bound. This paper is for people who work on practical post-training compression and want a hardware-agnostic SVD baseline that tries to stay loss-aware. A reader already following SVD-quantization hybrids would get the most out of the specific metric and allocation details. It deserves a serious referee because the idea is a direct, testable extension of existing techniques and the empirical scope is wide enough to produce useful feedback even if the approximation needs tightening.

Referee Report

2 major / 2 minor

Summary. The paper introduces IO-SVD, a post-training SVD-based compression technique for LLMs and VLMs. It constructs a double-sided whitening space by combining input-side activation statistics with an output-side metric derived from a second-order Taylor expansion of the KL divergence over top-K token probabilities. This metric guides heterogeneous rank allocation via first-order calibration loss estimates under a global budget, and the method is extended to loss-aware remapping for hybrid SVD-quantization. Experiments across model families report minimal degradation alongside inference speedups, with code released.

Significance. If the core approximations and empirical results hold, IO-SVD would offer a hardware-agnostic, loss-aware approach to adaptive-rank compression that improves upon input-only or homogeneous baselines. The public code and cross-family evaluation strengthen reproducibility and practical utility for deployment.

major comments (2)

[IO-SVD construction] IO-SVD construction section: the second-order expansion of KL divergence over top-K probabilities is used to define the output-side whitening metric and sensitivity scores for rank allocation. No explicit bound or empirical check is provided showing that higher-order terms remain negligible under the target compression ratios, where large singular-value truncation can produce non-local output shifts. This approximation is load-bearing for the central claim of minimal degradation.
[rank allocation] Heterogeneous rank-allocation paragraph: first-order calibration loss estimates are computed in the whitened space to prune components. If these estimates reuse the same calibration data or whitening transform that defines the metric, the procedure risks circularity; an explicit statement of data separation or a reduction showing the estimates are independent of the fitted parameters is needed to support the allocation strategy.

minor comments (2)

[Abstract] Abstract: the phrase 'minimal performance degradation' is repeated without quantitative qualifiers; adding a brief range of reported perplexity or accuracy drops would improve precision.
Notation: the input and output whitening matrices are introduced without an explicit equation linking them to the final low-rank factors; a single displayed equation would clarify the double-sided construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the IO-SVD method, particularly regarding the validity of the second-order KL approximation and the independence of the rank-allocation estimates. We address each major comment below and will revise the manuscript to incorporate additional validation and clarifications.

read point-by-point responses

Referee: [IO-SVD construction] IO-SVD construction section: the second-order expansion of KL divergence over top-K probabilities is used to define the output-side whitening metric and sensitivity scores for rank allocation. No explicit bound or empirical check is provided showing that higher-order terms remain negligible under the target compression ratios, where large singular-value truncation can produce non-local output shifts. This approximation is load-bearing for the central claim of minimal degradation.

Authors: We acknowledge the importance of validating the second-order Taylor expansion of the KL divergence. In the revised manuscript, we will add an empirical section that compares the approximated output-side metric against the exact KL divergence computed on a held-out calibration set for compression ratios ranging from 2x to 4x. We will also include a brief analysis referencing approximation bounds from loss landscape literature to discuss when higher-order terms remain small, thereby supporting the claim of minimal degradation under the evaluated settings. revision: yes
Referee: [rank allocation] Heterogeneous rank-allocation paragraph: first-order calibration loss estimates are computed in the whitened space to prune components. If these estimates reuse the same calibration data or whitening transform that defines the metric, the procedure risks circularity; an explicit statement of data separation or a reduction showing the estimates are independent of the fitted parameters is needed to support the allocation strategy.

Authors: We agree that explicit separation is necessary to avoid any appearance of circularity. The input whitening transform is computed solely from activation statistics on a first calibration subset, while the first-order loss estimates for heterogeneous rank allocation are performed on a disjoint second calibration subset that does not influence the whitening matrix. In the revision, we will add an explicit statement of this data separation protocol along with a short empirical check confirming that the sensitivity scores remain stable when the whitening transform is held fixed from the first subset. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs the output-side metric via an explicit second-order Taylor expansion of KL divergence over top-K probabilities (abstract and IO-SVD construction), which is an independent approximation rather than a self-definition or fitted input renamed as prediction. Heterogeneous rank allocation scores components using first-order calibration loss estimates on separate calibration data, a standard post-training technique that does not reduce by construction to the whitening space or target performance metrics. No load-bearing self-citations, uniqueness theorems imported from authors, or ansatzes smuggled via prior work are present in the core steps; the hybrid SVD-quantization remapping similarly relies on predicted loss change computed from the same expansion without circular re-use of fitted values. The derivation remains self-contained against external benchmarks and does not equate outputs to inputs by definition.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the assumption that a second-order KL expansion provides a reliable sensitivity metric and that first-order loss estimates suffice for pruning decisions under a global budget.

free parameters (1)

global rank budget
Controls total compression ratio; chosen to meet target size while minimizing predicted loss.

axioms (1)

domain assumption Second-order Taylor expansion of KL divergence over top-K token probabilities approximates output sensitivity
Invoked to construct the output-side whitening metric without full loss computation.

pith-pipeline@v0.9.0 · 5812 in / 1168 out tokens · 40592 ms · 2026-05-20T21:10:54.364427+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using a second-order expansion of the KL loss over the top-K token probabilities, IO-SVD constructs an output-side metric... ΔJℓ ≈ ½‖Cℓ^{1/2}(Wℓ − Ŵℓ)Rℓ^{1/2}‖_F² (abstract, §3.1, Eq. 4)
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean costAlphaLog_high_calibrated_iff unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

KL(pt ∥ softmax(zt + δzt)) = ½ δz_t^T H_t δz_t + O(‖δz‖³), H_t = Diag(pt) − pt pt^T (Eq. 2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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