The Functional Structure of Post-Compression Recovery in Low-Rank LLMs

Zishan Shao; Liang Tian; Georgiy Zemlevskiy; Kangning Cui; Lixun Zhang; Yixiao Wang; Ting Jiang; Jinhee Kim

cs.AIOct 4, 2026

The Functional Structure of Post-Compression Recovery in Low-Rank LLMs

Authors: Zishan Shao, Liang Tian, Georgiy Zemlevskiy, Kangning Cui, Lixun Zhang, Yixiao Wang, Ting Jiang, Jinhee Kim, +5 more

Organizations: Duke University · Wake Forest University · Carnegie Mellon University · University of Oxford · University of Florida

Abstract

Different low-rank compression methods can produce compressed LLMs that respond differently to the same post-compression recovery procedure, and relative advantages observed between methods at the compression endpoint may shrink, grow, or even reverse after recovery. We ask whether this recovery heterogeneity reflects functional structure beyond scalar loss evolution, and how that structure evolves throughout recovery. Our results establish that this heterogeneity reflects a reproducible compression-induced functional structure, which we formalize as recovery pressure. To characterize this structure consistently throughout recovery, we develop a standardized functional characterization within each backbone that is applicable across heterogeneous low-rank methods. The primary backward characterization reveals reproducible module-wise structure across independent probes, while a complementary forward-only characterization recovers related structure without loss or backpropagation. We further find that recovery pressure measured at the endpoint is associated with subsequent recovery response; during recovery, its module-wise structure is reorganized non-uniformly, and localized updates induce distributed responses beyond directly updated modules. Further evidence indicates that tracking this evolving structure provides a complementary functional view of recovery progress alongside scalar loss.

Figures & tables

Figure 1 : Endpoint rankings need not be preserved after recovery. Avg7 MCQ Performance ( 3.4 ) on Qwen3-8B-Base,

r=0.6

Figure 2 : Recovery reshapes comparative performance. Endpoint (hatched) and recovered (solid) Avg7 ( 3.4 ) across representative settings, with endpoint ranks annotated above each method.

Figure 3 : Fixed-coordinate recovery and backward characterization of recovery pressure. Recovery updates the trainable middle map while preserving compression-selected coordinates. Matched channel-wise perturbations at corresponding outputs of $c_{t}$ and $d_{0}$ define the dense-referenced response $p_{t}$ , summarized by $P_{t}$ and $q_{m,t}$ .

Figure 4 : Module-wise organization of endpoint recovery pressure. Colors show $\log_{10}q_{m,0}$ for $r=0.8$ endpoints on the independent Q128 probe, normalized over attention and MLP modules.

Backbone	$r$	$\rho_{F,B}$ median [range]	$\rho_{F}^{\mathrm{split}}$	Top-8
Llama-3.1 8B	0.8	.613 [.542, .910]	.987	5.5/8
Llama-3.1 8B	0.6	.611 [.293, .832]	.983	5.0/8
Qwen3-8B Base	0.8	.377 [.042, .701]	.603	3.0/8
Qwen3-8B Base	0.6	.471 [.295, .594]	.629	4.5/8

Table 1 : Forward-only grounding. Statistics summarize six compression methods within each backbone–retention setting.

\rho_{F,B}

is reported as median [range];

\rho_{F}^{\rm split}

and Top-8 are medians.

Backbone	$r$	$\rho(P_{0},H_{T})$
Llama-3.1 8B	0.8	1.000
Llama-3.1 8B	0.6	0.886
Qwen3-8B Base	0.8	0.771
Qwen3-8B Base	0.6	0.829

Table 2 : Endpoint magnitude and recovery. Spearman

\rho(P_{0},H_{T})

across six low-rank methods.

	Llama-3.1-8B $r=.8$			Llama-3.1-8B $r=.6$			Qwen3-8B-Base $r=.8$			Qwen3-8B-Base $r=.6$
Method	$P_{0}$	Endpoint	Recovered	$P_{0}$	Endpoint	Recovered	$P_{0}$	Endpoint	Recovered	$P_{0}$	Endpoint	Recovered
ASVD	1.314	33.55	55.14	1.285	32.55	44.86	0.241	59.61	62.40	1.464	34.60	52.35
SVD-LLM	0.192	51.94	58.80	0.567	36.81	50.92	0.336	55.93	60.57	0.430	43.12	54.57
Basis Sharing	0.374	51.17	57.97	1.425	33.67	48.47	0.450	53.97	60.42	0.680	44.03	53.23
MoDeGPT	0.074	61.10	62.45	0.159	45.94	53.80	0.421	54.48	59.35	0.550	41.50	49.77
GFWSVD	0.943	48.34	57.25	1.701	32.98	48.78	0.544	55.25	60.64	1.288	45.77	53.82

Table 3 : Recovery-aware comparison of low-rank compressed endpoints.

P_{0}

is measured on the Q128 probe before recovery. Endpoint and Recovered report zero-shot Avg7 (%) before and after 384 C4 updates under the shared fixed-coordinate intervention; recovered scores are averaged over three seeds. Bold marks the highest Endpoint or Recovered Avg7 within each backbone–retention setting. Full recovery gains and seed variation are reported in the Appendix A.2.9 .

Figure 5 : Nonuniform pressure reorganization under full and localized recovery. (a) Baseline-normalized module shares on Llama-3.1-8B ( $r=0.8$ , C4; mean $\pm$ range across 3 seeds). Flat line at 1.0 indicates uniform rescaling. (b) Share changes across unupdated modules after 64-step single-MLP recovery (SVD-LLM; 1.0 = no change).

		Recovery updates			Avg7 $\uparrow$			C4 PPL $\downarrow$
Backbone	$r$	Loss	Loss+P	Update Red- uction(%)	Full	Loss	Loss+P	Full	Loss	Loss+P
Llama-3.1 8B	0.8	341.3	256.0	25.00	58.351	58.309	58.331	16.70	16.70	16.83
	0.6	362.7	288.0	20.59	49.803	49.825	49.782	24.66	24.69	25.10
	0.4	960.0	522.7	45.56	40.640	40.520	40.516	44.22	44.52	48.49
Qwen3-8B Base	0.8	330.7	341.3	-3.23	61.029	60.794	61.017	16.57	16.58	16.59
	0.6	373.3	341.3	8.57	53.194	53.234	53.156	22.01	22.07	22.12

Table 4 : Controlled evaluation of pressure-informed stopping. Loss and Loss+P are evaluated on identical recovery trajectories and checkpoints, differing only by incorporating pressure dynamics. Each row averages six compression methods (Full uses 384 updates at

r=0.8/0.6

and 1024 at

r=0.4

). Saving is relative to Loss; C4 PPL uses 256 validation windows of 256 tokens.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

$r$	Method	Llama-3.1-8B				Qwen3-8B-Base
$r$	Method	Endpoint	+Recovery	Gain	$P_{0}$	Endpoint	+Recovery	Gain	$P_{0}$
0.8	ASVD	33.55 (6)	55.14 $\pm 0.12$ (6)	+21.60	1.544	59.61 (1)	62.40 $\pm 0.62$ (1)	+2.78	0.296
	SVD-LLM	51.94 (2)	58.80 $\pm 0.03$ (3)	+6.86	0.699	55.93 (3)	60.57 $\pm 0.92$ (4)	+4.64	0.398
	Basis Sharing	51.17 (4)	57.97 $\pm 0.09$ (4)	+6.79	0.868	53.97 (6)	60.42 $\pm 0.49$ (5)	+6.45	0.509
	MoDeGPT	61.10 (1)	62.45 $\pm 0.04$ (1)	+1.35	0.662	54.48 (5)	59.35 $\pm 0.35$ (6)	+4.86	0.448
	GFWSVD	48.34 (5)	57.25 $\pm 0.23$ (5)	+8.91	1.163	55.25 (4)	60.64 $\pm 0.11$ (3)	+5.39	0.609

Appendix

Table 5 : Model recoverability and endpoint pressure across low-rank baselines. Endpoint and +Recovery report zero-shot Avg7 (%) before and after 384-update BASIS recovery (mean

\pm

SD across 3 seeds), with Gain showing net improvement.

P_{0}

measures the initial pressure magnitude on the C4 probe. Superscripts denote rank within each setting; bold marks the best score.

	Llama-3.1-8B		Qwen3-8B-Base
Method	$r=0.8$	$r=0.6$	$r=0.8$	$r=0.6$
ASVD	19 / 10,725,120	21 / 11,137,728	31 / 10,963,770	20 / 11,261,960
SVD-LLM	14 / 10,958,080	19 / 11,151,936	13 / 11,191,752	17 / 10,975,608
Basis Sharing	18 / 10,892,160	25 / 11,083,200	17 / 11,122,488	22 / 10,794,960
MoDeGPT	4 / 10,878,704	5–6 / 10,994,530	4 / 10,826,464	5–6 / 10,990,644
GFWSVD	14 / 10,958,080	19 / 11,151,936	13 / 11,191,752	17 / 10,975,608

Appendix

Table 6: Native BASIS capacity for the 384-update C4 recovery. Each cell gives adapter rank

r_{a}

/ exact trainable parameter count for one frozen compression endpoint. Counts are identical across the three recovery seeds. Shared Basis Sharing adapters are counted once per shared basis.

A. Within-setting endpoint rank associations
Probe	Backbone	$r$	$\rho(P_{0},L_{0})$	$\rho(P_{0},H_{384})$	$\rho(L_{0},H_{384})$
Q0	Llama	0.8	1.000	1.000	1.000
Q0	Llama	0.6	0.886	0.886	1.000
Q0	Qwen	0.8	0.771	0.771	1.000
Q0	Qwen	0.6	0.943	0.829	0.943
Q128/V128	Llama	0.8	1.000	1.000	1.000

Appendix

Table 7 : Endpoint magnitude, scalar endpoint quality, and subsequent recovery. Panel A reports within-setting Spearman correlations across the six compression methods.

L_{0}

is held-out endpoint NLL and

H_{384}=L_{0}-L_{384}

is the mean held-out loss reduction over three recovery seeds. Q0 denotes the original 16-window pressure probe; Q128/V128 independently remeasures all endpoints using separate pressure and validation sets. Panel B reports a retrospective method-held-out prediction audit over all 24 Q128/V128 endpoints.

Figure 6 : Factor-update recovery shifts selected coordinates. Collapse-score ECDFs on Llama-3.1-8B show that directly updated factors in U-LoRA, V-LoRA, and UV-LoRA move away from the whitening reference, while BASIS keeps the protected outer factors fixed.

Recovery method	Endpoint Avg7 $\uparrow$		Avg7 projection diagnostic				PPL Avg $\downarrow$
Recovery method	Learned	Projected	Total	Projected gain	Coord.-shift	Shift share	Learned	Projected
BASIS	50.48	50.48	+17.29	+17.29	+0.00	0.0%	16.57	16.57
U-LoRA	50.15	39.17	+16.96	+5.98	+10.98	64.7%	17.34	223.96
V-LoRA	49.91	41.80	+16.73	+8.61	+8.12	48.5%	17.90	119.25
JFT	50.26	40.85	+17.07	+7.67	+9.40	55.1%	17.56	283.05

Appendix

Table 8: Controlled fixed-coordinate projection diagnostic for factor-update recovery (LLaMA-3.1-8B,

r=0.4

). Normal evaluates each method in its learned coordinates; Projected maps recovered back to Source-defined fixed-coordinate subspace.

	Llama-3.1-8B		Qwen3-8B-Base
Method	Spearman	Cosine	Spearman	Cosine
ASVD	0.996	0.963	0.993	0.996
SVD-LLM	0.968	0.983	0.978	0.997
Basis Sharing	0.988	0.996	0.991	0.999
MoDeGPT	0.358	0.337	0.493	0.997
GFWSVD	0.991	$>0.999$	0.988	$>0.999$

Appendix

Table 9: Reproducibility of endpoint module organization. Spearman rank correlation and cosine similarity compare module-share profiles from Q64 and independent Q128 probes over all attention and MLP modules.

	Llama-3.1-8B		Qwen3-8B-Base
Method	Spearman	Cosine	Spearman	Cosine
ASVD	0.958	0.477	0.948	0.986
SVD-LLM	0.913	0.739	0.936	0.999
Basis Sharing	0.960	0.905	0.948	0.998
MoDeGPT	0.201	0.338	0.539	0.986
GFWSVD	0.968	0.983	0.907	$>0.999$

Appendix

Table 10: Module-profile sensitivity to the original probe size. Module-share agreement between the original 16-window Q0 and independent Q128 probes at

r=0.8

	Llama-3.1-8B			Qwen3-8B-Base
Method	Spearman	Cosine	Top-8	Spearman	Cosine	Top-8
ASVD	0.940 ∗	0.971	5/8	0.662	0.783	7/8
SVD-LLM	0.614	0.690	6/8	0.811	0.813	5/8
Basis Sharing	0.738	0.983	7/8	0.877	0.822	6/8
MoDeGPT	0.934	0.992	7/8	0.714	0.481	3/8
GFWSVD	0.869	0.955	6/8	0.848	0.592	5/8

Appendix

Table 11: Forward–backward characterization alignment. Module-wise agreement on matched two-window inputs at

r=0.8

. This is a separate small-input/module-level audit and is not the source data summarized in Table 1.

Figure 7 : Absolute-energy responses to localized recovery. Normalized energy change ( $E_{m,64}/E_{m,0}$ on Q128; 3-seed mean) across 60 unupdated modules under single-MLP BASIS recovery (SVD-LLM, Llama-3.1-8B, $r=0.8$ ). Rows denote updated layers; blue indicates attenuation ( $<1.0$ ) and gray marks excluded sites. Relative share growth can coexist with absolute energy decay.

Method	Seed	TV	Ref.	$E\downarrow$
SVD-LLM	7711	0.479	0.210	64
SVD-LLM	7712	0.414	0.290	64
SVD-LLM	7713	0.369	0.311	64
Swift-SVD	7711	0.334	0.255	64
Swift-SVD	7712	0.436	0.290	64
Swift-SVD	7713	0.376	0.276	64

Appendix

Table 12: Full-profile change over steps 0–384 for each selected recovery seed. Ref. is the paired-document error reference.

E\downarrow

counts absolute-energy decreases out of 64 modules; these sign counts are descriptive.

Method	64	128	192	256	320	384
SVD-LLM	0.355/0.130	0.443/0.154	0.398/0.177	0.366/0.219	0.415/0.148	0.417/0.191
Swift-SVD	0.366/0.175	0.410/0.122	0.401/0.227	0.318/0.184	0.365/0.199	0.379/0.254
MoDeGPT	0.312/0.217	0.336/0.183	0.487/0.237	0.354/0.258	0.311/0.299	0.370/0.243

Appendix

Table 13: All measured checkpoints: seed-mean profile TV and its paired-document error reference. Cells contain TV / Ref.

Backbone	Method	TV	Ref.
Llama	ASVD	0.493	0.271
Llama	SVD-LLM	0.479	0.210
Llama	Basis Sharing	0.495	0.293
Llama	MoDeGPT	0.414	0.288
Llama	GFWSVD	0.768	0.265
Llama	Swift-SVD	0.334	0.257

Appendix

Table 14: Complete endpoint cohort at retention 0.8; one recovery seed.

Method	L1 A	L16 A	L31 A	L1 M	L16 M	L31 M
SVD-LLM	0.489	0.124	5.307	69.276	0.204	6.368
Swift-SVD	0.518	0.125	5.214	69.105	0.204	6.257
MoDeGPT	0.265	0.131	24.457	44.557	0.206	16.196

Appendix

Table 15: Initial module shares (percent) for the layers in the main figure. A and M denote attention and MLP. These baselines are shared across recovery seeds.

			Recovery updates			Avg7 $\uparrow$
Backbone	$r$	Seed	Loss	Loss+P	Update reduction (%)	Full	Loss	Loss+P	$\Delta$ Avg7
Llama-3.1-8B	.6	7711	362.7	288.0	20.59	49.803	49.825	49.782	$-0.043$
		7712	362.7	288.0	20.59	49.074	49.241	49.248	$+0.007$
		7713	384.0	320.0	16.67	50.118	50.118	49.648	$-0.470$
		Mean	369.8	298.7	19.23	49.665	49.728	49.559	$-0.169$
Llama-3.1-8B	.4	7711	960.0	522.7	45.56	40.640	40.520	40.516	$-0.004$

Appendix

Table 16 : Seed-level controlled stopping comparison. Each seed row averages six compression methods; mean rows aggregate all 18 method–seed trajectories within each setting. Full recovery uses 384 updates at

r=.6

and 1024 at

r=.4

. Update reduction is measured relative to Loss ;

\Delta

Avg7 is Loss+P minus Loss .

Backbone	$r$	Scale $s$	Loss	Loss+P	Update reduction (%)	$\Delta$ Avg7 (pp)	Within $\pm 1$ pp
Llama-3.1-8B	.6	$0.5\times$	362.7	352.0	2.94	$-0.112$	6/6
		$1.0\times$	362.7	288.0	20.59	$-0.043$	6/6
		$1.5\times$	352.0	266.7	24.24	$+0.047$	6/6
		$2.0\times$	341.3	245.3	28.12	$-0.012$	5/6
	.4	$0.5\times$	970.7	661.3	31.87	$-0.003$	6/6
		$1.0\times$	960.0	522.7	45.56	$-0.004$	6/6

Appendix

Table 17 : Controlled stopping comparison across nearby loss operating points. The loss threshold is scaled by

s\in\{0.5,1.0,1.5,2.0\}

while the remaining stopping protocol is fixed. Loss and Loss+P report mean recovery updates across six compression methods. Update reduction is measured relative to Loss ;

\Delta

Avg7 is Loss+P minus Loss in percentage points. The final column reports the number of methods whose Avg7 difference remains within

\pm 1

pp. We use seed 7711 that is also in Table 4

Explore similar work

UniRank: Unified Rank Allocation for Low-Rank LLM Compression
Jun 20, 2026Chao Han, Yongjie Du, Junjie Tan +1Large Language Model Compression Task-Aware Compression
Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo
Sequential Functional Structured Tucker Compression for Large Language Model Attentions
Sep 30, 2026Jiangfeng Chen, Xinyu Wang, Tianshuo Yan +4Large Language Model Compression Task-Aware Compression
University of Manitoba · McGill University · Simpleway +2
Understanding Calibration and Truncation Error Propagation in Training-Free Low-Rank Compression for LLMs
Aug 9, 2026Mohanad Odema, Gabrielle De Micheli, Dayin Gou +3Large Language Model Compression Low-Rank Structure
LG Electronics, North America

Jun 20, 2026, cs.LGJ/K move · Enter open · S save

UniRank: Unified Rank Allocation for Low-Rank LLM Compression

Chao Han, Yongjie Du, Junjie Tan, Zihao Xuan

Ningbo Institute of Digital Twin, Eastern Institute of Technology, Ningbo

Large Language Model Compression Task-Aware Compression Data Compression Methods Ranking

Open paper AskPDF arXiv

Low-rank decomposition is a promising compression paradigm for large language models (LLMs), yet its effectiveness hinges on rank budget allocation across weight matrices: uniform or hand-crafted rules ignore module-wise importance, while learning-based allocation incurs substantial training overhead. We formulate rank allocation as a global sorting-and-truncation pipeline that scores every singular component by combining local singular energy with global functional importance, estimated via layer-wise input--output cosine similarity on a tiny calibration set. We show, both geometrically and empirically, that high input--output cosine similarity implies low effective rank. We further propose rank-preserving fine-tuning (RPFT), which adapts only a small subset of retained singular components so that the allocated rank stays bounded without re-decomposition. Experimental results show that UniRank cuts zero-shot perplexity by up to 50%, improves average reasoning accuracy by 3.0% over LoRAP at 25% sparsity, and boosts four SVD-based decomposition methods as a plug-and-play module.

Abstract

Explore similar work

UniRank: Unified Rank Allocation for Low-Rank LLM Compression

Sequential Functional Structured Tucker Compression for Large Language Model Attentions

Understanding Calibration and Truncation Error Propagation in Training-Free Low-Rank Compression for LLMs