Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Authors: Jialin Zhao
Yingtao Zhang
Carlo Vittorio Cannistraci
Venue: ICML 2025 poster (2025)
Keywords: Large Language Model; Post-training Model Compression; Post-Training Pruning; Inference Acceleration
Primary area: deep_learning->large_language_models

Abstract: The rapid growth of Large Language Models has driven demand for effective model compression techniques to reduce memory and computation costs. Low-rank pruning has gained attention for its GPU compatibility across all densities. However, low-rank pruning struggles to match the performance of semi-structured pruning, often doubling perplexity at similar densities. In this paper, we propose **Pi**voting **Fa**ctorization (**PIFA**), a novel **lossless** meta low-rank representation that unsupervisedly learns a **compact** form of any low-rank representation, effectively eliminating redundant information. PIFA identifies pivot rows (linearly independent rows) and expresses non-pivot rows as linear combinations, achieving **24.2\%** additional memory savings and **24.6\%** faster inference over low-rank layers at rank = 50\% of dimension. To mitigate the performance degradation caused by low-rank pruning, we introduce a novel, retraining-free reconstruction method that **m**inimizes error accumulation (**M**). **MPIFA**, combining M and PIFA into an end-to-end framework, significantly outperforms existing low-rank pruning methods, and achieves performance comparable to semi-structured pruning, while surpassing it in GPU efficiency and compatibility. Our code is available at https://github.com/biomedical-cybernetics/pivoting-factorization.

Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Jialin Zhao 12 Yingtao Zhang 12
Carlo Vittorio Cannistraci 1
Abstract
1. Introduction
The rapid growth of Large Language Models (LLMs) (Rad- ford, 2018; Radford et al., 2019; Mann al., 2020; Touvron et al., 2023a) has revolutionized natural language processing tasks but has also introduced significant challenges related to memory consumption and computational costs. Deploy- ing these models efficiently, particularly on resource-limited hardware, has driven surge of interest in model compres- sion techniques (Wan et al., 2023; Zhu al., 2024). Among these techniques, semi-structurred pruning, specifically N:M sparsity, has emerged as a promising approach due to its fardware-friendly nature, enabling efficient acceleration on NVIDIA's Ampere GPUs (Mishra et al., 2021; nvi, 2020), However, semi-structured pruning suffers from two major limitations: it is restricted to specific hardware architectures, and it couldn't flexible adjust density.
The rapid growth of Large Language Models has driven demand for effective model compression techniques to reduce memory and computation costs. Low-rank pruning has gained attention for its GPU compatibility across all densities. However, low-rank pruning struggles to match the performance of semi-structured pruning, often doubling perplexity at similar densities. In this paper, we propose Pivoting Factorization (PIFA), a novel lossless meta low-rank representation that unsupervisedly learns a compact form of any low-rank representation, effectively elimi- nating redundant information. PIFA identifies pivot rows (linearly independent rows) and expresses non-pivot rows as linear combinations, achieving 24.2% additional memory savings and 24.6% faster inference over low-rank layers at rank 50% of dimension. To mitigate the performance degradation caused by low-rank pruning, introduce novel, retraining-fit reconstruction method that minimizes error accumulation (M). MPIFA, combining M and PIFA into an end-to-end framework, signifi- cantly outperforms existing low-rank pruning methods, and achieves performance comparable semi-structured pruning, while surpassing it in GPU efficiency and compatibility. Our code is available at https: /ttpt://github. http://ooedicll--cyyeeeeeee pivoting-factenn factorizatiiinn
In contrast, low-rank pruming methods, primarily based on Singular Value Decomposition (SVD), preserve the coher- ence of tensor shapes, making them universally compatible with any GPU architecture at any density. Recent works (Yuan et al., 2023; Wang et al., 2024) have demonstrated the potential of low-rank decomposition in compressing LLMMs However, despite their flexibility, these methods struggle to compete with semi-structured pruning in terms of perfor- mance, often resulting in 2x increase in perplexity (PPL) at the same densities. This performance gap stems primarily from two challenges: (1) Low-rank pruning introduces in- formation redundancy in the decomposed weight matrices, and (2) existing reconstruction methods accumulate errors across layers, leading suboptimal performance.
To address challenge (1), propose Pivoting Factorization (PIFA), novel lossless low-rank rep- resentation that eliminates redundancy and enhances computational efficiency. PIFA is meta low-rank representation, because it unsupervisedly learns a compact representation of any other learned low-rank representation. PIFA identifies linearly independent rows from a singular weight matrix, which refer to pivot rows, and represents all other as linear combinations of these pivot rows. PIFA achieves significant improvements in both speedup and memory reduction during inference. Specifically, at r = 0.5, the PIEA layer achieves 24.2 %
'Center for Complex Network Intelligence (CCNT), Ts- inghua Laboratory Brain and Intelligence (THBI), Depart- ment of Psychological and Cognitive Sciences Department of Computer Science DDepartment of Biomedical Engineering, Ts- inghua University, China. Correspondence to: Jialin Zhao <jialin.zhaan zhao97 @ comaiil.com... Carlo Vittorio Cannistraci <kalok- agathos. agon @ www.g..co........... com>>
Proceedings of the 42nd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Figure 1: Comparison of pa- rameter ratios.
Table I: Comparison of PIFA with other sparsity.
CPU Speedup
GPU Speedup
GPU Mem
Any Sparsity
GPU Support
Mothod
Reduction
Performance
175
1.50
Unstmactured Sparsity
x
x
x
1/1
21 1/225
C23
ND Dessity Necertio
Ssmi-Structures Sparsity
Ampere GPU
E100
Structured Sparsity
General
60.7%
SVD-Based Row-kakk Sparsity
General
PIFA Low-Rank Sparsity
General
0.0
0.2
0.4
LLaMA
(4) Ohtiie Error fna.kcccnddaalnndnnnniaaio Recomstruction [M]
(b) Pivoting Factorization (PIFA)
Dense
Low-tank
Next Linear Layer
PIFA Layer
Data
Data Flow
Denee Data
Low rank
FOw
Data flow
Forward
Mik
Dense
1 mark
Pivot rows
Paununt mm
rim
Coefficient matrix
Post Attestion Lapersom
W frank
W
NX
Seff Amoniion
Non-pivat rows
X
Pivot rows
LO Dea
DIEVE
Index
permutation
# #paran ams = nn + (m -
rrr
+
r
r +
n
Input Layeromm
Pa 13700
Pivot rows
Coefficient matrix
Flow rank params Further losslessly
reduced params
Figure 2:
Illustration of the low-rank pruning method MPIFA (Algorithm 3), which consists of: (a) Online Error- -inttiiin-iiimizztiionn Reconstruction (M). Block R solves the least-squares optimization problem. The improve- ments upon SVD-LLM'' full-batch reconstruction, highlighted in red, include using the dense data flow to minimize error accumulation, and processing each sample sequentially to avoid GPU memory overflow. (b) Pivoting Factorization (PIFA). For any singular matrix with rank r, Pivoting Factorization further reduces y2 parameters, with no additional loss induced.
additional memory savings and 24.6% % faster inference compared to SVD-based low-rank layers, without inducing any loss.
on LLaMA2 models (7B, 13B, 70B) and the LLaMA3-8B model 1, Further experiments show that MPIFA consistently achieves superior speedup and memory savings both layer- wise and end-to-end compared to semi-structured pruning, while maintaining comparable or even better perplexity. Notably, as shown in Table 6, at d = 32768, PIFA with 55% density achieves a 2.1x speedup, whereas various imple- mentations of semi-sparse methods are either slower than the dense linear layer fail to execute.
To address challenge (2), we propose an Online Error- Accumultion-Minniizaatiii
Reconstruction (M) algo- rithm that minimizes error accumulation a problem perva- sive in existing reconstruction methods for both low-rank pruning (Wang et al., 2024) and semi-structured pruning (Frantar& & Alistarh, 2023; Li et al., 2024). Existing meth- ods rely on degraded data flow, where accumulated errors from previous modules propagate through the reconstruc- tion process, leading to suboptimal performance. Our ap- proach addresses this issue by combining dense data flow and low-rank data flow, as reconstruction targets, effectively mitigating the errors carried forward from earlier layers. Furthermore, the method operates online, processing large numbers of calibration samples sequentially to stay within GPU memory constraints.
The two main contributions of this work can be summarized as follows:
1.
We propose Pivoting Factorization (PIFA), a novel lossless meta low-rank representation that unsupervis- edly learns compact form of any SVD-based low- rank representation, effectively compressing out redun- dant information.
2. We introduce an Online Error-Accumulationn Minimization Reconstruction (M) algorithm that mit- igates error accumulation by leveraging multiple data flows for reconstruction.
Combining M and PIFA, we present MPIFA- an end-to- end, retraining-free low-rank pruning framework for LLMs. MPIFA significantly outperforms existing low-rank prun- ing methods, reducing the perplexity gap by 40%%70%
'https:/ / www. 1 1lama. com/
2

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
2. Related Work
2.2. Low-rank pruning
2.1. Connection pruning
Low-rank pruning applies matrix decomposition techniques, such as Singular Value Decomposition (SVD), to approx- imate weight matrices with lower-rank representations, thereby reducing both storage and computational demands. This approach, compatible with any GPU, represents large matrices as products of smaller ones, improving computa- tional efficiency. Recent studies (Hsu et al., 2022; Yuan et al., 2023; Wang et al., 2024; Jaiswal al., 2024; Saha et al., 2024; Kaushal et al., 2023; Sharma et al., 2023; Qinsi et al.) highlight the effectiveness of low-rank decomposition in compressing LLMs. However, despite their flexibility, these methods lag behind semi semi-structured pruning in perfor- mance, often leading to a 2x increase in perplexity (PPL.) at the same densities.
Pruning methods We define connection-wist pruning as removing certain connections between neurons in the net- work that are deemed less important. To achieve this, a series of methods have been proposed. Optimal Brain Dam- age (OBD) (Le Cun et al., 1990) and Optimal Brain Surgeon (OBS) (Hassibi et al., 1993) were proposed to identify the weight saliency by computing the Hessian matrix using cali- bration data. Recent methods such as SparseGPT (Frantar & Alistarh, 2023), Wanda (Sun et al., 2024), RIA (Zhang et al., 2024), along with other works (Fang et al., 2024: Dong et al., 2024; Das al., 2024), have advanced these ideas. Wanda prunes weights with the smallest magnitudes multiplied by input activations. Relative Importance and Ac- tivations (RIA) jointly considers both the input and output channels of weights along with activation information. Fur- thermore, OWL (Yin et al.) explores non-uniform sparsity by pruning based on the distribution of outlier activations, while other works (Lu al., 2024; Mocanu al., 2018; Ye et al., 2020; Zhuang et al., 2018) investigate alternative criteria for non-uniform sparsity.
3. Lossless Low-Rank Compression
3.1. Motivation: Value Decomposition
Information Redundancy in Singular
For a weight matrix W E R=*x Low-rank pruning meth- ods (Hsu et al., 2022; Yuan et al., 2023; Wang et al., 2024) decompose the matrix into product of two low-rank ma- trices, W R UVF, where U E R and VT B forming a low-rank approximation of W. Consider naive SVD) pruning as an example. First, the weight matrix is factorized using SVD as W BEAAT, Next, the top-r sin- gular values and corresponding singular vectors are retained, denoted as B,, E,,, and AT" Finally, the singular values E, are merged into the singular vectors, yielding U = -B,E, and VT = A.
Pruning granularity (compared in Table l):
1. Unstructured pruning removes individual weights based on specific criteria. Today, unstructured pruning is critical technique for compressing large language models (LLMs) to balance performance and computa- tional efficiency. However, unstructured pruning can only accelerate computations on CPUs due to its un- structured sparsity pattern.
2. Semi-structured pruning, i.e., N:M sparsity, enforces that every group of M consecutive elements, N must be zeroed out. This constraint is hardware-friendly and enables optimized acceleration NVIDIA's Ampere architecture (Mishra et al., 2021). However, semi-structured pruning is constrained by its sparsity pattern, preventing flexible density adjust- ments and making it inapplicable for acceleration on general GPUs
The number of parameters in the dense weight matrix is mn, whereas the total number of parameters in the low-rank matrices is r(m + n). As shown in Figure 1, SVD-based low-rank representations fail to compress the weight matrix when exceeds half of the matrix dimensions. However, in low-rank decomposition, the orthogonality constraints among singular vectors reduce the effective degrees of free- dom. Specifically, for U and VT,, there are (2) = r(r-1) unique pairs of singular vectors for each matrix, and each pair imposes one linear constraint due to orthogonality. To- gether, these constraints reduce the total degrees of freedom by (()--)..
GPUs like
3. Structured pruning (Ma et al., 2023; van der Oud- eraa et al., 2024; Ashkboos et al., 2024; Lin et al., 2024; Song et al., 2024; Men et al., 2024; Gao et al., 2024) removes entire components of the model, such as neurons, channels, or attention heads, rather than individual weights. This method preserves tensor align- ment and coherence, ensuring compatibility with all GPUs and enabling significant acceleration on both CPUs and GPUs. However, in LLMs, structured prun- ing can lead to greater loss compared to unstructured or semi-stree structured pruning.
Thus, the actual degrees of freedom in the low-rank repre- sentation are r(m + n.) - (n²² -r). This reveals redundancy in low-rank representations and suggests the possibility of encoding the same information with fewer parameters by eliminating it.
3

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Algorithm 2 PIFA Layer
Question: Can we design a matrix factorization method that reduces parameters to r[m ((++n)) (r² - r) without losing representational power?
input Input X E R where b is batch size; pivot-row indices I E IR; pivot-row matrix W,, E IT coeffi- cient matrix C m))x)r for non-pivot rows
1: Define Ie (1, 2. m m) V representing non-pivot row indices
3.2. Pivoting Factorization
2: Compute output of pivot channels: Y, f W W
To address the previously discussed question, we propose
3: Compute output of non-pivot channels: Y f CY₃
Pivoting Factorization (PIFA), novel matrix factorization
4: Assign pivot channels to output: Y[I,:] :] + Y,
method. For any low-rank matrices, which can be obtained by any low-rank pruning methods, PIFA further reduces parameters without inducing additional loss.
5: Assign non-pivot channels to output: Y[7 f Ynp output Y
The process Pivoting Factorization is illustrated in Fig- ure 2(b). Given weight matrix already decomposed into low-rank matrices U ! and V grx=, we first multiply U and V to get the singular matrix, W = UVT. Since W has rank it contains r linearly independent rows, also referred to pivot rows. The set of linearly indepen- dent rows can be identified using LU or QR decomposition with pivoting (Businger& & Golub, 1971). Let I represent the set of row indices corresponding to the r pivot rows of W' linear combi-
3.3. Memory and Computational Cost of PIFA
Memory cost of PIFA. For each low-rank weight matrix W PIFA needs to store I, W, and C, totaling r(m + n) - +r. r. Figure I illustrates the relationship between the number of parameters in PIFA, traditional low-rank decomposition, and dense weight matrix (square).
Since r(m + n) > r(m + n)) - r² + for any rank PIFA consistently requires less memory than traditional low- rank decomposition. For the comparison with dense weight matrix, because r < min(m, n.), we have:
Thus, any non-pivot row can be expressed nation of these r pivot rows. Denoting the of non-pivot row indices I {1.2.... .... m II, then we define:
W :l,
W mp [
(m--)(n r) >
mn r(m - (3)
where Wp p E Rrx= is the pivot-row matrix, and Wnp E x( is the non-pivot-row matrix. With the definition of non-pivot rows, non-pivot-row matrix can be expressed as:
Neglecting the pivot-row index I, which has negligible memory overhead compared to other variables, PIFA could consistently consumes less memory than a dense weight matrix. In contrast, traditional low-rank decomposition may exceed the memory cost of dense matrices when m
W np CWp
where CE (m)- is the coefficient matrix. Algorithm 1 details the PIFA process, which generates the components of a PIFA layer: pivot-row indices I, the pivot-row matrix W ps and the coefficient matrix C. Algorithm 2 describes the inference procedure for the PIFA layer, which leverages W ps C and I to compute the output.
Computational cost of PIFA. We analyze the computa- tional cost of each linear layer for an input batch size b, where X E Ra We compute the FLOPs for each method as follows:
For the dense linear layer Y WX, where W mmx and X RR the computational cost is 2mnb FLOPSs
Algorithm Pivoting Factorization
input Low-rank matrix W' E mR with rank
For the traditional low-rank layer Y UVTX, where U E mmxr and VT E R the computational cost includes: VTx (2rnb) and U(V'xx) (2mrb). The total cost is 2rnb + 2mrb = 26r(m + n) FLOPs.
1: Use QR (or LU) decomposition with pivoting to find pivot-row indices: I +- QR Q (W")
2: Define Ie {1, 2, ..... m) I. the complement of I,
representing non- non-pi row indices
For the PIFA layer (Algorithm 2), the computational cost includes: Y, W,X (2rn6) and Y,p < CY, 2br(m r). The total cost is 2rnb 2br(m r)
3: Compute pivot-row matrix: W, + [I,
4: Compute ron- matrix: W np +- [T
5: Compute coefficient matrix C by solving matrix equa-
2br + r) FLOPs.
tion: C < linsolve W np = CW,)
output PIFA layer P: 1) Pivot-row indices E IR: 2)
PIFA's computational cost is proportional to its memory cost, differing only by a factor of 2b. As a result, PIFA con- sistently outperforms both dense linear layers and traditional low-rank layers in computational efficiency.
pivot-row matrix Wp p E Tr 3) coefficient matrix C I (m-))) for non-pivot rows

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
non-trivial params (not
preset as zero or one)
Q
W
L
Orthogonal vectors
jpermuted, rank
R
(Coeffideet matridl
W (Pivot rows)
W1
0
rows
24g
1
W2
2
91
X
C2
Non phot rows
Wmn
Singular Matrix
LU Decomposition
QR Decomposition non-trivial params m r[m= n) r(f 13/2
Pivoting Factorization
non-trivial params GPU usage m
non-trivaall params 7(8 + R.G usage - +(mm x))
Amon params r(m +:10
usage r(in 1))
&GP usage - r(m p
Figure 3: Pivoting Factorization vs. LU and QR decompositions. Applied to the permuted matrix (pivot rows at the top), Pivoting Factorization avoids the trapezoidal distribution of non-trivial parameters in LU decomposition, instead reorganizing them into rectangular pattern. This structure optimizes GPU memory usage and reduces computation overhead.
Comparison with LU and QR decomposition. Figure 3 compares the structure of LU and QR decomposition with Pivoting Factorization on permuted weight matrix, where pivot rows have already been moved to the top. LU decom- position retains the same number of non-trivial parameters (i.e., those not preset as zero or one) as Pivoting Factoriza- tion. However, the trapezoidal distribution of non-trivial pa- rameters in LU decomposition complicates efficient storage and computation on the GPU. In contrast, PIFA reorganizes all non-trivial parameters into a rectangular distribution, which is more GPU-friendly for storage and computation. Thus, Pivoting Factorization is more efficient for GPU com- putation.
samples is limited to maximum of 16 LLaMA2-7B (4 on LLaMA2-70B) with a 48GB A6000 GPU, leading to overfitting to the calibration data (see Section 5.3).
Applying the associative property of matrix multiplication, we reformulate Equation 4 into its online version:
U, = VVX"")v(V"(XXXX)
(5)
The term XXT can be computed incrementally as XXT x,x;", where x, represents the input of sample i. As XX E R"*", the memory consumption of the online least squares solution remains constant, regardless of the number of calibration samples.
4. Online Earoo-AccmmlationnNiiinnattooooo Reconstruction
@ Error Accumulation Minimization. Existing recon- struction methods in low-rank pruning (Wang al., 2024) and semi-structured pruning (Frantar & Alistarh, 2023; Li et al., 2024) rely solely on one data flow, i.c. low-rank data flow in Figure 2. This approach allows accumulated errors from previous modules to propagate through the re- construction process, potentially degrading performance, as each subsequent module optimized based on an already- degraded data flow.
In addition to PIFA, we propose novel Online Error- Accumulaionn-Mnimmzzan
Reconstruction (M) method (illustrated in Figure 2(a)) to reconstruct the low-rank matrix before applying PIFA.
A low-rank pruning step is first required to obtain the low- rank matrices U and VT before reconstruction. To achieve this, we adopt the pruning method from SVD-LLM (Wang et al., 2024), which has demonstrated superior performance among existing methods.
Our method mitigates this issue by correcting accumulated error at each module, realigning it with the dense data flow:
min WX, - UVTX,,l
(6)
SVD-LLM first introduced low-rank matrix reconstruction. It updates U using a closed-form least squares solution:
where X, represents the dense data flow input, produced by the previous layer's dense weight, and X,, represents the low-rank data flow input, produced by the previous layer's low-rank weight. This ensures that each module's output remains aligned with the output of original model, recovering the accumulated error in X.,
U U, = arg min || |WX- - UV F
min
(4)
WXD WXD'
VTX
where X is the calibration data. We improve Equation 4 in the following aspects:
However, in practical experiments, we observe that relying solely on the dense data flow output WX,, as the reconstruc- tion target tends to overfit the reconstructed low-rank weight to the distribution of the calibration data. Using a combi- nation of dense and low rank data flow outputs mitigates
① Online algorithm. Equation 4 requires loading the en- tire calibration dataset X into GPU memory to compute the least squares solution. As a result, the number of calibration
5

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
overfitting:
ASVD (Yuan et al., 2023) and SVD-LLM (Wang al., 2024). Vanilla SVD is also included for reference. SVD- LLM has two versions, as detailed in their original paper. Following their approach, we select the best best-performing version for each density. Results for both versions of SVD- LLM are provided in Table 5. MPIFA utilizes 128 cali- bration samples and sets À = 0.25. For all models except LLaMA-2-70B, MPIFA reconstructs both U and V For LLaMA-2-70B, MPIFA reconstructs only U.
Y, = + (1- - W
+
where À is the mix ratio. The optimization target becomes min| Y Uulr. The low-rank data flow output serves regularization term, minimizing the distance be- tween UVTX, and W Since W has been optimized on a much larger and more diverse pre-training dataset, using it as a constraint helps UVT generalize better and prevents overfitting to the limited calibration data. Empir- ically, we found that the mix ratio = 0.25 achieves the best performance (see ablation study in Section 5.3).
Table 2 displays the test perplexity (PPL) of each low- rank pruning method on WikiText2 under 0.4-0.9 density, which defined as the proportion of parameters remaining compared with the original model. The results show that MPIFA significantly outperforms other low-rank pruning method, reducing the perplexity gap by 66.4% (LLaMA2- 7B), 53.8% (LLaMA2-13B), 40.7% 7 (LLaMA2-70B),, and 72.7% (LLaMA3-8B) on average. The perplexity gap is de- fined as the difference between the perplexity of a low-rank pruning method and the original model.
② Reconstructing both U and VT. Equation 4 recon- structs only U. We find it beneficial to also update and provide the closed-form solution:
VT arg min Y - UVTX|IF V
(8)
-U'XX
The proof is provided in Appendix A. Updating VT can also be performed online by incrementally computing YXT and XXT.
Comparison with semi-structured pruning. We further compare MPIFA with 2:4 semi-structured pruning methods: magnitude pruning (Zhu & Gupta, 2017), and two recent state-of-the- works Wanda (Sun et al., 2024), and RIA (Zhang et al., 2024). According to (Mishra et al., 2021), for 16- bit operands, 2:4 sparse leads to <44 savings in GPU memory. Therefore, we compare 2:4 sparse method with MPIFA at 0.55 density to ensure that all methods achieve the same memory reduction (see Table for memory com- parison).
5. Experiments
MPIFA. We combine Online Error-Accumulation Minimization Reconstruction with Pivoting Factorization into an end-to-end low-rank compression method, MPIFA (illustrated in Figure 2). MPIFA proceeds as follows: Re-
Aaccumulation-MMniiizzaa construction is applied to obtain and refine the low-rank matrices U, and Vi:; @ Then, PIFA decomposes the sin- gular matrix W U, VT into I, W ps C < PIFA(W"), which are stored in a PIFA layer that replaces the original linear layer.
① First, Online
Table shows that MPIFANs outperforms 2:4 pruning meth- ods, reducing the perplexity gap by 21.7% for LLaMA2-7B and 3.2% for LLaMA2-13B.
Fine-tuning After Pruning. We investigate how fine- tuning helps recover the performance loss caused by low- rank pruning. The detailed experimental settings are pro- vided in Appendix B.3. As shown in Table 4, fine-tuned MPIFAss achieves the best performance among all fine- tuned pruning methods, bringing its perplexity close to the dense baseline at 55% density, Unlike semi-structured meth- ods, which cannot accelerate the backward pass due to trans- posed weight tensors violating the 2:4 constraint (Mishra et al., 2021), PIFA and other low-rank methods enable ac- celeration in both the forward and backward passes.
MPIFANS- Compared to semi-structured sparsity, MPIFA offers the advantage of allowing non-uniform sparsity for each module. We term the Non-uniform Sparsity version of MPIFA as MPIFANS- The detailed implementation of MPIFANs can be found in Appendix B.2.
Models and Datasets. We apply MPIFA to pre- pre-trained LLMs: LLLMA22 (7B, 13B, 70B) (Touvron et al., 2023b) and LLaMA3 (8B) (Dubey et al., 2024). Both the calibra- tion data and perplexity (PPL) evaluations are based on the WikiText2 dataset (Merity et al., 2022), with a sequence length of 2048 tokens for all experiments.
5.2. Inference Speedup and Memory Reduction
PIFA layer vs low-rank layer. The PIFA layer achieves significant savings in both memory and computation time, as shown in Figure 7, which compares its actual runtime and memory usage with those of a standard linear layer and an SVD-based low-rank layer. In FP32, at 50% density,
5.1. Main Result
Comparison with other low-rank pruning. We evaluate MPIFA against of-te-o-the-ar low-rank pruning methods:
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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Table 2: Perplexity (1) at different parameter density (proportion of remaining parameters relative to the original model) on WikiText2. The best- performing method is highlighted in bold.
Density
Model
Method
100%
90%
80%
70%
60%
50%
40%
SVD
16063
18236
30588
39632
53179
65072
ASVD
5.91
9.53
221.6
5401
26040
24178
LLaMA2-7B
5.47
SVD-LLM
7.27
8.38
10.66
16.11
27.19
54.20
MPIFA
5.69
6.16
7.05
8.81
12.77
21.25
SVD
2168
6177
37827
24149
14349
41758
ASVD
5.12
6.67
17.03
587.1
3103
4197
LLaMA2-13B
4.88
SVD-LLM
5.94
6.66
8.00
10.79
18.38
42.79
MPIFA
5.03
5.39
7.12
7.41
10.30
16.72
SVD
6.77
17.70
203.7
2218
6803
15856
ASVD
OOM
OOM
OOM
OOM
OOM
OOM
LLaMA2-70B
3.32
SVD-LLM
4.12
4.58
5,31
6.60
9.09
14.82
MPIFA
3.54
3.96
4.58
5.54
7.40
12.01
SVD
463461
626967
154679
62640
144064
216552
ASVD
9.37
275.6
12553
21756
185265
13504
LLaMA3-8B
6.14
SVD-LLM
9.83
13.62
23.66
42.60
83.46
163.5
MPIFA
6.93
8.31
10.83
16.41
28.90
47.02
Table 3: Perplexity (1) comparison with semi-structured pruning under the same memory reduction on WikiText2. The best performance pruning method is indicated bold. MPIFANS means MPIFA using non-uniform sparsity.
Table 4: Perplexity ()) of pruned models after fine-tuning on WikiText2 (LLaMA2-7B) The best cest-performance prun- ing method indicated in bold.
Method
1.MMA2-BB
Method
LLaMA2-7B
LLaMA2-13B
Dense
5.47
Dense
5.47
4.88
Magnitude 2:4
6.63
Magnitude 2:4
37.77
8.89
Wanda 2:4
6.40
Wanda 2:4
11.40
8.33
RIA 2:4
6.37
RIA 2:4
10.85
8.03
SVD 55%
9,24
SVD 55%
69128
24947
ASVD 55%
8.64
ASVD 55%
9370
2039
SVD-LLM 55%
7.36
SVD-LLM 55%
20.43
13.69
MPIFANS 55%
6.34
MPIFANs 55%
9.68
7.93
implemented using cuSPARSELI CUTLASSS library, with the reported speedup representing the higher value between the two implementations. The results span various dimensions on A6000 and A 100 GPUs. PIFA demonstrates consistently superior efficiency, achieving the highest speedup and lowest memory usage in all config- urations except d 4096. Notably, PIFA's accelcration increases with matrix dimensions, reflecting its scalabil- ity and computational effectiveness. As shown in Table 6, at d = 32768, PIFA with 55% density achieves 2.1 x speedup, while 2:4 (CUTLASS) is slower than the dense
PIFA achieves 47.6% memory savings and 1.95x speedup, closely matching the ideal memory and speedup. Addition- ally, at the same rank, PIFA consistently achieves higher compression and faster inference than the low-rank layer. For example, at r/d= = 0.5, PIFA losslessly compresses the memory of the low-rank layer by 24.2% and reduces inference time by 24.6%.
PIFA layer vs semi-sparse layer. Figure 4 and Table 6 compare the speedup and memory usage of the PIFA layer with 2:4 semi-sparse linear layers. The semi-sparse lay-
hnttps://dbss nvidde com/ htmm/cuda/aee coda/cusparsell httpps//giihhin com/ WWIDTA//C ADD.///c/llsssss

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
worsens perplexity compared to using only the prun- ing step (W). This highlights the drawbacks of full- batch methods, as overfitting to the limited calibration data can degrade performance.
A6000, 2:4
A100, 2:4
A6000, PIFA
1.8x
Dense
Spondup
1.6x
2. Our reconstruction method (W + M) consistently outperforms full-batch reconstruction (W U) and pruning alone (W) across all models and densities. This demonstrates the effectiveness of Online Error- Accumulation Aecumulation-Minimizateee Reconstruction in reduc- ing error accumulation and improving the compression of low-rank matrices.
1.2x
1.0x
0.8
4096
8192
16384
32768
Dimensian
3. PIFA further improves performance when com- bined with M. The W + M PIFA configuration achieves the best perplexity across all settings, vali- dating the advantage of applying PIFA for additional parameter reduction without inducing any additional loss.
Figure 4: Layerwise speedup of the PIFA layer and semi- sparse layers across various dimensions, compared to dense linear layers on the same GPU, with a sequence length of 2048 and a batch size of 32, using FP16 (FP32 is not sup- ported by 2:4 sparsity torch. sparse). PIFA shows increasing speedup as the dimension grows. Detailed values provided in Table 6.
These findings emphasize the significance of M and PIFA in achieving superior low-rank pruning performance.
linear layer, and 2:4 (cuSPARSELI) raises error.
LLaMA2- 7B| MPIFA
End-to-end LLM inference. Table 7 compares the end- to-end inference throughput and memory usage of MPIFANS with semi-sparsity (2:4 cuSPARSELI and CUTLASS) on LLaMA2-7B and LLaMA2-138 models in FP16. MPIFANS consistently outperforms semi-sparsity in both through- put and memory efficiency 55% density. Furthermore, the operations supported by semi-sparsity are limited in torch. sparse, resulting in errors when the KV cache is enabled, which further limits the application of semi-sparse for LLM inference.
18
LLaMA2-138 IMPIFA
16
1.4
10
0.4
0.6
0.8
1.0
Mix Ratio (A)
Figure Impact of mix ratio. With 0.5 density, MPIFA achieves lowest PPL when the mix ratio in Equation 7 is around 0.25.
5.3. Ablation Study
Impact of PIFA and M Table 5 presents an ablation study that evaluates the impact of our Online Error- Accumulation- Minimization Reconstruction (denoted as M) and Pivoting Factorization (PIFA) on perplexity across varying parameter densities. The methods compared include:
Impact of mix ratio À in M. The mix ratio À in Equation determines the proportion of the dense data flow in the reconstruction target. As shown in Figure 5, using mod- erate ratio À 0.25, MPIFA achieves significantly lower PPL compared to X O, where the reconstruction target relies solely on the low-rank data flow, as in previous stud- ies (Wang et al., 2024; Frantar & Alistarh, 2023) did. This demonstrates the effectiveness of our error- accrmuaccumlaaiion corrected strategy, in which the dense data flow output is beneficial as part the reconstruction target. In Figure 5, we also observe that an excessively large À increases PPL. indicating overfitting to the calibration data.
W: Using only the pruning step of SVD-LLM (truncation-wwarc data whitening).
W + U: Applying SVD-LLM's pruning followed by full-batch reconstruction.
+ M: Employing our Online Accummacccmllation- Minimization Reconstruction, which incorporates SVD-LLM's pruning as the initial step.
W M PIFA: Combining Online Error- Decumulation-Mnninizztiooo Reconstruction with PIFA (denoted as MPIFA).
Impact of Calibration Sample Size. M depends on cal- ibration data to accurately estimate U and VT. As shown in Figure 6. the perplexity of MPIFA decreases as the num- ber of calibration samples increases. We hypothesize that
The results reveal several key findings:
1. Full-batch reconstruction (W U) occasionally
8

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Table 5: Ablation: Impact of PIFA and M on perplexity (L) across parameter densities. on WikiTex12. W denotes using SVD-LLM's pruning only; W + U denotes using SVD-LLM's pruning and full-batch reconstruction; M denotes using our Online AerorrAccuulaation-- Aacuumulaton-Mininizzttooo Reconstruction, which incorporates SVD-ILM'' pruning as the initial step; Reconstruction followed by PIFA, i.e., MPIFA
W + M PIFA denotes using Online ietoo--ccuulaaioo-Miiinnn
Density
Model
Method
100%
90%
80%
70%
60%
50%
40%
W
7.27
8.38
10.66
16.14
33.27
89.98
W + U
7.60
8.84
11.15
16.11
27.19
54.20
LLaMA2-7B
5.47
W + M
6.71
7.50
8.86
11.45
16.55
25.26
W M + PIFA (MPIFA)
5.69
6.16
7.05
8.81
12.77
21.25
W W +
5.94
6.66
8.00
10.79
18.38
43,92
6.45
7.37
9.07
12.52
20.95
42.79
LLaMA2-13B
4.88
W + M
5.80
6.41
7.42
9.31
13.09
19.93
W M PIRA (MPIFA)
5.03
5.39
7.12
7.41
10.30
16.72
W W U
4.12
4.58
5.31
6.60
9.09
14.82
8.23
8.33
8.66
10.02
13.41
22.39
LLaMA2-70B
3.32
W + M W M + PIFA (MPIFA)
4.15
4.63
5.31
6.46
8.72
14.11
3.54
3.96
4.58
5.54
7.40
12.01
W W U
9.83
13.62
25.43
76,86
290.3
676.7
10.63
14.66
23.66
42.60
83,46
163.5
LLaMA3-8B
6.14
W + M W M PIFA (MPIFA)
9.16
11.25
15.27
23.55
36.
53.85
6.93
8.31
10.83
16.41
28.90
47.02
Impact of reconstructing U and VT. Figures 6a and 6b compare the effects of reconstructing only U, only V and both U and VT across different calibration sizes. The results indicate that with sufficient calibration samples, re- constructing both U and V achieves lower perplexity than reconstructing only U or only VT
with metrott
(a) LLaMA2-7B
(b) LLaMM2-13B
6. Conclusion and Discussion
Figure 6: Impact of calibration sample size. On MPIFA with 0.5 density, reconstructing both U and V is more sensitive to the number of calibration samples than recon- structing only U.
In this work, we proposed MPIFA, an end-to-end, retraining-free low-rank pruning framework that integrates Pivoting Factorization (PIFA) and an Online Error- CeeumulationMMnninzzatiit
Reconstruction (M) algo- rithm. PIFA serves as meta low-rank representation that further compresses existing low-rank decompositions, achieving superior memory savings and inference speedup without additional loss. While this allows PIFA to integrate seamlessly with various low-rank compression techniques, does not reduce matrix rank on its own and must be applied alongside low-rank compression method. Meanwhile, M mitigates error accumulation, leading improved per- formance. Together, these innovations enable MPIFA to achieve performance comparable to semi-structured pruning while surpassing it in GPU acceleration and compatibility. Future work could explore integrating PIFA into the pretrain- ing stage, as PIFA is fully differentiable, enabling potential efficiency gains during model training.
increasing the number of calibration samples reduces the condition number of the least squares solution, improving numerical stability.
To investigate this, we calculate the condition numbers of V''xx'V Equation and XXT Equation 8, as their inverses are required for reconstructing and VT. Figure presents the condition numbers for these matrices the first layer of LLaMA2-7B. The observed reduction in condition number indicates that the matrices become less singular as the calibration sample size increases, thereby improving numerical stability when solving the least squares equations. This increased stability ultimately results in lower perplexity in the reconstructed model.

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Acknowledgements
Frantar, E. and Alistarh, D. Sparsegpt: Massive language models can be accurately pruned in one-shot. In Inter- national Conference Machine Learning, pp. 10323- 10337. PMLR, 2023.
This work was supported by the Zhou Yahui Chair Professor- ship award of Tsinghua University (to CVC), the National High-Level Talent Program of the Ministry of Science and Technology of China (grant number 202417 100001, to CVC).
Gao, L.,, Tow, J., Abbasi, B., Biderman, S., Black, S., DiPofi,
A., Foster, C., Golding, L.,, Hsu, J., Le Noac h. A., Li,
Impact Statement
H., McDonell, K., Muennighoff, N., Ociepa, C.,, Phang,
J.. Reynolds, L., Schoelkopf, H., Skowron, A., Sutawika,
This paper presents work whose goal is to advance the field of Machine Learning. There are many potential societal consequences of our work, none which we feel must be specifically highlighted here.
L., Tang, E., Thite, A., Wang, B., Wang. K., and Zou, framework for few-shot language model evaluation, 12 2023, URL https: /ttps:///zenodoe zenooo org/ records/ 1 10256836.
Gao, S., Lin, C.-H., Hua, T., Tang, Z., Shen, Y., Jin, H., and Hsu, Y.-C. Disp-Ilm: Dimension Dimension-independent structural pruning for large language models. Advances in Neural Information Processing Systems, 37:72219-72244 2024.
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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Appendix
4.0
4.0
Realistic Low Rank Time
Realistic Low Rank Time
3.5
3.5
Realistic PIFA Time
Realistic PIFA Time
3.0
Ideal Low-Rank: 2nd
Ideal Low-Rank: 2rd
2.5
Ideal PIFA:
2.5
Ideal PIFA: 2rd
d.n
of
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2.0
densiy=0.55
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density=0.3
0.2
0.4
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
dd
rld
(a) FP32 Inference Speedup
(b) FP16 Inference Speedup
2.00
2.00 Realistic Memory
Low-Rank Memory
1.75
Realistic PIFA Memory
1.75
Realistic PIFA Memory
Normalized Memeey
1.50
Ideal Low-Rank: 2rd
1.50
Ideal Low-Rank: 2rd
Ideal PIFA: 2rd -
Ideal PIFA; 2rd
1.25
1.25
1.00
1.00
0.75
density-0.5
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density-O.5 0.5
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density=0.5
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density=0.5
0.00
0.00
0.0
0.2
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1.0
0.0
0.2
0.4
0.6
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rld
rid
(c) FP32 Inference Memory
(d) FP16 Inference Memory
Figure 7: Efficiency PIFA layer under various ranks, with sequence length 2048, batch size 32, and dimension 8192 FP32 and FP16 A6000 GPU. At 50% density, PIFA achieves 47.6% memory savings and 1.95x speedup FP32. These results guarantee that the overhead of both time and memory of the PIFA layer is quite low.
13

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
xXI
1046
vixX'v
1024
Condition Number
5.3e+13
4.9e+13
4.70+13
4.5e+13
10
1020
10
106
8.8e+04
7.1e+04
6.2e+04
5, 5.8c+04
x
20
40
60
80
100
120
Calibration Size
Figure 8: Condition number. Condition numbers of V V (Equation 5) and XXT (Equation 8) for the first layer of LLaMA2-7B, whose inverses are used in reconstructing U and V Larger calibration sizes reduce condition numbers, enhancing numerical stability and lowering perplexity.
Table Efficiency of PIEA layer and semi-sparse layer across different dimensions, compared to dense linear at same dimension on same GPU, with sequence length of 2048 and batch size of 32. using FP16 (FP32 sparsity in torch. sparse). The highest speedup and lowest memory are indicated in bold. PIFA shows increasing speedup as the dimension grows. For matrix multiplication with weight matrix shape 32768 x 32768, cuSPARSELA raises CUDA error
supported by 2:4
Dimension
GPU
Kernel
32768
16384
8192
4096
2:4 (cuSPARSELO)
Error
0.94x
0.97
1.09
A6000
2:4 (CUTLASS) PIFA 55%
0.79x
0.92x
1.15x
1.18x
2.10x
1.88
1.70
1.43x
Speedup
2:4 (cuSPARSELL)
Error
1.19x
1.31 x
1.68x
A100
2:4 (CUTLASS)
1.19x
1.12x
1.09
1.52x
PIFA 55%
1.98x
1.70
1.54x
1.37x
2:4 (cuSPARSEL:/ CUTLASS)
0.564
0.569
0.589
0.651
Memory
PIFA 55%
0.552
0.558
0.578
0.645
14

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Table 7: End-to-end efficiency of MPIFANS LLaMA2 models, FP16 (FP32 isn't supported by semi- sparse). The highest throughput and lowest memory indicated in bold. MPIFANS consistently outper- forms semi-sparse in both throughput and memory with 55% density. "Enabling KV cache for semi-sparse model will cause operations, can't t perform requested op (expand. default)
only supports specific set of
Model
Metrics
GPU
Use KV Cache
Dense
(cuSPARSELA)
2:4 (CUTLASS)
MPIFAss 55%
No
354.9
306.6
327.5
472.6
A6000
Throughput (token/s)
3409
Error
Error
4840
Ilama2-7b
No
614.8
636.2
582.3
822.2
A1000
Yes
6918
Error
Error
7324
Memory (GB)
12.55
7.274
7.290
7.174
No
190.0
163.2
180.0
268.7
A6000
Yes
2156
Error
Error
2721
Throughput (token/s)
Ilama2-13b
345.4
362.4
321.5
473.3
A100
Yes
4217
Error
Error
4532
Memory (GB)
24.36
13.90
13.99
13.69
15

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
A. Closed-Form Solution of V
aim to prove that minimizing the Frobenius norm || UV X following two-step optimization:
F with respect to V is equivalent to performing the
1. First, minimize || Y WX|| Ir with respect W.
2. Then, minimize W UV F with respect to V
We begin by directly minimizing |Y - x
l with respect to V
A.1. Direct Optimization
f(V) = IY - U|
= Tr -Tr( - -V - -V
=T:(Y-Y - Y'UV'' - +
= =T(YY" -
Let us define intermediate matrices:
A = XY"u
B= XX
C U²U
The objective function becomes:
f(V) = const - (TT((TT()))))) BV(TBBV))
where "const" denotes terms independent of V.
Compute the gradient f(V) with respect to V:
0f @V
22A 2BVC
Set the gradient to zero:
22 + 2BVC = = BVC A
Assuming B and C are invertible, we solve for V:
V = B--AC-
Substituting back the definitions of A, A, C:
V = (X (XYTU) (XX") (U*
Simplify:
VT = xx'
16

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
A.2. Two-Step Optimization
Now, we perform the two-step optimization and show it leads the same result.
First, minimize Y - - with respect to W.
Compute the gradient:
Y
WX 1 = - wx)x
oW
Set the gradient to zero:
(Y - -x =0 = YX- wxx.
Assuming XXT is invertible:
W W (XX
Next, minimize |W- - UVT | with respect to VT.
Compute the gradient:
w UV -22
Set the gradient to zero:
- U'UT
Assuming UTU is invertible:
V = --'
Substitute W:
V = (0* ' (XX-(
Simplify:
TT = (U"U) "UXX""""""""
A.3. Conclusion
The solution for VT obtained through both the direct optimization and the two-step optimization is:
V =
-X
Therefore, minimizing Y UVT X || with respect to VT is equivalent to first optimizing || Y WX]
F with respect to
W, and then optimizing W UVT || with respect VT
17

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
B. Experiment Details
B.1. MPIFA
The full method of MPIFA is outlined in Algorithm
Algorithm 3 MPIFA
input Original weight matrix W E Rmxx; calibration input from dense X. E Raxb; calibration input from low rank X E guxb; target rank r; original output ratio
Part 1: Online
Reconstruction
1: Compute dense output as dense input of next module: Y. WX,
2: Use SVD-LLM's pruning (truncation-ane canaee data whitening) to convert to low-rank matrix: U, ST < -VD--LL(()
3: Compute XXT accumulatively: XXT E x
- 4: Compute Y, X accumulatively: Y,X + + (Axx) + (1- -)-x,)
5: Reconstruct U as U,.: U, +- (Y,X VV
6: Reconstruct V as V,: V + ( (Y,XX (XX")-
7: Compute low-rank output as low-rank input of next module: Y, = WX,,
Part 2: PIFA
8: Compute low-rank matrix W W < U.V.
9:
Use Algorithm to build the PIFA layer P using low-rank matrix W
output PIFA layer P
potential issue that XXT can be singular some cases, leading to NaN values when calculating the inverse matrix during reconstruction. To address this, leverage prior knowledge that UVT should approximate W by adding regularization term to the original optimization target, modifying Equation 8 as follows:
V = arg min Y - UV* +o| - ||P VT
(9)
(0' (Y,x + (XX" + +
where a is the regularization coefficient, set to 0.001 in all experiments. Regularization is unnecessary for reconstructing U, as no singularity issues were observed for V (V.T)V.
B.2. MPIFANS
MPIFANs is the non-uniform sparsity variant of MPIFA, designed to leverage different sparsity distributions across model layers and module types for improved performance. It employs 512 calibration samples. This approach incorporates two key components to define module densities: Type Density and Layer Density, which are combined multiplicatively to determine the final density for each module.
Type Density. Type Density introduces non-uniform sparsity between attention and MLP modules. Based insights from prior literature (Yuan et al., 2023), MLP modules exhibit higher sensitivity to pruning compared to attention modules. To account for this, we search for the density of attention modules within (Global Density, Global Density 0. 1. optimizing for performance. The density of MLP modules is then calculated to ensure that the model's global density remains unchanged.
Layer Density. Layer Density accounts for non-uniform sparsity across layers. For this, MPIFANs adopts the layerwise density distribution from OWL (Yin et al.), which identifies layer-wise densities based on outlier distribution. By directly utilizing these precomputed layer densities, MPIFANs ensures that density is allocated more effectively across layers, balancing pruning across regions of varying importance.
18

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Table 8: C4 Perplexity (1) at different parameter density (proportion of remaining parameters relative to the original
model). The best- best-performing method is highlighted in bold.
Density
Model
Method
100%
90%
80%
70%
60%
50%
40%
SVD
18931
27154
37208
56751
58451
70567
ASVD
7.98
12.46
201.0
9167
25441
24290
LLaMA2-7B
7.29
SVD-LLM
13.95
19.89
33,02
61.97
129.8
262.9
МРIFA
8.15
10.20
14.68
25.43
52.01
97.71
SVD
1994
6301
37250
22783
18196
84680
ASVD
7.15
9.30
23.54
468.5
3537
3703
LLaMA2-13B
6.74
SVD-LLM
10.93
14.99
24.44
46.65
110.4
267.8
MPIFA
7.27
8.79
21.00
21.33
42.03
80.47
SVD
10.16
23.28
121.4
1659
7045
12039
ASVD
OOM
OOM
OOM
OOM
OOM
OOM
LLaMA2-70B
5.74
SVD-LLM
7.12
8.71
12.21
21.40
44.10
103.3
MPIFA
6.00
6.76
8.67
13.60
29.04
63.38
SVD
323597
461991
172968
70896
143573
271 176
ASVD
14.43
272.1
8511
18701
108117
9466
LLaMA3-8B
9.47
SVD-LLM
38.54
98.65
223.5
460.0
784.8
1416
MPIFA
14.76
22.45
44.62
123.0
257.4
429.2
Final Module Density.
The final density for each module in MPIFANs is calculated as:
Module Density = Type Density Layer Density
Global Density
This formulation ensures that the final density for each module accounts for both type- and layer-specific sparsity re- quirements, leading to a more effective pruning configuration optimizing performance while maintains the global density same.
In summary, MPIFANS combines the benefits of non-uniform sparsity across both types of modules and individual layers, achieving better performance while ensuring the global density of the model remains unchanged.
B.3. MPIFANS Fine-tuning
Fine-tuning is performed using a mixed dataset comprising the training set WikiText2 and one shard (1/1024) of the training set of C4 (Raffel et al., 2020). WikiText2's 's training set more aligned with the evaluation dataset but contains limited number of tokens, whereas the C4 dataset is significantly larger but less aligned with the test set. To balance these characteristics, the datasets are mixed at ratio of 2% Wiki'eex22 to 98% C4.
We limit fine-tuning to single epoch, which requires approximately one day single GPU (around 1000 steps). The line-tunig process updates all pruned parameters, including low-rank matrices and semi-sparse matrices, while keeping other parameters, such as embeddings, fixed.
The learning rate is set x 10-6, with warmup phase covering the first 5% of total steps, followed by a linear decay to zero. The sequence length is fixed 1024, with batch size of 1 and gradient accumulation steps of 128.
C. Perplexity Evaluation on C4 Dataset
To complement our evaluation on WikiText2, we assess MPIFA on the C4 dataset (Raffel et al., 2020), a widely used benchmark for evaluating LLM perplexity on large-scale, real-world text distributions. For consistency with the main text, we use WikiText2 as the calibration dataset for all methods in this evaluation. The results, summarized in Table 8, demonstrate that MPIFA consistently achieves the lowest perplexity across most density levels and model sizes, reducing the
19

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
perplexity gap by 47.6% LLaMA2-7B, 34.5% LLaMA2-13B, 55.3% LLaMA2-70B, and 62.6% LLaMA3-8B average across all densities, compared to the best-performing baseline.
D. Zero-Shot Evaluation SuperGLUE Benchmark
To provide more comprehensive evaluation MPIFA's performance across different compression levels, we conduct zero- shot evaluations at various compression rates on the SuperGIUUE benchmark (Wang et al., 2019) using the LLaMA2-7B model.
Evaluations conducted using the Im-evaluatioonharreee using accuracy (1), except for the ReCoRD task, which uses Fl score. The detailed accuracy results are presented in Table 9. The best- best-performing method at each setting is highlighted in bold. MPIFA consistently achieves the highest mean accuracy across all density levels, outperforming other low-rank methods across a wide range of tasks.
framework (Gao al., 2023). All tasks evaluated
Table 9: Zero-shot evaluations SuperGLUE datasets at different parameter density LLaMA2-7B. All tasks evaluated using accuracy (t), except for the ReCoRD task, which uses F1 score (f). The best-performing method is highlighted in bold.
Density
Method
BoolQ
CB
COPA
MultiRC
ReCoRD
RTE
WIC
WSC
Mean
100%
Dense
77.7
42.9
87.0
57.0
91.6
63.2
49.7
36.5
63.2
SVD
42.6
39.3
67.0
51.0
16.4
55.6
49.8
62.5
48.0
ASVD
55.9
37.5
69.0
47.1
42.5
53.4
49.8
41.3
49.6
90%
SVD-LLM
49.1
41.1
79.0
57.1
87.8
52.7
48.0
48.1
57.8
MPIFA
74.4
64.3
86.0
56.7
91.2
58.5
49.7
36.5
64.7
SVD
45.9
57.1
59.0
48.9
12.5
47.3
50.0
58.7
47.4
ASVD
41.6
33.9
58.0
46.8
24.6
55.2
50.0
60.6
46.3
80%
SVD-LLM
44.2
41.1
79.0
55.7
84.7
53.1
50.6
57.7
58.3
MPIFA
69.4
41.1
83.0
55.8
90.3
53.4
48.7
36.5
59.8
SVD
40.2
46.4
62.0
43.1
12.4
53.8
48.9
63.5
46.3
ASVD
39.2
46.4
59.0
48.2
14.0
49.1
50.3
63.5
46.2
70%
SVD-LLM
44.4
39.3
82.0
44.6
79.8
53.8
48.9
60.6
56.7
MPIFA
64.8
41.1
80.0
57.2
87.5
53.8
49.1
42.3
59.5
SVD
45.5
41.1
61.0
49.4
11.4
51.6
50.0
60.6
46.3
ASVD
48.9
37.5
59.0
46.6
15.2
50.2
50.0
40.4
43.5
60%
SVD-LLM
38.5
39.3
71.0
44.9
66.4
53.4
50.2
59.6
52.9
MPIFA
56.6
35.7
79.0
44.5
82.8
57.8
52.0
63.5
59.0
SVD
38.6
44.6
63.0
43.0
10.1
52.7
50.2
63.5
45.7
ASVD
42.2
39.3
63.0
45.7
14.4
52.7
50.0
63.5
46.3
50%
SVD-LLM
37.9
41.1
66.0
42.8
50.5
52.7
50.0
63.5
50.5
MPIFA
38.0
35.7
70.0
42.8
71.1
52.4
50.0
63.5
52.9
SVD
49,5
42.9
61.0
48.1
11.3
52.0
50.0
49.0
45.5
ASVD
42.6
42.9
59.0
47.7
14.5
53.1
50.0
64.4
46.8
40%
SVD-LLM
37.8
41.1
67.0
42.8
37.0
52.7
50.0
63.5
49.0
MPIFA
37.8
37.5
68.0
42.8
54.7
53.8
50.0
63.5
51.0
E. Comparison with Structured Pruning Baseline
To provide a fair comparison with structured pruning methods, we include additional evaluations of LLM-Prunnr (Ma al., 2023) as a baseline. All experiments are conducted on the WikiText2 dataset using the LLaMA2-7B model.
20

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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Perplexity Comparison. Table 10 shows the perplexity across various parameter densities. On average, MPIFA reduces the perplexity gap by 87.2% compared to LLMM-Prnner.
Table 10: LLM-Pruner MPIFA: Perplexity Comparison WikiText2 using LLaMA2-7B different densities. MPIFA consistently achieves lower perplexity across all densities.
Method
100%
90%
80%
70%
60%
50%
40%
6.58
8.81
13.70
40.49
126.0
1042
5.47
MPIFA
5.69
6.16
7.05
8.81
12.77
21.25
Inference Speedup and Memory Efficiency. Table 11 and Table 12 compare the inference speedup and memory usage (relative to dense linear layer) for PIFA and LLMM-Punner layers, across different hidden dimensions on an A6000 GPU.
Table 11: Inference speedup (x over dense) for PIFA and LLM-Prunne layers.
Method (density)
d=16384
d=8192
d=4096
PIFA (55%)
1,88x
1.70x
1.43x
LLM-Prunce (55%)
1.81x
1.77x
1,67x
LLM-Prunee (70%)
1.42x
1,41x
1.35x
Table 12: Memory usage (x over dense) for PIFA and LLMM-runer layers.
Method (density)
d=16384
d=8192
d=4096
PIFA (55%)
0,56x
0.58x
0.64x
M.MM-runer (55%)
0.56x
0.58x
0.65x
LLM-Pruner (70%)
0.70x
0.72x
0.75x
At the same density (55%), PIFA achieves similar speedup and memory efficiency as LLM-Pruner. When comparing MPIFA 55% density LLMM-Pruner 70% density, MPIFA consistently achieves lower perplexity, faster inference, and reduced memory usage.
F. Compression Time and Memory Usage Comparison
We provide a detailed comparison of the compression time and peak GPU memory usage during compression across different methods. These metries reflect the practical resource requirements of each approach.
Compression Time. Table 13 reports the time required to perform compression for each method, measured on single A6000 GPU. On A100 GPU, the compression time is approximately halved. ASVD requires significantly longer compression time, up to 10-20 hours, due to the need for sensitivity-based truncation rank searching. The M method involves only reconstruction.
Table 13: Compression time for different methods on single A6000 GPU,
Model
ASVD
SVD-LLM
PIFA
M
LLaMA2-7B
10h
30 min
min
min
LLaMA2-13B
20h
1h
30 min
1h
Peak Memory Usage During Compression. Table 14 reports the peak GPU memory usage during compression. Method M has a relatively low memory footprint, processes one layer and one sample at time.
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Pivoting Factorization: A Compact Meta Low-Rank Representation of Sparsity for Efficient Inference in Large Language Models
Table 14: Peak GPU memory usage (GB) during compression for different methods.
Model
ASVD
SVD-LLM
PIFA
M
LLaMA2-7B
15G
20G
0.3G
6G
LLaMA2-13B
30G
25G
IG
10G
For the M method, we report only the reconstruction time, excluding the time required for the initial low-rank pruning step, which could be SVD or SVD-LLM. The low memory footprint of M is primarily attributed to:
Online calibration: Only the current sample is loaded into GPU memory, while the rest of the samples remain in CPU memory until needed.
Layer-wise loading: Only the current pruning layer is loaded to the GPU at any time, with all other layers remaining on CPU.
G. Enhancing Other Low-Rank Pruning Methods with PIFA and M
To demonstrate the generality of our proposed methods, we evaluate how PIFA and M can be applied on top of existing low-rank pruning techniques beyond SVD-LLM, including the pruning strategies proposed in the ESPACE paper (Sakr Khailany).
For pruning-only comparison, we reproduce the pruning step of ESPACE and compare its performance when combined
with PIFA and M. ESPACE introduces six variants: MSE (Eq, 6). MSE-NORM (Eq. 7), GO-MSE (Eq. 8), GO-MSE-NORM (Eq. 8), NL-MSE (Eq. 9). and NL-MSE-NORM (Eq. 9). We exclude the NL-MSE variants as they rely backpropagation, which is infeasible on memory-constrained GPUs.
Table 15 reports the perplexity (PPL) on WikiText2 at 50% density using the LLaMA2-7B model, showing how PIFA and M further improve various low-rank pruning baselines.
Table 15: Perplexity (PPL) on Wiki'eex22 at 50% density using LLaMA2-7B.
Pruning Method (X)
X + PIFA
X+
X + MPIFA
SVD-LLM (W)
33.27
19.64
16.55
12.77
ESPACE (MSE)
280. 19
144.32
20.99
16.84
ESPACE (MSE-NORM)
172.30
113.82
21.73
17.42
ESPACE (GO-MSE)
41.75
24.17
17.47
13.55
ESPACE (GO-MSE-NORM)
37.45
23.19
17.47
13.63
Here, SVD-LLM (W) refers to the standalone pruning output from SVD-LLM without additional reconstruction.
PIFA and M consistently improve the performance of different low-rank pruning methods, including ESPACE, demonstrating that they are general- general-purpose purpose techniques applicable across various low-rank strategies. These results suggest that PIFA and M are not specific to SVD-LLM but can serve as plug-in modules for other pruning methods to improve accuracy while maintaining efficiency.
22


Reviews:
[{"review_id": "7RSJKIoSHb", "reviewer": "Reviewer_aFwh", "summary": "This submission addresses the significant performance degradation observed with low-rank pruning techniques. It proposes Pivoting Factorization (PIFA), a novel lossless meta low-rank representation that unsupervisedly learns a compact form of any low-rank\nrepresentation, effectively eliminating redundant information.  To mitigate the performance degradation caused by low-rank pruning, we introduce a novel, retraining-free low-rank reconstruction method that minimizes error accumulation (M). The authors mentions that their framework MPIFA for the first time, achieves performance comparable to semi-structured pruning, while surpassing it in GPU efficiency and compatibility.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your valuable and constructive feedback. We appreciate the opportunity to address your concerns.\n\n**Weakness 1: (Expand evaluation)** The authors have failed to ... existing tools like LMEvalHarness etc.\n\n**Reply:** Thank you for raising this concern. We have extended our evaluation in two directions:\n\n1. **C4 Perplexity (PPL) Evaluation**: We report perplexity on the C4 dataset across various model sizes and parameter densities. The results demonstrate that **MPIFA significantly outperforms existing low-rank pruning methods**, reducing the perplexity gap by  \n   - **47.6%** on LLaMA2-7B  \n   - **34.5%** on LLaMA2-13B  \n   - **55.3%** on LLaMA2-70B  \n   - **62.6%** on LLaMA3-8B  \n   on average across all densities, compared to the best-performing baseline.  \n   Full results: https://anonymous.4open.science/r/PIFA-68C3/c4_ppl.png\n\n   Example results at 50% density:\n\n   | Model        | SVD    | ASVD    | SVD-LLM | MPIFA    |\n   |--------------|--------|---------|---------|----------|\n   | LLaMA2-7B    | 58451  | 25441   | 129.8   | **52.01** |\n   | LLaMA2-13B   | 18196  | 3537    | 110.4   | **42.03** |\n   | LLaMA2-70B   | 7045   | OOM     | 44.10   | **29.04** |\n   | LLaMA3-8B    | 143573 | 108117  | 784.8   | **257.4** |\n\n2. **Task-Centric Evaluation**: We conducted **zero-shot evaluation** on the **SuperGLUE benchmark** using 8 downstream tasks via the lm-evaluation-harness (https://github.com/EleutherAI/lm-evaluation-harness) framework. All methods use WikiText2 as the calibration dataset, and follow the same configuration as in the main manuscript.  \n   Results: https://anonymous.4open.science/r/PIFA-68C3/zero_shot.png\n\n   MPIFA_NS achieves the best accuracy on **6 out of 8 tasks** and reduces the **average accuracy gap by 42.8%** compared to the best-performing low-rank pruning baseline.\n\nWe appreciate the reviewer’s suggestion. These additional evaluations have been added to the revised manuscript.\n\n**Weakness 2:** Wikitext2 dataset doesn't contain enough details (e.g. seqlen etc.).\n\n**Reply:** Thank you for the feedback. The sequence length used in all experiments, including both WikiText2 and C4, is **2048**. We have updated the manuscript to include this information.\n\n**Weakness 3:** In addition, it will be also interesting to see how the proposed method extend beyond only one single family of models (LLaMa 2/3) - may be on some MoE style models.\n\n**Reply:** Thank you for the suggestion. We are currently exploring this and will report the results here as soon as possible.\n\n**Weakness 4: (Expand related work)** Related work requires some upgrade ... arXiv:2502.02723.\n\n**Reply:** Thank you for the suggestion. We have updated the manuscript and included these articles in the related work section.\n\n**Weakness 5:** Authors introduce two variables r and d directly in abstract and introduction without explicitly defining it. \n\n**Reply:** Thank you for the suggestion. We have revised the original phrasing “at $r/d = 0.5$” to “at rank equal to half of the dimension” for clarity.\n\n**Weakness 6: (Improve the article's structure)** The writing of the paper ... in supplementary.\n\n**Reply:** Thank you for this valuable suggestion. We have moved Section 3.3 to the appendix while retaining the key conclusions about the computational and memory cost of PIFA in the main paper. Since the final version allows one additional page, we have also moved the previously referenced plots from the appendix (cited in Section 5.3) into the main body, ensuring the draft is more self-contained without exceeding the page limit.\n\n\n**Weakness 7:** The paper mentions that PIFA is fully differentiable, suggesting potential integration into the training stage. Have the authors explored this direction?\n\n**Reply:** Thank you for raising this interesting point. We are happy to share our preliminary findings. While PIFA ensures **lossless conversion in the forward pass**, it does **not guarantee identical gradient descent dynamics** compared to traditional low-rank training methods, as different factorizations can lead to different gradients.  \n\n**Weakness 8:** How can this method integrated and benefit some recent quantization techniques like ZeroQuant-v2.\n\n**Reply:** Thank you for the suggestion. We are currently exploring this and we will provide further updates as soon as possible.", "claims_and_evidence": "1. Comparative speedup of PIFA + MPIFA: Yes, the authors speedup evaluation across different GPUs + Kernels across different $d$ as well as end-to-end efficiency evaluation in Table 5, 6 are significantly important and convincing.\n2. Rich ablation experiments across the calibration sample size, mix ratio etc are interesting and bolster the claim of superiority of the proposed method.", "methods_and_evaluation_criteria": "The serious and most important concern of the submission is the evaluation strategy adopted in the paper.\n\nThe authors have failed to extend their evaluation beyond perplexity which have issues in true reflection of compressed model capabilities. Even the reported perplexity is only limited for Wikitext2 dataset without enough details (e.g. seqlen etc.). \n\nI strongly recommend the authors to extend their evaluation to other PPL datasets like C4 and incorporate task-centric evaluation using existing tools like LMEvalHarness etc. \n\nIn addition, it will be also interesting to see how the proposed method extend beyond only one single family of models (LLaMa 2/3) - may be on some MoE style models.", "theoretical_claims": "The authors have very well described related theory in simple way.", "experimental_designs_or_analyses": "Experimental design is very good and exhaustive to consider various cases such as impact of caliberation data, analysis of PIFA and MPIFA contriibutions, speedups etc. Evaluation strategy need to be improved to make the paper contribution/performance claims strong.", "relation_to_broader_scientific_literature": "The key contributions of paper will advance general hardware-friendly LLM compression community.", "essential_references_not_discussed": "Related work requires some upgrade. Some recent relevant literature need to be discussed in the Related work:\n\n1. Saha, R., Sagan, N., Srivastava, V., Goldsmith, A., & Pilanci, M. (2024). Compressing large language models using low rank and low precision decomposition. Advances in Neural Information Processing Systems, 37, 88981-89018.\n2. Kaushal, A., Vaidhya, T., & Rish, I. (2023). Lord: Low rank decomposition of monolingual code llms for one-shot compression. arXiv preprint arXiv:2309.14021.\n3. Sharma, P., Ash, J. T., & Misra, D. (2023). The truth is in there: Improving reasoning in language models with layer-selective rank reduction. arXiv preprint arXiv:2312.13558.\n4. Jaiswal, A., Yin, L., Zhang, Z., Liu, S., Zhao, J., Tian, Y., & Wang, Z. (2024). From galore to welore: How low-rank weights non-uniformly emerge from low-rank gradients. arXiv preprint arXiv:2407.11239.\n5. Wang, Q., Ke, J., Tomizuka, M., Chen, Y., Keutzer, K., & Xu, C. (2025). Dobi-SVD: Differentiable SVD for LLM Compression and Some New Perspectives. arXiv preprint arXiv:2502.02723.", "other_strengths_and_weaknesses": "The submission have significant innovation from both PIFA and MPIFA perspective. Evaluation is lacking and I am willing to increase score once authors provides some convincing experiments during rebuttal.", "other_comments_or_suggestions": "1. Authors introduce two variables r and d directly in abstract and introduction without explicitly defining it. Although simple to interpret, I encourage authors to first define any variable before using them.  \n2. The writing of the paper can be significantly improved. It important to understand what is important and what is not - to guide which sections should find a space in main draft and which can be moved in supplementary. For example, details of Computational cost of PIFA section in 3.2 can be moved to supplementary while listing the key finding. Many parts of the main submission in section 5.3 refers to results in supplementary and not standalone. I recommend authors to make the main draft as independent as possible while moving not-so-important ablations in supplementary.", "questions_for_authors": "1. The paper mentions that PIFA is fully differentiable, suggesting potential integration into the training stage.  Have the authors explored this direction?\n2. How can this method integrated and benefit some recent quantization techniques like ZeroQuant-v2.\nYao, Z., Wu, X., Li, C., Youn, S., & He, Y. (2023). Zeroquant-v2: Exploring post-training quantization in llms from comprehensive study to low rank compensation. arXiv preprint arXiv:2303.08302.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NgxThqfgME", "reviewer": "Reviewer_ADf5", "summary": "The authors propose a novel factorization method and reconstruction objective for LLM compression. Without requiring retraining, the method achieves perplexity performance comparable to semi-structured pruning at a 50% compression rate. Experimental results further demonstrate that the approach is efficient in both inference speed and memory usage.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful comments and suggestions.\n\n**Weak1:** The claim ... overstated.\n\n**Reply:** We have removed the phrase “for the first time” from the abstract.\n\n**Weak2:** Missing LLM Evaluation ...\n\n**Reply:** We have expanded our evaluation to include both **perplexity** and **downstream task accuracy** on widely adopted benchmarks.\n\nFor perplexity, we report results on the C4 dataset, a large-scale and diverse corpus commonly used for evaluating LLMs. Across various compression densities, MPIFA consistently outperforms existing low-rank pruning baselines. Specifically, MPIFA reduces the perplexity gap by:\n- **47.6%** on LLaMA2-7B\n- **34.5%** on LLaMA2-13B\n- **55.3%** on LLaMA2-70B\n- **62.6%** on LLaMA3-8B\n\non average across all densities, compared to the best-performing low-rank pruning method.\n\nTo further assess real-world utility, we conducted zero-shot evaluations on the SuperGLUE benchmark, covering 8 downstream tasks using the `lm-evaluation-harness` framework. MPIFA_NS achieves the highest accuracy on **6 out of 8 tasks**, and reduces the average accuracy gap to the dense model by **42.8%**, compared to the strongest low-rank baseline.\n\nThe full evaluation results are available here:\n- C4 PPL results: https://anonymous.4open.science/r/PIFA-68C3/c4_ppl.png\n- SuperGLUE zero-shot results: https://anonymous.4open.science/r/PIFA-68C3/zero_shot.png\n\nWe appreciate the reviewer’s suggestion and have incorporated these evaluations into the revised manuscript.\n\n**Weak3:**  Compression Resource Comparison ... practicality.\n\n**Reply:** We have added a detailed comparison of the **compression time** and **peak memory usage** during compression across different methods.\n\nCompression Time (on A6000 GPU):\n\n|Model|ASVD|SVD-LLM|PIFA|M|\n|-|-|-|-|-|\n|LLaMA2-7B|10h|30 min|15 min|30 min|\n|LLaMA2-13B|20h|1h|30 min|1h|\n\nPeak Memory Usage During Compression:\n\n|Model|ASVD|SVD-LLM|PIFA|M|\n|-|-|-|-|-|\n|LLaMA2-7B|15G|20G|0.5G|6G|\n|LLaMA2-13B|30G|25G|1G|10G|\n\nNotes:\n- All compression times are measured on a single A6000 GPU. On an A100 GPU, the compression time is approximately half.\n- For the \"M\" method, we report only the reconstruction time, excluding the time taken by the low-rank pruning step.\n\nAs for **memory efficiency** in method M:\n1. Online calibration is used, where only the current sample is loaded to GPU. Other samples remain on CPU until needed.\n2. Only the current pruning layer is loaded to GPU, while all other layers remain on CPU during processing.\n\nThese comparisons have been included in the revised manuscript to better reflect the practicality of our method.\n\n**Weak4:** Missing Structured Pruning ... time.\n\n**Reply:** We have conducted additional experiments to include **LLM-Pruner** as a structured pruning baseline.\n\nBelow is the perplexity comparison on WikiText2 using the LLaMA2-7B model at various densities:\n\n|Method|90%|80%|70%|60%|50%|40%|\n|-|-|-|-|-|-|-|\n|LLM-Pruner|6.58|8.81|13.70|40.49|126.0|1042|\n|MPIFA|5.69|6.16|7.05|8.81|12.77|21.25|\n\nOn average, MPIFA reduces the perplexity gap by **87.2%** compared to LLM-Pruner.\n\nWe also benchmarked the **inference speedup** of PIFA and LLM-Pruner layers (relative to dense linear layers) on an A6000 GPU across different hidden dimensions:\n\n|Method (density)|d=16384|d=8192|d=4096|\n|-|-|-|-|\n|PIFA (55%)|1.88×|1.70×|1.43×|\n|LLM-Pruner (55%)|1.81×|1.77×|1.67×|\n|LLM-Pruner (70%)|1.42×|1.41×|1.35×|\n\n**Memory usage** during inference, relative to dense linear:\n\n|Method (density)|d=16384|d=8192|d=4096|\n|-|-|-|-|\n|PIFA (55%)|0.56×|0.58×|0.64×|\n|LLM-Pruner (55%)|0.56×|0.58×|0.65×|\n|LLM-Pruner (70%)|0.70×|0.72×|0.75×|\n\nAt the same density (55%), PIFA achieves similar speedup and memory efficiency as LLM-Pruner. When comparing MPIFA at 55% density to LLM-Pruner at 70% density, MPIFA consistently offers **lower perplexity, faster inference, and reduced memory usage**.\n\nWe have included this comparison with LLM-Pruner in the updated manuscript. Additionally, we are evaluating other recent structured pruning methods and will report those results as they become available.\n\n**Weak5:** Retraining ... is applied.\n\n**Reply:** Fine-tuning experiments are already included in Table 4 of the original manuscript. All pruned models are retrained for **one epoch** on a mixed dataset consisting of 2% WikiText2 and 98% C4, to recover performance. This setup ensures a fair and consistent comparison across all methods.\n\nThe results show that **fine-tuned MPIFA continues to outperform other low-rank pruning methods**, and also achieves slightly better performance than fine-tuned semi-sparse models. Details of the fine-tuning setup can be found in Appendix B.3.\n\n**Weak6:** Structured Compression ... of semi-structured pruning.\n\n**Reply:** We have updated the manuscript to include discussions of SLEB, ShortGPT, and DISP-LLM in the related work section. We appreciate the reviewer’s recommendation to acknowledge them.", "claims_and_evidence": "The claim that this work is the first to achieve performance comparable to semi-structured pruning, while surpassing it in GPU efficiency and compatibility, appears vague and potentially overstated. Recent works such as MoDeGPT [1] and DISP-LLM [2] have demonstrated similar results. \n\n[1] Lin, Chi-Heng, et al. \"Modegpt: Modular decomposition for large language model compression.\" arXiv preprint arXiv:2408.09632 (2024).\n\n[2] Gao, Shangqian, et al. \"Disp-llm: Dimension-independent structural pruning for large language models.\" Advances in Neural Information Processing Systems 37 (2024): 72219-72244.", "methods_and_evaluation_criteria": "The LLM evaluation performances are missing in the experiments.", "theoretical_claims": "They seem correct to me.", "experimental_designs_or_analyses": "1. Missing LLM Evaluation Metrics: The paper does not report standard LLM evaluation metrics (e.g., perplexity or downstream task accuracy) on widely used benchmarks. Including these results is important for assessing real-world performance.\n2. Compression Resource Comparison: The paper lacks an analysis of the computational resources (e.g., runtime, memory, compression time) required to perform the proposed compression, which is essential for evaluating practicality.\n3. Missing Structured Pruning Baselines: Key structured pruning baselines are omitted from the comparison. In particular, methods like LLM-Pruner [1] and the SLEB layer pruning strategy [2] should be included. Although these methods may yield lower accuracy, they are highly efficient and result in fast, compact models at inference time.\n4. Retraining / Fine-tuning Comparison: How does the proposed method perform when retraining or recovery fine-tuning is allowed? A fair comparison with other methods should also include their best-case results when post-compression fine-tuning is applied.\n\n\n[1] Ma, Xinyin, Gongfan Fang, and Xinchao Wang. \"Llm-pruner: On the structural pruning of large language models.\" Advances in neural information processing systems 36 (2023): 21702-21720.\n\n[2]  Song, Jiwon, et al. \"Sleb: Streamlining llms through redundancy verification and elimination of transformer blocks.\" arXiv preprint arXiv:2402.09025 (2024).", "relation_to_broader_scientific_literature": "Model compression is important for efficient AI.", "essential_references_not_discussed": "1. Structured Compression via Layer Pruning: The paper omits comparisons with recent structured compression methods that use layer pruning strategies, such as SLEB [1] and ShortGPT [2]. While these approaches may trade off some accuracy, they are highly efficient in terms of compression speed and inference latency.\n2. Structured Methods Approaching Semi-Structured Performance: Recent works like DISP-LLM [3] have demonstrated that structured compression can approach or match the performance of semi-structured pruning.\n\n[1]  Song, Jiwon, et al. \"Sleb: Streamlining llms through redundancy verification and elimination of transformer blocks.\" arXiv preprint arXiv:2402.09025 (2024).\n\n[2] Men, Xin, et al. \"Shortgpt: Layers in large language models are more redundant than you expect.\" arXiv preprint arXiv:2403.03853 (2024).\n\n[3]  Gao, Shangqian, et al. \"Disp-llm: Dimension-independent structural pruning for large language models.\" Advances in Neural Information Processing Systems 37 (2024): 72219-72244.", "other_strengths_and_weaknesses": "Strengths:\n1. The proposed method demonstrates strong perplexity performance, outperforming many existing approaches.\n2. The decomposition technique appears novel.\n3. The paper includes thorough ablation studies and discusses method efficiency in detail.\n\nWeaknesses:\n1. The experimental results lack standard LLM evaluation metrics (e.g., downstream task accuracy or benchmark scores), which are important for assessing overall effectiveness.\n2. Comparisons with structured compression methods are limited, particularly regarding the accuracy-efficiency trade-off (e.g., inference speed, latency).", "other_comments_or_suggestions": "No minor suggestions.", "questions_for_authors": "1. Could you include standard LLM evaluation results (e.g., downstream tasks or benchmark datasets) to better assess the practical effectiveness of your method?\n2. Could you add comparisons with important structured compression baselines, and analyze their accuracy–efficiency trade-offs (e.g., perplexity vs. inference speed or memory usage)?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8KILdl9U6e", "reviewer": "Reviewer_8e78", "summary": "This paper is concerned with sparse inference, which aims to speed-up LLMs via sparsity. It is argued in this paper that previous methods either require specific hardware (e.g., semi-structured pruning) or yield degraded performance (low-rank pruning). This paper aims to propose a low-rank pruning named PIFA method that realizes decent performance yet impose no requirements to hardware. Specifically, PIFA firstly uses a pivot-row discovery to eliminate potential representation redundancy when conducting low-rank decomposition, and then utilizes a minimization-reconstruction estimator to alleviate potential accumulated errors across layers. The experimental results in terms of both effectiveness (perplexity) and efficiency (throughput) demonstrate the usefulness of PIFA. Essential ablation studies show the adequateness of design choices.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your detailed review and valuable insights. We are glad to address your concerns and provide clarifications.\n\n**Weakness 1:** It is not very clear how QR or LU decomposition serve as the backbone of finding pivot rows would differ from each other.\n\n**Reply:** The key idea behind PIFA is to select a set of **linearly independent pivot rows** that can be used to express the remaining non-pivot rows as linear combinations. Since a matrix of rank *r* contains multiple valid sets of *r* linearly independent rows, any such set can serve the purpose of PIFA.\n\n**QR decomposition with column pivoting** and **LU decomposition with row pivoting** are variants of QR and LU respectively, designed to improve numerical stability compared with original QR or LU. Their pivoting mechanisms reorder the columns (or rows), making the leading *r* columns (or rows) guaranteed to be linearly independent. The corresponding permutation matrix—produced as a byproduct of either decomposition—can then be used to identify the indices of the pivot rows (or columns).\n\nIn summary, both **QR and LU decomposition with pivoting** can be used to extract a valid set of pivot indices, and **they are mathematically lossless in the infinite precision setting**.\n\nTo further clarify their practical differences under limited numerical precision, we conducted additional experiments comparing the **residual error** when using pivot rows to reconstruct non-pivot rows under both **Float32** and **Float16** precision:\n\n| Precision | Method | Residual (±)|\n|--|-|-|\n|Float32|QR|8.80e-13 ± 3.51e-14|\n|Float32|LU|1.94e-12 ± 1.18e-13|\n|Float16|QR|9.56e-08 ± 3.71e-09|\n|Float16|LU|2.16e-07 ± 1.59e-08|\n\nThese results are averaged over 10 random singular matrices. While both methods yield very low residual errors, **QR with pivoting achieves lower numerical error than LU**. This suggests that **QR decomposition with pivoting is the preferred choice** for identifying pivot rows in PIFA.\n\nWe appreciate the reviewer’s question and have included both this explanation and the supporting experiment in the updated manuscript.\n\n\n**Weakness 2:** It would be much better to compare to structured pruning also as baselines.\n\n**Reply:** Thank you for the suggestion. To address this, we have included a structured pruning baseline, **LLM-Pruner**, in the updated version of the manuscript.\n\nBelow is the perplexity comparison on WikiText2 using the LLaMA2-7B model across various parameter densities:\n\n|Method|90%|80%|70%|60%|50%|40%|\n|-|-|-|-|-|-|-|\n|LLM-Pruner|6.58|8.81|13.70|40.49|126.0|1042|\n|MPIFA|5.69|6.16|7.05|8.81|12.77|21.25|\n\nOn average, **MPIFA reduces the perplexity gap by 87.2%** across these density levels compared to LLM-Pruner.\n\nWe also benchmarked **inference speedup** and **memory usage** relative to dense linear layers, using linear layers of different dimensions on an A6000 GPU:\n\n**Speedup over dense linear (higher is better):**\n\n|Method (density)|d=16384|d=8192|d=4096|\n|-|-|-|-|\n|PIFA (55%)|1.88×|1.70×|1.43×|\n|LLM-Pruner (55%)|1.81×|1.77×|1.67×|\n|LLM-Pruner (70%)|1.42×|1.41×|1.35×|\n\n**Memory usage relative to dense (lower is better):**\n\n|Method (density)|d=16384|d=8192|d=4096|\n|-|-|-|-|\n|PIFA (55%)|0.56×|0.58×|0.64×|\n|LLM-Pruner (55%)|0.56×|0.58×|0.65×|\n|LLM-Pruner (70%)|0.70×|0.72×|0.75×|\n\nAt the same density (55%), PIFA achieves similar speedup and memory efficiency as LLM-Pruner. When comparing MPIFA at 55% density to LLM-Pruner at 70% density, MPIFA consistently offers **lower perplexity, faster inference, and reduced memory usage**.\n\nWe have incorporated these comparisons into the revised manuscript.\n\n\n**Weakness 3:**  How about the memory footprint of PIFA and baselines.\n\n**Reply:** Thank you for the question. We provide a detailed analysis of the memory footprint for both **inference** and **compression**.\n\n**Memory Footprint During Inference:**\n- As shown in Figure 5, PIFA consistently achieves lower memory usage compared to low-rank layers at the same rank.\n  For example, at $r/d = 0.5$, PIFA **losslessly compresses** the memory of the low-rank layer by **24.2%**.\n- Table 5 demonstrates that PIFA at 55% density uses **slightly less memory** than a 2:4 semi-sparse layer.\n- Table 6 shows that MPIFA_NS at 55% density reduces **end-to-end memory usage** by **42.8%** on LLaMA2-7B and **43.8%** on LLaMA2-13B.\n\n**Memory Footprint During Compression:**\nWe report **peak memory usage** during compression for each method:\n\n|Model|ASVD|SVD-LLM|PIFA|M|\n|-|-|-|-|-|\n|LLaMA2-7B|15G|20G|0.5G|6G|\n|LLaMA2-13B|30G|25G|1G|10G|\n\nFor **memory efficiency** in method M:\n1. **Online calibration** is used, where only the current sample is loaded to GPU. Other samples remain on CPU until needed.\n2. Only the **current pruning layer** is loaded to GPU, while all other layers remain on CPU during processing.\n\nWe have included this analysis of memory footprint during both inference and compression in the updated version of the manuscript.", "claims_and_evidence": "The claims are supported by clear and convincing evidence.", "methods_and_evaluation_criteria": "The methods are mostly clear and the evaluation criteria is adequate. However, I still have several concerns:\n1) It is not very clear how QR or LU decomposition serve as the backbone of finding pivot rows would differ from each other.\n2) It would be much better to compare to structured pruning also as baselines.", "theoretical_claims": "The proofs for theoretical claims are correctly justified.", "experimental_designs_or_analyses": "The experimental designs and analyses are sound and valid. However, I still have several concerns:\n1) considering structured pruning also as baselines would strengthen the contributions of this work.", "relation_to_broader_scientific_literature": "N/A", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "How about the memory footprint of PIFA and baselines.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "N/A", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gVziOoWNv8", "reviewer": "Reviewer_92qm", "summary": "I'm providing my whole review here, rather than giving partial and discontinued comments here and there.\n\nThe paper proposes an extension to SVD-based low-rank approximation of weights. The insight being employed is that a rank r matrix has at most r linearly independent rows. Therefore, a UV decomposition, such as via a truncated SVD, causes the decomposition to be redundant since it employs 2r vectors overall. To bypass this redundancy, a pivoting factorization is used, where one of the matrices is pivoted such as its upper half is an identity matrix, which need not be stored, and the rest constitute the coefficient matrix which multiplies the pivot rows. This is a nice application of basic linear algebra.\n\nIn section 4, the weight decomposition is initialized using SVD-LLM, and then an online algorithm to refine the decomposition is proposed. SVD-LLM is not the SOTA for low-rank decomposition, and therefore, this may explain why significant accuracy loss is reported in the experimental section. I'd like to suggest for the authors to instead use a SOTA technique, such as ESPACE (NeurIPS 2024). At the very least, can we have a comparison with ESPACE which showed much better accuracy vs compression trade-off? E.g., they showed <1PPL increase at 50% compression for Llama2-7B as opposed to PIFA which increases the PPL from ~5 to ~12.\n\nIf the authors accept my suggestion above, then the following comment will not matter. However, if they chose to stick with SVD-LLM, then please address the following. In equations (4) and (5), a nice online algorithm for finding an optimal U matrix minimizing layer output Frobenius norm is shown. But note that this is fixes matrix V, which need not be optimal. This may explain why SVD-LLM performs so poorly anyway. If ESPACE is used instead, we'd find a provably optimal projection matrix which will reduce dimensionality of activations, and weights by multiplication associativity. Then the online accumulation error minimization reconstruction can still be employed to further refine the matrix resulting from weight projection; it would fall into place in lieu of matrix U in the presented setup.\n\nIf the above is rejected (I hope not). Does it then make sense to have an alternating optimization algorithm, such as an EM, rather than just refining V once in equation (8)?\n\nThe reliance on a mix of dense and compressed data-flows in eq. (7) is interesting. But setting Lambda = 0.25 means most of the information comes from the low-rank branch. Once the model becomes more accurate, does it make sense to increase the value of lambda progressively to allow for more fitting to the golden uncompressed baseline?\n\nPerplexity results are interesting (even though the number are not impressive, likely due to what I discussed above). But can we have more evaluations on more downstream tasks such as LM eval harness and MMLU?", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We greatly appreciate your thoughtful feedback and the opportunity to address your concerns.\n\n**Weakness 1:**  In section 4 ... PPL from ~5 to ~12. \n\n**Reply:** We have conducted additional experiments to clarify the comparison.\n\nIt is important to note that **ESPACE includes a fine-tuning stage with 200B tokens** (as stated in Section 4.4 of the ESPACE paper), which significantly contributes to the <1 PPL increase at 50% compression. In contrast, the results we reported in **Tables 2 and 3** of our paper are **without any retraining**. Fine-tuning results are presented in **Table 4**, where each pruned model is only retrained for **~128M tokens**—orders of magnitude fewer than ESPACE.\n\nTo ensure a fair comparison, we **reproduced the pruning step of ESPACE** and compared it against SVD-LLM. ESPACE proposes six variants: MSE (Eq. 6), MSE-NORM (Eq. 7), GO-MSE (Eq. 8), GO-MSE-NORM (Eq. 8), NL-MSE (Eq. 9), and NL-MSE-NORM (Eq. 9). We exclude the NL-MSE variants from our comparison as they rely on backpropagation, which is **infeasible on memory-constrained GPUs** due to their high resource demands.\n\n**Perplexity on WikiText2 at 50% density using LLaMA2-7B:**\n\n| Pruning Method (X)   | X      | X + PIFA | X + M   |X + MPIFA|\n|-|-|-|-|-|\n|SVD-LLM (W)|33.27|19.64|16.55|**12.77**|\n|ESPACE (MSE)|280.19|144.32|20.99|**16.84**|\n|ESPACE (MSE-NORM)|172.30|113.82|21.73|**17.42**|\n|ESPACE (GO-MSE)|41.75|24.17|17.47|**13.55**|\n|ESPACE (GO-MSE-NORM)|37.45|23.19|17.47|**13.63**|\n\n*SVD-LLM (W)* indicates the standalone pruning output from SVD-LLM.\n\nFrom the results:\n- **SVD-LLM** performs slightly better than ESPACE’s best variant (**GO-MSE-NORM**) under comparable conditions (i.e., without fine-tuning).\n- **PIFA**, **M**, and **MPIFA (PIFA + M)** consistently improve all ESPACE variants, demonstrating that our techniques are **general-purpose and complementary**, enhancing the performance of any low-rank pruning method.\n- Based on these observations, **SVD-LLM remains the strongest initialization** among all tested options when no fine-tuning is used.\n\nWe have included this new comparison and discussion in the updated version of the manuscript.\n\n**Weakness 2:**  In equations (4) and (5) ... presented setup.\n\n**Reply:** We agree that the quality of the initial low-rank decomposition, especially the choice of matrix $V$, can significantly influence the performance of the final reconstruction. A **better-initialized matrix $V$** generally leads to a **more accurate reconstructed matrix $U$**, resulting in lower final PPL.\n\nCurrently, **SVD-LLM remains the most effective low-rank pruning method** in our setup, but this remains an open question. If future methods such as ESPACE can produce even better projections, they can naturally be integrated with MPIFA to achieve improved performance. Our framework is flexible and designed to **enhance any low-rank pruning method**, not tied to any single initialization strategy.\n\nWe have added this analysis and discussion to the updated version of the manuscript.\n\n\n**Weakness 3:** Does it ... equation (8)?\n\n**Reply:** We assume that by \"EM\" you are referring to an alternating approach that iteratively updates $U$ and $V$ using Equations (5) and (8) with more than 1 round, similar to **alternating least squares (ALS)**. We are currently exploring this direction and will share the results as soon as they become available.\n\n**Weakness 4:**  The reliance on a mix ... golden uncompressed baseline?\n\n**Reply:** In our preliminary findings, increasing $\\lambda$ leads to **overfitting on the calibration data**, resulting in **low perplexity on the calibration set** but **high perplexity on the full WikiText2 dataset**.\n\nTo mitigate this overfitting, we explored several strategies:\n1. **Reducing $\\lambda$**, as mentioned in the right column of line 256, where the low-rank data flow acts as a form of regularization.\n2. **Increasing the size of the calibration dataset**, which could help improve generalization.\n3. Applying **additional regularization techniques** to control overfitting.\n\nWith **sufficient calibration data**, increasing $\\lambda$ may become beneficial. We are currently running experiments to test this hypothesis and will report the results as soon as they are available.\n\n**Weakness 5:** Can we have more evaluations on more downstream tasks such as LM eval harness and MMLU?\n\n**Reply:** We have incorporated **zero-shot evaluations on 8 downstream tasks** from the **SuperGLUE benchmark**, using the `lm-evaluation-harness` framework as recommended.\n\nThe results, available at https://anonymous.4open.science/r/PIFA-68C3/zero_shot.png, show that **MPIFA_NS outperforms other low-rank baselines on 6 out of 8 tasks**, and reduces the average accuracy gap by **42.8%** compared to the best-performing baseline. These evaluations have been added to the revised manuscript.\n\nWe are currently working on extending our evaluation to include **MMLU** and will provide updates as results become available.", "claims_and_evidence": "Please see main review.", "methods_and_evaluation_criteria": "Please see main review.", "theoretical_claims": "Please see main review.", "experimental_designs_or_analyses": "Please see main review.", "relation_to_broader_scientific_literature": "Please see main review.", "essential_references_not_discussed": "Please see main review.", "other_strengths_and_weaknesses": "Please see main review.", "other_comments_or_suggestions": "Please see main review.", "questions_for_authors": "Please see main review.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper proposes an approach for low-rank pruning of LLMs, where a dense matrix W is "pruned" into two rectangular matrices AB. Noting that SVD-based factorization introduces redundancies, it introduces a more parameter-efficient factorization scheme. This in an of itself is straightforward, but the paper has additional contributions, including a GPU-friendly factorization, as well as an online data-aware reconstruction approach. The method is validated on 7B-70B LLMs where it is found to outperform baselines such as SVD-LLM.

On the plus side, the approach is sensible and clearly improves upon baselines. On the negative side, there are still significant degradations even at 80% density. And while decent speed-ups are shown for 55% density (Figure 4), but at this point the perplexity degradations make the model is pretty much unusable (e.g., 6.14PPL to 28.9PPL for LLaMA3-8B). Nevertheless, while the method is unlikely to be usable in its current form, I do think the ideas and the algorithm itself, paired with improvements over baselines, push this submission above the acceptance bar.
Decision: Accept (poster)