Can MLLMs Absorb Math Reasoning Abilities from LLMs as Free Lunch?
Authors: Yijie Hu
Zihao Zhou
Kaizhu Huang
Xiaowei Huang
Qiufeng Wang
Venue: NeurIPS 2025 poster ()
Keywords: Multi-modal reasoning
math reasoning
model merging
Primary area: deep_learning

Abstract: Math reasoning has been one crucial ability of large language models (LLMs), where significant advancements have been achieved in recent years. However, most efforts focus on LLMs by curating high-quality annotation data and intricate training (or inference) paradigms, while the math reasoning performance of multi-modal LLMs (MLLMs) remains lagging behind. Since the MLLM typically consists of an LLM and vision block, we wonder: \textit{Can MLLMs directly absorb math reasoning abilities from off-the-shelf math LLMs without tuning?} Recent model-merging approaches may offer insights into this question. However, they overlook the alignment between the MLLM and LLM, where we find that there is a large gap between their parameter spaces, resulting in lower performance. 
Our empirical evidence reveals two key factors behind this issue: the identification of crucial reasoning-associated layers in the model and the mitigation of the gaps in parameter space. Based on the empirical insights, we propose \textbf{IP-Merging} that first \textbf{I}dentifies the reasoning-associated parameters in both MLLM and Math LLM, then \textbf{P}rojects them into the subspace of MLLM aiming to maintain the alignment, finally merges parameters in this subspace. IP-Merging is a tuning-free approach since parameters are directly adjusted. Extensive experiments demonstrate that our IP-Merging method can enhance the math reasoning ability of MLLMs directly from Math LLMs without compromising their other capabilities.

Can MILMs Absorb Math Reasoning Abilities from LLMs as Free Lunch?
Yijie Hu 2u Zihao Zhou 2; Kaizhu Huang", Xiaowei Huang?, Qiufeng Wang"
Xi'annJiaotong Liverpool University
2 University of Liverpool
Duke Kunshan University
Abstract
Math reasoning has been crucial ability of large language models (LLMs), where significant advancements have been achieved in recent years. However, most efforts focus LLMS by curating high-quality annotation data and intricate training (or inference) paradigms, while the math reasoning performance of multi- modal LLMs (MLLMs) remains lagging behind. Since the MILM typically consists of an LLM and vision block, we wonder: Can MLLMs directly absorb math reasoning abilities from off-the-shelf math LLMs without tuming? Recent model-merging approaches may offer insights into this question. However, they overlook the aligmment between the MILM and LLM, where find that there a large gap between their parameter spaces, resulting in lower performance. Our empirical evidence reveals two key factors behind this issue: the identification of crucial crasoning-associate parameter space. Based on the empirical insights, we propose IP-Merging that first Identifies the reasoning- eaasoning-associated parameters in both MLLM and Math LLM,, then Projects them into the subspace of MILM,, aiming to maintain the alignment, and finally merges parameters in this subspace. IP-Merging tuning-free approach since parameters are directly adjusted. Extensive experiments demonstrate that our IP-Merging method can enhance the math reasoning ability of MILMSs directly from Math LLMs without compromising their other capabilities.
layers in the model and the mitigation of the gaps in
Introduction
As one fundamental ability of large language models (LLMs), improving math reasoning abilities is crucial to handle complex problem-solving tasks (]]. By creating substantial amount of high-quality math reasoning data [47] and designing intricate training procedures, LLMs such as GPT-4 or Qwen have achieved remarkable progress in solving text-based math reasoning problems 2 Despite these advancements, the challenge of mathematical reasoning remains a significant obstacle for MILMS [9 23 57 54). Visual math reasoning tasks, which essential for real-world applications, require MILMS to extract image information, analyze problem constraints, integrate text and visuals, and perform complex reasoning.
Following similar training strategies for improving math reasoning abilities in LLMs, attempts have been made enhance MLLM's math reasoning skills (30]. Despite the notable progress, collecting and annotating high-quality multimodal reasoning data 88,56 expensive. Additionally, training large models demands extensive computational resources. which makes it costly improve their reasoning abilities. As MLLMs typically include a foundation LLM as the core component [00
"qqual contribution.
Corresponding author.
3 "Code Repository:
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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and share similar training processes with LLMs, we propose one intriguing question: Can MLLM directly absorb math reasoning abilities from off-the- the-shelf math LLMs to enhance the molli math reasoning without funing?
To answer this question, initial attempt is to adopt model merging techniques, which in- tend integrate task-specific knowledge into
one model by merging multiple fine-tuned mod- els without involving any training 15 4T. We illustrate this process in fig. where we aim to improve MLLM's s multi-moda math reasoning by integrating parameters from math LLM into ML..M. One representative method called task arithmetic [5] extracts the task vector of each model via the subtraction of the fine-tuned and pre-trained models (i.e., AW W ft- W pre). By adding all task vectors, the merged model is expected to perform well on all tasks. However, the direct merging of the reasoning aassoning-associteed associated LLM task vector and MLLMs does not improve the reasoning abilities of MLLMs (fig. (a)). We
Input Question
"The Answer 24"
MLLM
MILM
The Answer 22.5"
What the value of x?
Merga
MATH
MATHLLLM
Figure We aim to enhance multi-modal rea- soning skills of MLLM by merging parameters in MLLM and Math LLM without tuning or changing the size of the model.
observe that existing merging methods work ef- fectively when task vectors are similar [32 45 However, this does not hold between MILMS
2500
Accuracy
MAAAAAA
11.3
and Math LLMs. We argue that there exists gap between their task vectors. Without align- ing them properly, direct addition leads to con- flicts. MLLMs are designed integrate vi- sual and textual inputs, whereas math LLMs acquire reasoning skills from text-based mathe- matical problems. This fundamental difference creates substantial discrepancy their task vectors. To illustrate this, inspired by previous works [16] (39 that adopt trace value to quantify gradient change and learning difficulty in the fine-tuning stage, we plot the trace value of task vectors in 30 MLP blocks (fig. ()), and there is a large gap between the MLLM and Math LLM.
MILM
Tink Arismmtic
9.8-2.51
MahLLM-I
1.00
MathLLNN33
Tics Merging
10.7-0.4]
1000
Ous
15.3+4.07
Index of MLP MLLLarers
(a)
(b)
Figure 2: (a) Comparison of different methods of merging MLLM with math LLM on Math Verse. (b) Trace value of task vectors 30 ML.P layers in two math LLMs Tora-code-7B) and one MLLM (LLaVA-v1.5-7B))
and
To delve into the aligmment issue integrating reasoning abilities from math LLMs into MLLMs. we conduct empirical analysis and unveil two key challenges: (1) How to identify math reasoning abilities associated parameters in MLLM and LLM? We demonstrate that math reasoning-assoede associated parameters appear highly similar in the subspace; absorbing these parameters improves multi-moda math reasoning, detailed in section 4.1 (2) How to quantify and mitigate the gaps between models in the parameter space? ? We illustrate that parameter gaps between models can be quantified by singular values, detailed section (.22 Bridging these gaps can enhance the alignment between models and improve math reasoning performance.
Based our empirical findings, we propose IP-merging that Identifies the math reasoning parameters in the models, then Projects math reasoning parameters into the subspace of MLLM for better aligmment. In the parameter selection stage, the selection process is guided by the similarity between the subspaces of the task vectors from the two models. To achieve this, we first perform singular value decomposition (SVD) on the task vectors of both the LLM and the MLLM. Next, we compute the cosine similarity between the basis vectors extracted from the LLM and the MLLM. Parameters corresponding to the most similar subspaces are then selected for the subsequent merging process. To further align the selected parameters, these parameters are rescaled by computing the rescaling factor based on the eigenvalues of the selected parameters in MILM and math LLM. The rescaled math LLM parameters are projected into the subspace of the MILM, allowing the projected LLM parameters to be close to the MLLM parameters. Finally, the rescaled parameters are merged into the MLLM. The overall process allows the reasoning capabilities of the LLM to be effectively transferred

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and adapted to the multimodal context of the MLLM. Our method requires no data or any additional tuning and can be efficiently implemented.
We demonstrate the effectiveness of our method by merging MILM (e.g. LLaVA series and Qwen series) and different math reasoning L.Mss. We validate the performance of our method Math Math Verse, DynaMath and Math Vision for evaluating math reasoning abilities. We further show our method does not interfere with the model's other abilities by evaluating our method general knowledge datasets, i.e., MMMU (50) TexiVQA [3 and MMBench 22]. Our contributions can be summarized as follows:
We propose the problem of improving the math reasoning abilities of MLLMs by directly absorbing math LLMs without any tuning.
We reveal two key challenges to answer this question, including the selection of math- reasoning-assoocitted parameters and the reduction of gaps between models.
We propose the IP-merging method, which first identifies the crucial layers and then aligns the selected layers by projecting them into the MLLMs subspace.
Extensive experimental results validate that the IP-Merging method enhances the math reasoning abilities of MLLMs without compromising other capabilities.
2 Related Works
2.1 Math Reasoning of MLLMs
The community has made significant efforts to improve mathematical reasoning ability, which is regarded as fundamental capability of MLLMs. In the pre-training stage, previous works focus on enhancing math reasoning of base models by collecting math pre-training data [37 generating
synthetic data [5], and optimizing training strategies [18]. Furthermore, in the post-training phase, researchers significantly increase the scale and quality of math reasoning instruction data through data augmentation techniques 8
12
30 The introduction of variant fine-tuning (55), reinforcement
learning algorithms [48 49 38, and self-evolving frameworks [5T 21 greatly improves the efficiency of data utilization. Recently, O1-like models successfully leverage the deep-thinking chain of thought for inference scaling [6), substantially advancing reasoning capabilities. Different from these cost- intensive works (either training or inference), we aim to improve MLLMs math reasoning abilities by directly absorbing them from Math LLMs without any tuning.
2.2 Model Merging
Model merging has emerged as a promising technique for enhancing the capabilities of models without requiring raw training data or intensive computation, which offers a low-cost solution to elevate the abilities of LLMs. Model merging takes off-the-shelf task-specific models and fuses models into a single model with diverse abilities 43 Several advanced methods have been developed for model merging, which can broadly classified into pre-merging and during-merging strategies. Pre- merging methods focus merging model weights [40, [5] 41], architecture transformations [32 (52. or disentangling weight spaces to create optimal conditions for merging [33 25). During-merging methods address task conflicts using basic averaging, weighted strategies, subspace projections, dynamic routing, or post-merging calibrations [46] 45 44]. Previous methods focus on merging models trained on single modality 43 17, modalities.
while method aims to merge models from different
3 Model Merging Framework
3.1 Preliminaries of Task Vectors
Let Wo denote the parameters of a pre-trained language model, such as Llama-based model [34
The math reasoning model W Math is obtained by fine-tuning the pretrained model Wo on math reasoning data Derain mathhtzz- txt- Math task vectors are defined as the difference between parameters Wo. Here, the task vectors
of LMs before and after finetuning, i.c., AW Math W Math
3

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obtained by subtraction at each layer. For example, if the model has N layers, the task vector is AW Math = [-A Math AW3 Math..... ,, AW Math
For MILM such as LLaVA [20]. the model is trained by freezing the visual encoder 28 and
tuning the projection layers and the LLM using vision and language pairs Di Here, we denote the models' task vector of receiving
input AW MLLM W MLLM W o- The task vector of MLLM is computed between the LLM and the pretrained LLM, which does not include the frozen visual encoder and the visual projection layers. During the model merging process, the visual encoder and visual projection layers are not modified, only LLMs in the models are merged.
3.2 Model Merging Framework
Model merging techniques fuse several fine-tuned task-specific models into one comprehensive multi-task model without training the models, In our case, we aim to obtain one math reasoning MILM W MathMLLM by merging math LLM W Math with the MLLM W MLLM- For simplicity, we consider the case of merging two models here. We can formulate the general framework for merging the models as
W Math F(Wo. o, AW MILM; AW Math; M).
Prevailing approaches, such Task Arithmetic 115] Ties Merging 41. EMR Merging [1 can be further formulated as
W MathMLLM = Wo (AW MLLM, M,) + a22/2(A
Math M2),
(2)
where and (22 are scaling parameters, M1 and M2 can be regard as alignment matrices. (-) and f2(+) represent two mapping functions, such as the product of the element-wide product. For example, task arithmetic [15] set M1 and M2 to identity mapping. Ties merging [41] and DARE [46]
compute sparse matrices M1 and M2, and select parameters by element-wise product. EMR merging computes scaling factors a1, 0g based on the absolute values and derives task-specific masks M1.
M2. Our method follows a similar model merging framework. After merging the models, the model math-pi to evaluate its performance.
is then tested multi-man modal math reason dataset Diest
4 Methodology
In this section, we delve into the alignment issue in model merging by identifying parameters associated with math reasoning in section 4.1. and then we quantify the gaps between models section Finally, we describe our proposed IP-merging in section
4.1 Identify Math-RResoningg-Assiii Parameters
Experimental results in fig. provide one critical
observation: though limited, the MLLM already demonstrates an inherent capability for multi- modal mathematical reasoning. As the MILM is trained to align visual and textual input, the direct addition of task vectors in all layers de- rived from math reasoning LLM harms the learned alignment, yielding suboptimal perfor- mance. To address this issue, key question arises: Which parameters are associated with mathematical reasoning abilities in MLLM, and how can we effectively identify and prioritize them for merging?
litphot Smmllrit Scurs
Sanalarit Viluz Correpoodingg Anple
MMM by
Eigenvalue
Purmmeeers
Figure 3: (a) Absolute cosine value of the top 100 corresponding angles in one MLP layer. (b) The highest similarity score distribution among model parameters, orange bars represent the parameters with high similarity score.
To answer this question, we propose to quantify this correlation by calculating the subspace sim- ilarity between the parameters in Math LLMs and MILMM. Recent efforts 26 24 demonstrate that -sse-fic competencies, such as mathe-
matical reasoning abilitics, reside within particular subspaces of the model's parameter space. If the

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subspace of one parameter in MLLM has higher subspace similarity with math LLM, this parameter can be regarded as crucial for math reasoning abilities. Specifically, given the math reasoning task vector A AW Math and multi-modality task vector AW MLLM in the layer we can first factorize these weights using singular value decomposition (SVD) respectively:
AW Math UM EM VM T
AW MLLM = U, VV
which can be organized into a combination of singular values and orthonormal bases:
Math = E. UMMM.. T. AW
AW MLLM Σ
d
(4)
where Vyy and MM, represent the i-th singular value of n-th layer in MLLM and Math-LLM respectively, and d the number of dimensions. JMM, 2 0M..... 2 MM vice versa for ove "'M,, V8M, 2 represent corresponding orthonormal bases respectively. If the direction of two sets of orthonormal bases closely aligned, the similarity between the two subspaces is high. Meanwhile, we consider the singular value as the reference when computing similarity, the singular values determine the importance of each orthonormal basis. To this end, we propose to use the corresponding angle to measure the similarity between two subspaces 4
Definition 1 (Similarity Value of Corresponding Angle). Given two groups of eigenvectors: ( v al and (.v. UVa,,, the corresponding angle represents the angle between two eigenvectors corresponding the same eigenvalue index. The cosine value of the i-th eigenvec- tor's so's corresponding angle is
(NM,a , .... TT
ST
T 1V,3
UM
We visualize the distribution of the first 100 corresponding angles in selected MLP layer in fig. aa. We can see that the cosine value of the first corresponding angle is significantly higher than those of the other angles. This observation suggests that the basis associated with the largest singular value represents a subspace in MLLM that is strongly linked to math reasoning capabilities. Consequently, the parameters with high subspace similarity may play a more critical role in math reasoning abilities.
To explore this further, we plot the distribution of the high- est similarity value scores across model layers in fig. (3)... The figure illustrates that parameters have high similarity scores, which can be regarded as crucial math reason- ing. To validate their importance, we perform experiments where we selectively merge the corresponding layers of the LLM into these parameters of the MLLM with a high similarity score in table [] The gain of performance ver- ifies the critical role of the identified MLLM parameters in facilitating math reasoning. This finding shows that subspace similarity can be one key criterion for selecting and prioritizing math reasoning parameters. More analysis can be found in appendix D
Table 1: Performance comparison af- ter selecting reasoning-related layers to merge on Math Verse.
Approach
Average Accuracy
Base MLLM
11.3
Task Arithmetic
9.8 158
Task Arithmetic+Param Selection
13.4 #211
4.2 Gaps between Models in Parameter Space
As aim absorb math reasoning abilities from LLMS MILMs, the gap between models emerges as a significant obstacle. While both models are derived from the same foundational L.M (e.g.. LLaMA), the parameter changes within MLLMs and math reasoning LLMs can vary significantly. This disparity reflects fundamental differences in their learned representations, task objectives, and domain-specifts knowledge. To quantify these differences, we use the distribution of eigenvalues in the model parameters as an intuitive metric for understanding the magnitude and nature of these changes. Specifically, the eigenvalue distribution offers insight into the "scale" of parameter updates. We plot the top 1024 singular value distributions of one MLP layer in the model fig. aa), which reveals that LLMs and MILMS exhibit different eigenvalues, highlighting the inherent gap between LLMs and MILMs.. When combining models of a math reasoning LLM with large eigenvalues and an MILM with smaller eigenvalues by addition, parameters with larger eigenvalues can overshadow the contributions of parameters with smaller ones. Here, we take task arithmetic an example:
5

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Proposition 1. Let AW Math and AW MLLM. be two real matrices. The L2-norm of two matrices defined by JAW Math ll2 max (AW Math), ||AW M MLLMM 2 m (&W MLLAs), Model merged by task arithmetic satisfies the triangle's 's inequality 113
J M 2 A MuLMl22 S
A Moth + AW MLLM 2 S
AlWW Manll2 la + IAW MLZM
If we assume
AW MLLM|L2 S e (lA Math 12,
0<e< < 1,
then.
(1-e) || W M I WAW Math + A AW MLLM S (1 + E) | W Mach
Our proposition demonstrates that if the maximum singular value of the math reasoning parameters in LLM is far larger than the MLLM parameter, i.e.,
MLLM Mach1.LM
11.5
mmax (AW Math) > mmax (&W (AWALLL)) E 7 0, the MLLM parameter will overshadowed by math parame- (IAW Math + AWMLLMM ll (aa (AW Math)). To validate this point, we further plot the distribution of the max- imum singular across model parameters in fig. 4f)), indicating that the Math LLM model may overshadow MLLM during merging. Therefore, it is crucial to align the LLM and MLLM in the parameter space allow the smooth transfer of math reasoning abilities.
Singnlar Values
Maximamm Singulas
135
be
15
ters
mma
3M
180
Index
Layers
(a)
(b)
Figure 4: (a) Singular values distribution in one MLP layer. We plot the top 1024 singular values. (b) Maxi- mum singular values across all model parameters.
4.3 IP-Merging
To transfer reasoning abilities from LLMs MLLMs, we propose novel merging method called IP-merging. Our method first identifies the crucial parameters in both MLLM and math LLM, then projects the rescaled selected LLM parameters into the MLLM subspace for better alignment. Finally, the aligned parameters are merged with MILM. In the parameter identification stage, crucial parameters between L.MSs and MILMS are identified and selected based on their similarity in the subspace. In the parameter projection stage, these selected parameters are rescaled, aligned, and projected into the MLLM subspace to minimize the gap between the models. The overall process can be found in algorithm and appendix F
math-reasoning-asselll
4.3.1 Parameter Identification
Given that MLLMs share a larger number of layers with LLMs, we restrict the merging process to components shared by both models, such as attention layers and MLP layers, without modifying the visual encoder or projection layer. Let denote the total number of layers considered for merging. To identify compatible parameters, we compute the corresponding angle for each layer {ST S% , S between the MILMM and Math LLM using a cosine similarity metric (refer to definition [}. Parameters with similarity score higher than the threshold S, are selected for merging, while others are excluded. This selection process can be formalized follows
{ AWW (
ST 2 S.
W Math
Math
(6)
otherwise
4.3.2 Parameter Projection
Empirical observations in section 4.2 highlight the importance of aligning parameters across different
model modalities. facilitate the alignment, Ties Merging [1] or EMR Merging [14] offers a straightforward solution by selecting parameters with consistent signs (i.e., both positive or both negative) and discarding the rest. While these works reduce interference between the merged models, they risk discarding critical parameters, leading decline in overall performance shown in fig, 2

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Different from previous works, we aim to merge the parameters of Math LLM into the MLLM without harming other abilities of MILM. To achieve this, our key idea is reduce the gap between LLM and MILLM parameter spaces by rescaling and projecting the selected mathematical reasoning parameters from the L.LM into the subspace of the MI..M. By rescaling the layers of I.L.M we can reduce parameter gaps between the models. We further project the selected LLM layers into the subspace MLLM, which pushes parameters in LLM to lie close to the MLLM in the parameter space, fostering the alignment between the two models [29]
Rescaling. To normalize the parameter magnitudes between the two models, we compute rescaling factor An, defined as the ratio of the nuclear norms of the corresponding parameter spaces in the MLLM and LLM
Σi= T
An
(7)
> MM
where of and TMrs are the singular values of the corresponding parameters in the MLLM and LLM, respectively.
Projection. To align the rescaled parameters to the MLLM subspace, we calculate an importance score 7n for the subspace vectors based on their similarity ST. The importance score defined as
exp(S?)
(8)
D exp((;))
Using these importance scores, the selected mathematical reasoning layers from the LLM are projected into the MLLM weighted subspace
V = V/ VV T, AWMath Math-p = An AWW Math V V
(9)
This projection emphasizes the subspace basis vectors that are most relevant to mathematical reason- ing, thereby ensuring effective alignment of the LLM parameters with the MILM architecture. After projection, we can obtain the math reasoning MLLM by adding the projected parameter to MLLM as
Wh MathMLLM W + AW
MLLM + A,W Math V VV
(10)
By referring eq. (2.. our method also fits the general framework, where On is sets as and ag is set as An, fi(, M1) is the identity mapping, and f2(:, M2) is the projection matrix in eq. (9), the overall algorithm of the proposed method can found in appendix F
5
Experiments
5.1
Experiments Setup
We test our models on six benchmarks, i.e., Math Vista 23 Math Verse
53 DynaMath (DM)
59
Math Vision [6] and three general QA benchmarks MMMU (50, TextVQA [1 and MMBench 22
Math Vista can be divided into five subsets: Figure Question Answering (FQA), Geometry Problem Solving (GPS), Math Word Problems (MWP), Textbook Question Answering (TQA), and Visual Question Answering (VQA). Math Verse includes diverse set of math problems that require reasoning over both textual and visual information, such charts, diagrams, and equations, which can be divided into five subsets, i.e., Text Dominant (T-D), Text Lite (T-L., Vision Intensive (V-1), Vision Dominant (V-D) and Vision Only (V-O). DynaMath (DM) dynamic visual math benchmark designed for in in-depth assessment of VLMs. Math Vision (Math-V V) is a collection of high- mathematical problems with visual contexts sourced from real math competitions. The MMMU benchmark is suitable for assessing the general knowledge of MILM. TextVQA evaluates a model's general ability read and reason about text images, requiring joint understanding of visual content and language reasoning. MMBench is comprehensive multimodal benchmark that tests large models across diverse tasks to assess their general multimodal intelligence and robustness. We select the English split for evaluation. We compare our proposed methods with prevailing model merging techniques such as Task Arithmetic [55, Ties Merging [41] and EMR Merging 14]. More details on implementation and hyperparameters can be found in appendix
B and appendix B.3 respectively.
7

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Table 2: Comparison with other model merging approaches on the Math Verse and Math Vista. Bold represents the best performance. We merge LLaVA with fora-code-7B Qwen2-VL-77 Instruct with Qwen-2-Math-7B models, IntermV.3--B-Innstrmue with DeepSeek ---istilled-Qwen- 7B, respectively.
Appronch
Math Verse
Mita
DM Math-V
TQA
VQA
FQA
L- Base Model
Base Model
12.4
10.0
12.2
12.8
9.1
11.3
36.1
22.1
37.4
20.8
13.9
25.2
15.4
11.1
Task Arithmetic
12.4
11.0
12.6
11.9
0.9
9.8
31.7
33.6
29.6
21.6
11.3
25.2
13.9
11.0
Ties Merging
13.1
14.0
13.3
13.1
10.7
39.9
26.8
23.8
6.5
27.1
14.0
10.6
EMR Merging
10.8
10.7
11.3
12.6
6.7
10.4
36.7
25.9
30.7
21.6
13.9
25.0
13.8
11.6
IP-Merging
16.0
16.1
14.1
15.5
15.0
15.3
43.7
21.6
40.8
24.9
15.1
28.2 91
16.1
11.8
Qwen-2-
Vinn----L-BB-Iuttrrrt as
Rase Model
27.4
26.8
27.3
25.6
16.9
24.8
58.9
33.7
58.7
66.5
57.5
55.4
40.8
16.3
Task Arithmetic
15.7
37.3
11.3
Ties Merging
6.9
6.5
7.0
6.2
0.1
5,3
39.9
38.5
32.4
21.6
18.3
29.3
15.1
11.4
EMR Merging
20.9
18.4
18.3
18.4
17.6
50.6
43.1
30.1
40.8
21.8
14.5
IP IP-Merging
31,0
28.7
29.4
29.7
23.6
28.5
63.3
41.8
57.5
69.9
66.7
60,2
41.0
19,1
Intern MnterWVLL-BB-Inmie as Bave Model
Basc Model
47.6
39.1
37.8
38.5
65.2
65.8
66.1
50.7
24.9
Task Arithmetic
24.9
18.5
15.6
14.1
11.2
16.9
43.0
46.2
26.8
20.1
29.0
32.0
20.1
15.3
Ties. Merging
18.8
13.8
14.6
15.0
14.2
15.3
34.8
34.6
25.1
19.7
13.4
25.0
16.1
EMR Merging
35,9
30.7
29.4
27.3
27.3
30.1
59.5
51.4
42.5
45.0
50.5
49.2
39.2
18.8
Merging
39.0 S1
67.6 '1
51.4
25.2
5.2 Experimental Results
We conduct our experiments on Math Vista and Math Verse in table 2 and MMMU in table For LLaVA models, we select LLLVA-115-777 as the base MLLM and Tora-Code-7B [ as the math reasoning LLM, as it is trained logical reasoning-based math problems. For Qwen models, select Qwen-2VL-7B-BBl
as the base MLLM and Qwen-2-math-72 [2] as the math LLM. For as the base MLLM and DeepSeek-RI-
Intern models, we select Intern distilled-Qwen-7BB as the math LLM.
[58
Compared to existing merging methods, approach consistently improves performance across all sub-tasks. Notably, it achieves at least 3.7% average gain the original model on Math Verse, demonstrating its effective- ness in enhancing general math reasoning capabilities. For Math Vista, the biggest improvement for LLaVA models lies in the task of the TQA subset, where science and math knowledge are highly demanded, improving the base model by 7.6% The LLaVA model does not achieve performance gain in GPS after merging, where the multi- ple steps of geometry reasoning are required. On the other hand, the ties merging boost the LLaVA performance in GPS, owing to its ability to resolve model conflicts by selecting parameters with the same sign. The consistent sign may indicate shared abilities across models. In this case, the visual reasoning ability is retained and enhanced by merging with the text-based reasoning ability in the Math LLM, but other general abilities deteriorate (a drop of 6.9% in MMMU) or VQA tasks (a drop of 6.7%). The Qwen model obtains the performance gain of 4.8% on average, with the most notable performance gain in MWP, where task-specific reasoning skills are highly demanded, improving the model by 9.2%.
Table 3: Comparison with other merging approaches three general benchmarks.
IP-merging preserves general abilities.
Approach
MMMU
TextVQ
MMBench
LLAVA-1.5-7B Base Model
Base Model
34.2
47.5
62.7
Task Arithaetti
30.2
23.5
17.4
Merging
27.3
8.2
EMR Merging
34.8
48.0
57.1
IP-Merging
34.4
47.6
63.1
Owen-2-VL--B-Mmee 2--77B--nstruc as Base Model
Buse
Task Arithmettc
32.6
37.9
54.1
Ties Merging
30.8
4.3
43.7
EMR Merging
41.8
73.2
74.8
IP JerMing
S0.7
$4.0
80.8
Intern VI.3-8B-I-ustruct as Bave Model
Base Model
61.6
81.9
84.4
Task Arithmetic
Ties Merging
30.1
9.1
38.7
EMR Merging
47.0
67.6
67.8
IP MerMeggn
61.9
82.3
84.6
We also conduct experiments by merging larger models in appendix C. and merging multiple models in appendix C demonstrating the effectiveness of our approach. Our method can further improve the performance of math MLLM, and is effective when merging larger multiple models.

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To validate that our approach does not harm other abilities in the MLLMs, we further conduct experiments on evaluating the models on the general QA benchmarks in table Compared to the baseline model, our approach is able to maintain stable model performance across the general benchmarks, while other methods (e.g., Task Arithmetic and Ties Merging) decrease the performance. As our method only selects the crucial parameters related to math reasoning, our approach does not interfere with parameters that are important to other general abilities. Hence, our method improves the math reasoning abilities without interfering with the general abilities.
5.3 Merging Math MILM with Math LLM
We conduct the experiments of merging fine- tuned math MLLM (TableLLaVVA-155-7)) [56], G-
Table 4: We merge Math MILM with the math LLM model. Bold represents the best perfor- mance. "Avg" is the average performance.
LLaVA-7B [8] and the Tora model using our pro- posed merging method, validating the effective- ness of improving fine-tuned math MLLM further. G-LLaVA is obtained by further fine-tuning the LLaVA using geometry reasoning data. After merg- ing the math LLM, the geometry reasoning ability is further enhanced, which is validated by the 2.8% and 1.5% improvement on GPS and FQA tasks. LableLLa is obtained by further fine-tuning the LLaVA using table reasoning data. By merging math reasoning LLM, our method can further im- prove the model's performance on table reasoning tasks such as FQA by 3.7%. VL.-Rthnnker-71 [35 is one recent reasoning model that was tuned from Qwen2.5-7B via reinforcement learning. By merg- ing the Qwen2.5-Man model, the reasoning performance is further enhanced by 2.5 on Math Vista, with noticeable improvement of 5.4% on the MWP subset. All of these demonstrate that our method can further improve the math reasoning abilities of both base MLLM and fine-tuned math MILMM.
Approach
MathVista
TQA
GPS
VQA
FQA
MWP
Avg
G-LLaVA-7B Base Model
Base Model
29.1
48.6
33.5
19.3
11.3
28.0
Merging
32.9
51.4
Toble as Base Model
Base Model Merging
34.2
27.4
29.6
24.9
41.9
30.9
41.8
27.9
30.2
43.6
34.0
11
VL-Roskinker-7B as the Base Model
Model
70.9
76.0
54.2
72.7
+IP Merging
71.5
79.3
58.7
77.7
86.0
75.2
5.4 Ablation Studies
We conduct the ablation experiments of the proposed method us- ing the LLaVA model on Math Vista in table []] By selecting the
Table 5: Ablation results.
parameters and applying our method individually, the performance is elevated by 3.3%, 3.7% and 3.1 %, respectively. More param- eter selection analysis can be found in appendix D Combining Parameter Selection and Projection increases accuracy to 26.3%. suggesting that projecting the selected parameters into the sub- space provides better alignment. The combination of parameter selection and rescaling demonstrates that rescaling improves the utilization of selected parameters. The combination of rescaling and projection achieves 26.7%, highlighting the complementary benefits of parameter adjustment and alignment in improving per- formance. When all three components are applied together, the model achieves the highest accuracy. This improvement demonstrates the necessity of carefully selecting, rescaling, and aligning reasoning- related parameters.
Components
Ace.
Selection
Rescale
Projection
25.8
26.2
25.6
26.3
26.1
26.7
28.2
5.5 Selected Parameter Analysis
We visualize the selected parameters for merging in fig. Most of the selected layers are MLP components located in the middle and latter parts of the model, while only few attention layers in the early stages are chosen. This pattern aligns with recent findings that knowledge and reasoning skills in LLMs are mainly encoded in deeper MLP layers 373. The selected layers in layers 17-31, and a few in layers 3-4 indicate that our merging primarily operates on high-level semantic representations while preserving early-layer perceptual alignment. This supports the view that reasoning transfer mainly benefits from modifying deeper MIP pathways rather than early attention dynamics. We further analyze selected layers in MLLM before and after math reasoning in appendix D

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Selected Loyer Locasion in LLaVA
Highest Simillriiy Segre 4
model
124 modee lavns 17.
130 modde lovn 19 touslegst
188; modd lyyn 23 Skkilsaete prg
[831 model layor mit ger _en
250: model lagoe rala pry
200, moald loves s0p
gite
21Bomalll bayas self snnl 222: modd loyen Lint
233: model layer 31 only
Selected Layer Index
Figure 5: Selected layers in the LLaVA model for merging.
5.6 Case Analysis
We demonstrate two examples that the merged model improves reasoning ability compared to the base model in fig. 6 the first example, the base model fails to read the table and outputs meaningless text, showing that it cannot handle structured data. In contrast, the merged model correctly reads the numbers, clearly explaining the rate change. This shows that merging helps the model understand tables and perform basic numerical reasoning. In the second example, the task involves geometry problem. The base model guesses the answer without any explanation, while the merged model reasons step by step. This reasoning is both correct and interpretable. See more cases in appendix
Siaei Cyl B-E3
Question: Kyda woek at the beach and recorded the number sholte the fourd eoch According the
Quastion: in trangle AASC. the intomaa ingle biseoton of angles and interseet atppint O. or
Medrosde
Trusddy
Thurdday and Fnday??
indee7nndininn7ee
The answer is B, 140"
Base Model
Base Model
Since LA= 110", ZABC+ 2ACB 180 -LA- 70" Becauo internad ande biseciioss
change betwoon Thursday and Friday was shells per day.
change 10-7/1 change
VECABC EAC (c0B0 20CB) Honce, coneect answer IsC.
LAGB Intersect at point <800
Marjud Model
Merged
Figure 6: Case study of examples are correctly answered after merging the math LLM.
Conclusion
In this paper, we aim to enhance the math reasoning abilities of MLLMs by directly absorbing them from Math LLMs without tuning. However, it is challenging to merge models with different modalities due to the large gap in parameter spaces between models. To this end, we propose model merging method, namely IP-merging, which mainly consists of two parts: i.c., parameter Selection and Projection. Parameter selection identifies crucial parameters associated with math reasoning, while parameter projection aligns the MLLM and math LLM by rescaling parameters and projecting the parameters into the subspace of the MILM. The proposed approach is tune-free and efficient to implement. Experimental results demonstrate the IP (P-merging method can successfully enhance the math reasoning abilities of MLLMs without harming their other abilities. In the future, we will extend the method to merge models with different foundation LLM architectures.
Our proposed method is effective for merging the MLLM and math LLM with the same size and sharing the same foundation LLM architecture (c.g., LLaMA-2). Otherwise, it is challenging to find those associated layers to merge.
Limitation:
7 Acknowledgments
The work was partially supported by the following: National Natural Science Foundation of China under No. 92370119, 62436009, 62276258 and 62376113.
10

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A Border Impact
This work presents IP-Merging, a novel tuning-free method that effectively transfers mathematical reasoning capabilities from specialized math LLMs multi-modal LLMs (MLLMs). Our approach does not pose any potential societal impacts or ethical concerns. We rely solely on publicly available models and datasets, none of which require specific acknowledgment.
B Detailed Experimental Settings
B.1 Datasets
We test our models math reasoning benchmarks Math Vista and
(33] Math Verse (53), DynaMath (59)
[36), general QA benchmarks MMMU 50 Tex(VQA 3] and MMBench [22].:
Math Vista assesses the MLLMs' multimodal mathematical skills, the testing data can be divided into five subsets: Figure Question Answering (FQA). Geometry Problem Solving (GPS), Math Word Problems (MWP), Textbook Question Answering (TQA), and Visual Question Answering (VQA). For evaluation, following [23 30), we first employ GPT-4 to extract the predicted choices answers from responses, then report the answer accuracy, which determines whether the final answer matches the ground truth.
Math Verse includes diverse set of math problems that require understanding and reasoning over both textual and visual information, such as charts, diagrams, and equations. The testing data can be divided into five subsets, i.e., Text Dominant, Text Lite, Vision Intensive, Vision Dominant and Vision Only.
DynaMath is dynamic visual mathematics benchmark developed for comprehensive evaluation of vision- vision-language models. It consists of 501 carefully curated seed problems, each implemented as Python program, enabling the assessment of model's 's generalization capability by measuring its robustness across multiple input variations derived from the same underlying seed question.
Math Vision is a carefully constructed dataset comprising 3,040 high-quality visual mathe- matics problems collected from real-world math competitions. Covering 16 mathematical domains and organized into difficulty levels, offers a broad and balanced benchmark for assessing the visual mathematical reasoning capabilities of MLLMs.
MMMU includes 900 evaluation samples and covers six core disciplines: Art Design, Business, Science, Health & Medicine, Humanities & Social Science, and Technology & Engineering, making it suitable for assessing the general knowledge of MILLM.
TextVQA evaluates a model's ability to read and reason over textual information embedded within images. It requires integrating visual understanding with text recognition answer questions grounded in image content. The dataset comprises 45,336 questions spanning 28,408 images sourced from the OpenImages dataset, providing a large-scale benchmark for visual-text reasoning.
MMBench provides multiple-choiee questions in both English and Chinese, facilitating direct and fair comparisons of MI.LM performance across languages. Overall, serves systematically constructed and objective benchmark for comprehensive and reliable evaluation of MLLM models. We use English dev set in the experiments.
B.2 Models and Comparison Methods
We employ the LLaVA series 20) Qwen 2 series 37 and Intern [58 series our base model.
We use other fine-tuned Math LLMs such as Tora series models, MetaMath models [ 47 and Qwen2-Math models 42). The pretrained foundation LLM of LLaVA models, Tora models and
Metamath is the LLaMA-2 34 model. The pretrained foundation of Qwen series models the
Qwen-2 model 377 The IL model is trained based on the Qwen2.5 models. We use RTX 3090 GPUs for all of our experiments. We compare our proposed methods with prevailing model merging techniques:
22

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Base Model The performance of base MLLM on three tasks, we reproduce the results with the officially released code.
Task Arithmetic [15] combines all the task vectors extracted from the models into one multi-task model.
Ties Merging [41] addresses task inference by pruning redundant parameters. The process involves three steps: Trim, Elect Sign, and Disjoint Merge.
EMR Merging [14] computes one unified task vector and computes task-specific masks based on unified task vector. The final task vector computed by combining the masked task vectors and weighting all the task vectors by rescale parameters.
Table 6: List of MLLMs and LLMs.
Models
Type
Pretrained Base Model
Source LLM
ALaVA-V1.5-1
MLLM
Vincuna-v1.5-7B
Llama-2-7B
Table-LLaVA-VI
MLLM
Vincuna-vI.5-7BB
Llama-2-7B
MLLM
Vincuna-v1.5-77
Llama-2-7B
LLa
VVV-V..6-7B---mmaa.
MLLM
Llama-3-8B
Llama-3-8B
LLVVA-V1.55 1.B5-1BB
MLLM
Vincuna-v -1.-1-v
Llama-2-133
MiemVLL--BBIIm
MLLM
Qwen2.5-7B
Qwen2.5-7B
VL.-Rethinker
MLLM
Qwen2.5-7B
Qwen2.5-7B
Qwen2-VI. VI--BB-Instrec
MLLMM
Qwen2-7B
Qwen2-7B
Qwen2-Math. -a--aee---
LLM
Qwen2-7B
Qwen2-7B
Tora-7b
LLM
Llama-2-7B
Llama-2-7B
Tora-Code-7B
LLM
CodeLLaMA-7B
Llama-2-7B
Tora-Code-13B
LLM
CodeLLaMA-133
Llama-2-7B
WizardMatth7B-VII.
LLM
Llama-2-7B
Llama-2-7B
Tora-7b 7b
LLM
Llama-2-7B
Llama-2-77
MetaMath-7B
LLM
Llama-2-7B
Llama-2-7B
OpenO1-Llama3-883
LLM
Llama-3-85
Llama-3-88
DeepSeek-R1 -iii--R--distille---eeeee
LLM
Owen2.5-Math-
Qwen2.5-7BB
DeepSeek-RR --dittllee--llmm--3---
LLM
Llama-3-8B
Llama-3-8B
We list the models used in the experiments in table
B.3 Details of Hyperparameter Selection
Following previous works in model merging 41 14, we adopt the grid search for hyperparameters for the baseline methods, specifically, set the hyperparameters based the following range:
Task Arithmetic 15 involves the scaling coefficients for merged task vectors, which are set ranging from 10.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9].
Ties Merging [41] involves the scaling coefficient and ratio to retain large parameters, the scaling coefficients are set ranging from [0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9], ratio to retain parameters with largest-magnitude values: [0.1, 0.2, 0.3].
EMR Merging 114 does not involve specific hyperparameters.
IP Merging involves the similarity threshold to determine whether the layer should be selected for merging.
23

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C Further Experiments
C.1 Merge Larger Models
We conduct experiments on merging larger models in table The experiments are conducted using models with 13B parameters. When dealing with larger models, our approach can also boost the performance compared to other model merging methods by 2.8% on average. In subtasks such as VQA, where math-related knowledge is highly demanded, our method achieves 3.3% performance gain.
Table 7: Experiments of merging 13B models. We merge LLaVA-13B with Tora-code- 13B using different merging methods. Bold represents the best performance. "Avg" is the average performance.
Math Vista
Approach
Param
TQA
GPS
VQA
FQA
MWP
Avg
Base Model
13B
41.1
25.0
34.1
21.9
16.1
26.7
Task Arithmetic
13B
39.9
34.6
29.1
22.3
7.0
26.0
Ties Merging
13B
38.6
29.3
34.1
22.3
17.2
25,9
IP-Merging
13B
42.4
26.0
37.4
23.1
18.8
28.5
+1.8 1
C.2 Merge Multiple Models
We conduct experiments on emerging multiple models to verify the scalability of our method in table 8 We LlaVA-Next-7B as the foundation MILM,, then merge multiple LLMs, such as the Tora Model and MetaMath-based models. As is shown the table 8 by merging more math reasoning models, the performance of math reasoning MLLM can be further improved.
Table 8: Experiments of merging multiple models on Math Vista.
Math
Models
Params
Appreach
Merged I.I.M
TQA
GPS
VQA
FQA
MWP
Avg
MLLM
Base Model
None
44.9
26.9
32.3
17.8
30.7
MLLM+ LLM
7B
IP IP-Merging
Tora-code-79
46.2
32.2
30.2
33.8
18.3
31.9 +1.21
Tora-code-7B
MLLM+ 2 LLMs
7B
IP-Merging
47.5
26.0
32.4
35.7
23.7
32.7 +2.0
MetaMath-77
C.3
Merge Different System-1 LLMss and System-2 LLMs
By employing the proposed method, we conduct experiments on merging different reasoning pattern LLMs with base MILM LLaVA-7B in table 99] We compare system-1 thinking LLMs such as WizardMath, MetaMath, Tora and Tora-code models. Merging Tora-code yields the best performance. Different from other models, Tora-code process. We believe Tora outperforms others for two main reasons: (I) CoT & PoT Collaboration: Tora employs collaborative reasoning approach that integrates CoT and PoT solve problems. In contrast, WizardMath and MetaMath rely solely on COT. (2) Reflection Mechanism: Tora's training data incorporates reflection-basse correction process, where incorrect responses are analyzed and revised, contributing to improved reasoning abilities. Others merely filter incorrect reasoning data. By further fine-tuning the code Ilama. Tora-code is able perform code-like reasoning on math problems, which exhibits strong performance compared to other math LLM on text- based math reasoning datasets such as GSM8K and Math. This experiment also reveals one interesting observation: math reasoning abilities obtained by Tora series models are more transferable to MLLM. We further conduct experiments on merging system-2 thinking LLMs such as OpenO1 27 and DeepSeek-R R -ddistllleddI LaMA3 model [10], demonstrating the effectiveness of our method. Our method further improves the model performance by merging the long CoT L.MMs, obtaining 2.3% performance gain average.
the high-quality reasoning data involved with the critique

---

Table 9: Experiments of merging different System-1 LLMs and System-2 LLMs on Math Vista.
Math Vista
MathLLM
Params
TQA
GPS
VQA
FOA
MWP
Avg
LLa VA-1.5-7B with System-1 LLMs
Base Model
7B
36.1
22.1
37.4
20.8
14.0
25.2
Meta V1.0-7B
7B
36.1
22.6
34.1
23.4
15.6
25.7
+0.5
WizardMath-7B
7B
41.4
25.0
34.1
23.4
13.4
26.6
+1,4
Tora-V1.0-78
7B
39.9
26.9
34.6
26.7
12.9
27.4 +2,2
Cora-code-VI.0-11
7B
43.7
21.6
40.8
24.9
15.1
28.2
+3.0
LLaVA-I.--LLaMA3--88 with System-2 LLMs
Base Model
8B
50.6
29.3
38.6
46.5
23.1
37.8
OpenOI-LLaMA3--8
8B
51.9
29.8
40.2
46.5
24.7
38.7 +0.9 f
DeepSeat-RI-distile--LLaa
8B
51.9
32.7
40.8
45.7
29.6
40.1 +2.3
C.4 Similarity Threshold and Time Cost Analysis
We compare the time cost of different model merging algorithms in fig. a]). We conduct this experiment of merging 7B MLLM and one math reasoning LLM on GPU server with 8-card Nvidia RTX 3090. As task arithmetic does not involve operations that modify the models, this method takes the least time. Ties merging and EMR merging involve comparing the sign of each parameter. Thus, they go through all pa- rameters, which can be time-consuming. Since our method directly operates parameters of each layer (formatting as matrix), our method can be accelerated by GPUs and takes less time to merge the models.
Similarity
(a)
(b)
Figure 7: (a) Time cost of method compared to other methods. (b) The results of different simi- larity thresholds.
We show the results of different similarity thresholds in fig. [)),,, more results in ap- pendix C.5 The performance fluctuates within a small range with higher similarity. We further compare the time cost of different model merging algorithms in fig. a[). Task arithmetic is the fastest since it doesn't change the models. Ties and EMR merging are slower because they process all parameters. Our method is faster as it works on each layer's parameters and be accelerated by GPUs and takes less time to merge the models.
C,5 Results of Hyperparameter Experiments
We conduct ablation experiments for the hyperparameters in table TO We show the results of different scaling coefficients in the task vector, the proportion the retained parameters and the scaling coefficients in ties merging, and similarity thresholds (i.e., S,, in eq. (6) in our method. We can that for the Llava models, the threshold around 0.3 and 0.4 provides the best performance, balancing the number of layers associated with math reasoning be merged and the importance of the layers to math reasoning. For the Qwen model, 0.6 is the optimal choice. We can also see that with higher thresholds, the performance fluctuates within small range. We also show the comparative methods of ties merging and task vector, the performance is sensitive to the different scaling coefficients or the rate of the retained parameters.
25

---

Table 10: Results of different hyperparameters. Bold represents the best performance.
LLaVA-1.5-7B
Qwen-2-VL
Methods
Math Vista
Math Verse
MMMU
Math Vista
Math Verse
MMMU
Base Model
25.2
11.3
34.2
55.4
24.8
50.7
Task Vector
scale=0.1
21.0
6.5
24.4
25.8
6.0
29.3
scale=0.2
25.2
7.9
25.6
27.8
5.9
32.6
scale=0.3
22.2
8.5
26.6
30.8
8.8
29. 1
scale=0.4
24.5
3.6
26.3
31.1
9.3
31.0
scale=0.5
23.1
4.4
30.2
28.9
7.6
26.0
scale=0.6
24.8
5.4
25.8
28.1
4.9
26,9
scale=0.7
23.3
6.2
26.2
28.5
1,4
24.9
scale=0.8
23.9
8.0
24.4
29.2
0.0
25.2
scale=0.9
20.9
9.8
25.6
24.1
0.0
23.9
Ties Merging
22.9
5.9
25.8
24.9
1.5
27.6
retain=0.3,scal=002
22.9
4.8
24.2
26.5
1.2
27.0
rctain=0.3.xcalc=0.
24.8
7.3
25.8
26.2
27.2
relain=0.3.scalea2
23.1
6.9
25.6
27.9
0.6
26.4
(=:ale=0.3,caal==055
26.1
1.9
22.6
27.6
0.6
26.2
rctain=0.3,xcalc=dd)
25.7
1.8
26.2
26.7
0.4
27.1
tstain=0.3,ccal=00.7
26.6
0.0
25.7
26.8
0.1
25.7
26.6
0.0
22.6
27.0
24.8
retain=0.3,scalc=0.99
25.0
0.0
25.8
24.6
0.0
26.2
retain=0.2.scale=0.1
23.6
6.4
25.3
26.3
2.1
28.1
retain=:0.2.xaal==0.22
23.6
4.6
24.0
29.2
2.3
26.0
0
24.1
5.3
26.0
28.5
1.9
29.2
resaae=
25.6
7.1
26.0
27.9
1.2
26.9
petale--.2.scal=-0.55
26.1
0.5
21.0
27.5
3.4
25.2
retain=0.2.caal=0.66
26.8
0.0
27.3
27.2
0.3
23.9
retain==0)._ccale00..7
24.9
0.0
21.4
25.0
0.3
23.1
0
24.8
0.0
24.1
24.8
0.0
28.3
retain=0)2..cal==00.9
23.3
0.0
23.9
24.8
0.0
25.8
cctai==011ssaalee
23.2
6.2
25.7
27.5
1.5
30.0
retain=0.
0.
22.9
5.7
25.0
27.5
3.5
30.6
retain=(),
1,scale=03,
23.6
6.4
24.4
27.5
3,4
30,8
retain=0.1,seaa 0,s
22.6
10.7
23.4
29.3
5,3
28.8
1,saal=00.
24.3
6.7
25.0
27.0
27.3
retain=0. 1,6
24.8
0.3
24.3
25.0
0.9
27.3
1.scalo...c
27.1
0.0
26.9
26.3
0.5
24.4
(tain==0.1,scaeee-
27.1
0.0
20.4
25.8
0,3
27.1
retain=0.1,sem 1,sca=00
24.8
0.0
25.8
26.1
0.2
29.1
EMR Merging
25.0
10.4
34.8
40.8
17.6
41.8
IP Merging
Sim threshold=0.1
25.4
14.5
34.0
59.3
27.8
50.2
Sim threshold=0.2
26.2
14.8
59.7
28.0
50.2
Sim threshold=0.3
26.4
15.3
34.4
59.8
27.9
49.8
Sim threshold-0.44
28.4
14.7
33.9
59.7
28.0
49.8
Sim threshold=0.5
26.4
14.9
34.2
59.7
27.9
49.8
Sim threshold-0.6
27.3
14.6
34.2
60.2
28.5
50.7
Sim threshold-0.7
27.6
14.4
34.2
60.1
28.5
50.7
Sim threshold-0.8
27.0
14.
34.2
60.1
28.4
50.7
Sim threshold-0.9
27.3
14.3
34.2
60.1
28.4
50.7
26

---

D Analysis of Selected Parameters
We visualize the proportion and composition of the selected parameters in fig. As shown in the figure, the selected layers account for less than 10% of the total model parameters, with the majority concentrated in the MLP layers. This observation aligns with recent studies on knowledge storage in LLMs, which suggest that most knowledge and skills are encoded within the MLP layers 7 [5]] Since Table-LLaVA level of mathematical reasoning ability. Consequently, our selection process identifies higher proportion of reasoning-related layers Table-LLaVA compared to the base model, LLaVA. To further analyze the distribution of these selected layers, plot their locations in fig. [[c) and (d). The visualization reveals that most reasoning-assoonn associated layers are concentrated the latter part of the model, suggesting that deeper layers play crucial role in encoding mathematical reasoning skills.
fine-tuned on math reasoning datasets, it has already acquired certain
8.5%
Anrenia Lagur Lages
Highen Simianit Score
94.6%
Highos Similanyy Score
68.4%
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Number Model Layera
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Namber of Midel Layen
LLaVA-7B
TableLLaVA-77
LLaVA-7B
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Selected Liper Location LLAVA
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(d)
Figure 8: (a) The proportion of selected layer two MLLMs. (b) The proportion of attention layers and MLP layers in selected layers.(c) Selected layers' location in LLaVA. (d) Selected layers' location Lab.e-LLaVA.
27

---

E User Case Analysis
As illustrated in fig. 9 the merged model effectively solved a symbolic reasoning problem by following logical steps to infer intermediate relations and reach the correct answer, showing that mathematical reasoning knowledge from the text-based LLM was successfully transferred and integrated into the multimodal model. This demonstrates that our merging strategy not only enhances reasoning depth but also improves consistency across steps, leading to more interpretable and reliable outputs. In contrast, the failure case occurred in a geometric reasoning scenario where the merged model predicted an incorrect angle despite the visual clues being straightforward. This reveals that while the merged model excels at structured symbolic inference, still struggles to align textual reasoning with precise spatial understanding.
Question: seqaarn the figure
Question: A decrease reob would affect whfroan food source?
analyze given equetions stop by step:1. Circle Square 5. Trangie Triangle B. Circle Trangle 7. We need to find the value the square. Firsc, let' find the value Thereforre value
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Frannge Trangle - 8.3, Crice Triangle + From equation 2, we know that Trianda Tr Trianga +. This means that Triangle 44, Now, substitute the valtas of Triengle equation Circle 4 Circle subatitute
decresae in rabititt will affect mountain lion's food
Merged Model
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Circle equacion Square 5, Square Therrfore,,
Questioncin the (OO, and CD is chord. Given LADC 25 what is the digrea CCAB?
Question: average, how many people can commute this vehicte?
O
Prediction: 30
The correct ansiwer la 64
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Merged Model
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Figure Successful cases and failure cases studies.
28

---

Method Overview
IP merging firstly identifies key parameters in both the MLLM and the math LLM. It then projects the rescaled, selected parameters from the LLM into the subspace of the MLLM to achieve better alignment. Finally, the aligned parameters are merged into the MILM. During the parameter identification phase, reasoning-relates parameters are selected based on their similarity within shared subspace. discrepancy between the two models. The complete procedure is illustrated in fig. T0 process of IP-Merging as follows:
projection phase, these parameters are rescaled and aligned minimize
We visualize
Parameter Identification
Parameter Projection
Subspace
Selected Layers
Subspace
SVD
Rescale
Similarity Comparison
Rescaled
Projected
Selected Layer
Layer
Layer
SVD
LAT
Figure 10: The general process of IP merging.
Algorithm 1 IP-Merging
Parameters of MLLM W MLLLA and Math LLM W Math, Pretrained Model Wo: o: Thresh- old S.: Number of layers N.
Input:
Output: Math reasoning model W MathMLLM.
Compute task vectors: AW MLLM and AW Moth- for to N do
Compute SVD decomposition of AW M MLLM and AWn Math using Equation 3
Compute the similarity scores {ST, 5% ...., S if Sf Sa then
for the n-th layer using definition 1
Compute the scaling factor An using Equation Compute the importance score "/n using Equation
Project A W'
Math into the subspace obtain AW Matk-P using Equation
return W -WW
MLLM + AW
Math-P
else
return W AW MLLM
end if
end for
29


Reviews:
[{"review_id": "xsQQLM31ER", "reviewer": "Reviewer_F8Nd", "summary": "This paper addresses the challenge of enhancing the mathematical reasoning abilities of multi-modal large language models (MLLMs) by leveraging the expertise of specialized, text-only math LLMs. The authors explore whether this transfer of knowledge can occur without any expensive fine-tuning, proposing that a significant gap in the parameter spaces of these models is the primary obstacle to direct merging.\n\nTo overcome this, the paper introduces a tuning-free method named IP-Merging. This approach first identifies the most relevant parameters for the task through subspace similarity analysis and then projects them from the Math LLM into the MLLM's subspace to ensure better alignment. The authors validate their method on several visual math reasoning benchmarks, showing that it improves performance on the target task while successfully preserving the model's general capabilities.", "questions": "1. This work focuses exclusively on mathematical reasoning. Have you considered whether IP-Merging is equally effective for transferring other specialized, text-based abilities, such as code generation, legal knowledge, or medical question-answering? Does the success of the method depend on any particular properties of the task being transferred?\n\n\n2. The similarity threshold `Sω` is a key hyperparameter. Instead of a fixed threshold, have you considered a more adaptive selection strategy? For instance, one could select a certain top percentage of layers based on their similarity score distribution, which might make the method more robust across different pairs of models.\n\n3. The performance on the GPS subset is notably poor compared to Ties Merging, and your explanation for this is quite general. Could you provide a more detailed error analysis for this task? For example, does the model fail more on visual perception of shapes or on applying geometric reasoning? This would help clarify the root cause of this limitation.\n\n4. To better demonstrate the method's utility, it would be valuable to see experiments on more recent, powerful MLLMs like Qwen2.5-VL and InternVL3. Since these models already have strong baseline reasoning abilities, showing that IP-Merging can still provide a meaningful boost would be a critical test of its practical value.\n\n5. The claim that IP-Merging preserves general abilities is supported only by the MMMU benchmark. To make this claim more robust, could you provide evaluation results on other general multi-modal benchmarks, such as MMBench or TextVQA?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The work's main strength is its focus on a highly relevant and practical problem: finding low-cost ways to enhance powerful models with specialized skills. The diagnostic analysis presented is insightful, providing a clear and data-driven motivation for why simple model merging fails and why a more sophisticated approach is necessary. The experimental validation is also quite solid, especially the inclusion of a general-purpose benchmark to confirm that the method does not cause catastrophic forgetting, which is a critical aspect for any model merging technique.\n\nHowever, the paper has some notable weaknesses that temper its overall impact. The most significant limitation is that the proposed method is currently restricted to models that share an identical architecture and size. This severely narrows its practical applicability, as many real-world use cases would involve transferring knowledge between models of different scales or families. Furthermore, the methodological contribution feels somewhat incremental, as it primarily combines well-established techniques like SVD and task vector arithmetic in a novel application, rather than introducing fundamentally new concepts.\n\nBeyond these broader concerns, some specific experimental results raise further questions. According to Table 2, the merging of LLaVA-1.5-7B shows a very large performance gap (18.8 points) on the GPS subset of MathVista when compared to Ties Merging. This result casts doubt on the effectiveness of IP-Merging for geometric problems. The authors' explanation that this is because GPS requires multiple steps of geometry reasoning is insufficient; if the MLLM truly absorbed the Math LLM's reasoning ability, it should show improvement in this domain. Additionally, the method has not been validated on stronger MLLMs like Qwen-2.5-VL or InternVL 3. For example, InternVL 3 already achieves high performance of 39.8 on MathVerse and 71.6 on MathVista. Verifying that IP-Merging still provides a meaningful benefit to these highly capable models would be very helpful for testing the method's validity.", "was_revised": true, "final_justification": "I thank the authors for their detailed rebuttal and additional experiments. The paper's core idea of tuning-free knowledge transfer is well-motivated and presents a clear contribution. However, my primary concerns persist. The performance degradation on geometric reasoning tasks (the GPS subset) and the modest gains on more capable MLLMs are unresolved issues that temper the method's overall impact. For these reasons, I will maintain my original score.", "rebuttal": "We appreciate your insightful comments. Here, we address your concerns and questions point by point.\n\n- **Q1: Transfer other text-based abilities**\n\nThank you for the inspiring question. Math reasoning has been one crucial ability of LLMs and MLLM, hence we focus on math reasoning in our paper.  Essentially, our method relies on using a specialized model and its task vector to transfer capabilities from LLMs to corresponding MLLMs. This setup is not restricted to math reasoning and could be applied to other specialized domains such as code generation. However, we have not found an appropriate testing benchmark to evaluate the effect of the code generation ability of MLLMs. In our paper, we also made attempts to merge the code model to enhance math reasoning in Table 10 in the appendix, demonstrating its effectiveness. For the medical image question answering, we believe the proposed IP-merging might be useful, where both task-specific knowledge and visual understanding are required. In case that more benchmarks and corresponding specialized models are available, we are happy to investigate extending our method in these domains. \n\n- **Q2: Adaptive&fixed similarity threshold**\n\nWhile we fully understand your concern, we suggest setting the threshold directly (we find 0.4 performs well in our experiments).   Our justifications are as follows. Our goal is to identify crucial parameters between models using similarity scores.  We directly use a similarity threshold based on the two considerations: (1) **A fixed similarity threshold ensures that only highly related parameters are merged.** **In contrast, a top-k% approach selects a fixed number of parameters regardless of their similarity. Some layers might include low-similarity parameters, which may not benefit from merging.**  For some model pairs, the top-k% **parameters may have high similarity, while for others, they might be relatively dissimilar. This makes the threshold harder to tune across different model pairs. (2) **Similarity scores are normalized,** so the fixed threshold is naturally robust across different model pairs.  For your reference, we also conduct experiments of selecting the top-k percentage of parameters based on the score distribution. As shown in the following table, we find that the performance is not as robust compared to directly setting the threshold (0.4).  Although the two approaches are similar, directly setting the threshold is a more effective way for our method. \n\n| Adaptive threshold | 5% | 10% | 15% | 20% | 25% | 30% |\n| --- | --- | --- | --- | --- | --- | --- |\n| Corrspondingsimilarity threshold | 0.45 | 0.25 | 0.16 | 0.12 | 0.10 | 0.08 |\n| IP merging  | 26.1 | 26.1 | 25.0 | 26.7 | 25.9 | 26.8 |\n- **Q3: Poor performance on GPS**\n\nThank you for your comments.  Following your suggestions, we examine failure cases in MathVista,  and find that the IP-merged LLaVA model may struggle to recognize key visual information in the image, leading to wrong predictions. The cases below suggest that the IP-Merged model’s performance is hindered by weakened visual understanding, particularly in tasks requiring recognition of geometric information. In the revision, we will add one section of case studies for more discussions.\n\n| Cases | Questions | Comments |\n| --- | --- | --- |\n| MathVista-id:143,\nimage: images/143.jpg | In the parallelogram ABCD, line CE bisects ∠BCD and intersects edge AD at point E. Given DE = 3.0, what is the length of AB?  | **This question requires the model to recognize parallelogram image and use properties of angle bisectors and parallelograms to solve problem.** |\n| Ties-Merging | Prediction: 3 **(True)** |  |\n| IP-Merging | Prediction: 1 **(False)** | Failure on visual perception (The model fails to recognize which is edge AB） |\n| MathVista-id:59,\nimage:images/59.jpg | In the figure, AB is the diameter of circle ⊙O, and CD is a chord. Given that ∠ADC = 26°, what is the degree of ∠CAB?  | **This question requires the model to locate the right triangle in the circle image and make predictions use the relationships between diameters, chords, and inscribed angles.** |\n| Ties-Merging | Prediction: The correct answer is 64 **(True)** |  |\n| IP-Merging | Prediction : The correct answer is 26 **(False)** | Failure on visual perception (The model does not locate the triangle in the circle accurately） |\n- **Q4-Q5: Experiments of recent MLLMs and other general multi-modal benchmarks**\n\nWe are thankful for your suggestions. Following your kind suggestion, we further conduct experiments using recent MLLMs Qwen2.5-VL-7B-Instruct and InternVL3-8B-Instruct on more benchmarks including MathVision,  Dynamath, Wemath and one additional general VQA benchmark **TextVQA** (Due to the limited rebuttal window, we do not run MMBench). We use Deepseek-R1-distilled-Qwen-7B as the math LLM.  As shown in the table, our methods do enhance the reasoning performance of stronger baseline models while maintaining the general reasoning abilities. \n\n| Model | **Mathverse** | **MathVista** | **MathVision** | **Dynamath** | **Wemath** | **TextVQA** |\n| --- | --- | --- | --- | --- | --- | --- |\n| Qwen2-VL-7B-Instruct | 24.8 | 55.4 | 17.2 | 40.8 | 23.6 | 83.8 |\n| +IP Merging | **28.5** | **60.2** | **19.1** | **41.0** | **24.4** | **84.0** |\n| LLava-1.5-7B | 11.3 | 25.2 | 11.1 | 15.4 | 6.5 | 47.50 |\n| +IP Merging | **15.4** | **28.4** | **11.8** | **16.1** | **7.2** | **47.52** |\n| Qwen2.5 VL-7B-Instruct | 46.6 | 68.0 | 24.7 | 51.9 | 31.1 | 84.8 |\n| +IP merging | **46.8** | **69.7** | **26.3** | **52.5** | **33.5** | **84.9** |\n| InterVL3-8B-Instruct | 38.8 | 66.1 | 24.9 | 50.7 | 31.6 | 81.9 |\n| +IP meging | **39.0** | **67.6** | **25.2** | **51.4** | **33.0** | **82.3** |"}, {"review_id": "mPKj5hVMwx", "reviewer": "Reviewer_Vh1P", "summary": "This article attempts to address the issue of merging parameters between Large Language Models (LLMs) and Multimodal Large Language Models (MLLMs) to integrate the reasoning capabilities of an LLM focused on mathematical reasoning into a general MLLM, thereby enhancing the MLLM's ability to handle visual math-related problems. The authors identify two key challenges: how to select appropriate parameters and how to better align them. In response, the authors propose the IP-Merging method, which requires no additional tuning. Moreover, this method not only achieves promising results but also maintains the original capabilities of the MLLM across other tasks.", "questions": "Please see weaknesses.", "limitations": "N/A", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: The writing is fluent, and the analysis of the adjustments to the problem is insightful. The proposed method achieves excellent results without affecting the inherent capabilities of the MLLM.\n\nWeaknesses:\n\n1.Can this method be extended to models other than Math LLMs, such as code LLMs?\n\n2.There are many methods that have found training the visual encoder together can yield good results (e.g., Cambrian-1). Has the author considered this scenario where the visual encoder has been trained?\n\n3.You can consider the results of the benchmark mentioned in MathGlance (https://arxiv.org/pdf/2503.20745) to provide a more insightful assessment.\n\n4.There is some misalignment of text in Figure 2.", "was_revised": true, "final_justification": "The rebuttal addressed my main concerns with additional experiments on code LLMs, trained visual encoders, and the MathGlance benchmark, all showing consistent gains. The figure issue appears to be a rendering problem. Given the method’s simplicity, training-free nature, and positive results without harming general capabilities, I maintain my borderline accept rating.", "rebuttal": "Thanks a lot for your insightful comments. Here, we address your concerns and questions point by point.\n\n- **W1: Extend to other models**\n\n**Yes, the method can be extended to other models like code LLM.** In our paper, we tried merging CodeLlama and Tora-code with the LLaVA model, which improves MathVista performance from **25.2% to 26.3% and 28.2%, respectively**. It shows that code LLMs can contribute to multi-modal reasoning tasks. The experimental results are shown in  **Table 10 of Appendix.**\n\n- **W2: Trained visual encoder**\n\nWe are thankful for your insights. In our work, we focus on developing a train-free method to enhance the reasoning performance of VLMs, therefore we do not retrain the visual encoder in the paper. Nevertheless,  IP-merging also works  effectively when the base MLLM has a trained vision encoder.** We test the Cambrian-8B model and merge the Deepseek-R1-distilled model (LLama-3-based reasoning LLM). We illustrate the results on MathVista, showing that IP merging can further enhance the reasoning performance.\n\n| Model | **MathVista** |\n| --- | --- |\n| Cambrian-8B | 46.8 |\n| **+IP megring** | **48.8** |\n- **W3: MathGlance benchmark**\n\nThank you for your great suggestion. We conducted experiments on the MathGlance Benchmark using Qwen2.5-VL-7B-Instruct  as the base MLLM and the Deepseek-R1-distilled-Qwen-7B as reasoning LLM. After merging the LLM,  the geometry reasoning performance of the base MLLM is further enhanced.  \n\n| Model | **Plane Geometry** | **Solid Geometry** | **Graphs** |\n| --- | --- | --- | --- |\n| Qwen2.5-VL-7B-Instruct | 44.5 | 67.8 | 64.7 |\n| **+IP Merging** | **45.8** | **69.0** | **67.4** |\n- **W4: Misalignment of the text**\n\nThanks. However, we cannot find the misalignment in Figure 2. The figure appears correctly formatted on our end (We previously found that if an old version of a browser or a PDF reader is used, it may not show the figure correctly)."}, {"review_id": "tMIFgLFf7A", "reviewer": "Reviewer_XRHa", "summary": "This paper presents a method for improving the math reasoning capabilities of multimodal large language models (MLLMs) by directly merging parameters from math-focused LLMs, without additional training or data. The proposed IP-Merging approach identifies reasoning-related layers in both models and projects selected parameters from the math LLM into the MLLM’s subspace to address parameter misalignment. Experimental results on benchmarks such as MathVista and MathVerse, using models like LLaVA and Qwen2-VL, show that IP-Merging enhances math reasoning performance of MLLMs while maintaining their general abilities.", "questions": "**See W1&W2&W3&W4.** (The current evaluation mainly focuses on relatively small and outdated models. Can the authors provide results on more recent and larger MLLMs, especially those with stronger baseline performance? As plug-and-play methods often become less effective as baselines improve, it would be important to discuss whether IP-Merging still offers meaningful gains in these settings. If the improvements are marginal or absent with recent models, what are the main factors limiting its effectiveness?)", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "***Strengths***\n1. The paper introduces IP-Merging, a new parameter merging method that enables MLLMs to directly acquire math reasoning abilities from math-specialized LLMs without any additional training or data.\n2. The approach incorporates subspace similarity-based parameter selection and projection, effectively identifying and aligning reasoning-associated parameters between models to facilitate knowledge transfer.\n3. Experimental results demonstrate that IP-Merging consistently improves the math reasoning performance of MLLMs on benchmarks such as MathVista and MathVerse, while maintaining the general capabilities of the models.\n\n***Weaknesses***\n1. **Lack of evaluation on recent reasoning models:** Although the paper reports results on G-LLaVA and similar math-specific models, these are based on older architectures (2023). For example, after applying IP-Merging, G-LLaVA only achieves a MathVista score of 29.6, whereas recent 7B reasoning models can reach nearly 70 on the same benchmark. The lack of evaluation on recent reasoning models makes it difficult to judge whether the effectiveness of IP-Merging generalizes to current high-performing models.\n2. **Lack of diverse and strong general baselines:** The experiments only consider LLaVA-1.5-7B and Qwen-2-VL-7B as general MLLMs. The absence of more recent, larger, or more competitive models, as well as different training strategies and sizes, limits the assessment of IP-Merging’s robustness and general applicability.\n3. **Insufficient benchmark coverage:** The reasoning benchmarks used in the experiments are limited to MathVista and MathVerse. Results on additional benchmarks such as MathVision, WeMath and Dynamath would provide a more comprehensive assessment. In particular, WeMath and Dynamath can highlight reasoning mechanism deficiencies, while MathVision focuses on more challenging problems.\n4. **Experimental results and reporting require clarification:** There are inconsistencies in reported numbers, such as a MathVista score of 55.4 in the paper compared to the official 58.2. Additionally, for MMMU (Table 3), the baseline (Qwen-2-VL-7B) and IP-Merging results (including sub-categories) are exactly the same. These points require clarification, as they directly impact the credibility of the experimental conclusions.\n5. **Architectural restriction:** IP-Merging requires both the MLLM and the math LLM to have the same architecture and parameter size, significantly limiting its applicability in practical scenarios involving heterogeneous or larger-scale models.", "was_revised": true, "final_justification": "I appreciate the authors' efforts, and they have addressed my concerns during the rebuttal period.", "rebuttal": "We are thankful for the valuable comments. Here, we address your concerns and questions point by point.\n\n- **W1: Experiments of recent reasoning models**\n\nThanks for the suggestions. Due to the limited rebuttal window, we conduct experiments on one recent SOTA reasoning MLLM VL-Rehinker-7B [1], which still demonstrates the effectiveness of IP-Merging in enhancing the performance of reasoning MLLM on various math reasoning benchmarks.  We are happy to report results on more reasoning models in our final version.\n\n| Model | **MathVista** | **MathVerse** | **MathVision** | **WeMath** | **MMMU_val** |\n| --- | --- | --- | --- | --- | --- |\n| VL-Rethinker-7B | 72.7 | 51.9 | 28.5 | 37.6 | 59.4 |\n| + IP merging | **75.2** | **52.4** | **29.6** | **39.4** | **60.4** |\n- **W2&W3: Diverse baselines and benchmarks**\n\nFollowing your kind suggestions, we further conduct experiments using Qwen2.5-VL-7B-Instruct,  InternVL3-8B-Instruct, and extend the Math-Benchmarks to MathVision, DynaMath and WeMath. As shown in the table, our merged model can further improve the math reasoning performance across different benchmarks.\n\n| Model | **Mathverse** | **MathVista** | **MathVision** | **Dynamath** | **Wemath** |\n| --- | --- | --- | --- | --- | --- |\n| Qwen2-VL-7B-Instruct | 24.8 | 55.4 | 17.2 | 40.8 | 23.6 |\n| +IP Merging | **28.5** | **60.2** | **19.1** | **41.0** | **24.4** |\n| llava-1.5-7B | 11.3 | 25.2 | 11.1 | 15.4 | 6.5 |\n| +IP Merging | **15.4** | **28.4** | **11.8** | **16.1** | **7.2** |\n| Qwen2.5 VL-7B-Instruct | 46.6 | 68.0 | 24.7 | 51.9 | 31.1 |\n| +IP merging | **46.8** | **69.7** | **26.3** | **52.5** | **33.5** |\n| InterVL3-8B-Instruct | 38.8 | 66.1 | 24.9 | 50.7 | 31.6 |\n| +IP meging | **39.0** | **67.6** | **25.2** | **51.4** | **33.0** |\n- **W4: Clarification on the experimental results**\n\nWe are sorry for the confusion.  **(1) Inconsistencies in reported numbers on MathVista: For fair evaluation, we reproduce all the results using the public official evaluation code, parameters and prompts [2].** We also adopt the same setting in our merged models for fair evaluation. We applied the same setup to our merged models. Variations in hardware may also affect performance. For instance, we observed a score of 54.7 on an A800 GPU versus 55.4 on an RTX 3090. To ensure consistency, we used RTX 3090 and the same runtime environment for all experiments.\n\n**(2)** **Clarification for the same results on MMMU**: We are aware that the MMMU subsets have the same score. However, there do exist differences if we look into the statistics of each subset. For example, in Humanities and Social Science subsets of MMMU as shown below, we find that they are different in two subsets, including History and Sociology, although their sum is the same. \n\n**Base MLLM**: 'Overall-Humanities and Social Science': {'num': 120, 'acc': 66.667}, **'History': {'num': 30, 'acc': 70.0}**, 'Literature': {'num': 30, 'acc': 86.667}, **'Sociology': {'num': 30, 'acc': 56.667},** 'Psychology': {'num': 30, 'acc': 53.333}\n\n**Merged Model:** Overall-Humanities and Social Science': {'num': 120, 'acc': 66.667}, **'History': {'num': 30, 'acc': 66.667}**, 'Literature': {'num': 30, 'acc': 86.667}, **'Sociology': {'num': 30, 'acc': 60.0},** 'Psychology': {'num': 30, 'acc': 53.333}\n\nWe suspect this might be due to the limited evaluation samples (900 samples). For your reference, we add experiments of TextVQA (5000 samples), where the base model achieves a score of 83.8 while the merged model achieves 84.0.  In summary, we are confident that our results are reproducible, and all reviewers are welcome to run the code in the anonymous link to verify our results.\n\n- **W5: Limitation on architecture**\n\nAs noted in the limitations of our paper, our method is designed for merging MLLMs and math LLMs that share the same base architecture and size (e.g., LLaMA-2). This is necessary to reliably identify aligned layers for merging. Nonetheless, we believe this does not pose a major constraint in practice, as **Most existing MLLMs are built on top of their corresponding LLMs.** For example, LLaVA is based on LLaMA models, and Qwen2-VL is based on Qwen2 models, making such model pairs common in real-world scenarios. Currently, **this architectural constraint is a common limitation across train-free model merging methods,** which typically assume the same size and modality. Our work takes a step further by addressing cross-modality merging to enhance multi-modal reasoning capabilities. \n\n- **Questions:**\n\nWe provide extra experimental results of more recent baseline models (e.g., LLaVA and Qwen series, Fine-Tuned or RL-based training strategy) and benchmarks (e.g., math-related and general benchmarks), demonstrating the effectiveness of our approach even on stronger baseline models.\n\n[1] Wang et.al, VL-Rethinker: Incentivizing Self-Reflection of Vision-Language Models with Reinforcement Learning, 2025\n\n[2] Wang et.al, Qwen2-VL: Enhancing Vision-Language Model's Perception of the World at Any Resolution, 2024"}, {"review_id": "GkM2lTepbA", "reviewer": "Reviewer_hd9w", "summary": "This paper proposes IP-Merging, a tuning-free model merging technique to enhance the mathematical reasoning capabilities of MLLMs by merging them with math-specialized LLMs. It first identifies math-reasoning-associated parameters, rescales and projects them into a shared subspace, and merges the models. Compared with other model merging baselines, IP-merging is shown to more effectively improve math reasoning performance on MathVerse and MathVista without degrading general capabilities on MMMU.", "questions": "Please refer to the weaknesses section first.\n\n1. Can this method be applied to domains beyond math? It seems such a merging strategy could be generalizable to other domains—could you elaborate? If so, why only focus on math reasoning in this paper?\n\n2. Could you provide qualitative examples where the merged MLLM successfully solves a problem that the base model fails to address? This would offer intuition and insight about the improvements.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### **Strengths**\n\n1. The paper provides a rigorous empirical analysis of why conventional merging methods are ineffective, convincingly demonstrating the \"parameter space gap.\" This analysis offers a strong motivation for the proposed work.\n2. The proposed IP-Merging method is novel and effectively addresses the diagnosed issues. Its training-free nature makes it a practical solution for enhancing model capabilities.\n3. Experimental results effectively demonstrate that IP-Merging enhances the mathematical reasoning of MLLMs without compromising performance on MMMU, and IP-Merging is shown more competitive performance over baselines.\n4. Overall, the paper is well-written, clear, and easy to follow. (Though I still have some suggestions for improvement, please see below.)\n\n### **Weaknesses**\n\n1. As acknowledged in the limitations section, the method currently only supports merging models of the same size and base architecture. This indeed significantly limits its applicability and potential adoption.\n\n2. The \"free lunch\" claim is not sufficiently supported—evaluating only on MMMU is not enough. A broader set of evaluations is needed to verify whether the merged models truly retain their original capabilities.\n\n3. Beyond performance metrics, an analysis of error or failure cases before and after merging could offer additional insights and perspectives.\n\n4. Ablation studies on the hyperparameters $\\alpha_1$ and $\\alpha_2$ are missing.\n\n\n\n### **Writing & Presentation (Minor)**\n\n1. The experiments involving the merging of multiple models are valuable. I recommend moving them from the appendix into the main text and providing a more in-depth analysis.\n2. The readability of Table 9 is poor. Consider replacing it with line charts (e.g., a matrix of subfigures), which could illustrate trends more effectively.", "was_revised": true, "final_justification": "Overall, I believe that the article is generally a positive and innovative attempt. The author's additional experiments, to some extent, show that IP-Merge has not harmed the performance on TextVQA, so I give it a positive evaluation.\n\nConsidering the limitations in the model merging scenario and the fact that the experimental data is still not sufficient to completely eliminate concerns about the free lunch claims, I maintain a borderline recommendation.", "rebuttal": "We are thankful for your valuable comments. Here, we address your concerns and questions point by point.\n\n- **W1: Limitations of merging different structures**\n\nAs noted in the paper’s limitations, our method is designed for merging MLLMs and math LLMs that share the same base architecture and size. This ensures we can reliably identify aligned layers for merging. In practice, we believe this is not a major limitation, as most MLLMs are built on top of their corresponding LLMs. For example, LLaVA is based on LLaMA-based models, and Qwen2-VL on Qwen2-based models, making such model pairs common. **Currently**, **this architectural constraint is shared by most existing training-free merging methods**. which typically assumes the models have the same size and cater to the same modality. Our work goes a step further by enabling cross-modality merging to improve multi-modal reasoning.\n\n- **W2: More general benchmark results**\n\nWe are grateful for your comments. Following your suggestion, **we conduct extra experiments on TextVQA to support our claim.** These results are shown below. Our method allows the models to maintain their general capabilities. \n\n| Model | **TextVQA** |\n| --- | --- |\n| LLava-1.5-7b | 47.50 |\n| +IP merging | **47.52** |\n| Qwen2-vl-7b-instruct | 83.8 |\n| +IP merging | **84.0** |\n- **W3&Q2:  Failure / Successful case study**\n\nThank you for the valuable advice. We illustrate two cases in the MathVista below for your reference. As shown in the Failure Case, the merged model may fail on problems where visual common-sense reasoning is required. This suggests that merging may impair the model’s ability to reason over visual information and commonsense knowledge, potentially caused by math-focused weights after merging.  As shown in the successful case, the merged model is better in cases where statistical reasoning and arithmetic reasoning are demanded. In the revision, we will add one section of case study to show more qualitative results and analysis and provide a more intuitive understanding.\n\n| Cases | Question | Comments |\n| --- | --- | --- |\n| **Faliure Case:** \"MathVista-id:194\",\"image\":\"images/194.jpg\" | On average, how many people can commute on this vehicle? | **Note: This question requires the model to estimate the number of people that can be accommodated in a big green bus based on commonsense knowledge.** |\n| Base model(LLava): | Prediction: 50 **(True)** |  |\n| Merged model: | Prediction: 40 **(False)** | The model fail to do the commonsense reasoning of the people in a big bus. |\n| **Successful Case:** \"MathVista-id:359,\"image\":\"images/359.jpg\"  | Kylie spent a week at the beach and recorded the number of shells she found each day. According to the table, what was the rate of change between Thursday and Friday? (Unit: shells per day) | **This question requires the model to look at a statistical table and calculate the change rate of the item.** |\n| Base model(LLava): | Prediction: 7\\n0\\n0\\n7\\n0\\n0\\n7\\n0 **(False)**  |  |\n| Merged model: | Prediction: Rate of change = $\\frac{0 - 7}{1}$, ****Rate of change = $-7$. The rate of change between Thursday and Friday was -7 shells per day.  **(True)** | The model reads statistics in the paper, and calculate the results of change rate.  |\n- **W4:  Scaling parameter α1 and α2**\n\nSorry for the confusion.  **The scaling factors α₁ and α₂ are not hyperparameters**. As mentioned in line 296 of the main paper, α₁ is set to 1, and α₂  is computed based on the matrix norm using Eq. (7). **In our paper, the main hyperparameter is the similarity threshold, which we provide in Table 9 along with the hyperparameter results in the baseline.\n\n- **Wr&Pre:  Rearrange the experiments and optimize Tab 9**\n\nThank you for the valuable suggestion. We will move the results of merging multiple models to the main paper and plot line figures to better showcase the results in the revision.\n\n- **Q1: Apply to other domains**\n\nThank you for the thoughtful question. Math reasoning has been one crucial ability of LLMs and MLLM, hence we focus on math reasoning in our paper.  Essentially, our method relies on using a specialized model and its task vector to transfer capabilities from LLMs to corresponding MLLMs.  This setup is not restricted to math reasoning.  While prior works have explored model merging in text-only domains, our work extends this to the more challenging multi-modal setting. Our approach could potentially be applied to other domains where paired specific LLM and MLLM models exist. In case that more benchmarks and corresponding specialized models are available, we are happy to investigate extending our method in these domains."}]

Author rebuttal:

Meta-review: ## Summary

This paper studies whether multi-modal large language models (MLLMs) can gain mathematical reasoning ability from specialized math LLMs without additional training. The authors note that direct merging of parameters between the two types of models is limited by differences in their parameter spaces. To address this, they introduce a method called IP-Merging. This approach first identifies the parameters most related to reasoning in both models, then projects them into the MLLM subspace to reduce misalignment, and finally merges them. The method is tuning-free since it adjusts parameters directly rather than relying on gradient-based training. Experiments show that IP-Merging improves math reasoning in MLLMs while preserving their general performance.

## Decision

This paper addresses an important problem: finding low-cost ways to enhance powerful models with specialized skills. The experiments are thorough and well executed, and the results are convincing. The writing is clear, and the paper is easy to follow. In addition, the paper provides a detailed empirical analysis explaining why conventional merging methods are ineffective, clearly demonstrating the existence of a “parameter space gap.”

The proposed IP-Merging method is interesting, though somewhat incremental. During the review process, several concerns and criticisms were raised, but the authors effectively addressed them in the rebuttal. I recommend that the authors incorporate the additional results presented in the rebuttal to further strengthen the paper.

Overall, the reviewers were positive about this work. I recommend the paper for acceptance, as it should be of interest to the ICLR community.
Decision: Accept (poster)