Enhancing Geometric Perception in VLMs via Translator-Guided Reinforcement Learning
Authors: Hao Yu
Shuning Jia
Guanghao Li
Wenhao Jiang
Chun Yuan
Venue: ICLR 2026 Poster (2026)
Keywords: Vision Language Models
Geometry Perception
Reinforcement Learning
Primary area: reinforcement learning

Abstract: Vision-language models (VLMs) often struggle with geometric reasoning due to their limited perception of fundamental diagram elements. To tackle this challenge, we introduce GeoPerceive, a benchmark comprising diagram instances paired with domain-specific language (DSL) representations, along with an efficient automatic data generation pipeline. This design enables the isolated evaluation of geometric perception independently from reasoning. To exploit the data provided by GeoPerceive for enhancing the geometric perception capabilities of VLMs, we propose GeoDPO, a translator-guided reinforcement learning framework. GeoDPO employs an NL-to-DSL translator, which is trained on synthetic pairs generated by the data engine of GeoPerceive, to bridge natural language and DSL. This translator facilitates the computation of fine-grained, DSL-level scores, which serve as reward signals in reinforcement learning. We assess GeoDPO on both in-domain and out-of-domain datasets, spanning tasks in geometric perception as well as downstream reasoning. Experimental results demonstrate that, while supervised fine-tuning (SFT) offers only marginal improvements and may even impair performance in out-of-domain scenarios, GeoDPO achieves substantial gains: $+26.5\\%$ on in-domain data, $+8.0\\%$ on out-of-domain data, and $+39.0\\%$ on downstream reasoning tasks. These findings underscore the superior performance and generalization ability of GeoDPO over SFT. All codes are released at https://github.com/Longin-Yu/GeoPerceive to ensure reproducibility.

Published as a conference paper at ICLR 2026
ENHANCING GEOMETRIC PERCEPTION IN VLMs VIA TRANSLATOR-GUDDE REINFORCEMENT LEARNING
Hao Yu!.2
Shuning Jia'
Guanghao Li'
Wenhao Jiang?
Chun Yuan
Tsinghua University
@Guangdong Laboratory of Artificial Intelligence and Digital Economy (SZ)
ABSTRACT
Vision-language models (VLMs) often struggle with geometric reasoning due to their limited perception of fundamental diagram elements. To tackle this challenge, we introduce GEOPERCEIVE, a benchmark comprising diagram instances paired with domain-specific language (DSL) representations, along with an efficient automatic data generation pipeline. This design enables the isolated evaluation of geometric perception independently from reasoning. To exploit the data provided by GEoPERCEIVE for enhancing the geometric perception capabilities of VIMMs, we propose GRODPO, translator-guiddes reinforcement learning (RI.) framework. GEoDPO employs an NL-to-DSL translator, which is trained synthetic pairs generated by the data engine of GEOPERCEIVE, to bridge natural language and DSL. This translator facilitates the computation of fine-grained DSL-level scores, which serve as reward signals in reinforcement learning. We assess GEODPO on both in-domain and out-of-domain datasets, spanning tasks in geometric perception as well as downstream reasoning. Experimental results demonstrate that, while supervised fine-tuning (SFT) offers only marginal improvements and may even impair performance in out-of-domain scenarios, GEODPO achieves substantial gains: +26.5% on in-domain data, +8.0% on out-of-domain data, and +39.0% on downstream reasoning tasks. These findings underscore the superior performance and generalization ability of GEODPO over SFT. All codes are released at htt tps: //github: www. Yun Ntw.nuooeeooeeeeieeeeeeee to ensure reproducibility.
INTRODUCTION
VLMMs have made significant strides in multimodal applications (Team] Team et al.
2023 2024 Bai
ct al. Yu et al.) 2025a Zhu et al. [2033 Wei et al. 2025b] Xu et al. 2002 Wang et al. [2025] Yu et al. 2025b Wei et al.) 2025a Yue et al.) 2023). yet geometry problem solving (GPS) remains particularly challenging. This difficulty primarily stems from the need for accurate geometric perception: before engaging any symbolic reasoning, a model must correctly identify and ground geometric primi- tives-such as points, lines, and circles-and their spatial relations within diagram. In practice, even state-of-the-art VLMs frequently misinterpret these primitives, for instance, confusing an intersection with tangency, as illustrated in Table [T Moreover, widely adopted end-to-end benchmarks (Lu et al.
2023 Zou et al. 2024) tend to conflate perceptual errors with reasoning failures, thereby obscuring
the true source of model limitations (Cho et al. 2025 Zhang al. 2023 Chen et al. [021 Cao &
Xiao 2022 Ning et al., 22023] Chen et al. 2022 Liang et al.) 2023 Xia et al. 2024 [Li et al. 2023
Kosoy al. 2025). To address this, frame the problem around two questions: how can geometric
perception be measured, and how can it be improved?
To measure geometric perception unambiguously and scale, benchmark must provide both a pixel-level diagram and a formal scoring target that serves as the diagram's meta information. We adopt a DSL as this target and require it be (i) complete (able to reconstruct the original diagram) and (ii) canonical (one representation per diagram). ensuring that a given prediction does not receive different scores under semantically equivalent ground truths. Following this principle. we present GEoPERCEIVE: where every instance pairs a rendered diagram with its DSL program.
"Corresponding author.

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Published as a conference paper at ICLR 2026
Table 1: Natural language captions for different models. Green indicates correct content, while red denotes incorrect content.
Diagram
Caption
The diagram shows intersecting circles, external point U lies still farther to the right. red BN J2,
GPT-o3
Achiam al.
2025
The diagram cunsists two overlapping circles. The larger circle has center laheled as Z, and the smaller circle tangent the circle
Owen3 235B
Team 2025
The diagram depicts and two circles, the larger cen- lered at E, intersecting the smaller al point and N. The smaller circle passes through and U....
GEoDPO (Ours)
GEOPERCEIVE is produced by two engines: generation engine that samples diverse GEODSL programs, and solving engine that instantiates each program via gradient-based optimization and renders a pixel-level diagram. This design enables exact program-level evaluation, and yields an automatically generated, complexity-controlle
corpus for both evaluation and training low cost.
Leveraging GEOPERCEIVE, we can synthesize a large corpus for training geometric perception. However, directly fine-tuning such data is sensitive permutation-cquivaleni programs: many valid orderings describe the same diagram, while SFT optimizes a single instance and encourages surface-form patterns rather than invariants. We therefore adopt reinforcement learning paradigm. A straightforward alternative is to train the VLMS to emit DSL. directly. However, this departs from its NL--centric pretraining and thus requires bridge this gap without forcing the model off its NI. manifold, our proposed approach GEODPO keeps the VLMs in the NL regime and introduces an NL-to-DSL translator trained synthetic NL-DSL pairs generated by the GEOPERCEIVE pipeline. The translator assigns fine-grained DSL-level scores to model outputs, which we convert into reward signals for preference based alignment (Rafailov et al. 2024). This design avoids the order- permutation explosion that hinders sequence-leva SFT, mitigates distribution shift from NL pretraining, and yields perception-focuser supervision at scale without requiring human-labeled data.
data and training steps reach stable performance. To
Empirically, the translator achieves over 90% F1 on detecting points, lines, and constraints. Mean- while, GEODPO improves the primary perception score from 41.0 to 51.9 (a relative gain of +2).5%), with consistent improvements across out-of-domain settings: +8.0% on perception and +-990% on downstream reasoning. In contrast, SFT provides only marginal benefits and can even degrade performance under distribution shift.
Our contributions are threefold:
We introduce GROPERCHIVE, an automatically generated, cimpleexity-conrrolabb benchmark centered on GEODSL, a canonical and ambiguity- DSL that assigns a unique program to each diagram. This enables exact program-leve evaluation and supports scalable data pipeline for automatic training data generation, significantly reducing data-related costs.
We propose GEODPO, translator-guided preference learning framework in which an NL2DSL translator bridges natural language and GEODSL. The translator's soft scores are converted into reward signals for DPO-based alignment, addressing order order-permutatin permutation issues and distributional shift while keeping the model aligned with its NL pretraining distribution.
We conduct extensive experiments demonstrating strong and robust improvements over SFT across diverse settings: +26.5% in-domain perception, +8.0% OOD perception, and 4-39.0% downstream reasoning. Qualitative analyses further reveal substantially fewer geometric halluci- nations.
2 RELATED WORK
Formal Representation of Geometry and GPS Benchmarks. Recent progress in geometric perception has led to the development of various benchmarks and diagram representation frameworks. Krueger al. (202I) introduced the Geometry Model Building Language (GMBL) specify geometry problems and construction metadata; however, its sequential design introduces ambiguity by allowing multiple semantically equivalent representations. AlphaGeometry (Trinh et al. 2024
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Point
AB, AC Line(...)
A.. 6, roint
62 nerpendicularte. AG] at
Line [)
AD perpendiculartA, 503 at
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Linecc, R)
internection(() AC
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Constraint e. Parpend (Ala S0)
Constraint Perpendiccala
constratet
Figure 1: Pseudo-code of existing DSLs' common literals (Trinh et 2024 Krueger al. 2021 Lu et al. (021). The three DSL programs on the right are semantically equivalent, each describing the same diagram shown on the left. This one-to-many correspondence results in non-unique interpretations, thereby hindering rigorous and consistent evaluation.
proposed a formal language that facilitates symbolic reasoning, but its path-dependent formulation also results ambiguous descriptions, hindering precise evaluation. Inter-GPS (Lu et al.] 202T)
designed a predicate- sestte-nnd-paaaata parameter-basee representation to explicitly capture geometric relationships,
nevertheless, its verbose syntax and inconsistent granularity make stable model training difficult.
Other related works (Chen al., 2021] Zhang al.) 2023 Chen et al., [022] Lu al. (023) provide valuable datasets but largely rely on answer-aware metrics, lacking a unified and controllable framework for perception evaluation. Hence, the development of concise and unambiguous formal language, together with a standardized evaluation system, remains crucial for rigorous assessment of geometric perception.
Geometry Perception in Vision-Language Models. Multimodal GPS requires the joint processing of diagrams and textual descriptions (Cho et al., 2025] Zhang et al.) 2025). Current VLMs generally
fuse diagram features with textual prompts to perform end-to-end reasoning (Cho et al.
2025 Zhang
al.
2023; Chen al., 2021, Cao & Xiao 2022 Ning al., 2023] Chen et al. 2022 Liang et al.
2023
Lu et al. 2023). However, evaluation practices that rely solely final-answer accuracy often
conceal critical geometric perception errors. In particular, existing assessments cannot effectively disentangle errors arising from visual misinterpretation versus textual misunderstanding, thereby obscuring the underlying causes of multimodal hallucinations.
Reinforcement Learning in GPS. RL (Schulman et al. 2017 Rafailov al. 2024 Shao et al. 2024
has been extensively applied natural language tasks to better align model outputs with task-specific objectives. Recent research has further extended RL to mathematical reasoning, encompassing theorem proving (Kalisyyk et al. 2018), mathematical word problems (Wang et al. 2018 Lu et al. 2022), and algebraic computation (Chen al. 2018). Within the domain of geometry, GeoDRL (Peng et al. 2023) formulates problem solving as Markov Decision Process, generating complete solutions directly. In contrast, leverage RL to guide the translation from NL DSL, with a particular emphasis on enhancing geometric consistency in perception. Instead of adopting end-to-end problem-solving framework, our method employs RI to refine intermediate representations, thereby improving both the accuracy and reliability of the generated DSL.
3 THE CONSTRUCTION OF GEOPERCEIVE
In this section, we present GEOPERCEIVE, benchmark specifically designed to evaluate the geometric perception capabilities of VLMs. GEoPERCEIVE focuses exclusively on geometric perception, aiming to assess models' ability to accurately recognize fundamental geometric primitives such as points, lines, and circles. This focused evaluation is crucial for identifying and mitigating geometric hallucinations in VLMs. To overcome the limitations of aunaatandotated datasets, GEOPERCEIVE adopts an automated pipeline to generate large-scale and diverse training and testing data. The proposed GEOPERCEIVE consists of two key components built upon GeoDSL: (i) GeoDSL generation engine and (ii) a diagram solving engine. GeoDSL defines a Nomain-speccif language tailored for geometric perception. The generation engine samples a diverse set of DSL programs, while the solving engine renders each program into pixel-level diagrams. The following sections provide detailed description of these components.
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Point(
Treak LinetAA
Linef(e,
Innac Laneele, C.
Ineac Liner [a. K]>
Tnosa Lined(e. 6.
lseekl Liner k3)
Lineete.
ctre Cirelacs, [4])
Insdi Linellc, N. FJ)
Perzendicalare instn.
Tangent(iiieeec ctr
Illustrations of GEODSL syntax. A program may consist of up to four sections: points, lines, circles, and explicit constraints. Point-curve incidence relationships are automatically inferred within curve declarations.
Figure 2:
3.1 GEODSL: DOMAIN- SPECIFIC LANGUAGE FOR GEOMETRY PERCEPTION
Unlike existing formal languages such as Alpha Geometry (Trinh et al. 2024), Geo-model- (Krueger et al. 2021). and Inter-GPS (Lu et al. 2021). which susceptible semantic ambiguities (i.e., a single diagram can correspond to multiple DSL programs, as illustrated in Figure I and redundant expressions, propose GEODSL, concise domain- domain-specific language that establishes one-to-one correspondence between visual primitives in a diagram and their formal representations, thereby effectively eliminating these issues.
Design Principles. The design of GEODSL. follows two key principles: 1) Unambiguity. Each valid geometric relationship corresponds exactly one GEODSL. program, ensuring deterministic evaluation. More specifically. instead of constructive instructions (c.g., "connect AB and CD to form lines & and f2; and le intersect ET"), we use descriptive, relational statements (e.g., "points A, B, E lie on & and "points C, D, E lie on "z"). This avoids construction non-uniqueness and yields canonical, order-invariamt set of DSL literals. 2) Simplicity. The language adopts a minimal set of literals sufficient to describe standard Euclidean constructions, guaranteeing that program length grows linearly with the number of elements, thereby facilitating stable optimization during training.
Definitions. Following Lu et al. (2021), we define that: A primitive is a basic element in the geometric diagram, i.c., point, a line, or circle. 2) A predicate is a function that takes some parameters and optionally returns a value. 3) A literal an application of one predicate to sequence of arguments. GEODSL consists of four mutually independent literal categories: point, line, circle, and constraint declarations. Formally, GEODSL regards a geometry construction as a tuple G (P, f, C, R), in which P, represent set of Point, Line, Circle literals respectively, and R. represents a set of constraint literals. We call such tuple G a GEODSL program. Some examples shown in Figure 2 The details are listed in Appendix
3.2 BENCHMARK DEFINITIONS AND METRICS
Each instance in GEOPERCEIVE represented tuple (G, D). where G (P, C,C,R, denotes the GEODSL program, and D E x3xM* is the rendered diagram. A model M under evaluation
takes the diagram 1) as input and predicts GRODSL. program & = M(D) = (P, L, č, R)) The final score of perception is then calculated by the similarity between G and G which is denoted as Score(G, G).
To define Score(G, G) we independently compare the elements belonging each predi- cate category. Specifically, for each category, we consider the pair of multisets (X,X² (P.P), (c, 2), (C, (R, R)), where
(II...., 2m and X = ( f represent the multisets of ground-trutt and predicted primitives, respectively. Here, m = |X] and n | denote the number of primitives in each set. For each such pair, we construct a similarity matrix E R mxx where each entry 8ij s(z,, it)) E [0, represents the element matching score between primitive x; from X and ł. from X. The similarity function s(a, is defined recursively, detailed
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Viteratian
Grobeet
4..99
Data Engine
Main Generating dingrams solving engine
(5) Genenning
engine.
Genenting tminning translator.
(S,
J
(
X
VEM
Transaaorr
Caro
Finef
Gestos
(S.
Generaing poi lae preferences DPO training.
(c) Training the policy by algorithas.
Figure 3: Pipelines of GEOPERCEIVE and GEODPO.
Equation 1
[aabel(a) label(b)]
ifa, if
Fi (P(
s(a, b)
1/2 { ( + Fi (p(Q), if
(1)
Constraintsccree(a.))
i
Here, I(.) denotes the indicator function, returning if and only points a and share the same label, and 0 otherwise. It is assumed that points have unique labels within each GEODSL program (e.g., "A", "B"). For a line or circle x the function P(x) returns the set labeled points that lie on it. For circle, Center(() refers to its labeled center point. The F1-score between (e.g., P(a) and P (6)) is computed recursively by constructing a similarity matrix using label identity, applying the same assignment framework defined Equation T
point sets
and computing the Fl-score as in Equation 2 The function Constra rawaanntteeer Score b) defines similarity for constraint types, with formal definitions provided in Appendix
The optimal one-to-one matching between elements of and X is obtained by solving maximum- weight bipartite assignment problem, typically via the Hungarian algorithm (Kuhn 1955) or the Jonker- Volgenant algorithm (Crouse, 2016) (for m + n). Let < min(mn, n) be the number of matched pairs in this assignment, denoted by (Iiy (x(1))) The aggregated similarity for this
category computed as S(X, X) = Ei= s(yy) Based on the aggregated matching score, the per-category precision, recall, and F -scoe defined as:
n m==00 0, otherwise.
0
m =
if R-
P
R
F
(2)
g
S
otherwise.
[+++
+
otherwise.
To provide single scalar score for ranking systems, we compute weighted average of the Fl-scores:
Score(G, G) wyF (L, L) wc (C, FR(FRR R).
In our experiments, these weights are set equally: wp = Wh = wc = wR = 0.25.
3.3 GEODSL GENERATION ENGINE
To enable the creation of a large-scale dataset for GEOPERCEIVE, automated GEODSL generation engine serves as a central component. As illustrated in Figure B(),, this engine procedurally generates diverse GEODSL programs through two-stage process:
1. Primitive Initialization Stage: The process begins with the random selection of a fundamental geometric configuration. This may involve generating a basic set of primitives, such as a triangle, quadrilateral, or standalone circle, which defines the initial state the geometric scene.
2. Iterative Construction Stage: After initialization, the engine progressively expands the GEODSL program. At each step. randomly selects a geometric construction operation from predefined pool. These operations may include adding points, lines, or circles based
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their relationships with existing elements, introducing additional constraints. Through this iterative process, increasingly complex geometrie configurations are constructed.
complete list of the construction operations included in the pool provided in Appendix
3.4 DIAGRAM SOLVING ENGINE
Once a GEODSL program G (P,C,C, is generated, the next step is to render it as a visual diagram. To this end, we develop a diagram-solving engine that takes the GEODSL program input and produces the corresponding pixel-baud diagram D.
While prior work [Krueger et al. 2021) has introduced solving engines, they often rely different DSLs, which can cause ambiguities when interpreting our GEODSL. In addition, the complexity of their solving processes hampers the integration of new geometric predicates or primitive types, and reported technical issues in certain implementations (e.g., dependence on outdated TensorFlow versions) further restrict their applicability. To overcome these limitations, develop engine implemented in PyTorch. This design emphasizes usability, enables straightforward extension with new geometric elements and constraints, and offers a robust and accessible framework for diagram generation.
solving
As shown in Figure the core algorithm of our solving engine, described in detail in Appendix proceeds through the following key steps:
1. Primitive Parameterization:
The engine initializes variables corresponding to the geometric parameters of all primitives in P, L, and C. Specifically, each point P. P is parameterized by its coordinates (Ii, each line I.j E L is parameterized by coefficients (ag, bij, Cj) in the implicit form djæ: + byy Cj 0; and each eircle Ck E C is parameterized by its center (Tok, Bok) and radius Tk.
2. Constraint Formulation as Losses: Every explicit constraint Rg E R is converted into corresponding loss function. For instance, the constraint that point (1 yi) lies line Ljja by. ej) yields loss term proportional to (a,æ; + by3t + c,)². Implicit structural constraints, such as requiring that two points defining a line remain collinear with it, are similarly expressed as loss terms.
3. Visual Plausibility Penalties:
To promote clarity and visual coherence in the rendered diagrams, the objective function incorporates several penalty terms: density penalty to prevent excessive clustering of points, a distribution penalty to discourage unnatural scattering, and scale and boundary penalties ensure all elements remain within the canvas at reasonable size.
4. Iterative Optimization:
The engine combines all weighted loss and penalty terms into unified objective function. An iterative optimization algorithm is then applied to determine the primitive parameters that minimize this objective. The process terminates once the total loss falls below predefined threshold or the maximum number of iterations is reached.
the solver fails to converge to a satisfactory solution under the given constraints, the corresponding GEODSL instance is labeled as "unsolvable" and excluded from the dataset. Conversely, successfully solved instances yield parameter values that are subsequently used to render the final diagram.
4 GEODPO: TRANSLATOR-GUDE DIRECT PREFERENCE OPTIMIZATION
In this section, we present GEODPO, a translator-guided DPO approach to improve geometric perception. Instead of sequence-level SFT on GEODSL, which is sensitive to permutation-equivalen programs, GEODPO keeps NL outputs, uses an NL-to-DSL translator to score predictions, and turns these scores into preference signals for DPO training. Leveraging GEOPERCEIVE, we synthesize diverse training resources, enabling scalable alignment without human-labeled data.
4.1 NL.2DL TRANSLATOR
Training an NL2DSL translator requires a parallel corpus of natural language descriptions paired with their corresponding GEODSL. programs. Fortunately, while VLMS often struggle with direct

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Algorithm Preference Pair Generation for GEODPO
Require: Dataset D, Number of samples N Nampless Minimum score difference threshold 6min- pre-trained model fref
1: function CENERATERREEERNNGGPPRRR
N Nampless Omins Tref) > Initialize set of DPO training pairs b Walk through all pairs of diagram and ground-truth DSL
2: DDPO +
3: for each (D, Gerue) in D do
4: Let Samples f
5: for k to Naurpples do
6:
Sample Sk ~ Tree (r) (D)
> Sample NL description from the VLM Score using translator T and metric
7:
Sk < Scorcc (Grrue T(Sa))
8:
Add (Sk, 8k) to Samples
9:
end for
10:
Liorted f- DescendingSortel(SSSpppp such that 8, 2 8
idx +1
b Initial index for preferred (winner) sample
12:
iday <- [Namplees/? |2]
b Initial index for dispreferred (loser) sample
while idx S N samples/ /2] and idæ; N samplea
14:
(Sw, Sw) Lorted widxwl
15:
[t 5t) + Laorted [idzy]
16:
if Sw 81 Swin then
17:
Add (D, Si) to DDPO idr. idr
18:
Move the next potential winner D Move to the next potential loser for the new winner
19:
idra + ida; +
else
21:
idæ + idæs +
> Current loser not distinct enough, try next potential loser
end if
23:
end while
end for
25: return DDPO
26:
end function
NL2DSL translation, they tend be more adept the reverse task: generating NL descriptions from structured GEODSL representations. Building on this observation, we construct training data for the translator illustrated in Figure Specifically, we provide existing GEODSL programs as input to capable VLM and instruct it to generate descriptive NL paragraph, thereby producing the desired (NL. GEoDSL) training pairs.
Using the collected pairs, we tunettne& language model to serve as the NL2DSL translator T, which takes an NI. description S as input and outputs the corresponding GEODSL program G G== (SS). Leveraging the translator T, for model M, the complete workflow of generating GEODSL program from a diagram D can be expressed as G = T(M(D)) The resulting formalized G can then be seamlessly integrated into evaluation and reward signal computation.
4.2 SONNSLATOR-GUDDRR REINFORCEMENT LEARNING
The key insight is that our translator T, when combined with the GEOPERCEIVE scoring metric (Equation 33, can serve as an effective reward model. This composite system enables us to evaluate the quality of model-generated NL description S for given diagram D by translating S into â = T(S) and subsequently scoring Ô against the ground-truth GEODSL program Gtrae-
Given a diagram D with its ground-trutt GEODSL program Gtrue we prompt the VLM to generate diverse set of N Naampies natural language descriptions (S,, SNaamptez Each generated description S, is then evaluated using our reward function, yielding score s; = Score (CoreeGr T T(S)). Based on these scores, we construct preference pairs St). where S., denotes a preferred (winner) description and 5, dispreferred (loser) description. The procedure for generating these preference pairs illustrated in Figure d) and detailed Algorithm
Using the generated preference pairs (S... Si) for each diagram D. fine-tune the VLM with the DPO loss, as illustrated in Figure e... DPO trains a policy Ta (our VLM) to align with the specified preferences while simultaneously regularizing its deviation from reference policy gyef (typically
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the initial state of the VLM prior to DPO fine-tuning). The DPO objective for our task is:
log a [log To(Sw]D) Blg To ( D) Tre (Si )
( Fref) = -..DD
(4)
Trer(Su
This loss function drives the model Te to assign higher likelihoods to preferred responses S. and lower likelihoods to dispreferred responses Si, relative to the reference model Tref- In doing so, enhances the model's s ability to generate high-quality, geometrically accurate natural language descriptions.
5 EXPERIMENTS
5.1 DATA PREPARATION
The dataset used our experiments comprises two complementary subsets serving distinct purposes: the translator split and the main split. Each split contains 10,000 GEODSL samples. The translator split pairs NL descriptions with their corresponding GEODSL programs, facilitating the training and evaluation of the NL2DSL Translator. In contrast, the main split pairs GEODSL programs with rendered diagrams, enabling the assessment of visual grounding and geometric perception. Detailed statistics of the datasets used in our experiments are provided in Appendix D
5.2 NL2GEODSL TRANSLATOR
Experimental setup. The translators in our experiments are initialized from Qwen2.5-7D (Qwen al. 2024) and trained on the Translator split of GEOPERCEIVE. We employ LoRA (Hu al. 2021) adapters (rank 4) for parameter-efficient fine-tuning. Training is performed for 3 epochs with a batch size of 32, an initial learning rate of 10-1, and a cosine scheduler with linear warm-up over the first 10% of updates. All experiments are conducted on 4 H800 GPUs.
Results. Table reports the F₁ scores and validity across generation iterations 1 to 5, together with the averaged performance. The translator consistently produces syntactically valid programs and achieves an average F, above 95% for points and lines. As geometric complexity increases (iterations 4 ~ 5), performance degrades gracefully, primarily due to the greater difficulty of recovering circular primitives and compound constraints.
Table 2: Element-wise translation accuracy the GEOPERCEIVE Translator-test split. Validity denotes syntactic correctness, while P. R, and F2 correspond to precision, recall, and F, respectively.
Ite
Validity
Overall Score
Points
Lines
Circles
Constraints
F1
P
R
F1
R
F1
100.0
94.2
97.9
97.9
97.8
98.0
98.0
98.0
84.7
85.8
85,1
95.7
96.3
95.9
2
90.8
97.9
97.7
97.4
97.4
97.4
74.9
74.6
94.
93.4
93.7
3
100.0
87.4
95.2
94.2
94.7
93.4
93.4
93.3
71.9
71.7
71.7
90.8
89.4
89.8
87.0
90.2
5
100.0
83.2
94.0
92.9
93.4
92.3
90.7
91.4
67.6
65.8
65.9
84.9
30.7
82.1
Overall
100.0
89.2
96.4
95.9
96.1
95.1
95.0
95.2
75.0
74.8
74.6
91.7
90.4
90.8
5.3
GEODPO
5.3.
MAIN EXPERIMENTS
Experimental setup. We conduct experiments on Qwen2.5-Vi VL (Bai et al. 2025). Intern (Zuu al. 2025). and LLaVA-Next (Li et al. 2024). For each base model, we consider three settings: (i) the released model without additional supervision, (ii) a model finetuned directly on generated GEODSI., and (iii) a model optimized using our GEODPO method. In GEODPO, the original model is regarded the reference policy Tet with sampling parameters set to Nammpls 10 and 8min = 0.3, and preference pairs drawn from the Main-trait split. All finetuning is performed with epoch with batch size of 32, learning rate of 2 x 10 and a cosine scheduler with linear warm-up over the first 10% of updates. Unless otherwise specified, training is conducted on 4 H800 GPUs.
LoRA adapters (Hu et al. (021) of rank 8, for
Overall Results. As shown in Table 3 GEODPO consistently outperforms both the raw models and the SFT ones across three model families. In particular, the overall score improvements range from
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Table Scores on the GEOPERCEIVE Main-test split.
Model
Method
Overall Sovee
Points
Circles
Constraints
F1
75.19
64.15
$3.95
44.52
54.
54:18
Qwen1.5.VV. (7R)
w/ SET
64,02
83.81
82.18
X2.31
(6.30)
64.49
55.74
58,02
58,07
(928 (4
57.61
57.56
58.05
09.000
58.29
w/ GeaDPO
(411790
(900199)
Raw
58.44
82.07
76.09
78.59
60.52
53,47
54,61
43.70
48.02
48.16
52.45
52.37
52 20
InteraVIL3 (SB)
SFT
1402 1701
84
1x12 6/8
1-7,099
(425 3355175
16.251
w/ GosPPO
67.41
92.15
84.11
84
70.57
56.24
50.
50
58.5
59.1
62.5
62.5
61.5
Row
41,01
39.65
46.05
50.40
46.45
31,94
36.21
35.42
36,01
35.34
42.68
41.32
42.02
LAaVN-Ne (78)
w/ SFT
51.1
72:25
69.84
04
50.07
50.5
48. 31
44
44.85
44.5 19
41.31
42.31
41 96
w/ GeaDPO
(178 a/R)
51 86
74.35
(400 AM9
1-24 9MN
1442 ANR
2400 19N0
CLN 1709
1-0
(4949)
+ +14.2% to +26.46% relative to the raw models. These results confirm that GEODPO substantially enhances geometric grounding and overall model performance, independent of the model family.
Element-wise Performance. Across all perception types (Points, Lines, Circles, Constraints), GEODPO consistently improves upon the raw models. A notable trend emerges in the Constraints category: while SFT models exhibit slight performance degradation for models such as InternVI. and LLaVA-Next, GEODPO effectively alleviates this issue, achieving significant gains. For example, constraint performance increases by +9.9% to +19.27% across models with GEODPO, in contrast to the decline observed in SFT models. This indicates that GEODPO not only strengthens fundamental geometric features (e.g., Points and Lines) but also stabilizes the more fragile aspects of constraint reasoning, which tend to be negatively affected by SFT.
5.3.2 OUT-OF-DISTRIBUTION EVALUATION
Setup. To assess robustness under distribution shifts, we evaluate (i) perception using a hand- cand-crafted GFOPERCEIVE-OOD set consisting of 100 diagrams curated by 10 postgraduate annotators from public sources] with the same evaluation metric as in our main experiments, and (ii) reasoning on a 203-item Math Vista subset labeled "geometry diagram" and "multi-choice."
Table 4: Scores on the GROPERCHIVE-OOD.
Model
Overall Score
Points
Lines
Circles
Construints
P
R
F1
P
R
F1
P
R
F1
P
R
F1
Raw
58.14
50:42
75.11
77.67
67.53
49.74
57.29
43.82
44.63
44.22
57.26
49.95
53.35
OwenZ.5VVL (7B)
SET
58:41
50.63
74,15
77.25
67.53
50.37
57.7
46.17
45 65
45.92
56.5
49.52
52.78
GeoDPO
60
69
14.30%
81,89
79,96
04.199
040991
68.19
59.54
(41305)
44,42
(42.99)
45.0 69
45.05
506
02.891
69.49
(4295%)
$1,44
(4425)
55,0
Raw
58.74
76.05
76,90
76.48
63.92
$1.01
55.76
51.89
50.48
51.08
49.21
52.14
50.64
Intern VL3 (MB)
58.57
76.41
37.73
77.06
51.43
55.88
49.84
48.22
$9.02
49.97
$1.37
(0U5%)
(42265)
w/ GeoDPO
69.91
70.37
70.36
79.37
67.53
(97590
$2.33
58.97
$2.93
$2.17
$2.54
50.71
(4165%)
54.95
$2.76
42-48
96
46
33.12
37,03
35 65
36.33
96
39.21
40.07
LLaVA-Next (7B)
w/ SFT
(4225)
63.5
(41).00)
54,67
45.5
06. 99.
642
(3.01)
(4065)
37.43
(4.479)
36.5
(285)
(605%)
39
40,07
30P33
45
68.35
$4.09
60.39
52.5 37
37.27
43.55
39.79
36.43
38.04
42.2
40.87
41.55
Results. On GEOPERCHIVE-OOD (Table SFT proves fragile, often leading to marginal improvements or even regressions, whereas GEODPO consistently enhances overall OOD scores across all backbones, accompanied by concurrent category-wise gains. On Vista (Ta- 5). GEODPO yields substantial downstream improve- ments over both raw and SFT models (e.g., Intern VL3:
Table 5: Geometry subset of Math Vista.
Model
Method
Qwen2.5-VL
InteraVL3
VLaVA-Nest
Raw
30.05
29:06
26.11
w/ SFT
32.02
30.05
29.05
GeaDPO
40.39
33.50
29.06 00.06+40.0 40.399 corresponding to a gain of +39.0%), while SET offers only limited benefits.
Overall, our experiments demonstrate that trajectory-level RL reward optimization (GEODPO) achieves stronger performance and generalization than token-level logllicei SFT, with this advantage becoming particularly pronounced under distribution shifts.
Lu al. 2021
Lu al. 2023), and htpp://www.mo--offccala.gg

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Published as a conference paper at ICLR 2026
6 CONCLUSION
We addressed geometric perception in VLMs through two complementary directions. For measure- ment, we introduced GEOPERCHIVE, built upon GEODSL, canonical and ambiguity-free DSL that assigns each diagram unique program. This enables exact program-level evaluation and supports an automatically generated, complexity-controla cost. For improvement, we proposed GEODPO, a translator-guided preference learning framework in which an NL2DSL translator bridges natural language and GEODSL. Its soft scores are trans- formed into reward signals for DPO-based alignment, thereby mitigating order-permutation issues and distribution shifts while keeping the model close to its NL pretraining manifold. Experiments demonstrate strong and robust gains over SFT: in-domain perception +26.5% OOD perception +8.0% and downstream reasoning +39.0%, along with substantially fewer geometric hallucinations.
data pipeline, yielding scalable training data at low
ACKNOWLEDGMENT
This work supported by the National Key R&D Program of China (202YFBB-00144004700000)) SSTIC Grant (K7/D220230221115100001, KI/ID202309231 14916032, GJHZ222402181 01300224002211111000900
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A DETAILS OF GEODSL
A.1 DETAILS OF PREDICATES
Table lists the element predicates, each of which returns primitive geometric object. Table enumerates the constraint predicates together with their parameters and semantics.
Table 6: Predicates that return geometric primitives in GeoDSI..
Predicate
Parameters
Example
Point
label: str mull
Point ("A"), Point (null)
Line
through:
list<Point>
Line ([Point ("A"), Point ("B") )
center: Point null
Circle
Circle (Point ("O"), [Point ("A")
through: list<Point>
Table 7: Predicates that return constraint instances in GeoDSL.
Predicate
Operator
Purameters
Description
ConstreintParrallel
licel:
Constrain linel to be purallel to line2.
1ine2:
Sine
Const Perpeenn
1ine2: Jine
linel: sine
Constrain hiw to be perpendicular to neee.
Tee
circle: Circle
tongent ciiclee
Toc
circlel: Dircle
circle2; Bire'e
Constrain circlel to tangent to circle..
ConscraintEgual
dist
diat2:
TuglecPeint, Doint>
Tup" expleeheiing
Point>
equal length.
A.2 IMPLEMENTATION OF CONSTRAINT SCORE
The scoring function Constra wainttccore introduced in Eq. equation is implemented differently each constraint type as follows.
(C
FFI (F1 (1,
Fl (1(1),))), (1, (6,6)).
s(Tuc(l.c),
= [s(),) s(c,
(5)
(2) (c((t,
= F1 (2) (0).))))1..)))
d2)) d2))
2 max [i +1 F1(2,, Pi(d1, (((d),d)+FIi Fi(d1,
B DETAILS OF GENERATION ENGINE
B. PRIMITIVE INITIALIZATION STAGE
this stage, we bootstrap the seene with single, randomly-chosen geometric primitive sampled from a fixed probability distribution. A uniform random number r~~2((0, 1) decides whether we insert (i) triangle (50%), (ii) quadrilateral (30%), (iii) circle (20%). If circle is selected, optionally place random center (70% chance) before creating the circle object. The outcome is exactly fully-specified primitive, together with constituent points, lines, or ares, that serves as seed subsequent construction stages. The following pscudo code displays the whole process.
Uniform (0,
sandom value
if
create triangle probility
A, 5, C NowRandonnooitta (3)
bine ((A, BJ1
Line (IB, C11
CA Line ([2, All
Addshapa (A,
CA)
elseif
create a quadrilateral probility A, C, D NewRandomPoincsil
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AB Line ([A, B]1
Line ([B,
CD Line (IC, D11
Line ([D, A]1
Addshape (A,
Cb)
oreate cicle with 0.20 probility
Uni (0, ş
center NewRandonPoing AddShape (centez)
clae
center null
circle Circle (center) AddShape (circle)
end
B.2
ITERATIVE CONSTRUCTION STAGE
B.2.1
OVERVIEW
After the seed primitive is in place, the generator enters open-ended loop that augments the scene step-hy-step until the requested number of extra steps reached further operation is feasible. Every iteration follows the pipeline below:
1. Enumerate admissible operations. Each concrete subclass of Const ruct i on Processor exposes prepare () method that checks whether its pre-conditions can be satisfied the current scene (c.g., triangle must already exist before orthocentre before sampling two fresh points, etc.). Only the processors whose prepare returns True are kept as candidates.
drawn, least free point labels must remain
Randomly pick candidate. From the candidate set single processor is drawn uniformly; this adds controlled stochasticity while guaranteeing geometric validity.
3. Apply the operation. The chosen processor's 's app () method instantiates the necessary primitives (points, lines, circles), appends them the glohal construction, and registers any incidence perpendicu- larity constraints that logically follow. also returms structured CenstruutionDesrrreee later becomes part of the natural-language instruction sequence.
that
Record and continue. The formatted description appended to the running script; the loop then proceeds the next iteration.
Pseudocode overview. The essence of the loop is captured below; it mirrors the logic inside generate and generate_ster of the accompanying implementation.
for extra_steps do
operationa
op.prepare True
candi dates
1f C is empty then
move
end if
Uniformcheice op.apply()
pick operation
t apply the selected operation
for
B.2.2 AVAILABLE OPERATIONS
Construct Orthocentre. Given an existing triangle, draw its three altitudes and mark their concurrence point H, the orthocentre. Perpendicularity constraints between each altitude and its opposite side are added to the scene.
(A, e, C, BC, A2) Choosssenccmirrianng....
VawRandonPoint. with prob. 6.5 else. AnonymouaPoin::))
Al Line (A, H)
Bil Line (B, ())
CH Line (O, R)
AddShape
adéconstraint (Perpendicular, AE, BC)
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AddConsLraint |Pexpendicular, BE, AC) AddConstraint (eexpendicular, AB)
Construct Circumcentre, a chosen triangle, create circumeircle through the three vertices and. optionally, generate the centre O. additional constraints are required because the circle already passes through all vertices by construction.
(A, B, C, BC, AC, AB) ChoosssanncomTriinnng.llll NewRandomPoiat () with prob, 0.5 else null cire circle (0, AddShape (0, ciral
c))
Construct Incentre. Insert the incircle of triangle, enforce tangency to all three sides, and optionally expose the centre / as well as up three touch points where the circle meets the sides.
(nc. Cir (A, choosesenncmmiiinnngelll AddShape II, inc)
8, BC,
NawkandomPoint with prob, 0.5 clse. null
AddConstzaint flangent,
AddConatraint [langent, inc, BC)
AdeConat ITangent, ine, AC)
for each side in [AB, s2, do
with prob, 0.4:
Intersection end AddShape
Construct Segment. Connect two existing points that not already joined by segment lying the same line; this is the simplest way incrementally densify the point set.
(A, R) TwoRxi at nngood at!! thoutSegment ()
seg + Line (8, AddShape (seg)
Construct Two Points and Connect. Sample one new point on each of two (possibly identical) curves line or circle-and and link them with a segment, thereby coupling previously independent objects.
tourvet, curvez) Ne B NewPointOn (ouzve21
ChooseTwoCurves. () lines and circles are both regarded ourvea.
AB Line (A, 2) Addshape (A, 5, AB)
Construct Point and Connect Existing One. Place a fresh point on a chosen line or circle that does not yet contain the selected existing point, then connect the two points to form new segment.
curve) ChoosePointandCurvee NeeweiintOn (onnnttnnnnuueee (curvel AP teiee P) AddShape (2, AP)
C DETAILS OF SOLVING ENGINE
engine formulates every geometric construction as a continuous, constrained ootrrinndd-opimizzation problem. primitives are endowed with trainable real-valued parameters, and collection of differentiable loss terms is minimised by gradient descent. The global objective is
L =
Wk. Ca +
>Σ Um
Pm
>> 0,
(6)
REC
mEP
where C and P index the hard geometric constraints and the soft regularisation penalties, respectively, while and Um user-specified weights.
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C.1 PARAMETERISATION OF PRIMITIVES
Points P (z, y) R2 are free variables.
Lines (a, b, c) obey the implicit equation a.z + by + c with an appreximate normalisation a² + b² x enforced softly.
Circles (Cx, r. consist of centre C = (Cr, (yy and radius r >> 0.
C.2 INTERNAL CONSISTENCY LOSSES
These losses ensure that each primitive remains consistent with the incidence information stored inside it.
Point-Line Incidence. If point I' is declared to lie on line e, the squared (algebraic) distance
Lixe (P, E) (a - byp +)*
(7)
is accumulated.
Line Normalisation. To avoid degenerate parameterisations we penalise Luorm (2) = (v d² + 1)² Point-Cirele Incidence. For a circle y and one of its defining points P. Coneintte (P. 7) = (I) - ClI - r)² Fixed Centre (optional). If circle required to coincide with a reference centre Co.
(8)
(9)
Leentre (7) ||C Coll?.
(10)
C.3 EXTERNAL GEOMETRIC CONSTRAINTS
Equal Segment Lengths. For two segments P. P2 and P3P4.
C- Pe, Ps, Pa) (IP P2ll - 1|P Pal))
(11)
Perpendicular Lines. Given & (a1, bi, ei) and (a2, b2, 02).
(1(11,8) = (a102 + bybs)²
(12)
Parallel Lines.
C1(42, f2) (a1b2 02b1) )² Line-Cirele Tangency, For a line = (a, b, c) and circle = (C,r),
(13)
Lia(e. ccrr + bey +c
(14)
V + 6z
Prescribed Length. Enforcing || - Pall Li
Lia (P1, L); = PI Pall - 2)2
(15)
Prescribed Angle. Let LPVV be required to equal 0 (in radians):
(Pr V). (B2 - V)
P: P2 P2 P2-1| VII -
L2(V, Pi, P2:0) = aarcos
(16)
C.4 SOFT REGULARISATION PENALTIES
Point Density. For every unordered pair of points (P,, P,) with distance d,f = ||,, P,]l and a threshold т,
di <T.
Paeaa Pr:7)
d
(17)
0. de
T.
Global Spread. Let t4 & EN, P. be the centroid of points. tolerance radius p yields
Papeea(p)
Σ
max PpP all- =p)²
(18)
17

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Published as a conference paper at ICLR 2026
C.5 OPTIMISATION PROCEDURE
All free parameters are updated by Adam with step-decayed learning rate. Gradients are back propagated through the scalar objective equation Optionally, the solver halts early when every individual constraint term Le becomes smaller than user-defined threshold, guaranteeing that hard geometric relations are satisfied to the desired precision.
C.6 FAILURE SAMPLES OF SOLVING ENGINE
This section presents examples of failure cases encountered by our solving engine. As illustrated in Figure 4 these failures typically arise from issues such as high complexity unsolvability the given problem.
W
Figure 4: Failure Cases of Solving Engine.
The GEoDSLs corresponding to the example images are as follows.
point (label=fu*)
point (label=*ø")
point (label=*g*)
(label=*0")
(throoghho [2,
line_ 1ine (through- [N, B])
line_
(through:= [E, d, 0]1
circle_1 circle (Uhrough- H, E, 01)
circle_1)
tanjeea__inn_cccccelliiee__ circle_l)
Langent_line_circhle peint (1abel=""))
point
V point (1abeler)))))
point (1abel-*3))
T point (label-TT)
point (label-to*)
unlabeied__poinee point(
po (labal=***)
line 1
line (through-1U [U, 2, T]1 Ithrough-uh
line_ 3
(throughh [V, U, D]:
(through- D,
line_ 5
line Ithrougghg 0])
line_ 6
1ine (throgghhh-- untabeled_point_]
line_7
1ine/throuhh-12 [2, unlabelec_point_]]
line_ 8
line (throughhh uniabeled_point_]
[Lhroughhh unlabeled_poont__.)
circle_l (center=k, through=[U, 2,
secpendioulac line_2]
perpendicular (line_ line_3)
verpendiculas (ulineee.. line_1]
the right case
point (label=*Q")
18

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Published as a conference paper at ICLR 2026
8 point (label-"0*)
point (1abel-"1*)
unlabe_ed_ setee_dd_ooot____1_1 point point (label-*G")
line_1 1
line (G, K]]
line_ Ithrough--
line_ 3
line (through--h 21)
(through- [2, uniabeled_point__]
1ine_5 line [through- unlabeled__oont_]]]
1ine_6 1ine (through- uniabeles_point__]
line_7 line (through=
line_8 8 11ne Hine (through- G]1
1ine_9
G
line_10
line (through-(X, G])
circle_1 circle (throughh (Q, unlebeleed_pcit_.l))
circle_2 circle [Q, W, x1)
perpendiicllar|lnnee.. line_21
perpendicular (1ine_5, line_3)
clapendiclarrliine. liae_1)
corpandicular line_5)
pespendicular (line_3, line_4)
perpendicular line_1)
D DATASET STATISTICS IN MAIN EXPERIMENTS
As described in Section 5 the dataset used in experiments consists of two complementary subsets with distinet purposes: the translator split and main split. The statistics are shown Table ensure Translator receives a balanced and learnable distribution, the translator split is sampled using dleration-depender weights that mitigate the long-tail distribution of more complex programs. For the main split, we aim for approximately 2,000 examples per iteration stage. However, the final rendered count decreases at higher complexity levels, fewer advanced programs can be successfully solved and rendered.
Table Subsets of GEOPERCEIVE in our experiments. "Solving SR" (%) denotes the proportion of GEODSL programs that can be successfully rendered into valid diagrams.
Generation Iterations
Translator Split
Main Split
2
4
5
3
4
5
Train
3920
2548
1960
882
490
1524
1485
1333
1203
1034
Test
80
52
18
47
44
38
33
# Total
4000
2600
2000
900
500
1571
1523
1377
1241
1067
Solving SR (%)
78.5
76.2
68.8
62.0
53.4
It is important which is different to other henchmarks as shown in Table The statisties reported in this section reflect only the portion of data used in our experiments, not the upper limit of what GEOPERCEIVE can produce. Representative samples from the main split are shown in Figure
that synthesis pipeline supports unlimited data generation with controllable complexity,
Table 9: Comparison with existing benchmarks. Unlike reasoning-focuseee datasets with limited samples, GeoPerceive focuses on perception with infinite automated data generation.
Benchmark
Domain
Evaluation Gramularity
Automation
#Samples
GKK (Lu et 2021 202
Reasoning
Binary Answers
X
3,002
UniGieo Chen el al.
Reasoning
Binary Answers
x
9,543
MathVista (LU et al
GeoQA CChen 2025
Reasoning
Binary Answers
Reasoning
Binary Answers
X
4,998
GEOPERCEIVE (Ours)
Perception
Primitives Constraints
E MORE EXPERIMENTS
E. EXPLORATION OF ALTERNATIVE RL ALGORITHMS
It is important to note that DPO is not competing approach, but rather implementation choice within our proposed RI. framework. We selected DPO for its simplicity and direct applicability the problem at hand. However, framework allows for the of alternative RI. algorithms.
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Published as a conference paper at ICLR 2026
1 iteration
iterations
3 iterations
iterations
iterations
Figure 5: Randomly selected GEOPERCEIVE diagrams. For each generation iteration, sampled
figures are displayed to illustrate the progressive increase in geometric complexity.
explore this, we conducted additional experiments using PPO, which involves training both value model and a policy model. While we found that PPO tends to be slower due to the additional value model. performs on par with DPO in our framework (as shown in Table TOp. This highlights the flexihility of our framework in supporting different RL implementations.
Table 10: Performance of different methods across various metrics.
Points
Lines
Circles
Constraints
Method
Overall
P
R
F1
P
R
F1
P
R
F1
P
R
F1
Pretrained
$7.96
84.58
75.19
79.07
64.15
49.73
$3.95
45.16
44.82
44.63
54.30
54.12
54.18
SFT
64.02
83.81
82.18
82.21
64.49
55.74
58.02
58.07
57.61
57.56
58.05
58.76
58.29
w/ GcoDPO
66.19
93.97
85.96
89.26
68.82
60.65
62.30
49.38
48.54
48.70
64.50
64.50
64.50
w/ GeoPPO
65.54
91.13
86.18
87.89
69.05
59.81
61.58
56.51
55.70
55.66
57.12
56.98
57.04
E.2 ABLATION STUDY ON TRANSLATOR QUILITY
To isolate the effect of the NL2DSL translator from the DPO reinforcement Icarning signal, conduct an ablation study which train weaker translator, using less data, and thus achieving lower performance. As shown Table and Table [2] the performance of the GeoDPO framework significantly degrades when quality of the translator is compromised. This confirms that the quality of the translator directly influences the effectiveness of the reward signal in our method.
Table 11: Performance of Translator Trained Different Amounts of Data.
Points
Lines
Circles
Constraints
Training Samples
Validity
Overall
R
R
R
R
practice)
100.0
89.2
96.4
74.8
90.4
7500
99.5
89.1
94.9
94.7
94.8
94.3
94.2
94.2
76.7
75.9
76.2
91.9
91.0
91.3
5000
97.5
82.3
94,5
86.8
93.7
91.7
91.6
90.1
73.9
73.1
60.4
71.6
70.3
85.0
2500
95.5
73.8
80.4
83.2
84.6
65.6
56.7
$7.3
60.3
E.3 QUALITATIVE ANALYSIS OF PERCEPTION PERFORMANCE
To qualitatively illustrate the performance gap between the pretrained baseline and our GEODPO, we present representative case study. As shown in Figure a|dd the corresponding prediction logs shown in the following boxes, the pretrained model fails to correctly perceive nearly all geometric relations, resulting in severe halluci- nations (marked red). In contrast, the model trained with GEODPO accurately grounds the majority of visual elements, including the incidence of points on the cireunference and the arrangement of chords, One minor error persists: the triangle EFG is misidentified, likely due to the close proximity of point G to the chord FY, which creates visual ambiguity.
20

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Published as a conference paper at ICLR 2026
Table 12: Performance of GeoDPO with Rewards from Translators of Varying Quality,
Method
Training Samples Translator
Overall
Points
Lines
Circles
Constraints
R
F1
P
R
F1
R
F1
P
R
F1
Pretrained
GooDPO
9800 (ouns)
66.19
95.97
85.96
89.26
68.82
60.65
62.30
49.38
48.54
48,70
64.50
64.30
64.50
w/ GeoDPO
7500
66.27
93.08
84.48
86.61
62.39
60.01
61.35
58.71
56.71
57.67
60.15
58.94
39.45
GeoDPO
5000
65.48
88.70
82.01
84.53
63.12
59.42
61.04
56.98
56.17
56.29
60.08
60.08
60.08
82.09
56.87
58.73
55.99
Figure 6: Example Diagram for Perception Performance Difference.
Ferception of the Diagram before Training
The diagram depicts cyclic quarillateral EFGY Insoribed in circle. The const precess involves first drawing circle then marking points G, circumference such that the segments EG, GF, FY, and YE are chords of the circle. the formed by connacting points sequence.
quadrilateral
Ferception of the Diagram aftez Training
constructed drawing circle, Foints T, and Y are then marked the circumforance of this circls. The red linas connect these points, forming a triangle EFG withia the circle.
given dgaamam appears
addition, there extra chord
F LIMITATIONS
This study concentrates qualitative, diagram-level relationships in geometric problem solving. Purcly algehraic constraints receive far less attention. particular, do yet handle specifications such as forcing one segment to an integer multiple another. fixing an angle to a prescribed measure, constraining area of a polygon (e.g., triangle) to given value. Extending our framework to integrate such quantitative requirements remains important direction for future work.
G LLM USAGE STATEMENT
Large language models (LLMs) used solely for improving grammar and concision. The LLM do not contribute to the conception of ideas, methodological design, experiment execution, analysis interpretation of results, or formulation conclusions, All scientific contributions, insights, and conclusions are made by us.
21


Reviews:
[{"review_id": "AuL8hT7wDR", "reviewer": "Reviewer_MwSA", "summary": "This paper presents GEOPERCEIVE, a new benchmark and synthetic data pipeline for evaluating and training geometric perception in vision-language models (VLMs) using diagrams paired with a canonical domain-specific language (GeoDSL). To improve VLMs' geometric grounding, the authors propose GEODPO, a translator-guided reinforcement learning (RL) framework that uses an NL-to-DSL translator to provide fine-grained reward signals for DPO-based preference optimization. Extensive experiments on both in-domain and out-of-distribution setups show that GEODPO yields stronger and more robust improvements over supervised fine-tuning (SFT), especially in geometric perception and downstream reasoning tasks", "questions": "please see the weaknesses", "limitations": "", "rating": 2, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Benchmark and Synthetic Data Pipeline: The paper introduces GEOPERCEIVE, which systematically generates and renders diagrams with unambiguous geometric DSLs. The pipeline creates diverse complexity-controlled data at scale, providing a solid resource for both evaluation and training.\n\n2. Meaningful Visualizations: Figures such as Figure 1 (exposing ambiguity in existing DSLs), Figure 2 (illustrating GeoDSL syntax), Figure 3 (full pipeline including RL steps), and Figure 4 (showing data complexity across iterations), are thoughtfully constructed. Figure 3, in particular, makes the RL and preference generation workflow concrete, aiding reproducibility and interpretability", "weaknesses": "1. Lack of Rigor in the Task: The geometric perception task is not merely a simple perceptual task. Specifically, the geometric information obtained through basic image perception should undergo rigorous verification (at least ensuring that the perceived constraints are consistent and non-conflicting). For example, certain relationships in the image, such as perpendicularity or numerical data, must be strictly validated against predefined conditions.\n\n2. Insufficient Explanation of Geometric Relations: In the defined quadruplet (comprising points, lines, and circles), can these elements sufficiently cover all basic geometric entities? Additionally, what specific constraints are included? Further clarification is needed to demonstrate the rationality and completeness of the definition.\n\n3. Regarding OOD Data: Further clarification is needed on the distinction between Out-of-Distribution (OOD) data and the benchmark data. Specifically, why is the manually annotated data considered out-of-distribution? Is this due to differences in drawing style, or does it pertain more to the complexity and nature of the geometric relationships?\n\n4. Limited Analysis of Failure Cases and Limitations: The main text barely addresses where GEODPO (or GeoDSL itself) struggles. Section E (Limitations) is confined to algebraic constraints, but a frank, quantitative error analysis (e.g., visualizations of failed cases, confusion matrices on specific primitives or constraints, qualitative outlier examples) is absent. For instance, Figure 4 shows only successful, rendered samples; missing are diagrams or programs leading to failure, which could inform limitations of the framework or concrete bottlenecks in the translation pipeline or RL optimization.\n\n5. Ablation and Baseline Completeness: The experimental section (Table 3, Table 4) lacks comparison to several baseline RL/population-based approaches used in the field, including those explicitly referenced in missing related works. There is also a missed opportunity for ablation—e.g., separating out the effect of NL2DSL translator quality from the DPO RL signal, or evaluating against other forms of preference optimization.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gheS0BJEkD", "reviewer": "Reviewer_cHmt", "summary": "1. This paper proposes GEODSL to uniquely describe each specific diagram, and builds GEOPERCEIVE based on it, combining a generation engine and a solving engine to randomly generate DSL-diagram pairs. It provides low-cost and high-quality data for training and evaluating geometric perception models;\n2. They trains a NL-to-DSL translator based on GEOPERCEIVE data, and combines it with the scoring metric to form an effective reward model. It solves the difficulty of scoring caused by the equivalence of order-permutation or shift of natural language description, and makes the training for VLM remain in the original NL field;\n3. Experiments on Qwen2.5-VL, InternVL3, and LLaVA Next show that, compared with raw or SFT models, DPO training based on GEOPERCEIVE and the translator can effectively improve the perception ability, as well as achieve improvement in downstream reasoning tasks.", "questions": "1. How do you view the lag of DPO training compared to SFT in some specific indicators on the main-test split? (For example, the F1 score of circles perception, which serves as the aspect with the lowest score among all raw models)\n2. Does the GEODPO perform better than other methods, such as [Slow Perception](https://arxiv.org/abs/2412.20631) and [EAGLE](https://arxiv.org/abs/2408.11397)?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. This paper provides a rigorous and clear exposition of its methodology and experiment;\n2. The proposed GEOPERCEIVE pipeline provides an efficient automation scheme for the generation of geometric perception related data;\n3. It emphasizes the distinction between perception and reasoning, realizes the transformation from ambiguous natural language to strict DSL through NL-to-DSL translator, and provides a standard reward model for geometric perception results.", "weaknesses": "1. Lack of comparison with related methods in the experiment (e.g. [Slow Perception](https://arxiv.org/abs/2412.20631), [EAGLE](https://arxiv.org/abs/2408.11397));\n2. Partial experimental results show no significant improvement compared to the SFT method.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Eu0SX55FgB", "reviewer": "Reviewer_3WG9", "summary": "This paper presents two key components, GeoPerceive and GeoDPO, aimed at enhancing geometric perception in vision-language models. Specifically, the authors propose:\n1. GeoPerceive, a novel benchmark that explicitly disentangles geometric perception from reasoning. It employs a canonical domain-specific language (GeoDSL) to represent diagrams in a precise and unambiguous manner.\n2. GeoDPO, a translator-guided reinforcement learning framework that improves perception alignment by leveraging natural-language-to-DSL translation as a fine-grained reward signal, while maintaining consistency with the model’s natural-language pretraining distribution.\n\nThe authors conduct a relatively comprehensive set of experiments to validate the effectiveness of their proposed methods.", "questions": "1. Do the authors anticipate that GeoDPO could generalize to 3D geometric perception or other structured visual domains such as charts, circuit diagrams, or mechanical drawings? A discussion on this potential extension would strengthen the paper’s generalizability claims and highlight broader applicability.\n2. Did the authors attempt to apply PPO or DAPO using the same reward signal? If not, it would be valuable to discuss the expected behavioral differences, computational trade-offs, and design rationale behind choosing DPO over other preference-optimization methods. Clarifying this choice would help readers better understand the methodological positioning of GeoDPO within the broader landscape of reinforcement and alignment approaches.\n3. Regarding the downstream performance improvements, it would be helpful if the authors could provide additional results on benchmarks beyond MathVista, to demonstrate the model’s transferability and robustness across diverse geometric reasoning tasks.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The authors clearly identify geometric perception as a distinct subproblem within vision-language reasoning, separating low-level perception (recognition of geometric primitives and relations) from high-level symbolic reasoning (logical deduction). This represents a novel and well-motivated decomposition that has not been explicitly formalized in prior VLM research.\n2. The paper conducts extensive benchmarking across multiple major vision-language model backbones (Qwen2.5-VL, InternVL3, LLaVA-Next). The proposed method achieves substantial quantitative gains, and downstream reasoning tasks gains. Detailed element-level F1 analyses (covering points, lines, circles, and constraints) further confirm consistent improvements across all geometric subtypes.\n3. The GeoDPO formulation is mathematically well-specified, grounded in a principled DPO objective augmented by a translator-based reward model. The authors carefully address critical challenges such as permutation invariance, distributional shift, and the design of fine-grained reward signals, ensuring both theoretical rigor and practical stability.\n4. The implementation and codebase are open-sourced, providing transparency and reproducibility for future research.", "weaknesses": "1. In Section 3, the paper introduces the GeoPerceive Benchmark, but it does not provide sufficient details regarding its categorical composition, dataset scale, or comparative positioning relative to existing benchmarks. It is recommended that the authors include a summary table outlining the dataset’s structure, sample counts, and distinctions from related benchmarks to improve clarity and reproducibility.\n2. The experimental comparisons primarily focus on SFT-based fine-tuning, without incorporating other reinforcement or reward-based baselines (e.g., PPO, DAPO, or similar methods) that could leverage scalar rewards derived from GeoDSL F1 scores. Including such baselines would strengthen the empirical validation of GeoDPO’s effectiveness and robustness.\n3. The qualitative analysis of perception failures remains limited. Readers currently lack a clear understanding of the specific types of perception errors that GeoDPO mitigates. It is recommended to include before–after visualization pairs illustrating vision-language model outputs on representative examples, both before and after GeoDPO training, to better demonstrate qualitative improvements.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WgHeFct3XX", "reviewer": "Reviewer_nm2u", "summary": "This paper highlights the limitations of visual language models (VLMs) in geometry perception and introduces the GEOPERCEIVE benchmark along with the GEODPO training framework. By incorporating a standardized domain-specific language and a translation-guided reinforcement learning approach, substantial advancements in geometry perception are achieved.", "questions": "1. Could the authors provide comparisons with other reinforcement learning methods, such as ppo?\n2. How does the translator quality impact the downstream DPO training performance? \n3. The downstream evaluation is currently limited to 203 MathVista problems . Could the authors evaluate GEODPO on widely-recognized geometric reasoning benchmarks such as GeoQA , UniGeo , Geometry3K , and PGPS9k to demonstrate broader applicability?\n4. How would GEODPO-enhanced models perform when integrated into complete geometric problem-solving pipelines, such as existing systems like GeoX ?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. This paper accurately identifies a core limitation of existing VLMs in geometric problem solving: the inability to disentangle perceptual errors from reasoning failures. By decoupling geometric perception from downstream reasoning, the work provides a fine-grained evaluation perspective that is crucial for understanding model limitations in geometry tasks. \n\n2. This paper proposes an automated data generation pipeline centered on GEODSL, comprising a generation engine and a solving engine. This design enables the creation of large-scale, complexity-controllable training and evaluation data at low cost, addressing the limitations of human-annotated datasets. \n\n3. This paper proposes GEODPO, a translator-guided reinforcement learning framework that employs an NL-to-DSL translator to bridge natural language and GEODSL.", "weaknesses": "1. The experimental evaluation primarily compares GEODPO against supervised fine-tuning (SFT), but lacks comparisons with other reinforcement learning methods (e.g., PPO) and some recent geometric reasoning approaches (e.g., GeoX).\n\n2. The downstream evaluation is limited to a small subset of MathVista (203 problems ), without evaluation on widely-recognized geometric reasoning datasets such as GeoQA , UniGeo , Geometry3K , and PGPS9k .", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)