Counting Circuits: Mechanistic Interpretability of Visual Reasoning in Large Vision-Language Models

Liwei Che; Zhiyu Xue; Yihao Quan; Benlin Liu; Zeru Shi; Michelle Hurst; Jacob Feldman; Ruixiang Tang

cs.CVMar 19, 2026

Counting Circuits: Mechanistic Interpretability of Visual Reasoning in Large Vision-Language Models

Authors: Liwei Che, Zhiyu Xue, Yihao Quan, Benlin Liu, Zeru Shi, Michelle Hurst, Jacob Feldman, Ruixiang Tang,

Organizations: Rutgers University · UC Santa Barbara · University of Washington · Allen Institute for AI

Abstract

Counting serves as a simple but powerful test of a Large Vision-Language Model's (LVLM's) reasoning; it forces the model to identify each individual object and then add them all up. In this study, we investigate how LVLMs implement counting using controlled synthetic and real-world benchmarks, combined with mechanistic analyses. Our results show that LVLMs display a human-like counting behavior, with precise performance on small numerosities and noisy estimation for larger quantities. We introduce two novel interpretability methods, Visual Activation Patching and HeadLens, and use them to uncover a structured "counting circuit" that is largely shared across a variety of visual reasoning tasks. Building on these insights, we propose a lightweight intervention strategy that exploits simple and abundantly available synthetic images to fine-tune arbitrary pretrained LVLMs exclusively on counting. Despite the narrow scope of this fine-tuning, the intervention not only enhances counting accuracy on in-distribution synthetic data, but also yields an average improvement of +8.36% on out-of-distribution counting benchmarks and an average gain of +1.54% on complex, general visual reasoning tasks for Qwen2.5-VL. These findings highlight the central, influential role of counting in visual reasoning and suggest a potential pathway for improving overall visual reasoning capabilities through targeted enhancement of counting mechanisms.

Figures & tables

Figure 1: Overview of Main Contributions

Model	Dataset	Acc $\uparrow$	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$
Qwen2.5-VL-7B	SynDot	76.12	0.25	0.52	96.21
	SynPoly	53.74	0.94	1.75	82.27
	SynReal	73.48	0.79	2.13	91.01
Qwen3-VL-8B	SynDot	73.22	0.32	0.66	95.21
	SynPoly	70.34	0.52	1.49	94.02
	SynReal	88.79	0.15	0.47	98.88

Table 1: Performance comparison on synthetic benchmarks.

Figure 2: Counting Uncertainty Curve by Yes/No Answer of Qwen2.5 VL 7B.

Figure 3: T-SNE visualization of model hidden states based on black dots data.

Panel of Figure 4 — Figure 4: Left: Layer-wise Overwrite Rate of Different Input Tokens Patching Strategy; Right: Layer-wise Logit Lens Tracking Curve for Counting Number.

Figure 5: Visualization of two functional groups of counting heads . Left to right: Head importance (positive-only), attention ratio on image tokens, and top-10 HeadLens CTER across heads. We present the typical attention distribution and top-10 HeadLens results for Cross-modal Routing Heads (green boxes) and Counting Aggregation Heads (blue boxes), using L19H23(left) and L26H26(right). Both have "three" (三) in Chinese as the top-1 HeadLens token.

Figure 6: Visual Grounding Heads in Early Layers. Left: Head-wise heatmap based on Visual Grounding Score; Right: Layer-wise Visual Grounding Score.

Backbone	Method	SynReal				PixMo-Count
Backbone	Method	Acc	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$	Acc	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$
Qwen2.5-VL-7B	Baseline	73.48	0.79	2.13	91.01	58.79	0.84	1.79	84.00
Qwen2.5-VL-7B	Ours	84.83	0.74	1.47	97.75	64.15	0.62	1.46	87.10
Qwen3-VL-8B	Baseline	88.79	0.15	0.47	98.88	58.75	0.73	1.35	88.20
Qwen3-VL-8B	Ours	91.21	0.11	0.34	99.41	66.98	0.60	1.25	91.65
LLaVA-1.5-7B	Baseline	54.27	3.59	5.05	66.34	31.88	1.81	3.09	59.77

Table 2: Evaluation on Out-of-distribution Counting Benchmarks.

Backbone	Method	MMMU	RealWorldQA	MathVista	$\Delta$ (avg.)
Qwen2.5-VL-7B	Baseline	54.89	61.96	57.30	–
Qwen2.5-VL-7B	Ours	56.33	64.14	58.30	+1.54
Qwen3-VL-8B	Baseline	58.33	70.05	66.30	–
Qwen3-VL-8B	Ours	60.74	71.83	67.90	+1.93
LLaVA-1.5-7B	Baseline	44.44	56.21	23.90	–
LLaVA-1.5-7B	Ours	44.56	56.48	26.30	+0.93

Table 3: Evaluation on General Capability Benchmarks.

SFT	$\mathcal{L}_{\text{focus}}$	$\beta_{h}$	Synreal	PixMo-Count	MMMU	RealWorldQA	MathVista
Baseline			73.48	58.79	54.89	61.96	57.30
✓			79.78	61.67	56.00	63.14	58.10
✓	✓		82.34	63.32	56.11	64.05	58.20
✓	✓	✓	84.83	64.15	56.33	64.14	58.30

Table 4: Component ablation on Qwen2.5-VL-7B across specialized and general benchmarks.

Figure 7: Jaccard Similarity for the top-20 most important heads of different tasks.

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Data Samples of the three synthetic benchmarks.

Figure 9: Head Importance Heatmap for Qwen2.5-VL on SynDot

Rank	ID	Import.	Img_Attn	Obj_in_Img	GT@10	Top-1 Acc
1	L26H26	0.801	0.034	0.032	0.83	0.55
2	L26H24	0.792	0.301	0.005	0.83	0.53
3	L26H25	0.716	0.314	0.023	0.92	0.77
4	L18H10	0.521	0.937	0.129	0.13	0.03
5	L16H1	0.393	0.903	0.260	0.10	0.00
6	L22H13	0.289	0.538	0.094	0.68	0.48

Appendix

Table 5: Top-20 attention heads ranked by importance and categorized by HeadLens projection. Importance : importance score from head ablation; Img_Attn : fraction of attention on image tokens; Obj_in_Img : fraction of image attention on object-relevant regions; GT@10 : ground-truth count in top-10 HeadLens projections; Top-1 Acc : top-1 HeadLens projection accuracy. Rows in green denote Counting Aggregation Heads and rows in blue denote Cross-Modal Routing Heads .

Panel of Figure 10 — Figure 10: Layer-wise VAP on SynDot for LLaVA-1.5-7B, InternVL3.5-8B and Qwen3-VL-8B.

Figure 11: LLaVA1.5-7B: Head-wise HeatMap on Head Importance, Attention Ratio on Image Tokens and Text Tokens (from left to right).

Figure 12: Qwen3-VL-8B: Head-wise HeatMap on Head Importance, Attention Ratio on Image Tokens and Text Tokens (from left to right).

Figure 13: Layer-wise Probing Performance Curves for Binding and Counting.

Panel of Figure 14 — Figure 14: Comparison of HeadLens and AttentionLens on SynPoly. Top : Head-wise top-1 accuracy on model final output (not ground truth counting). HeadLens (left) produces a sparse signal where only a few heads in Layer 26 stand out, which is aligned with our head importance heatmap in fig. 9 . While AttentionLens (right) spreads high accuracy across the majority of heads, especially in layers 14–27. Bottom : KL divergence distribution across all 784 heads. AttentionLens concentrates nearly all heads into a narrow low-KL band ( $\text{KL}<3$ ), collapsing the dynamic range needed to distinguish critical from non-critical heads. HeadLens preserves a broad distribution that naturally separates the counting circuit from irrelevant heads.

	HeadLens	AttentionLens	Ratio
# Lenses trained	28	784	$28\times$
Parameters per lens	${\sim}$ 12.8M	${\sim}$ 0.46M	$0.04\times$
Total parameters	${\sim}$ 360M	${\sim}$ 362M	${\sim}1\times$
Training steps	14,000	235,200	$17\times$
Wall time (SynPoly)	37.9 s	2,385.1 s	$63\times$
Wall time (SynDot)	41.9 s	2,414.5 s	$58\times$

Appendix

Table 6: Computational cost comparison between HeadLens and AttentionLens.

Figure 15: Layer-wise emergence of numeric representations for both models. Both show zero numeric signal in the first $70\%$ of layers. Qwen’s emergence is more concentrated (layers 25–27) and achieves far higher correct-in-top-10 rates (up to $11\%$ ) versus LLaVA’s peak of $1.6\%$ .

Model	Method	SynDot				SynPoly
Model	Method	Acc	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$	Acc	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$
Qwen2.5-VL-7B	Baseline	76.12	0.25	0.52	96.21	53.74	0.94	1.75	82.27
Qwen2.5-VL-7B	Ours	89.20	0.11	0.33	100.00	79.10	0.21	0.47	99.60
Qwen3-VL-8B	Baseline	73.22	0.32	0.66	95.21	70.34	0.52	1.49	94.02
Qwen3-VL-8B	Ours	95.40	0.05	0.21	100.00	93.40	0.07	0.26	100.00
LLaVA-1.5-7B	Baseline	24.07	4.46	6.58	39.22	33.30	1.65	2.34	56.71

Appendix

Table 7: In-distribution counting benchmarks with full metrics.

Model	Method	Acc $\uparrow$	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$
Qwen2.5-VL-7B	Baseline	80.44	0.66	4.34	89.61
Qwen2.5-VL-7B	Ours	82.08	0.58	4.28	91.24
Qwen3-VL-8B	Baseline	87.98	0.23	0.86	95.52
Qwen3-VL-8B	Ours	90.22	0.20	0.85	95.93
LLaVA-1.5-7B	Baseline	42.16	1.38	2.08	59.67
LLaVA-1.5-7B	Ours	47.86	1.15	1.78	67.01

Appendix

Table 8: CountBenchQA results.

Model	Method	Range	Acc $\uparrow$	MAE $\downarrow$	RMSE $\downarrow$	OBO $\uparrow$
Qwen3-VL-8B	Baseline	11-20	35.00	0.90	1.25	80.02
	Baseline	21-30	10.42	2.49	3.02	32.37
	Ours	11-20	48.16	0.59	0.86	93.45
	Ours	21-30	18.27	1.61	2.05	52.81

Appendix

Table 9: Performance comparison over larger counting ranges for Qwen3-VL-8B. Ours are finetuned on the number range 1-10 only.

Panel of Figure 16 — Figure 16: Layer-wise ablation of the attention regularizer for three LVLMs.

Method	$\alpha$	MMMU	RealWorldQA	MathVista	Average
SFT + $\mathcal{L}_{\text{focus}}$	1.0	56.11	64.05	58.20	59.45
+ HTT	1.1	56.14	63.94	58.10	59.39
+ HTT	1.2	56.33	64.12	58.20	59.55
+ HTT	1.3	56.43	64.05	57.90	59.46
+ HTT + HIR	1.2	56.33	64.14	58.30	59.59

Appendix

Table 10: Ablation study of head temperature tuning (HTT) coefficient

\alpha

and head importance reweighting (HIR) on Qwen2.5-VL-7B. The baseline method is SFT+

\mathcal{L}_{\text{focus}}

with

\alpha=1.0

Figure 17: Data sample of SynColorShape.

Training Task	$\Delta$ MMMU	$\Delta$ MathVista	$\Delta$ RealWorldQA	$\Delta$ (avg.)
Color-Shape SFT	$-$ 0.1	$-$ 0.7	$-$ 0.2	$-$ 0.3
Color-Shape SFT + AttnReg	$-$ 0.2	$-$ 0.8	+0.2	$-$ 0.3
Counting (Ours)	+1.44	+1.00	+2.18	+1.54

Appendix

Table 11: Change in general reasoning benchmarks (

\Delta

from each method’s own baseline) for Qwen2.5-VL-7B. Color-shape recognition training shows no positive transfer, while counting training yields consistent gains.

Figure 18: SynPoly case studies at GT=5. Prompt: How many colorful polygons are in the image? Answer with only an integer. Observation: Base predicts 4 and Ours predicts 5. Base attention is more spread with weak peaks, while ours shows sharper peaks aligned with the dots already at Layer 2 and remains aligned in the all-layer average. Conclusion: Our method strengthens early visual grounding for counting, making the attention more instance-aligned even when both models output the correct number.

Figure 19: SynDot case studies at GT=6. Prompt: How many black dots are in the image? Answer with only an integer. Observation: Base predicts 6 and Ours also predicts 6. However, base attention is more spread with weak peaks, while ours shows sharper peaks aligned with the dots already at Layer 2 and remains aligned in the all-layer average. Conclusion: Our method strengthens early visual grounding for counting, making the attention more instance-aligned even when both models output the correct number.

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Counting to Four is still a Chore for VLMs

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Vision--language models (VLMs) have achieved impressive performance on complex multimodal reasoning tasks, yet they still fail on simple grounding skills such as object counting. Existing evaluations mostly assess only final outputs, offering limited insight into where these failures arise inside the model. In this work, we present an empirical study of VLM counting behavior through both behavioral and mechanistic analysis. We introduce COUNTINGTRICKS, a controlled evaluation suite of simple shape-based counting cases designed to expose vulnerabilities under different patchification layouts and adversarial prompting conditions. Using attention analysis and component-wise probing, we show that count-relevant visual evidence is strongest in the modality projection stage but degrades substantially in later language layers, where models become more susceptible to text priors. Motivated by this finding, we further evaluate Modality Attention Share (MAS), a lightweight intervention that encourages a minimum budget of visual attention during answer generation. Our results suggest that counting failures in VLMs stem not only from visual perception limits, but also from the underuse of visual evidence during language-stage reasoning. Code and dataset will be released at https://github.com/leduy99/-CVPRW26-Modality-Attention-Share.

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