1 Introduction↩︎

Vision-language models (VLMs) have achieved strong performance across a wide range of tasks, including image captioning [1], visual question answering [2], reasoning [3], and web navigation [4]. However, strong performance on these tasks does not imply reliable quantitative perception. Among such capabilities, counting is especially important because it requires a model to detect relevant objects, distinguish them from distractors, maintain consistent correspondences, and map visual evidence to an exact numerical answer. This makes counting a useful test of whether a VLM truly grounds its predictions in the image rather than relying on superficial correlations or language priors [5].

At the same time, counting exposes several core failure modes in VLMs, including hallucination, failures under clutter or occlusion, poor object individuation, and mistakes in translating perceptual representations into language. Recent work has shown that these weaknesses remain substantial even in strong contemporary models [5]–[8].

Most existing work documents that VLMs are weak at counting, but it does so primarily at the behavioral level, through benchmarks and aggregate performance comparisons [5]–[8]. These studies are important because they establish counting as a persistent failure mode, yet they largely leave open a more fundamental question: why do VLMs fail at counting? In particular, behavioral evaluations can show when a model is wrong, but they do not reveal whether the correct count is absent from the model’s internal representations, whether it is present but poorly aligned with the model’s eventual answer, or whether the failure emerges later during decoding.

A smaller body of work begins to examine counting more mechanistically [9], [10], and some methods attempt to improve performance directly. For example, [10] retrain parts of the model, while [11] use attention-based interventions and obtain modest gains. However these approaches primarily target better final behavior or what layers/tokens are involved in the counting process , rather than explaining how count information is internally represented, or why it sometimes fails to appear in the final prediction.

Our work addresses this gap by analyzing counting through multiple probes applied to intermediate VLM representations, with each probe trained to capture a different part of the model’s computation. Rather than asking only whether a model produces the correct count, we ask what information about the ground-truth count, the model’s predicted count, and the likelihood of error is present during the forward pass, and how these signals relate to one another across depth. This makes it possible to move beyond behavioral failure and study the internal structure of counting errors.

Specifically, we extend [12] to VLM counting and train separate probes on the same intermediate representations using different supervision: one to predict the ground-truth count, one to predict the model’s own output count, and one to predict whether the model’s answer is wrong. This multi-probe setup allows us to move beyond simple decodability and study how different counting-relevant signals are organized internally. Our results suggest that counting errors arise from at least two distinct sources. First, the model’s final generated answer does not always faithfully reflect information already present in intermediate representations. Second, even when intermediate representations support the model’s own eventual answer, they can remain more strongly aligned with the correct count, revealing a mismatch between internal quantity information and final prediction. To make this comparison explicit, we analyze the alignment of probe subspaces using an adapted version of SVCCA, which lets us test whether ground-truth-, output-related supervision gives rise to shared or distinct counting directions. We also perform causal steering intervention and prove that strengthening the count direction does improve model performance. Our contributions are as follows. (1) We construct four synthetic counting datasets paired with a real-world benchmark, enabling controlled evaluation across diverse counting regimes (Fig 1). (2) Building on [12]’s multi-probe framework for LLM arithmetic, we extend the paradigm to VLM counting, probing three supervision targets across both vision-encoder and text-decoder representations (Figs 2;2 (c)). (3) We show via SVCCA that ground-truth and output signals occupy substantially misaligned readout subspaces, especially for nonlinear probes (Fig 3), and confirm causal relevance via activation steering: strengthening the count direction improves accuracy while random directions degrade it. (4) We translate these insights into a detector-guided self-correction method that improves counting accuracy by up to \(15.6\) absolute percentage points without fine-tuning (Table [tab:synthetic95self95correction]).

2 Related Work↩︎

2.0.0.1 VLM counting and reliability.

Counting has long been used as a stress test for visual reasoning, from VQA-style benchmarks such as TallyQA [13] and everyday-scene counting [6], [14] to recent CLIP-era analysis [15]. Broader multimodal evaluations report systematic weaknesses in grounding and object fidelity, including poor counting performance [5], [16], hallucination [8], [17], and weak perception-reasoning alignment [5], [18]–[20]. Our work is complementary: rather than proposing a new benchmark, we test whether hidden activations already contain recoverable signals that can identify and correct counting failures.

2.0.0.2 Probing and Inference-time correction.

Probing methods are widely used to test which properties are encoded in activations [21], [22]. In NLP, probes have been used for syntax [23], pipeline-like linguistic structure [24], and factual associations [25]. Recent work also studies latent confidence and truth-related structure in activations [26], [27], as well as numeracy-specific representations [28], [29]. In VLMs, probes were used to both analyze the counting mechanism [9] and localize bottleneck components that limits counting performance [10]. Inference-time intervention has shown that internal signals can improve factuality without parameter updates [30]. In parallel, test-time reasoning and correction methods such as self-consistency [31], Tree-of-Thought [32], ReAct [33], Self-Refine [34], Reflexion [35], and StateAct [36] improve outputs through search or iterative feedback. Closest to our work is [12], which probes LLM hidden states for ground-truth, output, and binary-correctness signals during arithmetic reasoning and uses the resulting error detectors for selective re-prompting.

3 Analysis: Can VLMs internally count objects?↩︎

Counting is an important instance of reasoning; it has been proven to be challenging for VLMs [16]. In this paper, we investigate the counting mechanism in VLMs. We formalize the counting task as an input image \(I\) and a textual query \(q\), the task requires the model \(\mathcal{M}\) to produce a count estimate \(\hat{y}\) such that \(\hat{y} = \mathcal{M}(I, q)\), where \(y\) denotes the ground-truth count and \(y, \hat{y} \in \{1, 2, \ldots, 9\}\).

3.1 Probe Architectures↩︎

Let \(x_l \in \mathbb{R}^{d}\) denote the final text-token representation at layer \(l\), where \(d\) is the hidden dimension of the VLM. A probe is a mapping from \(x_l\) to a target label. We consider three probing tasks. First, a ground-truth count probe maps \(x_l\) to the true count \(y\). Second, an output-count probe maps the same representation \(x_l\) to the model’s own predicted count \(\hat{y}\). These two probes have the same architecture but are trained with different supervision (we illustrate below what architectures investigated), allowing us to compare what aspects of the hidden state align with the correct answer versus the produced answer. Third, for error detection, we train a separate error probe to predict whether the model’s answer is correct: \(e = \mathbb{I}[\hat{y} \neq y] \in \{0,1\}\), where \(e=1\) indicates an incorrect counting prediction. Thus, the count probes solve multi-class classification over count labels, while the error probe solves binary classification over correctness labels.

We employ four probe types of increasing expressivity [12]. The circular probe projects \(\mathbf{x}_l\) onto a learned 2D plane via \(\mathbf{w}_1, \mathbf{w}_2 \in \mathbb{R}^{d}\), encoding the digit in the angle: \(\hat{y} = \tfrac{10}{2\pi}\,\operatorname{atan2}(\mathbf{w}_1^\top \mathbf{x}_l,\, \mathbf{w}_2^\top \mathbf{x}_l)\). The linear probe applies an affine map \(\hat{y} = \mathbf{w}^\top \mathbf{x}_l + b\), trained with \(\ell_2\) regularization and rounded at inference. The logistic regression probe assigns per-class weights: \(\hat{y} = \arg\max_i\, (\mathbf{w}_i^\top \mathbf{x}_l)\). The MLP probe adds a single hidden layer (ReLU, 512 units): \(\hat{y} = \arg\max_i\, (\mathbf{W}_2^\top \operatorname{ReLU}(\mathbf{W}_1^\top \mathbf{x}_l + \mathbf{b}_1) + \mathbf{b}_2)\). All classification probes are trained with cross-entropy; the circular probe with smooth \(\ell_1\) loss.

3.2 Measuring Representational Alignment via SVCCA.↩︎

Probe accuracy tells us whether a representation contains information about a target. However, accuracy alone does not tell us whether two targets are decoded from the same internal directions. This distinction is central to our question. A layer may simultaneously contain information about the true count \(y\) and the model output \(\hat{y}\), yet the two may rely on different subspaces. In that case, the model may internally encode the correct quantity while organizing its final prediction around a different representational direction. Conversely, if probes trained on \(y\) and \(\hat{y}\) rely on highly similar subspaces, then the model’s final answer is more closely aligned with the internal count information available at that layer.

To test this, we compare pairs of probes with the same architecture trained on the same hidden states but with different supervision: one probe predicts the ground-truth count \(y\), and the other predicts the model output \(\hat{y}\). We then ask whether the two probes read out their information from similar directions (subspaces) in representation space.

We quantify subspace alignment using Singular Vector Canonical Correlation Analysis (SVCCA; [37]); full mathematical details are given in Appendix 8.1. Given the weight matrices \(W_{\mathrm{gt}}, W_{\mathrm{out}} \in \mathbb{R}^{c \times d}\) of two probes trained on ground-truth and model-output counts respectively, we extract the top-\(k\) left singular vectors from each, yielding orthonormal bases \(U_{\mathrm{gt}}^{(k)}, U_{\mathrm{out}}^{(k)} \in \mathbb{R}^{d \times k}\). We then compute the canonical correlations \(\rho_1, \dots, \rho_k = \sigma_1, \dots, \sigma_k\left({U_{\mathrm{gt}}^{(k)}}^\top U_{\mathrm{out}}^{(k)}\right)\) and summarize alignment as: \(\mathrm{SVCCA}(W_{\mathrm{gt}}, W_{\mathrm{out}}) = \frac{1}{k}\sum_{i=1}^{k} \rho_i.\)

This score has a simple interpretation. Values near \(1\) indicate that the ground-truth and output probes rely on nearly the same directions in the hidden representation, suggesting that the model’s final count is aligned with the internally available count information at that layer. Values near \(0\) indicate that the two probes depend on nearly orthogonal subspaces, suggesting a stronger mismatch between information aligned with the correct count and information aligned with the produced answer.

Our use of SVCCA is motivated by the need for a basis-invariant comparison. Directly comparing probe weight matrices is not reliable, because two probes can implement similar subspaces using different but equivalent bases for the same subspace. By comparing the subspaces spanned by the principal singular vectors instead of raw parameters, SVCCA provides a more robust measure of whether ground-truth-aligned and output-aligned counting information are geometrically similar inside the model. While SVCCA was originally introduced to compare activation matrices across layers or networks [37], applying it directly to activations is uninformative in our setting: both probes read from identical hidden states, so activation-level comparison is trivially maximal. The supervision-specific information instead resides in the probes’ readout matrices, whose rows live in the same \(d\)-dimensional coordinate system of the shared frozen representation and are therefore directly comparable. To our knowledge, we are the first to apply SVCCA to probe weight matrices as a tool for comparing supervision-specific readout subspaces.

This analysis gives us a mechanistic tool for locating where counting failures emerge. If a layer shows strong decodability of the true count but weak alignment with the output probe, this suggests that correct quantity information is present but not faithfully carried into the final answer. In contrast, if the ground-truth count is weakly decodable, the counting failure likely originates earlier in the visual or multimodal representation itself.

3.3 Experimental Setup↩︎

3.3.0.1 Datasets.

We carefully design and construct four synthetic datasets, each containing \(1170\) images (\(512\times512\), white background) with \(\{1, \ldots, 9\}\) non-overlapping objects placed on a shuffled grid at a fixed size. The four datasets vary the visual diversity of objects along color and shape (e.g., same color/different shape vs.different colors/different shapes, details in  8.2 Table 2). Samples are equally distributed across nine count classes with an \(80/20\) stratified train/test split. We additionally evaluate on CountBench [7], removing images with more than nine objects and using a \(50/50\) stratified split to match our experimental setup.

3.3.0.2 Models.

We evaluate four publicly available VLMs: InternVL2-1B, InternVL2-4B [38], Qwen3-VL-2B-Instruct, and Qwen3-VL-8B-Instruct [39]. All models are run with greedy decoding. Models are from HuggingFace Transformers [40].

All models are evaluated under zero-shot settings. All probes are trained for \(2000\) epochs using a weighted cross-entropy loss to mitigate class imbalance. The layer yielding the best performance will be used in the subsequent experiments of the corresponding probe type.

3.3.0.3 Training Probes.

For each (model, dataset, probe-type, task) combination, we train four probe architectures, sweep over all layers, and characterize each probe by the layer that achieves the highest held-out F1 score. We train each probe on three tasks: ground-truth count (gt_probe), output-count (output_probe), and error probe.

This process is repeated for all probes, datasets, layers, all four models, and averaged over three random seeds.

3.4 Probing Results↩︎

Figure 2 presents the results of training four probes on the activations of four models for two different objectives (ground truth and model output), aggregated across the four synthetic datasets, and three random seeds. Per-dataset results are provided in 9 8 7 6 in the Appendix 8.

The single scalar affine probe is insufficient to decode count reliably, especially in later layers, while multiclass linear probes remain effective. All other architectures exceed 80% F1, reaching near-perfect scores on Qwen3-VL-8B activations for both objectives. The fact that simpler probes such as Logistic and Circular achieve comparable performance to the MLP suggests that the recovered count information reflects genuine structure in the representations rather than an artifact of an expressive classifier learning the task independently [41]. A notable temporal asymmetry emerges: ground-truth probes plateau at earlier layers, while output probes peak near the final layers, suggesting the model encodes the correct count well before it commits to a verbalized answer.

The exception is InternVL2-1B, where output probes remain flat around \(60-65\%\), yet error detectors on the same activations achieve \(>90\%\) F1 (Figure 2 (c)). This indicates that predicting whether the model errs can be easier than predicting what it will say; the binary error signal may occupy a more accessible subspace than the full output distribution.

For error detection, we train four detector types: MLP, Logistic-Separately, Circular-Separately, and Circular-Jointly [12]. The “separately” variants flag an error when two same-architecture probes (one trained on \(y\), one on \(\hat{y}\)) disagree; the joint variant trains a single model on the angular distance between two circular probes. Detectors on InternVL2 activations yield near-perfect F1, while those on Qwen3-VL activations perform lower, consistent with fewer training errors being available. As we show in §5.1 and Figure [fig:f195vs95delta95scatter], detector F1 is strongly correlated with downstream correction gain. Similar training and analysis have been done at the level of the last token and average tokens extracted from the vision encoder, more details in appendix  10.

3.5 SVCCA: Subspace Divergence Between Ground-Truth and Output Probes.↩︎

SVCCA measures similarity between learned representations. In Figure 3, we report SVCCA scores computed between the weights (Figure 14 (a) for activations) of the same probe family trained under two objectives: ground-truth count prediction and model-output count prediction. In Figure 3 (a), nonlinear probes (Circular and MLP) lie in a near-orthogonal regime (low SVCCA). Linear probes show higher alignment than nonlinear probes, but alignment remains limited, especially for Logistic regression. These findings support a subspace-misalignment hypothesis: count-relevant information is internally encoded, but only weakly coupled to the representation subspace that drives final verbalized outputs. To test whether this pattern is merely a layer-selection artifact, Figure 3 (b) plots SVCCA against the best-layer gap. Low SVCCA persists even at small layer gaps, indicating that subspace divergence is not reducible to depth mismatch alone, as discussed in prior findings [27], [42]

4 Causal Steering Intervention↩︎

The probing and SVCCA analyses above establish that VLMs internally encode count information that diverges from their verbalized outputs. These results are correlational: the probe may recover a direction that is predictive of count without being causally involved in generating it. To distinguish these possibilities, we perform a direct activation-steering experiment on the full synthetic test split of all four synthetic datasets.

4.0.0.1 Method.

For each sample we first run the unmodified model to obtain a baseline prediction \(\hat{y}\) and record the baseline accuracy (Figure 4). We then rerun the same sample with a single forward-hook intervention applied at the best layer \(l^*\) of the saved Logistic gt_probe. Let \(x_{l^*}\!\in\!\mathbb{R}^{d}\) denote the final prompt-token hidden state at layer \(l^*\), and let \(w_c\) be the probe weight vector for class \(c\). We define the steering direction as the normalized pairwise difference between the ground-truth and baseline class directions: \[d = \frac{w_y - w_{\hat{y}}}{\left\| w_y - w_{\hat{y}} \right\|_2}, \qquad \tilde{x}_{l^*} = x_{l^*} + \alpha \cdot \mathrm{RMS}(x_{l^*}) \cdot d, \qquad\] \(\mathrm{RMS}(x) = \frac{\left\| x \right\|_2}{\sqrt{d}},\)

where \(\alpha\) controls steering strength and RMS-scaling ensures the perturbation is proportional to the local activation norm. As a control, we run the same experiment replacing \(d\) with a randomly sampled unit vector of the same dimensionality. Each \(\alpha\) is swept over a range from \(5\) to \(65\).

4.0.0.2 Results.

Figure 4 plots model accuracy as a function of \(\alpha\) for both the probe-derived (blue) and random-direction (red) interventions. Three observations emerge: (1) Probe-derived steering causally improves counting. On all four datasets, steering along the probe direction lifts accuracy above the unsteered baseline within a broad effective range (green region). The clearest gain appears on diff_col_diff_shape, where accuracy rises from a baseline of \({\sim}27\%\) to a peak near \(46\%\) at moderate \(\alpha\), corresponding to an absolute gain of roughly \(19\) percentage points. On sing_col_sing_shape and diff_col_sing_shape, more modest but consistent gains (\({\sim}6\)–\(8\) pp) persist across a range of moderate intervention strengths. These results demonstrate that the direction recovered by the probe is not merely predictive of count but plays a causal role in the model’s generation. (2) Random steering monotonically degrades performance. Across all four datasets, the random-direction control either stays near the baseline at small \(\alpha\) or steadily degrades as \(\alpha\) grows (red curves entering the pink region), confirming that the improvements in (1) are specific to the probe-derived direction rather than an artifact of generic activation perturbation. The widening gap between the blue and red curves as \(\alpha\) increases further corroborates the directional specificity of the causal effect. (3) Excessive steering is destructive. On all datasets, there exists a critical \(\alpha\) beyond which even probe-derived steering begins to hurt. On sing_col_diff_shape, the blue curve enters the destructive zone around \(\alpha\!=\!45\), and on diff_col_sing_shape, gains plateau early. This non-monotonicity is consistent with a local-linear interpretation: the probe direction is a good first-order approximation of the count subspace, but large perturbations push the hidden state beyond the regime where this linear approximation holds.

4.0.0.3 Implications.

The causal steering experiment indicates that probe-derived count directions can influence generation, but it also reveals that direct activation intervention depends on two factors: dataset geometry and steering magnitude \(\alpha\). This motivates a less invasive use of the same representational insight: rather than modifying hidden states directly, we use the activation-level error signal to decide when the model should reconsider its answer. Figure [fig:f195vs95delta95scatter] supports this choice by showing a strong positive correlation between detector F1 and correction gain, indicating that reliable internal failure prediction translates into effective downstream correction. We therefore turn to a detector-guided self-correction method that treats the probe as an intervention gate rather than as a steering mechanism. Next, we introduce detector-guided self-correction in Section 5

5 Detector-Guided Self-Correction Method↩︎

Let \(x _{l^*}\in \mathbb{R}^d\) be the activation extracted from last non-padding token from layer \(l^*\) of VLM \(\mathcal{M}\) for image-query pair \((I, q)\), where \(l^* = \arg\max_{l}\,\mathrm{F1}(D^{(l)}, \mathcal{D}_{\mathrm{val}})\) is the layer yielding the best detector performance on a held-out validation set \(\mathcal{D}_{\mathrm{val}}\). Let \(\hat{y}^{(1)} = \mathcal{M}(I, q)\) be the first-pass prediction. A detector \(D^{(l^*)} : \mathbb{R}^d \to [0,1]\) estimates the error probability:
\[s = D^{(l^*)}(x_{l^*}) \approx \mathbb{P}\!\left(\hat{y}^{(1)} \neq y \mid x_{l^*}\right)\]

and triggers a correction when \(s\) exceeds a threshold \(\tau\)¹: \[\hat{y} = \begin{cases} \mathcal{M}\!\left(I,\; f_{\mathrm{corr}}(q, \hat{y}^{(1)})\right) ;& \text{if } s \ge \tau \\ \hat{y}^{(1)} ; & \text{otherwise} \end{cases}\] where \(f_{\mathrm{corr}}(q, \hat{y}^{(1)})\) is a correction prompt that supplies the original response and asks the model to reconsider its answer. The correction prompt is in the Appendix 9.

5.0.0.1 Baselines.

We compare against two baselines. Always Reprompt applies \(f_{\mathrm{corr}}\) to all samples unconditionally. This is the maximum budget baseline. Random-\(K\) matches the detector’s intervention budget \(K = |\{i: D(x_{l^*,i}) \ge \tau\}|\) but selects samples at random, isolating the contributions of learned selectivity over random reprompting. An additional baseline (Entropy-guided) is reported in Table 9.2.

5.1 Proposed Method Results↩︎

2pt

Table [tab:synthetic95self95correction] reports the performance of the zero-shot raw model, the Always Reprompt baseline, the Random-K baseline, and our method across four synthetic datasets and four VLMs. In the Always Reprompt setting, every sample is reprompted unconditionally (i.e, \(100\%\) of the samples), whereas Random-K reprompts the same number \(K\) of samples as our probe flags, but selects them uniformly at random.

We observe that, for almost all models, the dataset with a single color and varying shapes is the most difficult. The only exception is the InternVL2-4B model.

While Always Reprompt outperforms Ours in several configurations, it is not a practical strategy for two reasons. First, it is computationally expensive, as it unconditionally requires a second forward pass on the entire dataset. Second, indiscriminate reprompting is unreliable: it can flip previously correct answers, degrading overall performance (Table [tab:synthetic95self95correction]). In contrast, Ours recovers most of the gains achieved by Always Reprompt while selectively reprompting only a small subset of samples, avoiding both the computational overhead and the risk of degradation. It consistently surpasses the Random-K baseline. Overall, our approach yields improvements of up to \(15.6\%\) (avg \(5.3\%\)). The largest gains occur for the InternVL2 family, while the smallest gains are observed for Qwen3-VL-2B; these trends align closely with the quality of the error-detection probe (Figure 2 (c)). Figure [fig:f195vs95delta95scatter] further illustrates how probe performance on error detection correlates with self-correction gains, showing that the largest improvements are obtained when the probe’s F1-score exceeds \(90\%\) (with strong correlation \(\rho = 0.803\)). Table [tab:detectors-stratified] provides a summary of correction gains across detector types. We can see several patterns. First, no single detector consistently dominates, suggesting that detector choice is not critical and the method is robust to error detector architecture. Second, the discrepancy between VLM families is larger than it between detectors: InternVL2 benefits substantially (up to \(+15.95\) pp) while Qwen3-VL-2B gains is (\(+0.43\) to \(+2.42\) pp) independent of probe choice, suggesting that the bottleneck for correction is the quality of the error signal in the activation rather than the detector architecture itself. Table A. 4 reports true-positive correction rates and false positive preservation rates for each detector. Additional prompts are reported in 9.1. By repeating the full pipeline on the CountBench dataset, we prove that our findings hold beyond synthetic datasets.

6 Conclusion↩︎

In this paper, we introduce the first study that examines how simple probes can predict the ground-truth count, the model-predicted count, and the model likelihood of error from VLMs’ internal representations. We further demonstrate that these simple probes can accurately anticipate when a VLM is about to generate an incorrect count. A complementary SVCCA analysis shows that probes trained on ground-truth counts and those trained on model outputs occupy misaligned readout subspaces, indicating a divergence between true and output count representations during generation. We further validate our findings using a causal steering intervention, proving that strengthening the count-identified probes’ direction does improve model counting performance. Finally, we integrate these insights into a method that uses simple probes for error detection and reprompting-based correction, achieving up to a \(15.6\%\) improvement in counting performance.

7 Limitations and Future Work↩︎

The limitations are that we study only models up to 8B parameters due to computational constraints, use synthetic data whose transfer to real-world settings is still unverified, cover only two VLM families, conduct a causal intervention experiment with only one model, and restrict counting tasks to the \(1\)–\(9\) range. Future work should evaluate larger models, real-world generalization, additional VLM families, and harder counting regimes beyond \(10\) objects.

8 Detailed Results↩︎

8.1 SVCCA Details↩︎

We quantify this using Singular Vector Canonical Correlation Analysis (SVCCA) [37]. Consider a pair of trained linear classifiers of the same architecture, represented by weight matrices \[W_{\mathrm{gt}} \in \mathbb{R}^{c \times d}, \qquad W_{\mathrm{out}} \in \mathbb{R}^{c \times d},\] where \(c\) is the number of classes and \(d\) is the hidden dimension. Each row of a probe weight matrix corresponds to a class-specific readout direction, so the row space of the matrix characterizes the subspace used by that probe for prediction.

To extract a stable low-dimensional basis for each probe, we compute the singular value decomposition \[W_{\mathrm{gt}}^\top = U_{\mathrm{gt}} \Sigma_{\mathrm{gt}} V_{\mathrm{gt}}^\top, \qquad W_{\mathrm{out}}^\top = U_{\mathrm{out}} \Sigma_{\mathrm{out}} V_{\mathrm{out}}^\top.\] We then retain the top-\(k\) left singular vectors from each decomposition, yielding orthonormal basis matrices \[U_{\mathrm{gt}}^{(k)},\, U_{\mathrm{out}}^{(k)} \in \mathbb{R}^{d \times k}.\] These bases span the principal readout subspaces used by the ground-truth and output probes, respectively.

Next, we measure how well these two subspaces align by computing the singular values of their cross-projection matrix: \[\rho_1,\dots,\rho_k = \sigma_1,\dots,\sigma_k \left( {U_{\mathrm{gt}}^{(k)}}^\top U_{\mathrm{out}}^{(k)} \right),\] where each \(\rho_i \in [0,1]\) is a canonical correlation. Intuitively, \(\rho_i\) measures how similar the \(i\)-th most aligned directions are across the two probe subspaces. We summarize alignment by the mean canonical correlation: \[\mathrm{SVCCA}(W_{\mathrm{gt}}, W_{\mathrm{out}}) = \frac{1}{k}\sum_{i=1}^{k} \rho_i.\]

8.2 Synthetic Datasets Details↩︎

4pt

3pt

9 Correction Prompt Template↩︎

The following prompt is used during the second pass of the detector-guided self-correction pipeline (§5).

The system prompt used for all passes is:

9.1 Experiments on Additional Prompts↩︎

To evaluate the sensitivity of detector-guided self-correction to the specific framing of the intervention prompt, we compare three distinct reprompting strategies. The exact text for each prompt variation is defined as follows:

As shown in Table 5, the counting prompt yields the highest absolute gains on three of the four datasets, peaking at a \(+16.67\) improvement on sing_col_diff_shape. However, stronger remains the most conservative and stable approach. When evaluated by preservation-oriented metrics, stronger maintains the highest average false-positive preservation rates (ranging from \(71.95\%\) to \(95.21\%\) across the synthetic datasets) and achieves the highest combined effectiveness on a majority of splits. This highlights a clear architectural trade-off between aggressive error correction and the preservation of baseline coherence.

9.2 Comparison of Entropy and Probe-Guided Self-Correction↩︎

The results demonstrate that probe-guided self-correction strictly outperforms entropy-based heuristics across all evaluated datasets. Relying purely on output entropy to trigger self-correction is highly unreliable; it actively degrades model performance (yielding net-negative accuracy deltas) on two of the four datasets (diff_col_diff_shape and sing_col_sing_shape).

In contrast, the probe-guided approach consistently yields positive gains, achieving a massive +16.24% absolute improvement on the sing_col_diff_shape split. Mechanistically, the probe-guided method succeeds by accurately targeting and correcting flawed reasoning paths that generic, uncertainty-based entropy metrics fail to effectively address.

10 Vision Encoder Probing Results↩︎

Testing was conducted across two distinct settings: probes were trained on activations extracted from the last vision token of the VLM vision encoder (results shown in Figures  19, 20, 21 and  22 and on the mean activations across all tokens in each image (results shown in Figures  15, 16, 17 and  18).

The results of the layered analysis of the DC-DS, DC-SS, SC-DS and SC-SS data sets reveal a persistent mechanistic gap between the internal perception of the model and its final verbalized result. Specifically, multilayer perceptrons (MLPs) reach peak performance very early - often by the 5th or 10th layer and maintain this plateau until the final layers. This suggests that the "true" answer is successfully extracted from the visual input and represented within the model early in the forward pass. The model "knows" the answer long before it reaches the final output layers, where the decision to verbalize it is made. It is also worth noting that the ground-truth probes of Qwen-3-VL models begin to fall after the middle layers, which is not observed in the InternVL-2 models. Unlike ground-truth probes, which plateau early in the process, output probes typically reach peak F1 values only in the final layers of the model, indicating that the representational subspace governing actual count generation is fully formed only at the end of the processing pipeline. This trend persists across varying visual representation complexities, although the performance of these samples is generally lower than that of their ground-truth counterparts.

Furthermore, high F1-score values obtained by error detection algorithms trained on vision encoder activations are likely due to a spurious correlation between scene complexity and the model’s error rate, rather than genuine error detection. Because the vision language model (VLM) makes more errors on images with a large number of objects, which also tend to be visually denser and cluttered, the vision encoder activations for "complex" scenes (many objects) and "simple" scenes (few objects) systematically differ. The error detection algorithm exploits this: it learns to predict "error" for activations corresponding to complex scenes and "correct" for simple ones, effectively acting as a scene complexity classifier rather than an error detector. Because complexity and error coexist in the dataset, this leads to inflated F1-score varies that do not reflect the model’s true ability to determine whether the model succeeded or failed on any given image.

Comparative analysis shows that probes trained on average activation values consistently outperform probes restricted to the last visual token across all four levels of visual complexity. This performance advantage likely stems from the fact that the critical spatial information required for accurate counting is not concentrated in a single patch or token, but is distributed across the entire image grid. By aggregating features across all patches, averaged representations allow for a more holistic representation of the scene, reducing the information loss inherent in extracting individual tokens and providing a more robust signal for probes to identify internally represented values.

References↩︎

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1. here, \(\tau\) is fixed to a constant equal \(0.5\)↩︎