SIM-CoT: Supervised Implicit Chain-of-Thought — part 4/4

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SVAMP ( Patel et al., 2021 ) (Simple Variations on Arithmetic Math Word Problems) is a benchmark dataset designed to test the robustness of math word problem solvers to superficial changes. It contains 1,000 elementary-level arithmetic word problems (grade 4 and below), each involving a single unknown and solvable by an arithmetic expression with no more than two operators. The problems are transformations of existing datasets (such as MAWPS and ASDiv-A) with controlled variations in wording, structure, and number values to reduce artifacts and superficial cues. SVAMP’s average number of reasoning steps required is around 1.2, similar to the base datasets, but model performance drops significantly when tested on SVAMP, showing that many models rely on heuristic patterns rather than deep understanding.

GSM-Hard ( Gao et al., 2022 ) is a more challenging variant of the GSM8K dataset, intended to test models’ ability to cope with harder numerical values. It retains the same problem statements as the original GSM8K, but replaces many of the numbers with larger and less common numerical values, making superficial arithmetic computation and reasoning harder. The dataset has about 1,319 examples (matching the GSM8K test set size).

MultiArith ( Roy & Roth, 2015 ) is a dataset of multi-step arithmetic word problems designed to challenge systems to correctly sequence multiple operations. It contains 600 problems collected from educational sources, each solvable by one equation involving two or more of the four basic operations (addition, subtraction, multiplication, division). The problems require reasoning over multiple sentences to extract and combine numeric quantities, understand the implied operations, and compute the result. MultiArith has been widely used as a benchmark for evaluating arithmetic reasoning generalization, particularly for models that go beyond single-operation problems. Analyses show that many models struggle on these examples compared to simpler datasets, highlighting the importance of handling compositional and sequential numerical reasoning.

#### Training Data.

GSM8K-Aug ( Deng et al., 2024 ) is the only training corpus we use. It is an augmented dataset derived from GSM8K ( Cobbe et al., 2021 ) , expanding the original 8.5k training problems to roughly 385k examples through paraphrasing, numerical resampling, and synthetic generation with GPT-4. The distribution of reasoning steps in GSM8K-Aug is illustrated in Fig. 5 , where the majority of problems require two to four steps, while a long-tail of six or more steps persists. This balance of common and complex instances makes GSM8K-Aug particularly suitable for training models that need to generalize across reasoning difficulty levels.

## Appendix G SIM-CoT Training Implementation

We provide pseudocode for the SIM-CoT training process, which illustrates how continuous latent embeddings are aligned with explicit supervision at the step level. In particular, each reasoning step in the explicit chain is mapped to a corresponding latent representation, and the training objective enforces consistency between the predicted latent tokens and the ground-truth step annotations. This design ensures that the model learns to represent intermediate reasoning steps in a compact latent space while still retaining interpretability through explicit alignment. By supervising at the step level rather than only at the final answer or trajectory level, SIM-CoT enables finer control over the reasoning process and reduces instability that often arises when scaling to longer chains or larger numbers of implicit tokens.

## Appendix H Geometric Diagnostics of the Latent Space

We analyze the geometry of latent representations with two metrics.

Inter-latent distance. Dist ( z 1 : K ) = 2 K ⁡ ( K − 1 ) ∑ 1 ≤ i < j ≤ K ∥ z i − z j ∥ 2 . \mathrm{Dist}(z_{1:K})=\frac{2}{K(K-1)}\sum_{1\leq i<j\leq K}\big\|z_{i}-z_{j}\big\|_{2}. (9) A larger value indicates better separation, reducing the risk of collapse.

Distance to vocabulary center. Let μ = 1 | 𝒱 | ​ ∑ v ∈ 𝒱 E v \mu=\tfrac{1}{|\mathcal{V}|}\sum_{v\in\mathcal{V}}E_{v} denote the mean embedding. Then, DistVC ( z 1 : K ) = 1 K ∑ k = 1 K ∥ z k − μ ∥ 2 . \mathrm{DistVC}(z_{1:K})=\frac{1}{K}\sum_{k=1}^{K}\big\|z_{k}-\mu\big\|_{2}. (10) Moderate values indicate that latents remain close enough to the lexical manifold for stability, while avoiding collapse toward the center. These diagnostics are not used in training but serve as indicators of diversity and stability in the learned latent space.

## Appendix I Additional Details for Interpretability Analysis

### I.1 Making Implicit Reasoning Visible

Continuous thoughts produced by implicit reasoning models are represented as latent embeddings that do not correspond to discrete vocabulary tokens, and therefore cannot be directly decoded by a tokenizer. This makes it difficult to interpret how the model internally organizes multi-step reasoning. To address this, we reuse the decoder that was employed for step-level supervision during training, and apply it at inference time to map each latent embedding into a human-readable token sequence.

As illustrated in Figure 4 , the process begins with a natural language problem (e.g., a math word problem) that is embedded and passed into the large language model. The model generates a sequence of implicit latent tokens, which capture intermediate reasoning steps in continuous space. These latent tokens are then fed into the optional decoder, which translates them into interpretable expressions. Each latent corresponds to one reasoning step, and the autoregressive generation order encodes the dependency structure across steps.

For example, in the GSM8k case study shown in the figure, the first latent is decoded as 0.3 × 120 = 36 0.3\times 120=36 , representing the number of watermelons harvested initially. The second latent builds upon this result to compute 120 − 36 = 84 120-36=84 , the remaining melons. The third latent then calculates 3 4 × 84 = 63 \tfrac{3}{4}\times 84=63 , and the final latent derives the answer 84 − 63 = 21 84-63=21 . For clarity and to save space, the figure merges the second and third steps into a single box, but the actual implicit reasoning unfolds across four distinct latent steps. This sequence of decoded latents mirrors the logic of explicit chain-of-thought reasoning, while being produced implicitly within the latent space.

By projecting implicit tokens into interpretable space, we gain direct visibility into how the model structures multi-step reasoning. This not only enables analysis of the correctness and consistency of intermediate steps, but also highlights the dependencies across steps that underlie the final prediction. The visualization confirms that SIM-CoT can encode semantically meaningful and logically ordered reasoning steps in its latent space, bridging the gap between implicit and explicit reasoning.

### I.2 Distance Analysis

We analyze the geometry of latent representations by measuring two quantities: the average pairwise distance among latent tokens (Dist.) and the average distance from latent tokens to the vocabulary center (Dist. to VC). Table 4 (b) reports the numerical outcomes under different configurations, while Figure 6 provides a visualization of the same phenomena.

As the number of latent tokens increases from 1 to 5, Dist. gradually grows from 20.30 to 28.34, suggesting that the model distributes the latent representations more sparsely in the embedding space, which improves separability. However, in the failed case with 5 latents, Dist. collapses to 4.21, reflecting a degeneration where all latent tokens converge to nearly identical points, losing their ability to represent distinct reasoning steps. By contrast, SIM-CoT pushes Dist. to 32.81 under the same configuration, showing that step-level supervision effectively enforces stronger separation and prevents collapse.

For Dist. to VC, we observe the highest value with 1 latent (36.20). As the number of latents increases, Dist. to VC decreases toward approximately 28, meaning that the representations become better aligned with the vocabulary manifold rather than drifting away. The failed 5-latent case instead shows an abnormal increase to 39.39, a clear signal of instability where the model drifts away from the vocabulary space. SIM-CoT stabilizes this measure at 29.80, achieving a balance: the latent tokens remain distinct from each other while staying anchored close enough to the vocabulary space to preserve semantic interpretability.

The visualization in Figure 6 corroborates these trends. Normal implicit tokens (first row) are well separated and positioned near the vocabulary manifold. Failed tokens (second row) collapse together and drift outward, losing both separation and grounding. After applying SIM-CoT (third row), the tokens regain a structured configuration: distances between latents expand, while their proximity to the vocabulary center remains stable. This alignment between quantitative and qualitative results highlights the effectiveness of SIM-CoT in maintaining a structured and interpretable latent space.

### I.3 Summary

Overall, the results demonstrate that SIM-CoT establishes a balance between diversity and stability in the latent space. Larger inter-latent distances mitigate representation collapse, while moderate distances to the vocabulary center prevent excessive drift. This equilibrium supports stable implicit reasoning and provides robustness when scaling to more latent tokens.

## Appendix J Future Directions

We outline several promising directions for extending this work: (1) Multimodal extension. Incorporating intermediate supervision from images ( Chen et al., 2023 ; Sun et al., 2024 ; Liu et al., 2024 ) or videos ( Chen et al., 2024a ; Wei et al., 2025c ) could generalize implicit CoT to multimodal reasoning tasks ( Shen et al., 2025a ; Zhang et al., 2025a ) , an especially timely direction given the rapid progress of multimodal learning ( Liu et al., 2023 ; Chen et al., 2023 ; Zhang et al., 2025c ; Liu et al., 2025a ; Wei et al., 2025b ; Zhang et al., 2024 ; Dong et al., 2024 ; Zhao et al., 2025 ; Ding et al., 2025 ; Chen & Xing, 2024 ; Xing et al., 2025 ; Xing et al., 2024 ) . (2) Multi-path implicit reasoning. Exploring branched latent structures, inspired by Tree-of-Thought ( Yao et al., 2023a ; Wang et al., 2023 ; Zhang et al., 2023 ) methods, may enhance the diversity and robustness of implicit reasoning. (3) Integration with RLHF. Step-level supervision can complement preference optimization and reinforcement learning ( Guo et al., 2025 ; Rafailov et al., 2023 ; Liu et al., 2025b ; Zang et al., 2025 ) , leading to a more reliable and adaptive reasoning framework. (4) Theoretical foundations. Future work may also study implicit reasoning from the perspectives of information theory and representation learning ( Zhu et al., 2025 ; Xu & Sato, 2025 ) , providing a principled explanation for why step-level supervision stabilizes the latent space.

## Appendix K Additional Case Studies on GSM8k

In practice, implicit reasoning continues to produce latent tokens even after the correct answer has been reached. These trailing latents no longer introduce new steps but simply repeat the final prediction. For clarity, we omit such redundant tokens in the visualizations. As a result, only the latents that correspond to meaningful intermediate steps are displayed in Figure 7 , while those mapping directly to the final answer are hidden. This choice improves readability without changing the underlying reasoning process.

Notably, the decoded reasoning steps consistently match the semantic structure of explicit chain-of-thought annotations, while being generated implicitly within the latent space. The final predictions align with the ground-truth answers, demonstrating that SIM-CoT is capable of encoding interpretable and step-ordered reasoning without requiring explicit supervision at inference time.

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