Mitigating Overthinking in Large Reasoning Models via Manifold Steering

We greatly appreciate you for recognizing the importance of addressing overthinking in LRMs, as well as the success of our method in mitigating it in R1-based models. Below, we address your concerns in Weaknesses (W), Questions (Q) and Flaws (F). We sincerely hope that you can find our response satisfactory.

W1&Q1-F1. Paragraph "Linear Low-Dimensional Manifold Verification" states that the steering direction is composed of an overthinking ($r_{\text{overthinking}}$) and an error ($r_{\text{other}}$) term by performing a PCA on joint redundant and concise reasoning traces. However, I don't see how a PCA shows orthogonality between the two by doing a joint PCA. The PCA just shows the concentration of the variance in the first $k=10$ dimensions (of the PCA), no separation between different traces.

For the orthogonality of $r_{\text{overthinking}}$ and $r_{\text{other}}$, it is a property arising from the principles of PCA. For a centered data matrix $A^{(l)} \in \mathbb{R}^{d \times N}$, where $d$ is the ambient dimension (i.e., the hidden size of activations) and $N$ is the number of activation vectors, the covariance matrix is $C^{(l)} = \frac{1}{N-1} (A^{(l)} - \bar{A}^{(l)}) (A^{(l)} - \bar{A}^{(l)})^\top$. Its eigenvalue decomposition is $C^{(l)} = U^{(l)} \Lambda^{(l)} (U^{(l)})^\top$, where $U^{(l)} = [u_1^{(l)}, \dots, u_d^{(l)}]$ is orthogonal ($(U^{(l)})^\top U^{(l)} = I$), and $\Lambda^{(l)} = \text{diag}(\lambda_1^{(l)}, \dots, \lambda_d^{(l)})$ with $\lambda_1^{(l)} \geq \dots \geq \lambda_d^{(l)} \geq 0$. The low-dimensional subspace $\mathcal{M}$ is spanned by $U_{\text{eff}}^{(l)} = [u_1^{(l)}, \dots, u_k^{(l)}]$, the top $k$ eigenvectors, satisfying $(U_{\text{eff}}^{(l)})^\top U_{\text{eff}}^{(l)} = I_k$. The projection matrix is $P_{\mathcal{M}} = U_{\text{eff}}^{(l)} (U_{\text{eff}}^{(l)})^\top$. The overthinking component is $r_{\text{overthinking}} = P_{\mathcal{M}} r^{(l^\star)} = U_{\text{eff}}^{(l)} (U_{\text{eff}}^{(l)})^\top r^{(l^\star)}$, and the error term is $r_{\text{other}} = r^{(l^\star)} - r_{\text{overthinking}} = (I - P_{\mathcal{M}}) r^{(l^\star)}$. Their inner product is $\langle r_{\text{overthinking}}, r_{\text{other}} \rangle = (r^{(l^\star)})^\top P_{\mathcal{M}}^\top (I - P_{\mathcal{M}}) r^{(l^\star)}$. Since $P_{\mathcal{M}}$ is symmetric ($P_{\mathcal{M}}^\top = P_{\mathcal{M}}$) and idempotent ($P_{\mathcal{M}}^2 = P_{\mathcal{M}}$), we have $P_{\mathcal{M}} (I - P_{\mathcal{M}}) = P_{\mathcal{M}} - P_{\mathcal{M}}^2 = 0$. Thus, $\langle r_{\text{overthinking}}, r_{\text{other}} \rangle = 0$, confirming their orthogonality due to PCA’s orthogonal projection.

Finally, we appreciate your valuable feedback, and we will incorporate a clearer explanation in the revised version.

W1&Q1-F2. Moreover, the comparison between the magnitudes of $r_M$ and $r\_{\text{other}}$ are confusing. Step 2 in the proof of Theorem 4.2 states that the magnitude of the error term is much larger than the $r_M$. However, $r_M$ contains 70% of the eigenvalues, hence it has a larger norm!

We are sorry for this mistake and the resulting confusion. The direct comparison in Step 2 of the proof for Theorem 4.2, stating that $d \gg k, \|r\_{\text{other}}\|_2^2 \gg \|r_M\|_2^2$ leading to $\|r^{(l^\star)}\|_2 \approx \|r\_{\text{other}}\|_2$, may not always hold, particularly given that the subspace associated with $r_M$ captures 70% of the eigenvalues (variance), which could result in a larger norm for $r_M$ under strong alignment with activation differences.

To be fortunate, this does not impact the core conclusion of Theorem 4.2, which establishes that the mean activation shift $\Delta \mu^{(l)}$, induced by the intervention along the over-thinking direction $r^{(l^\star)}$, has an unexpected norm proportional to $\alpha \|r\_{\text{other}}\|_2$. This shift undergoes layer-wise amplification through the effects of attention mechanisms, non-linear activations, and weight matrices, leading to disruption of the model's normal abilities.

For a more rigorous description, we will revise Step 2 to remove the direct magnitude comparison and instead clarify the actual noise contribution from $r\_{\text{other}}$ and how it drives a shift component that propagates and amplifies across layers.

W1&Q1-F3. Theorems 4.1 and 4.2 are based on empirical evidence; hence, they cannot be stated as formal theorems. When being framed as a formal theorem, all assumptions have to be stated inside the theorem.

We appreciate the reviewer’s valuable feedback. However, Theorems 4.1 and 4.2 are truly formal theorems, not empirical findings. Here, empirical evidence is used solely to further validate the correctness of these analytical solutions.

For Theorem 4.1, it derives the analytical expression for the expected noise norm of the intervention component, based on algebraic properties of covariance and projection matrices. Theorem 4.2 similarly computes the mean activation shift $\Delta \mu^{(l)}$ and its layer-wise amplification in transformers, derived from mathematical properties of attention mechanisms, GeLU non-linearities, and residual connections, rather than empirical evidence.

The reviewer’s concern may stem from less precise phrasing elsewhere, such as the case in W1&Q1-F4. To avoid any misunderstanding and enhance readability, we will revise Theorems 4.1 and 4.2 to explicitly include the assumptions and premises (e.g., low-dimensional manifold, transformer architecture specifics) in their statements. Thanks for your reminder again !

W1&Q1-F4. Step 1 in the proof of Theorem 4.1 shows equality between an expectation and an empirical estimation of the covariance matrix.

We apologize for the imprecise expression. The correct statement should clarify that the empirical covariance estimate converges to the theoretical expectation as the datasets $D_{\text{redundant}}$ and $D_{\text{concise}}$ grow sufficiently large and representative, i.e., $\mathbb{E}[r^{(l^\star)} r^{(l^\star)\top}] \approx \frac{C^{(l)}}{|D_{\text{redundant}}|} + \frac{C^{(l)}}{|D_{\text{concise}}|}$ in the limit of large sample sizes. We will revise the manuscript to include a clearer and more rigorous description.

W1&Q1-F5. Eq. (4) is missing the cross terms.

As stated in our response to W1&Q1-F1, $r_{\text{overthinking}}$ and $r_{\text{other}}$ are orthogonal due to PCA's orthogonal projection, satisfying $\langle r_{\text{overthinking}}, r_{\text{other}} \rangle = r_{\text{overthinking}}^T r_{\text{other}} = 0$, and $\langle r_{\text{other}}, r_{\text{overthinking}} \rangle = r_{\text{other}}^T r_{\text{overthinking}} = 0$. As a result, the cross terms vanish, allowing us to eliminate them in Eq. (4).

W1&Q1-F6. The covariance matrix in Eq. (5) is missing the average on the LHS: $\frac{1}{N-1}(A-\overline{A})(A-\overline{A})^T$.

We apologize for the missing the average on the LHS and will correct it in the revised version instantly. Thanks for your reminder again!

W2&Q2. Apart from the formal flaws, the paper only focuses on DeepSeek-R1-based reasoning traces.

Thanks for your suggestion. To address this concern, we have extended our evaluation to include three additional models: Skywork-OR1-7B, Qwen3-8B, and QwQ-32B. Here, we also compare our manifold steering against the baseline SEAL method on GSM8k and Math500. The results are as follows, where we could find that our approach consistently outperforms SEAL across all tested models, exhibiting superior performance and robust model generalizability.

Q3. Line 150 states that the intervention is applied across all layers. However, Lines 264-265 mention only single layers to be adjusted. Please clarify.

Actually, in Lines 264-265, the metioned layer refers to the layer used to compute the steering direction, not the layer where steering is applied. During the inference stage, steering is indeed applied to all model layers, consistent with the approach described in Line 150. We will further clarify it in the revised version. Thanks for your reminder!

We once again appreciate your time reviewing our work and rebuttal, and we’re glad to see that our explanation has addressed your concerns! We will include the above explanation in the revision properly.

We greatly appreciate your recognition of our innovative discovery of the "low-dimensional manifold" within a steer vector, fairly comprehensive experiments, as well as the paper’s clarity. Below, we address your concerns in Weaknesses (W) and Questions (Q). We sincerely hope that you can find our response satisfactory.

W1. The font size used in the figures is too small to read

Thanks for your suggestion. We will increase the font size of figures in the revised version to ensure clarity and better readability.

W2. Decomposing activations or parameters and performing steering using selected low-dimensional components has been extensively studied in prior works, for example [1, 2]. However, the discussion of these related studies is insufficient.

Prior work, such as [1], uses spectral filters based on singular value decomposition (SVD) of embedding and unembedding matrices to partition the residual stream spectrum in transformer-based LLMs. While their method suppresses noise, they mainly focus on explaining attention sinks through "dark signals" in tail-end subspaces. In contrast, we specifically address intrinsic mitigation of overthinking in reasoning models. In addition, we also provide a theoretical analysis on the expected noise norm in high-dimensional spaces (Theorem 4.1).

For [2], it innovatively utilizes confident directions derived from user history activations and filtered through probabilistic classification with logistic regression for noise reduction, emphasizing robustness in multi-preference alignment scenarios. Different from it, our approach mainly relies on the low-dimensional manifold projection to eliminate interference noise, targeting intrinsic activation patterns for reasoning efficiency rather than behavioral preferences, thereby enabling precise interventions across diverse LRMs.

Lastly, thanks for your suggestion and we will add detailed discussion with these related studies to the revised version.

Q1. The Implementation Details section (Lines 264–265) specifies the layers used to implement Manifold Steering. However, it lacks an explanation for why these particular layers were chosen or a discussion on the sensitivity of layer selection. Could you provide some insight or illustration to justify your choice?

We are sorry for the lack of implementation details for layer selection. In this work, we follow prior work SEAL and conduct a hyper-parameter search on the validation set rather than the test datasets to identify the optimal layer. We apologize for not including this detail in the manuscript. (we will add the whole search results in the revised version) The results show that the second-to-last layer of the model typically yields the best steering direction. This is reasonable, as representations in later layers tend to become more linear due to the cumulative effects of residual connections and low-norm block contributions (also corresponds to [1,2]), which facilitate stable and interpretable steering while preserving high-level semantic information.

[1] Razzhigaev et al. Your Transformer is Secretly Linear

[2] Hernandez et al. Linearity of Relation Decoding in Transformer Language Models

Q2. You manually select the values of $\alpha$ for different models based on test performance and generation length. This is a form of data leakage and gives your method an unfair advantage over the baselines. I remain unconvinced by the experimental fairness unless a method for selecting that is independent of test accuracy is proposed.

Actually, the way we use for selecting $\alpha$ is following the baseline method SEAL, which selects $\alpha$ on MATH500 and the $\alpha$ is not tuned specifically for other test datasets. As shown in Tab. 1 and Fig. 3, our method demonstrates strong generalization across different datasets with the same $\alpha$ value.

To further improve clarity, we will include a specific note in the revised version to highlight that $\alpha$ is selected solely based on MATH500, reinforcing that our approach is independent of test accuracy on other datasets.

Q3 & Update. Could the underlying mechanism by which your method reduces overthinking be the simple suppression of transitional tokens such as "wait" or "alternatively"? Figure 4 shows that as α decreases, the token "wait" appears more frequently. [1] noted that such tokens tend to carry high entropy. They may share significant features on a low-dimensional manifold—which may coincide with the one your method identifies. Or, could you report the performance of a model that simply removes "wait" and "alternatively" from the vocabulary? Would it be comparable to your method?

Thanks for your suggestion. First, we want to clarify that Figure 4 is used to illustrate the bidirectional effect of the $\alpha$ parameter, i.e., positive $\alpha$ values drive our method to mitigate overthinking, producing concise outputs, while negative $\alpha$ values enhance overthinking, leading to verbose generations. Your observation that "as α decreases, the token 'wait' appears more frequently" aligns with our demonstration of amplified overthinking in the negative $\alpha$ regime. Howover, when the $\alpha$ changes to be positive, in the mitigation direction, we will instantly observe a significant reduction in high-entropy transitional tokens like "wait" and "alternatively," consistent with [1]'s findings, effectively reducing unnecessary deliberation.

As you suggest, We also test the performance of removing "wait" and "alternatively" from the vocabulary on R1-7B with GSM8k. This approach reduce the tokens by 31%, which is less effective than ours, 62%. This may be because suppressing specific tokens shifts probability to other redundant words, failing to address all overthinking patterns. In contrast, our method more fundamentally captures the essential overthinking modes.

We greatly appreciate your recognition of the importance of mitigating overthinking, as well as your positive feedback on our mainfold steering's simplicity, effectiveness, and broad applicability. Below, we address your concerns in Questions (Q). We sincerely hope that you can find our response satisfactory.

Q1. In (2), does $x$ refer to "prompt + response" or only "prompt"? Also, is (3) applied to each decoding step or only at the first decoding step?

In Eq. (2), $x$ refer to "prompt + response" and Eq. (3) applied to each decoding step.

Q2. In (5), why $C = \frac{1}{N - 1}A(A - \bar{A})^T$ instead of $C = \frac{1}{N - 1}(A - \bar{A})(A - \bar{A})^T$ as the latter one is the definition of the covariance matrix?

Thank you for pointing this out. We are sorry for this mistake and will revise it instantly. The expression in Equation (5) should indeed be $C = \frac{1}{N-1}(A - \bar{A})(A - \bar{A})^T$.

Q3. In Theorem 4.1, it states $C^{(\ell)}$ is for the redundant dataset, but in line (195) it says $A^{(\ell)}$ is calculated using all reasoning data. Which one is correct?

We sincerely thank the reviewer for pointing out this mistake. The correct statement, as described in line 195, is that $A^{(\ell)}$ is calculated using all reasoning data. We will revise Theorem 4.1 in the revised version. Thanks for your reminder again !

Q4. In Line 240, $P = UU^T$ where $U \in \mathbb{R}^{d \times k}$ and $k << d$, then how could $P = I$ (it cannot be full-rank)?

We sincerely appreciate the reviewer for identifying the issue with the statement $P_M = I$ in Line 240. We must acknowledge that stating $P = I$ is a mistake, as $P = UU^T$ with $U \in \mathbb{R}^{d \times k}$ and $k \ll d$ cannot be full-rank.

Our intended explanation is that when $r^{(l^\star)}\_{\text{new}}$ is transformed by the projection matrix to become $r^{(l^\star)}\_{\text{new}}=P_M r^{(l^\star)} = r^{(l^\star)}\_{\text{overthinking}}$, the new $r^{(l^\star)}\_{\text{new}}$ satisfies $r^{(l^\star)}\_{\text{new}} = P^{\text{new}}_M r^{(l^\star)}\_{\text{new}}$, i.e., $(I-P^{\text{new}}_M)r^{(l^\star)}\_{\text{new}} = 0$, since $r^{(l^\star)}\_{\text{new}}$ lies within the subspace $M$. However, we erroneously imply that $I - P^{\text{new}}_M = 0$. In fact, the reason $E[|r\_{\text{other}}||_2^2] = \text{tr}((I - P^{\text{new}}_M) \Sigma^{(l)}{\text{noise}}) = 0$ holds is because $\Sigma^{(l)}\_{\text{noise}}$ is primarily supported in $M$, resulting in zero components under $M^{\perp}$. We will revise the statement in the revised version. Thanks for your reminder again !

Dear Reviewer 9Mtj,

We sincerely apologize for an ordering error in our rebuttal. While carefully proofreading each response to ensure accuracy, we inadvertently arranged our responses in character order rather than the original review sequence. This means the responses to you and Reviewer AXAi have been reversed in order.

We deeply regret this mistake and are truly sorry for any trouble this may cause. If you could forgive us for this oversight, we would be immensely grateful.

With our sincere apologies,

Authors

We greatly appreciate your recognition of our approach’s innovative perspective on LRM overthinking, its impressive experimental results, and the paper’s clear, logical presentation. Below, we address your concerns in Weaknesses (W) and Questions (Q). We sincerely hope that you can find our response satisfactory.

W1&Q1. The evaluation is limited to the DeepSeek-R1-Distilled model family, which raises questions regarding the generalizability of the proposed method.

Thanks for your suggestion. To address this concern, we have extended our evaluation to include three additional models: Skywork-OR1-7B, Qwen3-8B, and QwQ-32B. Here, we also compare our manifold steering against the strongest baseline SEAL on GSM8k and Math500. The results are as follows, where we could find that our approach consistently outperforms SEAL across all tested models, exhibiting superior performance and robust model generalizability.

W2. The method intervention strength is sensitive to real-time task complexity, which might limit the practicality of the method.

Thanks for your comment. We fully understand your concern about its potential impact on practicality. To address your concern, we want to clarify that the intervention strength, determined through hyperparameter tuning, have exhibited impressive task generalization. As shown in Table 1 and Figure 3, the intervention strength optimized on Math500 achieves robust performance across diverse mathematical datasets of varying difficulty, including GSM8k, AMC2023, and AIME2024, as well as in code-related and subject-knowledge question-answering tasks, which verify our method’s adaptability and practical utility in real-world scenarios.

Q2. In section 3.2, the steering is applied to all model layers. However, in section 5.1, model steering is only applied to a single specific layer, why? How do you choose this layer, does it involve expensive hyper-parameter search?

We are sorry for the confusion caused by our explanation in the paper. To clarify, in Sec. 5.1, the "specific layer" refers to the layer used to compute the steering direction, not the layer where steering is applied. During the inference stage, steering is indeed applied to all model layers (following [1]), consistent with the approach described in Sec. 3.2.

Regarding the choice of the specific layer for computing the steering direction, we follow the baseline method SEAL and conduct a hyper-parameter search on the validation set rather than the test set to identify the optimal layer, which means that we do not need very expensive search cost. We apologize for not including this detail in the manuscript. We will add it in the revised version. The results show that the the second-to-last layer of the model typically yields the best steering direction.

[1] Arditi et al. Refusal in Language Models Is Mediated by a Single Direction

Q3. Table 1 shows that Manifold Steering can sometimes improve the accuracy on challenging math benchmarks like AIME24, could you share some insights behind this phenomenon?

Thanks for your suggestion. We believe this phenomenon can be attributed to two key features of Manifold Steering. First, by mitigating overthinking, our approach reduces redundant reasoning steps, thereby decreasing the number of tokens required. This allows the model to find correct answers within given token limits. Second, Manifold Steering enhances the clarity of the model’s reasoning process, as shown in Fig. 4. For complex problems, this clearer logical structure enables the model to identify correct solution paths more effectively.

Dear Reviewer AXAi,

We sincerely apologize for an ordering error in our rebuttal. While carefully proofreading each response to ensure accuracy, we inadvertently arranged our responses in character order rather than the original review sequence. This means the responses to you and Reviewer 9Mtj have been reversed in order.

We deeply regret this mistake and are truly sorry for any trouble this may cause. If you could forgive us for this oversight, we would be immensely grateful.

With our sincere apologies,

Authors