Mitigating Overthinking in Large Reasoning Models via Manifold Steering
Authors: Yao Huang
Huanran Chen
Shouwei Ruan
Yichi Zhang
Xingxing Wei
Yinpeng Dong
Venue: NeurIPS 2025 poster ()
Keywords: Large Reasoning Models
Overthinking
Mechanistic Interpretability
Manifold Steering
Primary area: deep_learning
TL;DR: We propose manifold steering that projects the steering direction of model overthinking on the low-dimensional activation manifold, effectively reducing output tokens while maintaining accuracy.

Abstract: Recent advances in Large Reasoning Models (LRMs) have demonstrated remarkable capabilities in solving complex tasks such as mathematics and coding. However, these models frequently exhibit a phenomenon known as *overthinking* during inference, characterized by excessive validation loops and redundant deliberation, leading to substantial computational overheads. In this paper, we aim to mitigate overthinking by investigating the underlying mechanisms from the perspective of mechanistic interpretability. We first showcase that the tendency of overthinking can be effectively captured by a single direction in the model's activation space and the issue can be eased by intervening the activations along this direction. However, this efficacy soon reaches a plateau and even deteriorates as the intervention strength increases. We therefore systematically explore the activation space and find that the overthinking phenomenon is actually tied to a low-dimensional manifold, which indicates that the limited effect stems from the noises introduced by the high-dimensional steering direction. Based on this insight, we propose **Manifold Steering**, a novel approach that elegantly projects the steering direction onto the low-dimensional activation manifold given the theoretical approximation of the interference noise. Extensive experiments on DeepSeek-R1 distilled models validate that our method reduces output tokens by up to 71\% while maintaining and even improving the accuracy on several mathematical benchmarks. Our method also exhibits robust cross-domain transferability, delivering consistent token reduction performance in code generation and knowledge-based QA tasks. Code is available at: https://github.com/Aries-iai/Manifold_Steering.

Mitigating Overthinking in Large Reasoning Models via Manifold Steering
Yao Huang 1.2, Huanran Chen', Shouwei Ruan', Yichi Zhang", Xingxing Weil Yinpeng Dong 2.3"
Institute of Artificial Intelligence, Beihang University, Beijing 100191, China College of AI, Tsinghua University, Beijing 100084, China Shanghai Qi Zhi Institute 4 Sttate Key Laboratory of Virtual Reality Technology and Systems, Beihang University x ky_huang, xxwei][buaa. en, dongyinpengdmail. www.illssiinnhaeeeeoee
Abstract
Recent advances in Large Reasoning Models (LRMs) have demonstrated remark- able capabilities in solving complex tasks such as mathematics and coding. How- ever, these models frequently exhibit phenomenon known overthinking during inference, characterized by excessive validation loops and redundant deliberation, leading substantial computational overheads. In this paper, we aim to mitigate overthinking by investigating the underlying mechanisms from the perspective of mechanistic interpretability. We first showcase that the tendency of overthinking can be effectively captured by single direction in the model's activation space and the issue can be eased by intervening the activations along this direction. How- ever, this efficacy soon reaches plateau and even deteriorates as the intervention strength increases. We therefore systematically explore the activation space and mani- find that the overthinking phenomenon is actually tied to a low-dimensions fold, which indicates that the limited effect stems from the noises introduced by the high- dige-dinensional steering direction. Based on this insight, we propose Manifold Steering, a novel approach that elegantly projects the steering direction onto the low-dimensional activation manifold given the theoretical approximation of the in- terference noise. Extensive experiments on DeepSeek-R distilled models validate that our method reduces output tokens by up to 71% while maintaining and even improving the accuracy on several mathematical benchmarks. Our method also exhibits robust cross-domain transfcrability, delivering consistent token reduction performance in code generation and knowledge-baat based QA tasks. Code is available at: https://github.com/ coom/Aries-ii/Maniiff iai/Manifodd _Sterring.
Introduction
Building the versatility of Large Language Models (LLMs) in text generation, particularly their emergent ability in cfain-of-thought (CoT) reasoning [41], the field is now undergoing a transition toward Large Reasoning Models (LRMs). Exemplified by the OpenAl o-series [28] and the DeepSeek- RI series [14], LRMs acquire internal capabilities for long-horizon reasoning through reinforcement learning with verifiable rewards. These models are able to explore diverse solution paths, reflect on potential errors, refine intermediate steps, and validate final outputs, mimicking the process of human problem-solving by scaling inference-time computations [28]. As result, they excel in domains such as mathematics [1, 23, 39] and coding [27, 43, 44]. This makes them well-suited for tasks that demand deep logical analysis and paves the way for their applications in complex scenarios, including web search [19] and research assistance [49].
*Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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However, despite the remarkable reasoning capabilities, LRMs often suffer from a critical efficiency issue known as overthinking [35], where they generate excessive and unnecessary reasoning steps. even for simple questions. For example, when tasked with a straightforward calculation, like "2 + 3" [9]. an LRM might redundantly validate its approach or explore irrelevant alternatives, significantly increasing the computation overloads. This overthinking not only impacts inference latencies, posing great challenges for time-critical applications, but also risks degrading performance by entangling the model in repetitive verification loops or unproductive reasoning paths [9, 16, 40J. To mitigate such overthinking in LRMs, several approaches [3, 8, 16, 22] have recently been proposed. They often utilize external mechanisms regulate reasoning and prevent overthinking, which can incur additional computational overhead for probing [16] or be susceptible to performance degradation due to the reliance external models [22]. While these methods address overthinking from external and behavioral perspectives - relying on human-designed workflows and interventions, the underlying mechanism remains underexplored, posing significant challenges achieving intrinsic mitigation.
In this paper, we address the overthinking problem of LRMs through mechanistic interpretability [50], based on an in-depth analysis of their internal states. Specifically, we attribute this phenomenon to the distinctive activation patterns in the deeper layers of the model and identify a single, interpretable direction by comparing the differences in the activations between overthinking and concise reasoning. By manipulating the activations along this direction, we can effectively steer the model away from overthinking tendencies. However, this intervention is insufficient to fully resolve the problem. As in Fig. 2(a). the reduction in output tokens does not consistently scale with increasing intervention strength. This suggests that the computed steering direction is not accurate enough and introduces unintended interference noise.
To address this issue, we further analyze the model's 's activation patterns and find that the overthinking phenomenon is intrinsically tied to a low-dimensional manifold, which can be well approximated by a linear subspace. This result sheds light on why high-dimensional steering directions often introduce noises, as they fail to align with the underlying structure of model activations. To more effectively mitigate overthinking, we introduce Manifold Steering method to align the steering direction with the reconstructed oiw-dimensional manifold. We first theoretically derive a linear approximation of the amplitudes of the interference noise and then project the steering direction by nullifying this approximated term. In this way, we can effectively purifyy the steering direction and better mitigate the overthinking issue with larger intervention strength, as depicted in Fig. 2(a).
Extensive experiments on multiple DeepSeek-RR RI distilled models [14] of different sizes verify the effectiveness of our manifold steering method. We first test it on mathematical datasets of varying difficulty, including GSMSK [10], Math500 [20], AMC2023 [24], and AIME2024 [25]. Our method achieves up to 71 %% tokens reduction while consistently maintaining improving accuracy. Moreover, exhibits robust cross-domain transferability, delivering consistent mitigation effects in tasks such as LiveCodeBench [18] (code generation) and Diamond-GPQA [32] (knowledge- based QA), surpassing existing methods in both overthinking mitigation and accuracy preservation.
2 Related Work
Mechanistic Interpretability. Mechanistic interpretability [5, 11, 26, 29, 31, 34, 37, 50] seeks to reverse-engineer the internal computations of LLMs uncover the causal mechanisms underlying their behavior, offering fine-grained insights into learned representations and decision processes. A key technique within this framework involves identifying steering directions [29, 34, liner vectors in the activation space that correspond to specific model behaviors. By manipulating these directions during inference, researchers can precisely control outputs, such as ablating refusal behav- iors in safety-critical scenarios [2, 45], Similarly, Cao et al. [7] proposed Bi-directional Preference Optimization (BiPO), leveraging steering vectors derived from contrasting human preference pairs to customize attributes like truthfulness and hallucination. These approaches highlight the versatility of steering directions in manipulating models' behaviors. Additionally, some efforts have explored dimensionality reduction in activation spaces: [6] use SVD-based spectral filtering to suppress noise in residual streams, while others derive steering directions from probabilistic classification of user history for multi-preferent in L.LMs [9, 16, 35), phenomenon characterized by redundant or divergent reasoning trajectories. By analyzing the latent space, we identify steering direction that encapsulates overthinking and
alignment [33]. Our work extends this paradigm to address overthinking
2

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further propose manifold steering, novel method that projects this direction onto low-dimensional manifold to mitigate interference noise, thereby improving its performance.
Overthinking Mitigation. Efforts [3, 8, 16, 22, 30, 42] to mitigate overthinking in LRMs have gained traction means to enhance inference efficiency and output quality. Among them, the griiiin-bbsed method [30] tend to modify the reward function for length control in reinforcement learning. However, these incur significant computational costs and are orthogonal inference-time interventions like ours, thus warranting no direct comparison in this work. Existing training-free methods mainly rely on external mechanisms to regulate reasoning. For instance, Dynasor [16] employs periodic monitoring to detect and halt redundant reasoning, incurring computational over- head, while Thought Manipulation [22] uses auxiliary models to guide inference, limited by the external model's 's performance. These shortcomings suggest that a more fundamental solution lies in understanding and modifying the model's internal reasoning processes. Though some concurrent works [3, 8] have tried to leverage mechanistic interpretability for achieving it, they only partially reduce overthinking, quickly encountering bottlenecks due to interference noise in bigh-dimensional steering directions. In contrast, we propose manifold steering to project the steering direction onto low-dimensional manifold, effectively eliminates interference noise, achieving superior overthinking mitigation and substantial token reductions across diverse tasks, as demonstrated in Sec. 5.2.
3 Mechanistic Analysis of Overthinking
In this section, we investigate the phenomenon of overthinking within the activation space of Large Reasoning Models (LRMs) and identify a general mechanism by which ablating a single direction in the activation space can reduce redundant reasoning steps some extent.
3.1 Background
Transformers. Decoder-only transformer language models [21, 38] map an input token sequence x [..,...,, IT] to a probability distribution over the vocabulary for next-token prediction. Each token x4 is associated with sequence of residual stream activations h()) (2;) ) E IRd across layers, initialized by the token embedding h(") (x.) Embed(x;;). At each layer (z:) is updated by combining the previous layer's activation h hd'-)(x,,) with two components: (i) a multi-head self-attention mechanism, which computes (0) (z1x) by attending to prior tokens {I, : I i} using a causal mask to enforce autoregressive context flow; and (ii) multi-layer perceptron (MLP), which applies non-linear transformations to the post- post-attention state h h(--)(().) + + (x1:) and produces m
{I....,L), L), the
residual stream (1))
m (xr.). The whole update is expressed as follows:
h (x:) = (x-)) + (II.) + m (1) (x:), m (x:) = (LP(h""-)(x.)) + (x1a)) (I)
Through autoregressive aggregation, each (0())) aggregates context from prior tokens, with the final token's residual stream (5) (z) : h() (IT) encapsulating the entire input's 's context.
Large Reasoning Models. LRMs are tailored for complex problem-solving and mnstruction-followiigg
which leverage structured templates handle user inputs:
<lbegin_of_ > User (
where the content following <think>> comprises the model's reasoning process and final answer, separated by </think>, Despite the excellent reasoning capabilities of these models, they often exhibit the overthinking phenomenon [9, 12] during the reasoning process, characterized by repetitive validation or redundant deliberation. As a high-level cognitive phenomenon, overthinking may manifest in the model's 's residual stream activations, similar to other abstract concepts such as safety [5] and honesty [50], as widely studied from the perspective of mechanistic interpretability. This suggests that overthinking and concise reasoning exhibit distinct activation patterns. the next section, we systematically examine this hypothesis and investigate whether these activation differences are sufficient to identify specific direction that characterizes overthinking - one that, if isolated, could be ablated to improve reasoning efficiency.
3

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Loyer
Layer
Layer
Layer
Layer
Layer
-10
-14
X
Comparati
Componet
Companee
Figure I: Visualization of residual stream activations ( (x)) for Dredundant and conniss across different layers of BaoSSee-RR--istil----0 redundant and concise data, while middle-to-lat layers exhibit distinct separation.
(R1-7B) Early layers show considerable overlap between
3.2 Extracting and Ablating an Overthinking Direction
Before extracting overthinking direction, we first investigate whether the residual stream activations corresponding to redundant and concise reasoning are separable in the model's activation space, as this is a necessary condition for identifying meaningful and controllable direction. Drawing on [16], we also focus on mathematical problems, where the overthinking phenomenon is particularly pronounced. To construct representative data, we first randomly sample questions from the OpenMathhnstruct-2 training set [36]. For each model, five responses per question are independently generated. Based on these responses, we construct two model-specifie datasets 2 as follows:
Redundant set - consists of questions for which all five responses exceed 16k tokens and contain hesitation keywords (e.g., "waif", "alterratively", etc.) surpassing specified number.
Concise set D D consists of questions for which all five responses are under 1 tokens and contain none of the hesitation keywords.
As demonstrated in Fig, 1, we visualize the distribution of residual stream activations (x)) for both D Dredundant and Deancis across different layers of RI-7B. We observe that, while early layers exhibit substantial overlap between the two distributions, the middle-to-late layers display clear separation. This separation indicates that the overthinking phenomenon is more prominent in specific layers and provides empirical support for identifying a meaningful overthinking direction.
We use the difference-in-menne in-manns technique [4] for extracting the steering direction, which computes the mean difference in residual stream activations between the redundant and concise data for each layer The overthinking direction r!)) is then defined as:
Σ
(x)
Σ h (x),,)
(2)
Dredundant
Deoncise
2EDostanar:
IE Demine:
where (x)) denotes the residual stream activation of the final token of input x at layer 1, with x being the prompt concatenated with the response for x E Dreduumamt but only the prompt for
Deoncise. The direction (!) is normalized to unit length, i.e., ((()) =r())) rr"//|/"" /||""""""] I|2- Following [2],
we also select the single most effective direction r!!") and apply for intervention across all layers.
Finally, to further explore the role of the overthinking direction ((") ) in the model's computations, we ablate the component aligned with r ((") each residual stream activation h. Specifically, the modified activation h' computed as:
h a x (x")) Th
where a controls the intervention strength. We apply this ablation to every activation h (x.,) across all layers / and token positions i. The parameter allows adapting the extent of overthinking mitigation, balancing the reduction of redundant reasoning with the problem-solving accuracy.
Manifold Steering for Robust Intervention
Following our mechanistic analysis in Sec. 3, which identifies single direction capturing overthinking in the model's activation space, we proceed to explore whether increasing the intervention strength a further reduces redundant reasoning, as expected.
Details on data selection and dataset composition provided in the Appendix A.

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4.1 Low-Dimensional Manifold Analysis
To rigorously evaluate the effect of increasing intervention strength a for the direction r((), derived via Eq. (2), we test R1-7B's performance the Math500 dataset [20], diverse mathematical test set, different from the anchor dataset used in Sec. 3.2.
Formulation of Interference Noise. As illustrated in Fig. 2(a), increasing the intervention strength a initially reduces the token count. However, beyond a certain threshold, the token count ceases to decrease, and continues to increase beyond 1.5, it rebounds, even nearly returning to levels observed without intervention. This suggests that the intervention direction r((") may be imprecise and introduces unintended noise in the model's activation space, which defined as interference noise. Mcanwhile, we confirm the model collapse caused by interference noise, as below:
Instruction: What power of is equal to 8? Express your answer Response: 10000000000000000
Thus, we can hypothesize that the r r!'") computed via the difference-in-meen method in R com- prises both the overthinking direction r Foverthinking and an orthogonal interference component Tothers such that p(())
Foverthintingg + Fother- The Eq. (3) actually modifies the activation as follows:
h h (x:) h0 a
(Toverlinnknng) T
"h" (x) Noverchinking
(Fother) "h (2x: (xi))ooter
overthinking component
interferrncee companent
The Torher term perturbs h (d (xxx) potentially disrupting unrelated capabilities such as normal expres- sion, especially for large This interference explains the token count rebound beyond 1.5, as the intervention affects dimensions irrelevant to overthinking.
Linear Low-Dimensional Manifold Verification. As shown above, the direct computation of difference in digh-dimensional activation space leads to noisy estimation due to the existence of the interference part Fother- straightforward solution is to estimate the amplitude of Fother and remove its influence from the steering direction r r((") Hooeeer, it is orthogonal to the overthink- ing direction r Fovernhinking and is decided by space of Toverthinking Inspired by prior work [15] that the activations in LLMs reside on cin-dimeniional manifold M IR is reasonable as-
Diod Stariig
1766
2007
Tokens
134
(o) Perfonmances on Math500
sume that I ovvrrhinking. representing the shift between activations of redundant and concise reasoning paths, also falls into this mani- fold. To verify this, we employ simple linear method Principal Component Analysis (PCA), on the activations from the complete reasoning dataset D Dreasoning D Dredundant U Deonsise at layer l, Let (1) (x)) .... (xrw) RdxN denote the matrix of activation vectors h (x:)) Rd for inputs I; Dreasoning- We compute the covariance matrix and its eigendecomposition as:
Commiattee Vantoce Hartes
6:728
e 2 8 2
Numther Principal Compananti
(b) Cumulative Variance Activation Space
1 COO N - (A (A AA = U A ( (U"),,
Figure (a) Performance of R1-7B with varying for di- (5) rect and manifold steering on Math500. (b) Cumulative vari-
where A (1) N EN (x:), U E Raxd contains the prin-
cipal components, A = diag(A (X))), ..... \ and VR(k)
ance ratio of R1-7B's 's activation space Dredundant across differ- is the variance ratio. As Fig. 2(b) shows, the top k = 10 ent hidden layers.
Σ (0
D4 X
components account for over 70% of the variance, indicating that the effective dimension of M. denoted derr. is significantly smaller than the ambient dimension This confirms the low-dimensional structure of M. This also suggests that the linear manifold composed by the orthogonal basis effec- tively captures the activations of reasoning trajectories and therefore overthinking direction r Toverthinking
Thhe orthogonality of Fovemiimting and Father property arising from the principles of PCA.
5

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can be estimated using simple linear dimensionalitty reduction in this subspace. Eventually, this finding supports our earlier hypothesis (Eq. (2)) that the steering direction r((') Foverthinking + Kother includes an orthogonal interference component Torher, which falls into M-
4.2 Theoretical Analysis of Interference Noise
As discussed in Sec. 4.1, the overthinking phenomenon is tied to the low-dimensional manifold structure of the activation space. The overthinking direction r(("), computed via Eq. (2), introduces an orthogonal interference component rotker due to computation in high-dimensional spaces. When activation dimension d far exceeds the sample size N (d > N), interference noise even accumulates in M- inflating Lother and disrupting the model's other normal abilities. To further clarify the potential effects, we quantify this interference noise in the following theorem.
Theorem 4.1. (Proof in Appendix B) Lei PM = U 1 (, 1 kJ)" be the projection ma- trix onto the low-dimensional manifold M, where U [. k] contains top-k principal components of the activation covariance (() for D Dredundant and D Deoncise- As the sample sizes grow sufficiently large, the expected noise norm of Yother is:
C(
CC
E Tother 11 =t -r PI- noise)
! Hoise
(6)
concissl The trace is significant, indicating that the interference noise is substantial and is greatly likely to disrupt the model's 's normal abilities.
D Predundional
The significant noise norm of Tother, established in Theorem 4.1, suggests that interventions using r!(") introduce considerable perturbations in MM. Moreover, these perturbations can propagate through layers, amplified by attention mechanisms, non-linear activations, and residual connections, leading to more substantial shifts in the activation distribution. To understand this, we further analyze the mean activation shift and its layer-wise amplification in the following theorem.
Theorem 4.2. (Proof in Appendix B) Let (")) and Tother be as in Theorem 4. and let the intervention be applied as in Eq. (4). The mean activation shift at layer I and its amplification through layers are:
ap = -a
| (1) x ax ac|/ooher (T (x:))
(7)
N
(A 112 2 x11 Colam Yo o min (( "h [I
(8)
where is the intervention strength, h (x:)) is the activation at layer 1. W(++) denotes the combined MLP and attention weights, Nau and %o are the minimum amplification factors of the attention softmax and GeLU non-linearities, amin (Wi())) is the minimum singular value of the weights, and 7 > is the layer-wise amplification factor.
The shift in M is significant and grows through layer-wise propagation, driven by attention and non-linear transformations, severely disrupting the model's other normal abilities.
4.3 Manifold Steering
The interference direction Toreers quantified Theorem 4.1, causes activation shifts that amplify through transformer layers and disrupt reasoning (Theorem 4.2). To eliminate this interference, simple but effective approach to set Eq. (6) to O. Based on this insight, we propose Manifold Steering, which projects the direction y!!") onto M to mitigate Pather-
Formally, let UOO) E Rd denote the top-k principal components of the activation covariance in Eq. (5), spanning M. The manifold direction is obtained by:
(!)
overnhinking = -PMr(")) = Ull (1)
(())
r (!)
overthinking
overthinkknng
overthinking ll2
where P. = (()) (U!!) is the projection matrix onto M. The reason why the interference norm
off off (!)
E|[|r oother [2] = tr (I P PMM
holds is because (1)
noise now primarily supported in M,
noise

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Table 1: Performance of Manifold Steering compared to Vanilla, Dynasor, and SEAL on GSM8K, MATH500, AMC2023, and AIME2024 for varied LRMs. Metries include Pass @ (1) and #Tokens (L). Changes relative to Vanilla are shown in yellow for Pass I and blue for #Tokens.
AMC2023
Methods
Pase (f. S)
#Tokens (1)
Passet (f. 5)
#Tokens (.)
Pass <f. %))
#Tokens (i)
Pask <f. %))
#Tokens (3)
Vanilla
76.7
2005
76.4
4762
70,0
7089
26.7
11352
R1-1.58
Dynasoe
SEAL
76.9 (40.2)
77.1 (60-4)
1055 (-49%)
1076 (-47%)
77. (+1.9)
(10.8)
3721 (-225)
3694 (-2218)
72.5 (12.5)
70.00-0.0)
6418
(-8%)
26.7 (-0.0)
26.7 26000)
10864 (-75))
(89)
(42.2)
3458 (-27%)
72.5 (:2.)
6236 (-12%)
360
Vanilla
87,5
1143
88.2
3824
87.5
5871
50,0
10784
R1-78
Dynasoe
SEAL
(4D.)
K7.7 (40.2)
839 (-325)
732 (:36%)
88 $9.2 (10.0)
87.8 (40.)
2651 (:345)
(-2918)
(-.0(22.))
(-.0
(-19)
(-139)
46.
467(-333
9394 (339)
9864 (-9%)
440 (462%)
2239 (-42%)
(-B.0)
4440 (-24%)
(-22%)
Vanilla
82.7
1217
87.8
4009
85.0
5723
33.3
L1278
R--8
Dynasee
SEAL
82.9 (40.2)
K2.7 (+0.0)
749 (-38%)
826 (:3258)
SK (10.))
87.4 (-0.)
3171 (-219)
3091 (-25%)
(-.0--00)
82.5 (-2.)
4731 (-175)
(-12(-
46.7 (+13.9)
46.7 46.7 (613.4)
9789 (-13%)
9901 (-12%)
(:55%)
2873
850 (10.0)
4400
(-23%)
(-16.7)
Vanilla
93.2
742
92.8
3496
90.0
5484
66.7
9986
R1-148
Dynasee
SEAL
93.6 (40.2)
(-0.2)
996 (-2088
583 (-219)
92.6 (-1.2)
92.8
3139 (-18)
(-85))
92.5 (12.5)
87.5 (2.5)
4817 4 (-12%)
(-188-
63.3 (3..))
8563 (-148)
8941 (-11%)
92.8 (10.0)
2074
4061 (-26%)
OOrgiel Stoerig
14:00
13:08
B1:15
R1-73
61-83
81:14
LiveCodeBeach
GPQA--Homan
Figure 3: Cross-domale performance of Manifold Steering for overthinking mitigation on Live- CodeBench (code generation) and GPQA-Diamond (disciplinary knowledge).
resulting in zero components under MM Le., M successfully eliminating perturbations.
h (x)) = h 0 (x) a x r (1]
"verthinling" (r" overnhinkking! (h(").)).
(10)
The performance of manifold steering is shown in Fig. 2(b) and Sec. 5.2, where we find that, unlike the original paradigm, our manifold steering enables sustained reduction in token count.
5 Experiments
5.1 Experimental Setups
We begin by briefly outlining the baseline methods, target LRMs, evaluation datasets, and metrics. For more detailed descriptions of the experimental settings, please refer to Appendix A.
Baseline Methods. We compare our manifold steering with two latest baselines, including Dyna- sor [16] and SEAL [8], both chosen for their ability maintain the model's original accuracy. For their settings, we both adopt its official setting. To be aware, Dynasor's 's early stopping often omits the -oolvig- process the final answer, which is impractical for real-world applications. Thus, we require the model to provide a complete solution upon stopping.
Target LRMs. For comprehensive evaluation, we select the DeepSeek-RI -DDsttilled series [14], comprising models of varying scales and architectures: DeepSeek-R --------lll--eeaar (R (R1-1.5B, R1- 7B, R1-14B) and -iirrSeck-R--Disii--LLAL- temperature of 0.6, top-p of 0.95, and a maximum token limit of 16,384.
(R1- All models use recommended settings:
Evaluation Datasets & Metrics. To evaluate the effectiveness of Manifold Steering, we include mathematical datasets of varying difficulty: GSM8K [10]. MATH500 [20], AMC2023 [24], and AIME2024 [25]. To further verify the transferability,
use BineCodeBench [18] for code generation
and GPQA- A-Damoon [32] for expert-level disciplinary knowledge. All datasets are evaluated using Pass 1 as the task-solving metric and the average token count (# (#Tokens) for overthinking mitigation.
Implementation Details. The data for computing steering directions filtered using the method outlined in Sec. 3 on the OpenMathInstruut22 dataset [36]. For each model, we specify the layer
7

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Convert the point (0.3) in rectongular coordinates polor coordinotes. Enter onswer the form (r. 8), where? Dand S < 3/4
Standard
Tokemss: 1475
Ocyy, convert rectongutar coordinate (0.3) poir coordinates, Himm, remember that poior cocrdinttes.... people
remember sometimes....
So, pretty
(3. correct
representation point
0.3
Tolkess: 538
Histl need to deteriin the polar coand nates 6J for the rectangudar point 10, 31. findina use the.... Next.) calculate late the angle.... la convert the rectangudar coordinats... Combining the valuee of and (o 0) m/]) Finsl Answer....
11.5
Takens: 501
detormine the...
calculate ingle... convort the cotongubrr coordiates.... Combining values 8, polar coontinates n/2)
Tokems: 1708
Al Alniht, voi need to compent the rectongsfar coordinattn (s, 31 polar coordiuatis. itmm, along, remeebber... Hmm, arcteng319) Wiett distion by zeno undelined, timm,
that might be problum. .... Wait, that correct? Let me double-check-.. Warf, another may think about a:ll wore to dhew lin.... But just to actras sure, lst me recall the convension formulak... Wast, just make sure didettt mis anything let recall another way to find B.... think confident that the polar coordinatos are (5, n/2). Walt, another thought. 111 were pier this.... Thus, the polor coordirates are (3, x/2)."
Tokems: 228-1
Ainght, Hmm, just plag
rethoggiie
poinr
making
Alversttiell,
camptte.... Albemativelly. male Idddn't any mistalees me recall another to... Okay, double that.... Walt, to thorough think about another This, polor coordinates are: m/2)
Figure 4: An example of steering overthinking in model outputs. Forward steering yields concise, confident responses, eliminating hesitant phrases, while reverse steering induces verbose outputs.
used compute the steering direction and the intervention strength follows: R1-1.5B (layer 27, a = 0.7), R1-7B (layer 27, a = 0.3), R1-8B (layer 31, a = 0.5), and R1-14B (layer 47, a = 0.3).. During inference, this direction is applied to all layers as stated in Eq. (3).
5.2 Performance of Manifold Steering in Overthinking Mitigation
We conduct experiments on four mathematical datasets of varying difficulty using four LRMs with different parameter sizes and architectures. Table presents the results, where models are evaluated for accuracy and redundancy reduction. Based on Table 1, we draw the following observations:
Manifold steering achieves the best performance across all models and datasets. Our method consistently outperforms baselines on four out-of cit-off-istte oi-distribution mathematical datasets, with particularly strong results on GSM8K, where it achieves token reduction of 41 % 71% while preserving accuracy. This is reasonable, as unlike Dynasor, which relies external monitoring of the model's 's certainty, our method modifies model outputs at the more fundamental feature level to mitigate overthinking behavior. Moreover, Dynasor's reliance on external monitoring brings extra computational overhead. We include a comparison of average time cost to demonstrate this drawback in Appendix E, while our method incurs nearly no latency. Compared to SEAL, our manifold steering effectively reduces interference noise, yielding a more precise direction. Additionally, we observe that the overthinking phenomenon diminishes to some extent as model parameter size increases, which is expected, as some overthinking stems from models' inability to solve complex problems.
Overthinking is more pronounced in simpler problems. As presented in Table 1, all methods exhibit more effective overthinking mitigation on simpler datasets, with GSM8K and MATH500 (~ 40%) showing greater token reduction compared to the more complex AMC2023 and AIME2024 datasets (~ 20%). This suggests that overthinking is more pronounced in simpler problems, which is reasonable, as complex problems inherently require larger token budgets and may exceed the models' internal capabilities, thereby constraining mitigation effectiveness.
5.3 Cross-Domain Transferability for Overthinking Mitigation
To further investigate the transferability of manifold steering for overthinking mitigation, we assess its performance across two distinct domains: code generation and discipline-specife knowledge, both separate from the mathematical domain used for steering direction extraction. We utilize two representative datasets: 1) BiveCodeBench [18], benchmark of coding challenges that probe algorithmic and programming expertise, and 2) GPQA-Diamond [32], carefully curated dataset of challenging multiple-choie questions targeting expert-level disciplinary knowledge across various

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fields. As shown in Fig. 3, our manifold steering achieves token reduction of 12% ~ 27% across both datasets while maintaining accuracy, demonstrating the generalizability manifold steering to diverse domains. This cross-domain effectiveness offers multiple benefits: it incurs no additional computational overhead, adapts seamlessly to varied problem structures, and effectively mitigates overthinking without requiring domain-specific fine-tuning.
5.4 Directional Analysis and Hyperparameter Tuning
In this section, first explore the precise impact of the direction computed by manifold steering on model outputs through case studies, analyzing how steering and its reversal affect response characteristics to better understand directionality's role. Then, we conduct hyperparameter tuning.
Directional Analysis. As shown in Fig. 4, applying the steering direction for overthinking mitigation leads to model outputs that are significantly more concise and confident. Specifically, overthinking behaviors, such as hesitant phrases (e.g.. "wait"), frequent shifts in reasoning (e.g., "alternatively". and repetitive self-checking, are largely eliminated. The model generates streamlined responses with clear, focused reasoning, delivering direct outputs. This effect strengthens as the intervention strength increases 0.5. In contrast, when reverse steering is applied, the model becomes markedly more hesitant, often repeatedly checking. This leads to verbose outputs filled with excessive caution. Thus, it is crucial to underscore the role of directionality for overthinking.
Hyperparameter Tuning. We use R1-7B model on MATH500 for this analysis, with results for other models in Appendix C. As shown in Fig. 5, our manifold steering direction demonstrates ef- ficacy at much lower strength of a 0.1. As increases, token counts continue to decrease, with a remarkable 77.1% 1% reduction observed at 0.5. This substantial token reduction highlights the purity and effectiveness our steering direction in mitigating overthinking. However, excessively rapid reasoning, induced by intervention strengths, can hinder the model's ability to thoroughly address complex problems, a phenomenon also observed in human cognition, leading to a decline in accuracy. To balance the trade- off between overthinking mitigation and maintaining accuracy, select an intervention strength of 0.3 as the optimal value for robust performance.
Talees
P1A
Figure 5: Hyperparameter tun- ing for strength a in R1-7B
5.5 Cross-Tass Transferability of Manifold Steering
-aseanl -
In this section, we investigate the applicability of manifold steering to tasks beyond overthinking mitigation, such as refusal feature ab- lation, to assess its cross-task transferability. Prior studies [2, 45] demonstrate that while steering directions can suppress refusal fea- tures in models, some instances persist unless intervention strength increased, which risks model collapse. Here, we apply our man- ifold steering method using the Qwen2.5-7B-Ine -nsstruct the target LLM, computing the steering direction with the same data as [2]. As shown in Fig. 6, method achieves a 100% jailbreak success rate (JSR) on AdyBench [51] while the baseline [2] obtains a JSR of 74% (a 2.0), with all responses verified valid through manual check, which further validates the robust transferability of manifold steering across diverse tasks and underscores the urgent need for enhanced safety efforts [13, 17, 46, 47, 48] to ensure responsible AI.
131
50
[SR (%)
60
54
34
20
0.5
1.2
1.6
Figure 6: JSR of baseline and ours on Advbench.
Discussion and Limitations
Our proposed manifold steering method has demonstrated robust effectiveness in mitigating overthink- ing, as evidenced by significant token reductions across varying LRMs. However, its applicability to multi-modal large language models remains unexplored. Additionally, while our approach excels in controlling overthinking with minimal accuracy trade-offs, its interaction with highly specialized tasks, such as domain reasoning, e.g., legal medical analysis, warrants further investiga-

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tion. Moreover, the sensitivity of our method to varying intervention strengths suggests potential for optimizing dynamic steering strategies, where the strength adapts task complexity in real-time.
7 Conclusion
In this work, we propose manifold steering, a novel method to address overthinking in LRMs while preserving task performance without additional computational cost. Specifically, by aligning the steering direction with the low-dimensional activation manifold, our approach effectively eliminates the interference noise based on theoretical analysis. Extensive experiments across diverse models and datasets confirm substantial token reductions and robust cross-task transferability. These findings underscore the potential of manifold steering enhance model efficiency and adaptability, opening new avenues for improving LRMs.
Acknowledgments and Disclosure of Funding
This work was supported by NSFC Projects (62576020, 62276149) and was also supported by the Fundamental Research Funds for the Central Universities.
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A
Implement Details
In this section, we will introduce the implementation details of our approach, focusing on the data selection process for computing the steering direction to mitigate overthinking in each model and the specific configurations of the baseline methods used for comparison.
A.1 Data Selection for Direction Computation
To construct representative datasets for computing the steering direction to mitigate overthinking, we begin by randomly sampling 20k questions from the OpenMatthnstruut-22 training set [36]. For each model, we generate five independent responses per question using the official sampling configuration: a temperature of 0.6, top-p of 0.95, and maximum length of 16k tokens. These responses form the basis for constructing two model-specific datasets, the Redundant set (D (Dreundant) and the Concise set conncis)) as described below:
Redundant set (Dredundant:: This dataset includes questions where all five responses exceed 16k tokens without terminating and contain more than 20 times of hesitation keywords (e.g.. "wait", "alternatively". etc.). capture meaningful overthinking behavior, we process the responses using the following template, truncating the response the occurrence of the hesitation keyword:
< bbegin_ ><>< (asp///eeee/stttrtcooooaaaaaaaaa (partial_responsq) (hesitation keyword}
\n
The truncation hesitation keyword is reasonable because overthinking typically emerges after a certain point in the response, rather than immediately upon encountering the question. Moreover, through activation visualization, observe no significant differences in the activation patterns of different hesitation keywords. Thus we choose "wait" as a consistent marker here.
Concise set (Deoncise): This dataset includes questions where all five responses are under Ik tokens and contain none of the hesitation keywords. The template for these responses includes only the instruction without the response, as they inherently represent concise and focused outputs:
< Obbeginoo_ssntenenn User ([antttt/( Assin))aasssssssstlaas /><think>>>>
These selection criteria ensure that Dredundant captures responses exhibiting excessive verbosity and hesitation, while Deoncise represents efficient and direct responses, providing a clear contrast for computing the steering direction representing overthinking. To ensure high-quality data, we retain only 500 samples for each dataset after applying the selection criteria and double checking. For computing the steering direction, we follow [50] to sample 100 samples from each dataset and employ the IsolationForest algorithm to filter out outliers. For manifold subspace estimation, we utilize the entire set 500 samples from each dataset to capture the full representational structure.
A.2 Baseline Methods
As stated in Sec. 5.1, we select two latest baselines, Dynasor [16] and SEAL [8], for their ability to preserve the original accuracy in reasoning tasks. Below, we detail the specific settings for them:
General Setting. All large reasoning models adopt the official recommended settings with a temperature of 0.6, top-p of 0.95, and maximum length of 16k tokens.
Dynasor. We adopt the official settings for Dynasor. The configuration probes the model every 32 tokens with "Probe-In- In-oo--I-Thhee The-Middle" technique and injects "Final Answer" prompt each iteration to ensure complete solutions upon early termination. Generation stops when the Certaindex metric (H) exceeds a predefined confidence threshold. To be aware, Dynasor's early stopping often omits the problem-solving process in the final answer, which is impractical for real-world applications. Thus, we require the model to provide a complete solution in the final answer upon stopping.
SEAL. We adopt the official settings for SEAL [8]. using 1k training samples from the Math dataset [20] to extract the reasoning steering vector. Reasoning processes are segmented into thoughts using "nln" delimiters, classified execution, reflection, transition via keyword-bate rules
21

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(e.g., "Alternttvely"" for transition, "Wait" for reflection). The steering vector is computed at layer 20 as S H HT, where Hæ and H nr are average representations of execution and reflection/rransition thoughts, respectively. During greedy decoding, hidden states of "An'n" tokens at layer 20 are adjusted as H H 1.0.5.
B Proofs
B.1 Proof of Theorem 4.1
Proof. We derive the expected noise norm of the interference component Pothers the part of the overthinking direction r(l") in the orthogonal complement M of the low-dimensional manifold M. The theorem states:
C
oother [2) = tr ((-- (I-PMI PMM
E!)
Σ (1)
noise
Predundant
Deoncise
where PM = [ : (E, 1 A and [: : are the top-k principal components of the activation covariance C We build on prior findings that M is low-dimensional, identified via
PCA on Dreasoning = Dredundant U D concises with k = 10 capturing over 70% of the variance, validating the linear manifold assumption.
Step 1: Define the overthinking direction ,(),, Per Eq. (2), ((()) = Foverthinkinn + Tothers where
T Toverthinting E .M captures the shift between redundant and concise reasoning, and Fother E MM is interference. We model:
(wxs))
(x,).
Predundant
Conncise
2. Drnantoo
2s Dramanee
Assume activations (xx)) NH C(O), estimated over Dressoning- The covariance is:
with Predundant Monncise for each dataset, and C!!)
C(
C
+
Dredundant
[Deoncise
Step 2: Define M and derive P.M The manifold M is spanned by the top-k eigenvec-
tors of C (A - T where A (x1),,,,,.... (((,)))...,,(())),))))))). (xx)), N N Di=1 h())))))) The eigendecompositionn C(O) ( A (1) (UO)T yields [; k). and: [:, 1 T.)T.
and A
PM [:, 1: [
The projection onto M orthonormal, PM is idempotent and symmetric, so:
P.M. it removes the M-component. Since (. is
PMP
PMN PT) - PM
PCA's linear basis ensures M captures the d - k dimensions of noise, critical when d > k.
Step Define Fother- Since Foverthinking E M, the interference is:
Yother (I PM))
This isolates noise in M which disrupts normal abilities due to high-dimensional computation.
Step Compute the squared norm. Calculate:
|| [[I - PMrr"" 1 - = (() PP))
using the idempotence of P.M
Step 5: Take the expectation. Compute:
E r other (T PMDr(")) = tr((I (("))))
Substitute:
(!
(
E/ (1'))))
poise
D Dredundant
Deoncise
Thus:
E3 Yother ||] = E
(I-PM)
22

---

B.2 Proof of Theorem 4.2
Proof. We derive the mean activation shift Au at layer I due the intervention (applied in Eq. (4).) along the overthinking direction r(t'), showing its norm is proportional to or oollroueer 22 and establish the layer-wise amplification of the shift at layer ++ 1. The theorem builds on Theorem 4.1.
Step 1: Derive the mean activation shift. The intervention layer is:
h 1 (I,) = h (z,) al(r"))) T h (xx))r
with a > 0. The mean activation before intervention is:
(0 ho
and iost-intervention:
h
(x;)
1
h (xx) - (x;))r())
N
Compute:
10
11
1
((((t))Thi (h:)):))r)
N
The mean shift is:
N N
o = 100
1e
-0
(r (1))
( (2:))rrx
matching the first part of Eq. (7).
Step 2: Decompose the shift and isolate Tother contribution. Since r(()) TM + Kother with
TM 1 Fother, we can decompose:
(r T (x,) =(x) h (x:)) + (Nother) T hh (2x).
Thus:
N
1
(Tm + (Fother)
(T:) + Tother)
00
N
N
N
Σ (h0)
(rM) Th (ho)) Fother
0
N
N
1=1
[(oahher)
Th (x:))
[Koahher) Th (1) Fother.
N
Let:
S. N > (FM) (1 Sather
1
1
(rother) T (I:))-
N
Then:
O ()) WSMTM Fother
O.Shhher
Sttheerroother
Since FM L Fother:
A a² MMMMM + Sother M 1 + (a2llssroother + Sother
as + Sother) 2 || a + Sother) ) Totherl 1
Let s = S. + Sother- Then:
Au = als/y |I + || 112.
23

---

By Theorem 4.1, when the error component Father is present (i.e., ||Tothelll > 0), it contributes to the total norm. The dominant term depends on the relative magnitudes of ||r ll2 and Fowker 2 Assuming h (x;) ~ (H C(O) and [s] is positive constant, we obtain:
|
( ll2 x a: ao]]odherll2
when Fotherll2 dominates, completing Eq. (7)-
Step 3: Derive the layer-wise amplification from Tother- For layer ( + 1, the activation is: h (zz) = (wrl-))ttthh
Attn(h (x:))
and post-intervention:
(1+1) (x:) = (wil+1) Aut (m:))
where W(++1) combines MLP and attention weights, Attn the attention mechanism, and a GeLU. The mean shift is:
N
(p!l+1] (141 (41, (
h u+1 (z+),
((+11 1
(1+1) (x;).
To isolate the Fother contribution, decompose the single-input shift at layer I:
(a.) = hh (a:)
= -a((((()) h (<:))=")
-a((rm) " hh (Father) Th (Z: Tother)
The norm is:
| (z))ll2 al(r(11)) h (zx) (|T 112
= =a((rm) T (I:)) + (I (Father) Th (x:)ly || + + ||
The component from Fether be isolated by considering its contribution:
Ah 1) (x:))l2 a (Fother) Th" ( (ax)ll[Fonherll)
when the other term is significant. Propagate to layer 1 + 1:
Ah Ah(+1) (z:) = h (++1) (xr) (xx) v o' (w(+)) Aun' (x:)))())))
where Attn' and o' are the Jacobians of attention and GeLU. The attention softmax and GeLU have minimum amplification factors Yatas Yo > 0, and the linear transformation by (++1) satisfies:
[[(+1) x|22 2 amin )[x+))
Thus:
(h 2 You min W ((+ Ah Ah (x:)ll.
Focusing on the Fethe contribution:
Ah (x:))l 2 Yamo Fmin (w(++)) (Kother) T (1) (xx) Tother
The mean shift norm
N Σ (ht))
1
A
N
Assume the layer-wiss wise propagation amplifies the previous shift by 1, reflecting attention and non-linear effects across layers. Combining the amplification of the existing shift and the new rother contribution:
A (1+1) 2 r/1 A1l + Coaaan miin ( (r+t))((rrree)))) ( (1x)) | l2z
matching Eq. (8). This shows that the Fother component causes layer-wise amplification through both the accumulated shift (first term) and the direct contribution at each layer (second term).
Step 4: Analyze the amplification mechanism. The amplification factors 7 1, Vatt To 0. and non-zero Omin (++11) ensure that perturbations from Fother grow across layers. The first term 7 ( ||2 represents the propagation of accumulated shift, while the second term represents the fresh perturbation introduced at layer + 1 due to Porher- This dual mechanism ensures the shift grows across layers, disrupting the model's normal abilities.
D

---

C Hyperparameter Tuning
In this section, we present the results of tuning the intervention strength a across four models: DeepSeek- Peppeeek-R--iisee -i--iisll--wwen-I.... DeepSeek-RI RI-Distill DhistillQQeen-hee 7B, DeepSeek-R
and DeepSeek-Ree -iitt-RR---itt--eeseeee balance between efficiency and accuracy, we ultimately select 0.7 for R1-1.50 0.3 for RI-7B, = 0.5 for RI-8B, and a for RI-14B.
14B on MATH500 [20]. As shown Fig. 7, to achieve an optimal
A3-78
RE--44
Figure 7: Impact of intervention strength on the token reduction and accuracy of RI-1.5B, R1-7B,
R1-8B, and R1-14B on the MATH500 dataset
D Layer Selection for Manifold Steering
The selection of intervention layers is critical for the effectiveness of Manifold Steering. We conduct a layer-wise analysis across multiple model sizes determine the optimal intervention points. As shown in the tables below, we evaluate the performance across different layers by measuring accuracy and tokens on the MATH500. The results demonstrate that later layers consistently achieve better performance: Layer 27 for R1-1.5B and R1-7B, Layer 31 for R R1-8B, and Layer 47 for R1-14B.
Table Layer-wise performance analysis for R1-1.5B on MATH500.
Vanilla
Layer
Layer
Layer 10
Layer 15
Layer 20
Layer 25
Layer 27
Accuracy (%) Tokens
76.4
76.6
77.0
76.4
57.6 1469
74.8 3930
67.4
78.6
4762
4472
4434
4223
1179
3458
Table 3: Layer-wise performance analysis for R1-7B on MATH500.
Vanilla
Layer
Layer
Layer 10
Layer 15
Layer 20
Layer 25
Layer 27
Accuracy (%)
88.2
88.4 3685
88.0 3665
88.2 3701
84.4 2713
80.6 1906
72.2
88.4
Tokens
3824
1070
2239
Table 4: Layer-wise performance analysis for R1-8B on MATH500.
Vanilla
Layer
Layer
Layer 10
Layer
Layer
Layer 25
Layer 30
Layer
Accuracy Tokens
87.8
87.2
87.8
88.2
75.6 2950
86.4 3280
71.8 1856
87.6
88.0
4009
3896
3820
3654
2975
2873
Table 5: Layer-wise performance analysis for R1-14B on MATH500.
Vanilla
Layer
Layer
Layer 10
Layer
Layer
Layer 25
Layer 30
Layer 35
Layer
Layer
Layer 47
Accunccy (%)
92.8
92.4
02.4
92.0
92.2
92.6
80.8
90.4
87.4
84.6
82.4
92.8
Toleess
3:96
3384
3420
3095
2958
2857
2398
1814
2207
1836
1625
2074
E Time Latency Analysis
In this section, we analyze the time latency for the DeepSeek-R1-De -aaieee--R---issil-----e------- model on the Math500 dataset [20], comparing our approach with Dynasor [16] and SEAL [8]. All experiments conducted an Ubuntu 22.04 system with A800 GPUs. We find that Dynasor exhibits the
25

---

significantly longest time latency, which is reasonable due to its frequent probing of intermediate states and its unsuitability for parallel processing of large reasoning models. For SEAL, although both SEAL and our method introduce negligible additional computational cost, SEAL's token reduction is less effective than ours, resulting in higher time latency.
Table 6: Average Time Lateney Math500 for different overthinkingg methods in R1-7B.
Methods
Original
Dynasor
SEAL
Ours
Time Latency (s)
1.74
39.89
1.37
1.05
26


Reviews:
[{"review_id": "HRNECHHs6c", "reviewer": "Reviewer_AXAi", "summary": "The paper proposed a novel method to mitigate overthinking in large reasoning models (LRMs) while maintaining the performance. First, by using the OpenMathInstruct-2 training dataset, the authors obtain a difference vector that represents the direction of the difference between concise and redundant responses. While directly steering the model along this direction too much might even increase the length of the response, the authors utilize a low-dimensional manifold obtained by principal component analysis (PCA) to effectively mitigate overthinking. They conducted extensive experiments across different models and different datasets, corroborating the effectiveness of the manifold steering method. They also showed that the manifold steering method can be effectively transferred to other tasks such as jailbreaking.", "questions": "1. 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?\n2. In (5), why $C = 1/(N-1) A(A-\\bar A)\\^T$ instead of $C = 1/(N-1) (A-\\bar A)(A-\\bar A)\\^T$, as the latter one is the definition of the covariance matrix.\n3. 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?\n4. 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)?", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strengths:**\n\n1. The problem of Mitigating Overthinking is important.\n\n2. The authors propose a simple yet effective method to solve the problem.\n\n3. The proposed method is applicable to various tasks.\n\n**Weaknesses:**\n\n\nThere is no obvious weakness of this paper. There might be some minor issues in the theoretical derivation (see the question section).", "was_revised": true, "final_justification": "The authors addressed my questions in the response. I think this paper has a good contribution and decided to keep my score.", "rebuttal": "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.\n\n---\n\n***W1&Q1. The evaluation is limited to the DeepSeek-R1-Distilled model family, which raises questions regarding the generalizability of the proposed method.***\n\nThanks 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.\n\n| Model  | Methods | GSM8K Pass@1 (↑, %) | GSM8K #Tokens (↓) | MATH500 Pass@1 (↑, %) | MATH500 #Tokens (↓) |\n|---|---|---|---|---|---|\n| | Vanilla | 93.2            | 2377       | 95.8   | 5033      |\n| Skywork-OR1-7B| SEAL    | 93.4 (+0.2)  | 1639 (-31%)    | 95.6 (-0.2)    | 3775 (-25%)         |\n| | Ours    | 93.4 (+0.2)     | 951 (-60%)        | 96.2 (+0.4)      | 2986 (-41%)   |\n|      | Vanilla |       97.7          |       1583       |          96.2        |      5280          |\n| Qwen3-8B      | SEAL    | 97.9 (+0.2)         | 1123 (-29%)       | 95.8 (-0.4)           | 4384 (-17%)         |\n|      | Ours    | 98.0 (+0.3)    | 612 (-61%)   | 96.4 (+0.2)    | 3273 (-38%)    |\n|       | Vanilla |     96.7     |  2040     |    95.4     |    4447   |\n| QwQ-32B       | SEAL    | 96.8 (+0.1)         | 1714 (-16%)       | 94.8 (-0.6)           | 3602 (-19%)         |\n|        | Ours    | 97.1 (+0.4)     | 1265 (-38%)    | 95.6 (+0.2)      | 2668 (-40%)         |\n\n\n--- \n***W2. The method intervention strength is sensitive to real-time task complexity, which might limit the practicality of the method.***\n\nThanks 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.\n\n----\n\n***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?***\n\nWe 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.\n\nRegarding 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. \n\n[1] Arditi et al. Refusal in Language Models Is Mediated by a Single Direction\n\n\n\n---\n\n***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?***\n\n\nThanks 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."}, {"review_id": "jV9by8Vair", "reviewer": "Reviewer_9Mtj", "summary": "This paper introduces **Manifold Steering**, a novel method to mitigate \"overthinking\" in Large Reasoning Models (LRMs). The authors begin by identifying a single \"overthinking direction\" in the model's activation space but find that steering along it introduces interference noise that limits its effectiveness. They further discover that the overthinking-related activations reside on a low-dimensional manifold. Manifold Steering leverages this by projecting the initial steering vector onto this manifold to remove the noise. This technique leads to token reductions of up to 71% on math benchmarks, maintains or improves accuracy, and demonstrates strong transferability to other domains.", "questions": "- Do you plan to test this method on non-DeepSeek-distilled models?\n- 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?\n- 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?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths\n- The paper's proposed approach efficiently addresses LRM overthinking from the perspective of mechanistic interpretability, offering a new perspective on the \"overthinking\" problem.\n- The proposed method achieves impressive experiment results, consistently reducing token counts while maintaining or even improving accuracy across evaluated math benchmarks.\n- The paper is well-written and logically organized, presenting a sound derivation and theoretical justification for the proposed method.\n\nWeaknesses\n- The evaluation is limited to the DeepSeek-R1-Distilled model family, which raises questions regarding the generalizability of the proposed method.\n- The method intervention strength is sensitive to real-time task complexity, which might limit the practicality of the method.", "was_revised": true, "final_justification": "The author provide extra strong results of their proposed methods on some latest open-source models, and also provide clear clarifications on their methods. There're no remaining issues, and I keep my score at 5.", "rebuttal": "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.\n\n\n---\n\n\n***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?***\n\nIn Eq. (2), $x$ refer to \"prompt + response\" and Eq. (3) applied to each decoding step.\n\n---\n\n***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?***\n\nThank you for pointing this out. We are sorry for this mistake and will revise it instantly. The expression in Equation (5) should indeed be $C = \\frac{1}{N-1}(A - \\bar{A})(A - \\bar{A})^T$. \n\n---\n \n***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?***\n\nWe 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 !\n\n---\n\n***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)?***\n\nWe 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.\n\nOur 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 !"}, {"review_id": "Ics1L2tgv1", "reviewer": "Reviewer_Z6Zt", "summary": "This paper presents Manifold Steering, an Inference Time Intervention (ITI) strategy to mitigate overthinking in Large Reasoning Models (LRMs). The authors first sample \"redundant\" and \"concise\" responses to derive a steering vector that assocates with the overthinking direction. Given that, the authors perform ITI on LRMs at various intensities，and observe that the steer contains noisy components that interfere with reasoning. Through PCA on the activations, they reveal that these distruptive components reside in a subspace spanned by residuals. By removing the redudance via a simple linear transformation, Manifold Steering(MS) achieves effective and robust overthinking mitigation than simple ITI.\n\nThe authors conduct comprehensive experiments to demonstrate MS: \n\n(1) consistently reduce generation tokens while maintaining reasoning performance, and\n\n(2) demonstrates strong cross-domain generalization.", "questions": "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?\n\nQ2: 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 $\\alpha$ that is independent of test accuracy is proposed.\n\nQ3: 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?\n\nUpdate: Today I came across a new paper [2], which explicitly removes \"wait\" from the vocabulary, leading to shorter generations and improved performance. I’m not suggesting that your work should be compared with this paper, especially since your submission predates it. Rather, I mention it to support my hypothesis raised in Q3.\n\n[1] Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning\n\n[2] Wait, We Don’t Need to “Wait”! Removing Thinking Tokens Improves Reasoning Efficiency", "limitations": "yes", "rating": 2, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. The discovery of the \"low-dimensional manifold\" within a steer vector is interesting.\nThe interpretability tool introduced to identify such a subspace is both intutive and easy to grasp.\nIt resonates with me, as residual components that preserve core language abilities can indeed hinder precise reasoning. Overall, I find these operations and insights valuable.\n2. The experiments are fairly comprehensive and effectively address concerns about the robustness of the ITI method in cross-domain transferability. The experimental results also confirm the method’s effectiveness in token reduction.\n3. This paper is mostly well-written and easy to follow.\n\n\nWeakness\n\n1. The font size used in the figures is too small to read\n\n2. 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.\n\n3. See Questions.\n\n[1] Spectral Filters, Dark Signals, and Attention Sinks\n\n[2] Effectively Steer LLM To Follow Preference via Building Confident Directions", "was_revised": true, "final_justification": "After rebuttal, I continue to have concerns about this paper. The selection of hyperparameters for the paper’s core method is based on a preprint (titled SEAL) that has not undergone peer review. Specifically, the authors determined the optimal α by evaluating performance on Math-500, yet they include Math-500 results in their main experiments as evidence of the method’s effectiveness—this is test-data leakage.\n\nAdditionally, since the paper focuses on mathematical tasks, hyperparameter tuning on Math-500 (an in-domain dataset) benefits other math tasks.\n\nI raised these issues, but the authors defended their hyperparameter selection by citing SEAL, which failed to address my concerns, because I find SEAL's hyper-parameter selection is also problematic. As a result, I kept my rating to reject this paper.\n\n**I found that while other reviewers have consistently gave high ratings to this paper, none have raised the issue of test data leakage. I kindly request that ACs examine this concern and take it into account as part of a comprehensive evaluation. Thank you.**", "rebuttal": "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.\n\n---\n\n***W1. The font size used in the figures is too small to read***\n\nThanks for your suggestion. We will increase the font size  of figures in the revised version to ensure clarity and better readability.\n\n---\n\n***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.***\n\n\n\nPrior 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). \n\n\nFor [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.\n\nLastly, thanks for your suggestion and we will add detailed discussion with these related studies to the revised version.\n\n---\n\n***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?***\n\nWe 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. \n\n[1] Razzhigaev et al. Your Transformer is Secretly Linear\n\n[2] Hernandez et al. Linearity of Relation Decoding in Transformer Language Models\n\n---\n\n***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.***\n\nActually, 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.\n\nTo 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. \n\n---\n\n***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?***\n\nThanks 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.\n\nAs 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."}, {"review_id": "jURQAeru2T", "reviewer": "Reviewer_nrvD", "summary": "This paper aims to mitigate overthinking in large reasoning models (LRM) by adjusting their internal activations. Based on the assumption that activations corresponding to overthinking reasoning traces can be distinguished from concise ones, the paper introduces a correction term derived from the difference-in-means from overthinking (redundant) and concise reasoning traces (in math). However, the paper finds that such a correction doesn’t always guarantee redundancy reduction. As a remedy, the paper introduces manifold steering, where the correction term is projected to a low-dimensional space (manifold). Experimental results on math (in-domain), code generation, and disciplinary knowledge (out-of-domain) demonstrate a reduction of reasoning trace lengths while achieving the same or better accuracy.", "questions": "- I would appreciate it if the formal flaws could be addressed in a revised version.\n- Moreover, experiments on another family of reasoning models (e.g., Qwen 3) might be interesting. \n- Line 150 states that the intervention is applied across all layers. However, Lines 264-265 mention only single layers to be adjusted. Please clarify.", "limitations": "Limitations are provided in the paper (Section 6).", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "*Strength* \nThe paper addresses an important issue in LRMs. Indeed, empirical results show that the proposed method can mitigate overthinking in R1-based models. \n\n*Weakness*\nI see the main weakness in the major flaws of the formal derivations of the paper, which might impact the actual results of the paper. In that regard, the paper needs major rewriting. \n- Paragraph “Linear Low-Dimensional Manifold Verification” states that the steering direction is composed of an overthinking ($r_{overthinking}$) and an error ($r_{other}$) 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. \n- Moreover, the comparison between the magnitudes of $r_{M}$ and $r_{other}$ 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!\n- 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. \n- Step 1 in the proof of Theorem 4.1 shows equality between an expectation and an empirical estimation of the covariance matrix. \n- Eq. (4) is missing the cross terms $r_{overthinking}^T h^{l}(x_i) r_{other}$ and $r_{other}^T h^{l}(x_i) r_{overthinking}$\n- The covariance matrix in Eq. (5) is missing the average on the LHS: $\\frac{1}{N-1}(A^{l}-\\overline{A}^{l}) (A^{l}-\\overline{A}^{l})^T\n\nApart from the formal flaws, the paper only focuses on DeepSeek-R1-based reasoning traces. It is not given that the finding will also hold for other families of LRMs.", "was_revised": true, "final_justification": "After the rebuttal clarified some of my concerns regarding fundamental steps in the method, I have updated my score to a weak accept.\nThe paper presents an effective method supported by theoretical insights and practical experiments. As noted by other reviewers, remaining weaknesses include the approach to hyperparameter selection and the question of generalizability to non-mathematical design tasks.", "rebuttal": "We greatly appreciate you for recognizing the importance of addressing overthinking in LRMs, as well as the success of our method in mitigating it in R1-based models. Below, we address your concerns in `Weaknesses (W)`, `Questions (Q)` and `Flaws (F)`. We sincerely hope that you can find our response satisfactory.\n\n\n---\n\n***W1&Q1-F1. Paragraph \"Linear Low-Dimensional Manifold Verification\" states that the steering direction is composed of an overthinking ($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.***\n\nFor 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.\n\nFinally, we appreciate your valuable feedback, and we will incorporate a clearer explanation in the revised version.\n\n---\n\n***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!***\n\nWe 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. \n\nTo 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. \n\nFor 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.\n\n\n\n---\n\n***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.***\n\n\nWe 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.\n\nFor 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.\n\nThe 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 !\n\n---\n\n***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.***\n\nWe 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.\n\n\n---\n\n***W1&Q1-F5. Eq. (4) is missing the cross terms.***\n\nAs 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).\n\n---\n\n***W1&Q1-F6. The covariance matrix in Eq. (5) is missing the average on the LHS: $\\frac{1}{N-1}(A-\\overline{A})(A-\\overline{A})^T$.***\n\nWe apologize for the missing the average on the LHS and will correct it in the revised version instantly. Thanks for your reminder again!\n\n---\n\n***W2&Q2. Apart from the formal flaws, the paper only focuses on DeepSeek-R1-based reasoning traces.***\n\nThanks 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.\n\n| Model  | Methods | GSM8K Pass@1 (↑, %) | GSM8K #Tokens (↓) | MATH500 Pass@1 (↑, %) | MATH500 #Tokens (↓) |\n|---|---|---|---|---|---|\n| | Vanilla | 93.2            | 2377       | 95.8   | 5033      |\n| Skywork-OR1-7B| SEAL    | 93.4 (+0.2)  | 1639 (-31%)    | 95.6 (-0.2)    | 3775 (-25%)         |\n| | Ours    | 93.4 (+0.2)     | 951 (-60%)        | 96.2 (+0.4)      | 2986 (-41%)   |\n|      | Vanilla |       97.7          |       1583       |          96.2        |      5280          |\n| Qwen3-8B      | SEAL    | 97.9 (+0.2)         | 1123 (-29%)       | 95.8 (-0.4)           | 4384 (-17%)         |\n|      | Ours    | 98.0 (+0.3)    | 612 (-61%)   | 96.4 (+0.2)    | 3273 (-38%)    |\n|       | Vanilla |     96.7     |  2040     |    95.4     |    4447   |\n| QwQ-32B       | SEAL    | 96.8 (+0.1)         | 1714 (-16%)       | 94.8 (-0.6)           | 3602 (-19%)         |\n|        | Ours    | 97.1 (+0.4)     | 1265 (-38%)    | 95.6 (+0.2)      | 2668 (-40%)         |\n\n\n---\n\n***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.***\n\nActually, 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!"}]

Author rebuttal:

Meta-review: This paper describes an approach to mitigating the problem of LLMs overthinking in the course solving reasoning problems. The paper presents an interesting and novel insight that the phenomenon of overthinking can be related to activation along specific directions in the models activation space. The paper proposes an approach that involves encouraging the model to steer its activations to a desirable subspace identified during training.

Strengths:

The reviewers were generally accepting of the analysis presented in the paper and appreciated the novel approach to mitigating the overthinking problem

The results in the manuscript show meaningful improvements on four mathematical reasoning benchmarks. During the rebuttal process the authors presented results on additional non mathematical datasets that strengthened the claims.

Weaknesses:

Reviewer Z6Zt raised issues regarding the selection of an important hyper parameter which governed the extent of the intervention applied in the manifold steering process. There was a specific concern about this parameter being tuned on one of the mathematical benchmark datasets that was also used for evaluation. The authors attempted to address this concern by testing the system on non-mathematical reasoning benchmarks where they also demonstrate an improvement.

Three of the four reviewers were favorably disposed to this paper and felt that it represented a promising novel approach to addressing the problem of overthinking in reasoning systems.
Decision: Accept (poster)