Layer-Wise Modality Decomposition for Interpretable Multimodal Sensor Fusion
Authors: Park Jae Hyun
Konyul Park
Daehun Kim
Junseo Park
Jun Won Choi
Venue: NeurIPS 2025 poster ()
Keywords: Interpretable AI
Deep Taylor Decomposition
BEV Perception
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: This paper proposes Layer-Wise Modality Decomposition method that enables transparent attribution of output predictions to individual input modalities in perception models

Abstract: In autonomous driving, transparency in the decision-making of perception models is critical, as even a single misperception can be catastrophic. Yet with multi-sensor inputs, it is difficult to determine how each modality contributes to a prediction because sensor information becomes entangled within the fusion network. We introduce Layer-Wise Modality Decomposition (LMD), a post-hoc, model-agnostic interpretability method that disentangles modality-specific information across all layers of a pretrained fusion model. To our knowledge, LMD is the first approach to attribute the predictions of a perception model to individual input modalities in a sensor-fusion system for autonomous driving. We evaluate LMD on pretrained fusion models under camera–radar, camera–LiDAR, and camera–radar–LiDAR settings for autonomous driving. Its effectiveness is validated using structured perturbation-based metrics and modality-wise visual decompositions, demonstrating practical applicability to interpreting high-capacity multimodal architectures. Code is available at https://github.com/detxter-jvb/Layer-Wise-Modality-Decomposition.

Payer-Wiss Modality Decomposition for Interpretable Multimodal Sensor Fusion
Jaehyun Park
Konyul Park
Daehun Kim
Junseo Park Jun Won Choi"
Seoul National University
(jhpark, kypark, dhkim, jspark) sondr.suu.co kr junwchoi@smu kr.
Abstract
In autonomous driving, transparency in the decision decision-making making of perception models is critical, as even a single misperception can be catastrophic. Yet with multi-sensor inputs, it is difficult to determine how each modality contributes to predie- tion because sensor information becomes entangled within the fusion network. We introduce Layer-Wirs Wise Modality Decomposition (LMD),, post-hoc, model- agnostic interpretability method that disentangles modality-peecific information across all layers of a pretrained fusion model. To our knowledge, LMD is the first approach to attribute the predictions of a perception model to individual in- put modalities in sensor-fusion system for autonomous driving. We evaluate LMD on pretrained fusion models under camera-radat, camera-LiDAR and cam- -ra-AR settings for autonomous driving. Its effectiveness is validated using structured perturbation-basee metrics and modality-wi visual decompo- sitions, demonstrating practical applicability to interpreting high-capacity multi- modal architectures. Code is available at fattttts:////t thub. com/ /dddxte--ee-..b.. Layer-Wiss Modality-
Introduction
There is a growing move toward perception models that leverage multimodal inputs to improve downstream task performance. In autonomous driving, for example, modern systems increasingly fuse diverse modalities to produce more accurate and contextually relevant outputs. Incorporating different sensor streams such as camera, radar, and LiDAR has consistently improved reliability. Yet this integration complicates interpretability and transparency: when the model makes decision, it becomes hard to discern to what extent each modality contributed to the final outcome.
This challenge is particularly pronounced in cafety-critical domains such as autonomous driving, where model outputs directly affect safety and reliability. Consequently, improving the interpretability of multimodal perception models is essential. Although interpretable Al has advanced in various domains, its application autonomous driving remains underexplored. To bridge this gap, is necessary to explore how existing interpretability methods can be applied to sensor-fusion models.
Current work in interpretable AI be grouped into categories: intrinsically interpretable models and post-hoc interpretation methods. Intrinsically interpretable models have been widely adopted to directly interpret input feature attributions model predictions. Generalized Additive Models (GAMs) [I] represent outputs as sums of feature-wise functions, ensuring intrinsic inter- pretability. Recent variants 2 3 improved flexibility while preserving this property. However, they require structural constraints, which make it difficult to model complex patterns. This inter-
Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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limits their applicability to multimodal perception tasks, where capturing cross-modal interactions is essential.
Deetablity-accuuree trade-off [6]
Post-hoc interpretation methods are particularly advantageous for high-capacity models, as they maintain predictive accuracy without necessitating architectural modifications. Local surrogate models, such as LIME and LORE 7 g enhance interpretability by approximating complex decision boundaries with low-capacity models. However, this simplification limits their ability capture the intricate, high-dimensional dependencies inherent in large-scale perception networks. Global interpretation techniques, such as functional ANOVA (fANOVA) [9] are effective for tabular or low-dimensional data, but they encounter substantial challenges in scalability and decomposition fidelity when applied multimodal perception models.
Consequently, there is a need for a post-hoc interpretability method that can deliver layer-wise, modality-specific attributions for high-capacity models without compromising performance re- quiring architectural modifications. In this study, we propose Layer-Wise Modality Decomposition (LMD), framework that enables the interpretation of complex multimodal perception models while fully preserving their original performance.
Our approach locally linearizes neural network operations to separate modality-specific information while forwarding it through the model to the final prediction. In particular, we formulate LMD within the Layer- Wise Relevance Propagation (LRP) [10 [3] [14] [6) and Deep Taylor Decomposition (DTD) [8] frameworks, which are widely adopted post-hoc bost-hoc methods in Explainable AI (XAI).
Our LMD grounded in rigorous theoretical framework that separates modality-specife compo- nents at each layer. We validate its effectiveness on multi-sensor perception models using the widely adopted nuScenes benchmark [[9]. More broadly, LMD is generalizable and can be readily applied to other multimodal architectures.
Our key contributions are as follows:
1. We propose LMD, novel post-hoc interpretability method that decomposes the contribution of each sensor modality across all layers of a complex multimodal model. The decomposition process operates without any modification the original architecture.
2. We conduct both extensive qualitative and quantitative experiments to validate the proposed method. We introduce novel perturbation-based metrics assess interpretability of our model.
3. We investigate several LMD variants employing different bias-splitting strategies and assess their effectiveness from a modality separation perspective.
2 Related Work
2.1 Interpretation Methods
Interpretable methods aim provide transparency in model predictions by designing architectures in which feature contributions are explicitly traceable. Such models are inherently understandable, removing the need for external post-hoe explanations. Broadly, interpretable approaches can be grouped into two categories: intrinsically interpretable models and post-hoe interpretation methods.
Intrinsically interpretable sondels-including Generalized Additive Models (GAMs) []] Neural Additive Models (NAMs) [2]. NODE-GAM [4], and Gaussian-NAM (5) represeeent model outputs as the sum of feature-wise functions. This additive formulation enables fine-grained attribution and clearer interpretation of individual feature effects. However, the structural constraints that ensure interpretability also limit these models' capacity to capture complex feature interactions and contextual dependencies.
Post-hoe interpretation methods aim to explain the behavior of pretrained black-box models. The methods such as Functional ANOVA (fANOVA) ( decompose the output variance of trained model into contributions from individual features and their interactions. While FANOVA is effective for tabular low-dimensional data, it faces scalability challenges in high-dimensional settings, where decomposed features may become unidentifiable (20).

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2.2 Post-hoc Explanation Methods
Post-hoe explanation methods aim interpret complex models without modifying their original architecture. By operating directly on pretrained black-box models, these approaches preserve predictive accuracy while providing interpretability.
Backpropagation--aeed methods generate explanations by leveraging gradients or gradient-related signals within the model. For instance, Layer-Wise Relevance Propagation (LRP) [10] assigns relevance scores to input variables by decomposing layer-wise computations, thereby producing attention maps that can be interpreted input-level contributions. The propagation rules in LRP are grounded in the Deep Taylor Decomposition (DTD) framework (18), which requires local linearization of each layer remove non-linear biases. Building on this foundation, subsequent studies have proposed refindd attribution schemes to ensure faithful relevance propagation across diverse network components such as residual connections, attention mechanisms, and normalization layers in modern deep architectures
[T] 15
16 12
2.3 Multi-sensor Perception
Multi-modality perception leverages the complementary strengths of different modalities to achieve robust scene understanding in autonomous driving. Cameras offer rich semantic context, LiDAR provides precise 3D geometric information, and radar ensures reliable velocity and range estima- tion under adverse conditions. To harness these complementary signals, recent models explore
intermediate fusion strategies that align features from different modalities [2] 22 23
24 25
For instance, BEVFusion 22 employs modality encoders for camera and LiDAR, followed by fusion via spatially consistent representations. RCBEVDet [25) introduces radar-specific encoder and integrates output with camera features using deformable cross-attention 26]. facilitating effective moos alignment.
Despite their strong performance, these models often lack interpretability. During the fusion process, features from different sensors are aggregated, making it difficult to disentangle the contribution of each modality to the final prediction. This lack of transparency complicates failure diagnosis and system validation, highlighting the need for interpretable techniques in safety-critical applications.
3 Background
In this section, we briefly review the Deep Taylor Decomposition (DTD) 18 and the Layer- Wise Relevance Propagation (LRP) 10) before introducing our LMD method.
Let R - IR M be a vector-valued function consisting of deep network with the input neurons x { f be the j-th output neuron. DTD decomposes fú into contributions from N input
variables using first-order Taylor expansion of fイ(r)
fx(x (x)
EJ R.) + Σ JRC fx()
Je +c,
by
0f is the Jacobian X and is the first-order approximation error. The summation term containing x; represents the first-order contributions of input variables to the output (x). With x treated as a constant, the bias by consists of constant terms and which represents high-order interactions determined by the order of differentiability of the function along the line segment joining and X.
where X is a reference point, J,:
developed local linearizations of non-linear network layers to attribute predictions to input variables. Within the DTD framework, two core design choices arise: (i) selecting the root point which the local Jacobian JJs is computed, and (ii) specifying how the bias term is redistributed to the inputs. For element-wise non- activations, Jjs was typically set to the identity [ K to enforce the conservation property-i.e., that the sum of input attributions is preserved across layers. For LayerNorm, prior work notes that the true Jacobian entries are near zero, rendering direct linearization uninformative [I] [3]. common workaround treats LayerNorm effectively clement-wise by assuming (local) variance to be constant.
Prior LRP studies 18
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Bias handling has been central to maintaining conservation and differentiates LRP variants. For LayerNorm other local renormalization operations, some methods distributed the bias uniformly over inputs [14 15]. whereas others applied the identity rule and excluded the bias from redistribution 17 13 An alternative was to omit the bias altogether, as [1 argues, based on observed numerical instabilities in prior schemes.
4 Method
In this section, we describe the details of the proposed LMD method. The overall process of L.MD depicted in Figure ! For clarity, derive the formulation for two modalities (e.g.. camera and radar) without loss of generality; the extension to an M -modalitt setting is provided in Appendix
4.1 Formulation of Layer-Wise Modality Decomposition
We first show that, using first-order Taylor approximation, the features of a fusion model at each layer can be decomposed into modality-speciffi terms. Let f', .......f f be of N layer functions consisting of the camera-radar sensor fusion model performing perception tasks. Suppose the composition of layer functions from the first layer to the 1-th layer given by
(' X+) - ....o f² (xxx
where I (1,...,,N NJ. X ( and x, { denote D₁ and D2 dimensional inputs from the camera and radar modalities, respectively. Notice that j-th output neuron of the perception model becomes PN (Xe, xy).
4.1.1 Handling Fusion Layer
Beginning at the first fusion layer (1 = 1), where aggregation of each modality information takes place, the first-order Taylor expansion of f) is applied
[X Xr) Σ
Σ
mmi +
(G Σ
Σ mE (e.r)
Inssamm tej
(3)
mE (c.r)
where m indexes the modality, Z.. 2, represent reference points for camera and radar data, respec- tively, and Jny denotes the Jacobian calculated at the reference points for the modality
To assess the contribution each modality to a function output, we try setting the input values of camera and radar data to zero, yielding the following results
f) (Xe. or) Jojse + +],, (o,, xx)
En +), f) (oc. o,) =b}.
We let Mags My and hhy indicate the bias at I-th layer respectively, i.e., h'r = Σ Jejs
features from the camera data, radar data and h1 Σ Jrys and hy = bj. Then, we
can show that
(X), = Σ Myj.
(5)
4.1.2 Handling Subsequent Layers
For subsequent layers with = 2.... N, the 1-th layer function takes Emele.r.t h/ and produces the output
f
1-1
Σ
Σ
Anggh 1oi++00 +b/
(6)
my
ml
mm..r)))))
mEle.r)))
It shows that each layer output results from the summation of the camera, radar, and bias components.

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From 6 we can show inductively that the following equation holds
Fx(e = f
1-1 mý
hm
(e.r,))))
mff..r))))
Through modality decomposition, is strictly satisfied across all layers. To effectively decouple the contributions of different modalities, two conditions must be met. First, the decomposition should preserve the functional behavior of the original model. Second, each modality-speciie feature must remain unaffected by the others-a condition we refer to as the separation property.
4.2 Lincarization Deep Networks
we showed that LMD can be derived using first-order Taylor decomposition. However, when higher-order terms remain, the modality decomposition difficult to achieve. To circumvent this, we linearize every layer of the fusion model. For the linearized layer functions f)...., faN our linearization process should preserve the functional behavior of the original network,
In Section 4.1
(Xe: x+) = (x,,
for l E I.....,,, where (xc: designed to satisfy the constraint in ensuring that the linearized network behaves identically to the original one.
x+) o...o o (xe.x))]5 In other words, our decomposition
In typical neural networks, non-linearities primarily arise from activation and normalization layers. Section 4.2.1 I and 4.2.2 describe how these layers are lincarized for activation and normalization operations, respectively.
4.2.1 Linearizing Activation Layers
We first linearize the activation layers. Specifically, we transform the activation operation into (linear) element-wise multiplication operation which enables modality decomposition. This process consists of two forward passes.
From the first forward pass, we the output-to-input ratio df for I-th layer before the linearization. For ReLU [27], this ratio degenerates to a binary mask ef E {0, 1} that records whether j-th neuron in the ReLU function I was active or not. The main purpose of the first forward pass is to record the activation behavior under the full multimodal input (e.g., camera and radar), which the target configuration for decomposition.
Then in the second pass, we replace the activation layer f with the linearized layer ft which multiplies the imput neuron by c/. The ultimate goal the second forward pass is follow the f's behavior even when the input from the specific modality is set to zero to compute the other modality contributions. other words, the linearization enables the decomposition of the modality features for the target function output which we want to interpret.
Proposition 1. Let fl be the locally linearized activation function,
(xe, Xx)
(( = cr. 1 &
F. (Xe: Xx) +
This construction is equivalent to setting the diagonal entries of the J/. in (3) with the slope of the line segment joining the two operating points (F,- xx), F) (Xex (x)). thereby satisfying (8) exactly.
The proof can be found in Appendix B.I.
4.2.2 Linearizing Normalization Layers
The non-linearity in BatchNorm 28 originates from the bias term. the case of BatchNorm, statistics stored during the training phase are used fixed values in the evaluation mode. Thus, aside
5

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from the activation layer, it is not needed to store the behavior of the layer f' from the forward pass. Consequently, we can linearize the BatchNorm simply by excluding the bias term from the modality features while satisfying (8)
Proposition 2. If f is BatchNorm, linearizing f yields the following decomposition rule
f3 (M/))) = h mmj 8mth 1/ + 8mL} = Vo,² +
23
14
n + 6m (3) - Vos" +
Vog² +e
(10)
modality spocific term
biax term
where m E (c,rr,). E, Hiy, 13. R. nlj and o', denote the batch mean and standard deviation, and 1y, B3 are the affine parameters. In our LMD configuration, des 8, 0 and 6, 1, satisfying (8)
The proof can be found in Appendix B.2.
In contrast to BatchNorm, LayerNorm [29] computes normalization statistics from the current input in both training and evaluation. As noted in [13]. LayerNorm be decomposed into centering step and rescaling step. Since the Jacobian in the centering step of LayerNorm a linear projection matrix I - 1 11 where is the identity matrix and 1 is all-one vector, additional linearization is required.
For the rescaling step, the variance d²² = Var[F"(x, x,) is assumed to be constant, as the Jacobian entries rescaling step are near zero, as shown [33 Hence, the variance saved from a single forward pass, and used for the linearized LayerNorm. Even if the interaction effect between modalities remains, modality features maintain their meanings since the rescaling process is applied across all modality features, not altering the importance relationship of modality features. We refer to this approach as the ratio rule, which is adopted in the variants of LMD.
Proposition If f is LayerNorm, linearizing f yields the following decomposition rule
13 (M(3) M-11 B(hmn 1/2 8m
(h
(m
8m B
(11)
Vot² te
Vo,² +E
bias
modality specific term
where m E fc,r, b), e. Yz. 3) ER, of denote the standard deviation, and 7/, 3) are the affine parameters. In our LMD configuration, the expectation is computed over the spatial dimensions (height and width) for each modality and de, 8, = and 8, = 1, satisfying 8).
The proof is given in Appendix B.3. In a similar way, also linearize InstanceNorm as shown in Appendix B.3.
4.2.3 From Linearization to Modality Decomposition
In this subsection, we formally show that the output of each layer in LMD be decomposed into modality components. This modality decomposition is possible because all non-linear operations the network have been replaced with the linear operations described in Scction 4.2.1 4.2.2. As result, the LMD framework becomes fully linear, allowing the exact separation of modality contributions across all layers in the entire network.
Here, f denotes the linearized layer function obtained from the original non-linear function f. Eq. 12) thus represents the unified layer formulation after linearization, where activation and normalization operations are already expressed in their linearized forms. Let h = (Xc, 0r), h! = f) (oc. x,).
and hi f) (oc, or) be the first layer's modality features. Then, with all the linearized layers in network, the following equation holds.
Kms = -2 (b
(12)
for all / E 2,..., N and
{c, r..

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LMD
Non Lineer Layer
1- Place
Camera Encoder
Forward
Image feature
Image output
Image
Dutipt feature
Image
Unearized Layer At
Radar output
Radar Encodirr
Image feature
Pass
Roder Toature
Radar feature
Radar Points
SabortarTron
Bist
Fusion Module
Figure 1: Overall Process of LMD. LMD decomposes the multimodal features into -peccfi components through two-stage process.
Since each layer function f becomes linear after applying the linearization process described in the superposition property holds as shown in
Section 4.2.1
4.2.2
E
(13)
mj
(crr)))
mE(ar)))
for all E N.
When the entire network is fully linearized as F= = PN ....... of the decomposition extends to the whole network, leading to 14
Xx), (xe, + (oc, xx) Ooc.
(14)
for all E I.
This guarantees that the LMD framework satisfies both the constraint (8) and the separation property throughout the whole linearized network.
4.2.4 Exploring LMD Variants
This section presents linearization schemes for normalization layers that satisfy our constraints (8) and the separation property. For the BatchNorm layer, Proposition 2 exxcludes the bias term from each modality feature that we call the identity rule. Alternatively, the uniform rule can be applied by uniformly distributing the bias term across all modality features, defined by setting 8n 1/M where m {c, b), and M = [e, b)] from (1O). To linearize the LayerNorm, in addition to the ratio rule (TT. one can adjust them to operate similarly to BatchNorm from stored statistics calculated from a single forward pass of the original fusion model, where the statistics become
x,)]. Then the identity and uniform rules can also be applied. In Section 5.3 we quantitatively evaluate separability across these variants.
n² EF (xe, xx)] and d²² Var F' (xe:xx)]
4.3 Post-hoc Interpretation through LMD
In practice, the LMD implemented with two forward passes. We store the behaviors of non-linear layer, d; for activation layer and variance in LayerNorm. Then in the second forward pass, we replace the original layer into linearized layer with the cached values. The overall process LMD described in Figure
We present the post-hoc interpretation method that LMD for the multi-sensor perception model. Specifically, the visualizations in Figure 2 represent Eo (Xe: xr), f:N (0., xx). foN (Xey 0r), and f.N (Oe: 0r) for the prediction from fused modality and prediction from each modality. Since the prediction from the original model is perfectly decomposed into the prediction from each sensor data,
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(a) Ground Truth
Prediction Fusion Model
[C) Prediction from Camera
(d) Prediction from
from
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Ground Truth
Prediction Fusion Modil
Prediction from Camera
(c) Prediction
[el Prediction Blas
100
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Figure 2: Post-hoc Interpretation through LMD: In the first row, a comparison between (c) and (d) shows that the model successfully detects the vehicle using radar data, as indicated by the green marker, whereas the camera-based prediction lacks confidence. Similarly, in the second row, the green marker in (d) highlights either a correct or incorrect prediction made by one modality that was not captured by the other. The prediction from bias in (e) exhibits a certain degree of perceptual capability. This component includes constant effect and high-order interactions largely originated from linearization of activation layers.
Figure 2 shows not just an explanation but an exact description of how pretrained model produces its output exploiting each sensor data. Note that visualizations in Figure 2 extract the positive components of each prediction and then normalize to maximum values to effectively visualize the high-confidence regions. Additional visualizations containing negative values are included in the Appendix D.3.
5 Experiments
In this section, we validate the proposed LMD method built on SimpleBEV [2]] where modalities are integrated to perform vehicle perception tasks. To demonstrate the generalizability of our approach, we apply LMD to radar-camera, LiDAR-camert and radar- regarding model configurations and the dataset [19] are provided in Appendix A. Also, quantitative results with three- prce-modality settings are provided in Appendix D.I.
fusion models. Details
Table 1: Comparison of Pearson Correlation and Mean Squared Error for Linearization Experiments.
Pearson Correlation
Mean Squared Biror
Modality
Mithon
(1)
R,)
(T
C,7C())
R,R
Cyc
0,22 à: 0.10
0.76 0.07
0.22 a: 0.07
0.09 :: 0.06
1 d: 0.86
851 19.73
1.46 = 1.18
40.06 :: 23.14
Bracline
0.0M
0000 0007
1122
2.26 à: 0.8
753 16.59
3/75
34.9
15:3
Camera
0.08
0.99 0.00
0.13
0.74:0099
41.06 à: 20.79
LMD (aat Rule)
0.85 +0,02
1,00: 0.00)
1 : 0.00
0.15 6: 0194
9,8 d: 2.34
IJ08 8 0.00
0,00 -1100
$1,09 :: 13.15
1,40
Bracline
0,56 à: 0.08
0.90 0.05
0.22 :: 0.07
0.12 d: 0.05
2.26 d: 6.77
7.53 A 6.83
3.75 = 6.81
34.90 :: 34.17
LIDAK Camera
0.99 0.00
0.93
8.40 2:
0.74:00333 0.33
41,66 31.79
LMD (Rotio Role)
8.89 0.05
1,00 (100)
1 0.00
0:44 0.06
16.17 3.99
IL06 0.00
0,09 0.00
96,18 +: 14.53
5.1 Metrics to Evaluate the Constraints in LMD
Existing studies rarely attempt modality decomposition in fusion networks, and there is no standard evaluation method to validate such a decomposition. To bridge the gap, we introduce a metric designed to assess how each modality contributes to the model prediction.
We evaluate the model by replacing the input for one modality with an uncorrelated sample while keeping the other modality fixed. Then we compare the linearized predictions before and after
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substitution to assess whether the model reflects the change and remains stable for the unchanged modality.
We define two criteria for the assessment: (1) The prediction associated with the perturbed modality should change noticeably, ensuring that the model output reflects the new input rather than being dominated by should remain unchanged, indicating that each modality decoupled from the other modality.
bias. (2) The prediction associated with the unperturbed modality
For example, when the radar input is perturbed, the radar-based prediction should present low correlation with its original output, while the camera-based prediction should remain highly correlated with the original prediction. To quantify these effects, we employ the Pearson Correlation Coefficient 30 3J 32] and Mean Squared Error (MSE) to measure both similarity and distance between modality-specine predictions. Detailed metric formulations and sampling strategy are provided in Appendix D.3.
5.2 Evaluating the Linearization Effects in LMD
We evaluate the impact of linearization by comparing the poodality-specifie predictions before and after the input perturbation. Specifically, Rp/ R measures how much the radar-based prediction changes when the radar input is replaced. Since the baseline model lacks decomposition, it requires separate forward passes for clean and perturbed inputs. In contrast, LMD applies a single linearized model derived from clean inputs, ensuring consistent behavior across perturbations. In partial variants, only specific components such as activation layers or normalization parameters are fixed, while the rest of the model remains responsive to input changes. This setup enables a fair comparison, despite architectural differences.
As shown in Table LI LMD with the ratio rule satisfies both evaluation criteria. When the radar input is replaced, the radar- based output shows a Pearson correlation of 0.05 with its original output and a mean squared error of 9.05, indicating strong response to the perturbation. Meanwhile, the camera- based output remains unaffected, with a correlation of 1.00. Similar results are observed when perturbing the camera input. These findings demonstrate that LMD effectively isolates the influence of each modality, as each output responds only to changes its corresponding input. In contrast, baseline models show residual correlation across modalities, implying that the predictions remain entangled and fail to achieve modality separation.
Table 2: Comparison of Pearson Correlation and Mean Squared Error using different LMD variants.
Peurse Correlation
Mean Squared Error
Modality
BN-LN
R,R (1)
RyC
CJR (1)
Ry (1)
C.)
CJR (1)
(,10(()
Unifoom Menitty
0.50 + 0.04
1.00 0.00
1.00 = 0.00
0.2 + 0.05
9.58 + 2.71
0.00 + 0.00
0.00 + 0.00
38,89 + 10,80
Ideatity - Ideaiity
0.15 0.22
1.00 0.00
1.00 = 0.00
0.04
2.71
= 0.00
0.80 0.00
38,89 10,80
Rodar Camera
Identity Unifomm
0.00
1.00
0.99 0.00
2:71
0.00
0.04
38.55 10.71
Mantity - Ratio
0.05 + 0.02
1.00 0.00
1,00 + 0.00
0.15 + 0.04
9.05 2.34
0.00 + 0.00
0.00 + 0.00
54,49 22.15
Uniform- Ideniiy
0.41 1. 0.05
1.00
1.00..00 0.00
1.00 1 0.00
0.48 1. 0.07
15.82 1 3,63
0.08 1 0.02
0.00 1 0.00
31.77 1. 15.37
Ideatity - kdemitt
0.08 + 0.04
1.00 0.00
1,00 = 0.000
0.00 0,07
15.82 # 5,63
0.08 #: 0.02
0.00 = 0.00
31.77 = 15.37
LIDAR Camera
Identity- Unidons
0.09 0.04
=: 0.00
15:82 3.61
0.01
0.01
31.92 15.39
Maniry- Ratio
0.09 0.00
1.00 0.00
16.17 3.39
0.00
0.00 0.00
36.38 14.52
5.3 Quantitative Analysis of Separation Property on LMD Variants
Table 2 presents the quantitative results of applying different variants of LMD in both cader- and LIDAR-camera fusion models. We evaluate four combinations: aniform-identiry. identity-identiry. identity and identity-ratio, while the same metrics applied as in Section 5.2
Among the tested configurations, the combination of the identity rule for BatchNorm and the ratio rule for LayerNorm exhibits the clearest separation effect. In the padar-camers setting, this configuration yields the lowest correlation for the perturbed modality, with values of 0.05 for Rp/ R and 0.15 for CC, while maintaining full stability in the unperturbed modality, with R₂/C and C,/ R both equal to 1.00. It also produces zero mean squared error for the fixed modality. Similar behavior is observed in the LiDAR-camen fusion case. Other configurations such as using mniform-idention and rules result in higher modaltty correlations and smaller error between clean

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and perturbed inputs. This indicates that the outputs are not well separated. These findings suggest that applying the ratio rule essential for preserving the separation property in LMD.
Discussion
Computational Efficiency and Scalability Table summarizes the computational complexity and memory consumption from a single forward pass. LMD requires two forward passes maintaining (1) auxiliary state by caching information for linearization, which are discarded immediately after decomposition. LRP-based explanations necessitate an additional backward pass under gradient checkpointing. retain OV)N)) activations across Ni layers. Shapley-based approaches preserve low memory but incur exponential computation in the number of modalities, requiring ((24) forward passes to enumerate all coalitions over M modalities.
LMD + SHAP As SHAP 33 provides lo- cal explanations under an independence sumption among features, stability under cross- modal interactions is examined by the modality- replacement experiment and measuring cross- modality correlations of the resulting predic- tions, thereby quantifying the consistency of SHAP interpretations in the presence of interac- tions.
Table Computational Complexity and Memory Consumption (Single Forward-pass. Measurement). M number of modalities, N₂ number of layers
Computationnl Complexity
Memory
Methods
Passes
Consumption
BWD
0(1) 0124)
O(vN7)
Shapley based 33
2 FWD
O(L)
LMD
FWD
0(1)
O(1)
A combination of LMD and SHAP was also eval- uated. LMD first decomposes the model predic- tion into modality-specifie components; SHAP is then applied to the bias term, and the result- ing attribution is redistributed to the modality- specific predictions. In experiments, this LMD + SHAP better satisfied modality-level separability than SHAP alone. Quantitative results are summarized in Table
Table 4: Correlation under Modality Replacement: SHAP vs LMD + SHAP.
Modality
Metrie
SHAP
LMD SHAP
Radar Camera
R,
0.6909 = 0.11
0.9385 1 0.01
C,/R
0,6746 = 0.07
0,8942 1: 0.05
LIDAR Camera
L/ C,/L
0.7062
0.12
0.9210 + 0.04
0.7206
0.02
Radar LiDAR Camera
R,L,/C
0.7108
0.11
0.8790 + 0.05
C,L,/RR 0,7066
0.08
0.9574 1: 0.01
CpRp/L. 0.7025 0.07
0.9389 4: 0.02
Applicability to Attention-based Model LMD can also be applied an attention- based fusion model. For example, based on the Appendix D.2, we demonstrate LMD on an attention- radar-camera fusion network, CRN (2]] and extract the main term for each modality. Appendix D.2 details how the non- nonelinnesssies of the attention block's- (bilinear) matrix multiplication. and softmax, can be treated. We also note a limitation: the attention module inevitably introduces bilinear cross-modal terms that cannot be fully assigned to single modality under LMD framework. Hence, the assessment of the bilinear component requires deeper investigation by future works.
7 Conclusions
In this study, we introduced LMD, a framework designed enhance the interpretability of multimodal fusion models, with particular focus on multi-sensor perception in autonomous driving context. LMD provides model-agnostic approach that clearly attributes model predictions to individual sensor modalities. By disentangling modality contributions across network layers, LMD offers deeper insights into how each modality influences the fusion process. Our qualitative and quantitative evaluations demonstrated the effectiveness of LMD in identifying behaviors, revealing when and where fusion improves-on potentially degrades-modes performance. Overall, this work underscored the critical role of interpretability in autonomous driving systems, contributing to greater transparency in model decision-making and laying the groundwork for future research interpretable multimodal learning.
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Acknowledgments
This work was supported by 1) Institute of Information communications Technology Planning Evaluation (IITP) grant funded by the Korea government (MSIT) INO.RS-2021-111111 Intelligence Graduate School Program (Seoul National University)] and 2) the Ministry of Trade, Industry and Energy (MOTIE, Korea) Al Computing based AI Fusion Mobility Device).
Artificial
Development and PoC of On-Device
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
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Question: Does the paper describe the usage of LLMs if it is important, original, non-standard component of the core methods in this rescarch? Note that if the LLM is used only for writing, editing, or formatting purposes does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Answer: [NA]
Justification: The research does not involve the use of LLMs in its methodology.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, or non-standard components.
Please refer to our L.M policy (https://eunn http:///eeerpppec//ooeeeeereeeee 2225/LLM for what should or should not be described.
19

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Appendix
A Implementation Details
This section introduces the details utilized to implement LMD. First, descrihe the dataset used for the BEV perception model employed to demonstrate our method, as well as configuration of the fusion model utilized in the experiments.
A.1 Dataset
The nuScenes dataset [9] is large-scale, multimodal dataset designed for autonomous driving tasks. encompasses 1000 scenes, segmented into 700 training. 150 for validation. 150 for testing, each lasting about 20 seconds. The datasct captures 360 horizontal field view (FOV) through the use six surround-vies cameras, one lidar, and five radars, providing comprchensive environmental perception.
Notably, the nuScenes dataset stands out due to its high annotation frequency of every 0.5 seconds, translating to a 2 Hz rate, which includes over 1.4 million 3D bounding boxes across ten object categories. This extensive annotation facilitates advanced tasks like 3D object detection and tracking.
For evaluating detection performance, the dataset introduces unique metrics, including (mAP) and five true positive (TP) metrics (ATE, ASE, AOE, AVE, and AAE) to measure errors translation, scale, orientation, velocity, and attribute, respectively. These metrics are aggregated into the nuScenes Detection Score (NDS) to provide comprehensive performance overview.
Average Precision
Furthermore, for the task like the Bird's Eye View (BEV) segmentation. the dataset follows specific settings proposed in prior research, leveraging its rich radar point cloud data among other modalities. The nuScenes dataset not only facilitates the development advanced autonomous driving technologies but also sets benchmark for evaluating the performance of these systems through its detailed and nuanced evaluation metrics.
A.2 Model Architecture & Hyper-Parameten
The basic settings of the camera-radar fusion model follow those of SimpleBEV 23
The camera branch processes input from six view images, each of which is passed through backbone network, such ResNet to extract image features. These image features, along with radar point cloud information, are subsequently mapped onto the BEV coordinate system to facilitate perception tasks in the BEV space. The camera features and radar features in the BEV representation are concatenated and processed using convolutional operations generate latent features. In this paper, this convolution process is defined the fusion operation performed by the layer function (I 1). The latent features are then passed through a U-Net (35) decoder, after which the prediction head carries out segmentation task focused on identifying vehicle regions. table (5) presents radar-camera and I LiDA R-cmerra configuration details used for the SimpleBEV framework.
Table 5:
Detailed configuration of modality-specifi fusion setups
Parameter
Radar LiDAR Camera
Backbone
ResNet101
Fusion Operation
Concat & Conv 5
Sweeps
Input Size
(224, 400) (200, 200)
BEV Coordinate
A.3 Experiments Compute Resources
We employed pre-trained encoders for camera. radar, and LiDAR modalities. Our method was evaluated for 6019 iterations with a batch size of on 4 NVIDIA RTX 3090 GPUs, which required approximately hours in total.
20

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B Proofs on LMD
B.1 Proposition 1: Handling Activation Layers
Let A, = (xe, [y) denote the total input to the activation function for neuron j layer I. Let O2 (xe, æ.) denote the original output of this activation function for neuron j in layer The linearized activation function for a modality-specifff component Knj defined as ()(m)) = eg h The constant oj is given by Cj
The total input to the activation function is the sum of its decomposed modalityysseccini components: A, Emg(er.)) M/jj The output of the linearized layer (at,, xy) obtained by summing the outputs of linearized activation function applied each component (due the linearity of h ++ ryh and implied by eq. 12 and eq. 13) which state Kims (i(hlm))) and f (((Eme( Kmms)) >mm (e.r(e.r.)) F5(Kmn))
(Z),
Σ ((
Σ
eylimn =
Σ Wmng ez
(c.nd)
mE (e.n))
me
Substituting the definition of ej:
(00))
Ag + -11=) Aj.
(x),
eq. requires that the linearized model's output equals the original model's output at the operating point:
F TTe x.) = (2,, which is (Ee, x.) = O,. Therefore, we must have:
Oj = Aj + Aj-
We analyze equation in different cases:
Case 1: Oj + O. In this case, we can divide both sides by O5:
A₁
Aj +
This implies AÚ Aj, which means
Thus, if O2 # eq.
is satisfied when 0.
Case 2: O, 0. The equation becomes:
0
A, + A5.
This simplifies O. provided that A, + + 0. This condition typically holds as is small constant used to prevent division by zero. This case covers scenarios such as:
A, and O, 0 (c.g., ReL.U())=0, GELU(()=0). Here с. 0/(0 x) 0. Then
(Ce, 2.) = 0 0 0, which equals O5. eq. 8 holds.
A, # 0 but O,
(e.g., ReLU applied to a negative input). Here ej = 0/(A, + 0
(assuming A, + E 전 0). Then (Ec, = 0 A, O, which equals Oj. eq. 8 holds.
The connection to DTD, where ey is interpreted as the slope of the segment (5 (and possibly the origin, if the activation passes through it), underpins this choice. If c, O,/A, (for A, 0), then FJ = (0;/4,444
Oj. directly satisfying eq. (8). The formulation with makes this robust.
Thus, the proposed construction for the linearized activation function J ensures that eq. (8 is satisfied under the conditions discussed.
B.2 Proposition 2: Handling BatchNorm Layers
linearized output of the layer is the sum over modality components:
1 (Fe.
Σ
me m)(()))))
Camera (m c) and radar (m r) components. Because = 8,
7/
(l0))))) =
(m E (c,r)).
Vlof)=+6
21

---

Bias component (m b). With 8, =
7/ (hip -1²) 03.
(((b))))
V(o))2+e +E
Summing the three parts yields
7
fj xx)
(No 1155 hij 111 - 11) $3. (off2 +
Because the LMD framework guarantees ( EmmE(enn) 1/53 we can rewrite the above as
(x), 1
(F) (sre 11)) +
Vioty+
which is the BatchNorm forward pass F (Re, 2x) in evaluation mode. Hence F F3). satisfying eq. (8)
B.3 Proposition 3: Handling InstanceNorm Layers
Let (x,, x,) denote the input of the feature map to the InstanceNorm layer in layer l. For a specific channel j within this feature map, let H represent the activations for the channel j. The standard InstanceNorm operation for this channel is defined as:
FJ H E[H Var(H [
where "B[H) and Var[H are the mean variance of H computed over spatial dimensions, respectively. 7/ and (3, are learnable scaling and shifting parameters for channel j.
In the LMD framework, the input to the I-th layer, fd v.). is decomposed into codallty-speciine compo- nents:
Fl- (2c,
1-1
me(c.r.8)
where 1 represents the feature map attributed to modality (camera, radar, or bias) layer channel j. this means:
H
(c.r.b))
The LMD framework requires that the linearized layer function f satisfies the property that the sum of the decomposed outputs equals the output of the lincarized function applicd to the sum of inputs, which in turn must match the original function output at the operating point from eq. (8).
For InstanceNorm, we state that the variance of. Var [ "1 is treated as a constant, saved from a single forward pass of the original fusion model.
By considering proposed decomposition rule for ((
1-11 Blh
7/ + 8m
+
To sum these decomposed outputs over all modalities m E (c,r,b) to see if they reconstruct the original InstanceNorm output F
[rn E/h 1-1 7 + 8m
Σ )((S))
E
Voj² to
me(c.r.))
me(c.r.8)
22

---

By splitting the summation:
(Kin)
Σ
8mp
Vo + mE(c.n))
me (e.r.)))
For the first term, we the lincarity of expectation:
Σ (Wim) E(hmg)))
E
Kmj
E/hins
mE (c.r.)
mE(ann))
me (c.m,)
11
-H -
For the second term, within the definition of 6m: 5m 0 for m E rc and 6m for E (b). Therefore:
Σ
bm
8, + 6.53 + 5.33 03 + B} + B} 83
Substituting these back into the summed expression:
( -BIH 'I) + 18
Σ
(((m))))
00 +E
(E(C.)))
Since of is taken the fixed variance Var(H]] from the first forward pass, this expression is the original InstanceNorm output Fj. Since each modality is processed independently except for the shared, fixed scale, the resulting features adhere to intended separation property. The proposed decomposition rule for InstanceNorm layers ensures that the sum of the modality-specinne outputs from the linearized layer perfectly reconstructs the output of the original InstanceNorm layer, satisfying eq. (8) of the LMD framework.
C Variants on Activation Layers
Necessities of considering the bias-splitting rules in activation layers, aside from cases where constant term is inevitably introduced (e.g., normalization layers), is follows. When model linearized, bias' prediction can obtained by inputting zeros into all data points. As shown Figure is evident that bias features also exhibit perception capabilitics. Therefore, the process of appropriately distributing the bias contribution between the camera and radar can be considered. Particularly in the activation layers, if a specific neuron is activated by the camera, adding positive bias to the camera feature at that neuron could enhance the camera's 's contribution. Conversely, neuron is activated by the radar, adding a positive bias the radar feature could strengthen the radar's contribution. This idea to the development the sum rule introduced appendix C. The modality predictions based on various bias-splitting rules need to he extensively explored in future experiments.
While the sum rule adds the entire hias to whichever modality actually triggens a ReLU, it disregards how strongly the two modalities contribute in magnitude, Empirically this may still blur modality separation when both camera and radar deliver noticeable- but differently scaled signals. The ratio splitting rule therefore apportions the bias term h proportionally to the absolute pre-activations of camera and radar. The impact of sum and ratio splitting rules can be assessed the four perturbation based metrics (R,p R. Rp/C. Cp/R, C,/C). These rules modulate how sensitively and independently each modality responds to perturbation, offering an empirical basis for bias handling in activation layers.
C.1 Sum Splitting Rule
For the most common activation function, ReLU, if camera feature value is less than zero but a radar feature value is greater than zero, resulting in the activation of specific neuron, it reasonable to interpret that the neuron was activated by the radar. In such cases, a splitting rule be applied where the bias feature is added to the radar feature when bias feature value is greater than zero. Conversely, if both camera feature value and a bias feature value are greater than zero while the radar feature value is less than zero, distributing the bias feature to the camera feature strengthen the camera's 's contribution to that neuron. The sum condition tested in experiments derived from this idea. The related equations
follows:
23

---

Com - Condition = h-r < 0, h > 0, 1/ <
and hbj
1-1 > 0, 1-1 -
k,
Rad Condition
1-1 0, 1
> 0, 1 > 0
and Nij > 0, nrj < 0, Naj < 0
[ + tbog ((hbe
for Cam Condition,
(((g)))
otherwise.
(hi7 + 11g5))
for Rad Condition, otherwise.
((155))
c ch()))
(15)
0, for Cam &Rad Condition,
P(
N153"
otherwise.
F. (xcc
where
F Xx)
+ +E
C.2 Ratio Splitting Rule
In activation layers bias term hby can enhance the modality responsible for the neuron's 's activation. The ratio splitting rule redistributes that bias the camera and radar features in proportion their absolute magnitudes. while leaving neurons that fail the ratio conditions unchanged. This preserves conservation property eq. generalizes the identity and uniform rules introduced earlier.
Ratio Condition
(he > 0, h > 0, Ki > 0) or (li < 0, | < 0, 1 < <0),
Ratio Condition2 = (hi 1-1 > 0, h 1 > 0, h 1 < 0)
or (hi
1-1 > 0). 1-1 < 0, hi < 0, m
|M
E [].
+ 1b +
with small for numerical stability.
(, x.)
c F3. (Te, (+) +
be the slope from proposition
Then
(he +(1 - (1-aa))
Ratio Condition,
f((bl
dhl a 11 chi-1
Ratão Condition2, otherwise,
dhi + aa (1
Ratio Condition,
dhit7 (1 ax) 11
Ratio Condition2.
chiy
otherwise,
0, (((() = cchy otherwise.
Ratio Conditionor Ratio - Condition2,
Because 11, is only re- paratitiioned the eq.
8 strictly preserved.
The ratio rule thus interpolates smoothly between existing bias-handling strategies while adapting to the magnitude of each modality's evidence, leading the stability improvements reported in section

---

Table 6: Comparison of Pearson Correlation and Mean Squared Error for selected LMD rules (Radar + Camera)
Peurson Currelation
Squared Error
Modality
Melhod
R,
R,
C,R
C,C
R2/R(())
K,CCO)
()/R())
C,/C(t)
Activation
Rudar + Camera
Sum Spliting
0.12 0.04
1.00 0.00
1.00 a: 0.00
0.23 0.05
9.63 2.34
0.00 d. 0.00
0.00 =: 0.00
40.43 :: 19.35
Ratio-Splittng
0.08 0.06
1.00 0.00
1.00 + 0.00
0.18 0.04
4.46 = 1.45
0.00 + 0.00
0.00 = 0.00
30.35 +: 4.19
D Further Experiments
D.1 Three Modality Experiments
We extend LMD to three-sensor setting (camera radar + LiDAR) and performed comprehensive variant study on bias-splitting strategies. This experiment is expected to further demonstrate the generality of LMD. For brevity, we share the correlation results in tables tables[ usiin with the main text, we verify that perturbing modality does not affect the other. You refer to the relevant thcoretical formulation in Appendix F.
juiig the same metries as in the main paper. Consistent
Table 7: Threc-modality (R+L+C) ablation of LMD versus baselines
Metual
Nerm
C, c
C2 (2
R.t./Cm
Bucckac2
115499 2.11
BU0K0 2 1.08
0.5455 2. 317
0.399/21008
0 02022
013350 211012
1MHh Borts Bides
1.009 6:40
011366 3665-5-11
6 9-232
0.029 à 8.01
10000-0 0.00
61 088
-0 0.130
0:2300 8807
10000 0.00
B:0073 6.mm
0 0/06661 06.097
Table 8: Three-modality (R+L+C) ablation of LMD variants.
Muthed
C,
C,
C,T,//
Cgt,/C
C, (1)
Ry :1]
R, (1)
Unifeca Mensiy
1.00000: =: 0.000
0,2224 = 0.148
0 0.124
1,0000 = 0.000
0,2064 a D.805
0,3162 a 0.94
10000 =: 0 000
0,3572
0.134
0.3522 4: 0.089
lamilyy - Mentiy
0000
1 0.000
0.3315
0.170
0:061
0.06
1,000
0.000
0330
0.16
00000 e 0.13
10000
0.0
0.000
0.000
Bolan 1 0.115
Identy Unitom
0,505%
= 0.000
0,831
= 0.10
0.007
= 8008
0.20
03182
+ 0.14
0.087
= 0.3125
3,m9
= 0:200
0.057= = 80075
00085 :: 0.118
Identiy
1.0000
0.000
0.1385
0.302
0.0379
0.042
0.000
0.3310 01667
100000
0.000
0.3023 0.097
D.2 Application to Attention-based Model
Here, we present the experimental results and limitations applying LMD to attention-based model 21 In attention operation, two major sources of nonlincarity must be considered: the (bilincar) matrix multiplication and the softmax. For matrix multiplications, we treat term modality feature only when it multiplics same-modee made-itd features modality fcature with constant; otherwise, it assigned to the bias features. For the softmax function, use the same procedure as for linearizing activation layers.
We successfully decomposed modality-specific terms in CRN and observed that high-order term remains due matrix multiplication between input variables.
Table 9: Comparison (Pearson Correlation) of LMD Variants with CRN
21
Modality
Method
R (4)
RC)
C,/RR (1)
C2/C(L)
Uniform Identity
0.6123 + 0.043
1.0000 + 0.000
1.0000 + 0.000
0,5912 4: 0.052
Identity- Identity
0.2184 + 0.125
1.0000 + 0.000
1.0000 + 0.000
0.3891 + 0.043
Camera + Radar
Identity- Uniform
0,2572 + 0.067
1.0000 + 0.000
0.9882 + 0.010
0.3827 0.046
Identity Ratio
0.0552 1 0.0213
1.0000 + 0.000
1.0000 + 0.000
0.1571 + 0.045
Table 10: Comparison (Mean Squared Error) of LMD Variants with CRN 21
Modality
Method
Rg/R (t)
R₂/C ())
C,/RR (1)
C,/C (1)
Uniform Identity
8.9732 + 2.142
0.0000 0.000
0.0000 +: 0.000
39.1135 = 19.802
Identity Identity
9.4676 + 2.295
0.0000 0.000
0.0000 :: 0.000
28.8149 18.501
Camera + Radar
Identity Uniform
10.5438 + 1.714
0.0000 0.000
0.0087 + 0.041
27.5582 16.711
Identity Ratio
11.5812 + 2.712
0.0000 0.000
0.0000 + 0.000
54.4890 + 22.151
D.3
Further Post-hoc Interpretation
This section presents additional examples of applying LMD to the BEV perception task. Through further visualizations, we expect that our proposed method is validated with sufficient consistency to reliably interpret the model.
25

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fig Bgad fig. show further visualization results for the reliability of the proposed LMD on the BEV perception benchmark. lig. normalizes each modality-spenn specific feature map with sigmoid transform before projection. so only positive activations remain; bright regions therefore mark locations where a single sensor delivers decisive, supportive cues to the detector, making the saliency of true objects immediately apparent. By contrast, fig. Frtains the full signed output and colour-codes positive and negative responses, allowing us to see not only where modality reinforces the final prediction but also where actively suppresses prediction.
Prediction from Fusion Model
Prediction from Camera
Prediction from Radar
Prediction from Bias
150
150
250
150
250
250
250
250
350
350
350
350
300
Figure Visualizations of using Sigmoid.
Prediction from Fusien Model
Prediction from Camera
Prediction from Radar
Prediction from Bios
150
150
130
150
250
250
250
359
150
350
258
100
200
200
302
Figure 4: Visualizations of both Positive and Negative Values.
Interpreting the Perceptual Capability of the Bias Component In our setting, the prediction from bias generated solely from the internal constants without any modality input. perceptual capability mainly stems from the linearized activation layers: internal constant components follow the pre-determinet activation patterns, yielding structured responses. On the other hand, linearized normalization layers are responsible only for scaling of features and thus they unlikely to contribute to perceptual expressivity of the hias term.
D.4 Metric Formulas
To evaluate the separation property of the modality-specifie decomposition, we introduce four perturbation-base metrics. Here, perturbation refers to replacement of one modality input with uncorrelated sample. These metrics assess how much each modality-specific output responds to changes in its own input and remains invariant to others. Each metric is computed using the Pearson Correlation Coefficient (PCC) and Mean Squared Error (MSE).
Pearson Correlation Coefficient (PCC). Briefly, for pearson correlation coefficient, we define:
Σ, (Prli) - Pt)) (P(++5000 [i] Pikk P(k+ P moo
PCCk.m
(16)
V V.. Pa) Σ, (Pee Pcl4500mmmmod - Pcc4:000mmmod
where N 6019. Pt and P(e+500mmmmd the predictions for the sample spaced by 500m from k, Pr and P (1++-500mmmod are their respective mean values, and 21, 12) ensures 12 distinct comparisons. PCC measures the similarity between two prediction vectors by first subtracting their respective means (mean- centering), computing the dot product of the centered vectors, and normalizing it by the product of their standard deviations. The resulting value lies in the range [-1, 1], where I indicates perfect correlation, O indicates no correlation, and -1 indicates perfect inverse correlation.
26

---

Mean Squared Error (MSE). In addition to PCC, compute the mean squared enror to quantify absolute differences:
MSE
M Σ (Pr P0k+500m)mggl [i])
(17)
where M denotes the total number of pixels in the BEV feature map.
Perturbation- Metrics. Let øe and 2,, denote the clean camera and radar inputs, and x= x their respective perturbed versions. Let Feamera(Tes (T)) and (Ta, denote the modality-specifer predictions.
We define the following four metrics:
Rp/ R. Radar prediction change under radar perturhation:
PCCk.m Fe of (ee, MS (Frnar (Te, "f"",, Faauu Fr))
Lower PCC and higher MSE are better, indicating sensitivity to radar input.
*
Rp/CC. Camera prediction change under radar perturbation:
PCC (aumera (Te, xf Feumera zr)), MSEA.m (Feamerl (Taeeeaaeeeaa (ee orr"" Feamera (Te, xr))
Higher PCC and lower MSE are better, indicating invariance of the camera modality.
Cp/ Radar prediction change under camera perturbation:
PCC (F Faia (x _x+), (ee, 2r)), MSEA. (Frnur Den ,.,), Paauu(Zes xr))
Higher PCC and lower MSE are better, indicating invariance of radar modality,
.
Cp/C. Camera prediction change under camera perturbation:
PCCk, ( (Feamera (
(xlt ,rr), Feamenaeer 2,)),, MSE.... (Fiamen (xc (r Femmeralal 2.,))
Lower PCC and higher MSE are better, indicating sensitivity to camera input.
These metrics are averaged over the entire test set using uniformly sampled perturbations. Together, they verify whether the decomposition satisfies: (1) sensitivity to the perturbed modality and (2) invariance to the unperturhed modality - key indicators of successful modality separation.
E
Further Discussion
E.1
Broader Impact
LMD introduces the first hoc framework capable of attributing the predictions of each layer in a multi-sensor fusion model its respective modalities, without modifying the original architecture.
Positive societal impacts may include:
Enhanced safety in autonomous systems. By revealing the sensor modality (camera, LiDAR, or radar) primarily responsible for cach prediction, LMD enables identification of single-sensor failures before they propagate into critical errors.
Accelerated certification and debugging. Modality-level attributions support system audits and facilitate the generation safety evidence, line with emerging regulatory requirements for ADAS systems.
Extension of interpretability to other multi-modal domains. Since LMD is agnostic to model architecture, it can applied to range multimodal perception systems beyond autonomous driving, such as medical image- text fusion or surveillance vidco analysis, to clearly attribute the role of cach input modality in the model's decisionnmaaing process,
Commitment open science. We will publicly release our codebase, pretrained checkpoints, and evaluation scripts under an open-source license to promote reproducibillity and rigorous external validation.
Potential risks include:
Misleading reassurance. Users may over-trust visual explanations; clear modality-wise attribution does not guarantee that fused decision is accurate.
Bias amplification. LMD uncovers but does not correct dataset imbalances, and naive use of its explanations may perpetuate structural biases.
27

---

Residual opacity. The interpretability of high-order interactions and root-point selections in activation layers remains open challenge for future work.
To mitigate these risks, we recommend pairing LMD with the perturbation- based metrics introduced in this paper to quantitatively assess explanation fidelity prior deployment. We also encourage conducting independent reproducibility audits geographically and demographically diverse datascts, and suggest applying the same level of scrutiny augmented by domain-sponn speciiii expertise operational context when extending LMD to other multimodal domains, such as medical imaging or service robotics. LMD sets a new benchmark for interpretability in multimodal domains and provides a solid foundation for future research and real-world deployment.
E.2 Model- Wise Explanation
Understanding the extent which we can perform model-wise explanations by comparing the camera-only model and the fusion model is a critical issuc. Interpreting the contributions cach modality based on the results of these two models for specific data is highly challenging. This difficulty arises, for the two models are trained on different input distributions.
For instance, if a fusion model successfully detects an object that camera-only model fails detect, it would be clear enror attribute this success radar data. Predictions from the camera-only model should be utilized only for references. For example, the camera-only model successfully detects particular vehicle, can be inferred that the information captured by the camera sensor for that region is sufficient. This insight can be considered when analyzing the fusion model using the LMD approach.
Nonetheless, it is crucial to recognize that LMD is designed to provide hoc interpretations for single trained model, not to be tool for interpreting two different models trained on different data distributions.
E.3 GradCAM Experiments
Since LMD enables access to internal activation maps, is possible to compute activation maps radar data and subsequently apply the Grad-CAM B6] approach.
Figure 5: Visualizations
Using Radar's Activations from LMD.
Grad-CAM leverages gradient information to highlight relevant areas, Grad CAMM+CCM+++ introduces modified approach by taking positive gradients of target functions, and Score-CAM 38 bypasses the need for gradient information altogether, instead relying on activation scores to determine feature significance. Seg-Gree Grad-rad- CAM 39 is adaptation of the Grad-CAM method, applied to the task of image segmentation by treating the segmentation output the target function while maintaining the original mechanism of Grad-CAM for visual explanations. This approach enables focused analysis regions relevant to segmentation tasks, leveraging the gradients of the target output with respect to the feature maps. The Grad-CAM is calculated by
LaMAcAM ReLU Σ as Al
(18)
where LEaSCAM is the class activation map for class of are the weights for the k-th feature map in 1-th layer " computed as the global average of the gradients flowing back from the output unit for class ReLU is
28

---

Table 11: Comparison of LRP and LMD methods
Method
Target
Result
Direction
Computation
Element-wise Op.
Property
LRP
Output Segment
Input Segment
Backward
2 FWD +1 BWD
Direct pass
Conservation rule
LMD
Inpul Segment
Ouipul Segment
Forward
FWW
Scaled pass
Eq S
applied to focus features that have positive influence the class of interest. The weight of k-th feature map for class computed follows.
ΣΣ
oy
Ok Z
DA
(19)
where yf,, A and Z denote the score for class c, (-th) entry in k-th activation map, and normalization factor respectively. In Reg-Grad-CAM, yf replaced by ((,))E Biy. where is set of pixel indices of interest in the output mask.
Through obtained activations which are derived solely from the radar data with LMD, it becomes possible to use the conventional CAM method based the activations the intermediate layer, which call The Madar-GradCCMM calculated by:
Σ Lal(A" - - (0.: (oc:
Liadar GMdCAM = ReLU
(20)
where A (Oe, Xx) and A (Oe, Oz) the activations of linearized fusion model f and of. is calculated in the same manner as in eq. (19..
Σ (ssiee (xe, xx) aA (xe
1 of Z
ΣΣ
(21)
Based on this calculation, the Radar-GradCAM shown in Figure 5 reveals which parts of the fusion model are relevant the radar data points, by taking the radar-only activations that crucial to the perception task.
E.4 Positioning LMD
In [I we compare LMD with LRP to clarify its positioning. In LRP, for element wise operations, such normalization activation layers, relevance scores are directly assigned to input neurons with distinct bias
splitting rules [14 16
retaining the scale information used seement-wise operations. Also, I,P requires two forward passes and one hackward pass with gradient checkpointing 440). while LMD requires one forward pass to store the information. allowing to be implemented with a total of two forward passes. Lastly, LRP must satisfy the conservation rule during correctly interpreted. In contrast, LMD follows
17 [1] contrast, LMD adheres original function behavior
the backward pass ensure that the output relevance original function behavior, focusing on modality decomposition adheres (8. These different constraints arise from the distinct goals of each method.
F Extension to General Multimodal Model
this section, demonstrate that LMD method can be generalized to multimodal models. achieve this by extending the formulation to accommodate M modalities.
Beginning at the fusion layer (1 1), the first-order Taylor expansion of f} hecomes:
f [Xm1s" Xmnga
Σ
Σ
ImyeTmet
me mm?
f (āmps Amms Σ
Σ
Imsidma +e
(22)
where indexes the modality, ........ Doner represent reference points for M modalities, and Jujs denotes the Jacobian caleulated at the reference points for modality
29

---

For subsequent layer with / = .., .N, suppose i-th layer function takes Em (m Knn as inputs. the following holds: :
Σ
Kim
mm(mm m... be
(23)
Σ
Σ
Imyt (b/ma)) + +b}
Σ
hmn
(m
me (mp
We define all modality features that all layers fusion model be decomposed into them.
Fj (Xmi) XmmM
1-1
(24)
me me (mi.... mm
himy + Bemass 165
From eq. (24). we can confirm inductively that the following holds. For the linearized layer functions ʃ ....... f
with bias-splitting rules applied, F (Xm1...... Xmma) F) (Xm1 ..... Xmma) for E (1...., .V. where Ftr (Xma Xmaa) ) be f' ....o o f² "(xmi Xmaa forcing the epsilon in eq. 22 to zero. Based on all the layer functions, f !"...."" the modality features are computed as follows:
hmn (((m)) m E {mi,.... mM. b}
(25)
Furthermore, the following arithmetic is satisfied for layer function f
(26)
(1 +
From eq.
25 and eq. (26). the following property holds for every layers:
(Xmi (Xmi..... Xmaa
F Xmms Omz:
Omn
M
(omp....
Omps
Xmy1
(27)
+ ma (Omy Ompa
The lincarization and bias splittini rules applied to the normalization and activation layers are consistent with those described in the main paper.
30


Reviews:
[{"review_id": "t6pIJ7ACmf", "reviewer": "Reviewer_UztF", "summary": "This paper introduces Layer-Wise Modality Decomposition (LMD), a post-hoc interpretability method designed to disentangle and attribute the contributions of individual sensor modalities (such as camera, radar, and LiDAR) across all layers of multimodal perception models, particularly in autonomous driving. Motivated by the opacity of high-capacity fusion models, LMD locally linearizes neural network layers to decompose their outputs into modality-specific components without altering the original architecture.", "questions": "1. Have the authors tested LMD on settings with more than two modalities? While Appendix F theoretically extends the method to an arbitrary number of modalities, it remains unclear whether any practical challenges arise in such cases, for example, in the selection of reference points or maintaining modality separation when scaling beyond two modalities.\n\n2. Can the proposed method be readily applied to more complex fusion architectures, such as those incorporating cross-modal attention mechanisms — for example, the network used in TransFuser [1]?\n\n3. Figure 2 shows \"perceptual capability\" from the bias term, which is described as representing constant offsets. However, what exactly does this component capture, residual cross-modal interactions, constant activations, or something else? If it truly represents constant information, does the bias-based prediction remain stable or show minor variations when presented with similar inputs?\n\n[1] TransFuser: Imitation with Transformer-Based Sensor Fusion for Autonomous Driving, TPAMI 2023.", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths:\n\n1. Advancing interpretability in multimodal sensor fusion is highly valuable, particularly for safety-critical applications such as autonomous driving.\n\n2. The proposed method is well-grounded in theory, building on established frameworks like Layer-Wise Relevance Propagation and Deep Taylor Decomposition, and shows promising results in modality attribution.\n\nWeaknesses:\n\n1. The current evaluation and demonstrations appear limited to two-modality settings, without experiments or evidence showing its scalability to scenarios involving three or more modalities. While Appendix F is said to discuss generalization to >2 modalities, no experiments are shown for such settings. It’s unclear how well LMD scales or handles interference when more than two modalities are used.\n\n2. While the paper discusses bias components extensively, the semantic meaning of bias remains underexplored (Please see question 3 below).\n\n3. The paper doesn’t compare against other post-hoc multimodal interpretability methods, such as SHAP[1] or attention-based attribution techniques in transformer-based models [2], which are frequently adopted in multimodal fusion settings (e.g., [3])\n\n[1] A unified approach to interpreting model predictions, NeruIPS 2017\n\n[2] Transformer Interpretability Beyond Attention Visualization, CVPR 2021\n\n[3] TransFuser: Imitation with Transformer-Based Sensor Fusion for Autonomous Driving, TPAMI 2023.", "was_revised": true, "final_justification": "After reading the rebuttal and reviewing the authors' responses to other reviewers, I believe this work has demonstrated its value from a practical usage perspective. Therefore, I have decided to raise my score. I hope the authors will include the additional experiments in the revised version of the paper, as they provide essential evidence of the advantages of LMD.", "rebuttal": "We appreciate your thoughtful feedback.\nYour recognition of the value of interpretability for safety-critical fusion models and the theoretical soundness of LMD is encouraging. Below we address each concern in turn and outline changes we will include in the revised submission.\n\n## **W1 & Q1** \n> **&nbsp;The current evaluation and demonstrations appear limited to two-modality settings, without experiments or evidence showing its scalability to scenarios involving three or more modalities**\t\n\nTo address the concerns, we have extended LMD to a three-sensor setting (camera + radar + LiDAR) and performed a comprehensive variant study on bias-splitting strategies. This experiment is expected to further demonstrate the generality of LMD. For brevity, we share the correlation results using the same metrics as in the paper. To ensure reproducibility and enable other researchers to verify, we plan to release the code for all experiments presented.\n\n\n\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n**Table A.** Three-modality ablation of LMD versus baselines  \n| **Method** | **Act.** | **Norm.** | **CₚRₚ/L (↑)** | **CₚRₚ/C (↓)** | **CₚRₚ/R (↓)** | **CₚLₚ/R (↑)** | **CₚLₚ/C (↓)** | **CₚLₚ/L (↓)** | **RₚLₚ/C (↑)** | **RₚLₚ/R (↓)** | **RₚLₚ/ L(↓)** |\n|---------------------|:--------:|:---------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|\n| Baseline 1 | ✗ | ✗ | 0.3606 ± 0.095 | 0.0679 ± 0.069 | 0.1103 ± 0.095 | 0.1957 ± 0.092 | 0.0955 ± 0.069 | 0.2265 ± 0.094 | 0.6794 ± 0.099 | 0.4755 ± 0.092 | 0.4663 ± 0.104 |\n| Baseline 2 | ✓ | ✗ | 0.9764 ± 0.018 | 0.5459 ± 0.118 | 0.6565 ± 0.134 | 0.9717 ± 0.029 | 0.5488 ± 0.117 | 0.5360 ± 0.103 | 0.9865 ± 0.009 | 0.6722 ± 0.107 | 0.5384 ± 0.092 |\n| Baseline 3 | ✗ | ✓ | 0.3685 ± 0.089 | 0.0669 ± 0.072 | 0.1245 ± 0.086 | 0.2165 ± 0.084 | 0.0570 ± 0.073 | 0.2289 ± 0.098 | 0.6880 ± 0.098 | 0.4751 ± 0.092 | 0.4678 ± 0.104 |\n| **LMD (Ratio Rule)** | **✓** | **✓** | **1.0000 ± 0.000** | **0.1385 ± 0.232** | **0.0279 ± 0.042** | **1.0000 ± 0.000** | **0.1488 ± 0.132** | **0.2310 ± 0.057** | **1.0000 ± 0.000** |  **0.0373 ± 0.091** | **0.2023 ± 0.097** |\n\n----------------------------------------------------------------------------------------------------------------------------------------------------------------\n\n**Table B.** Three-modality ablation of LMD variants  \n  | **Method**            | **CₚRₚ/L (↑)** | **CₚRₚ/C (↓)** | **CₚRₚ/R (↓)** | **CₚLₚ/R (↑)** | **CₚLₚ/C (↓)** | **CₚLₚ/L (↓)** | **RₚLₚ/C (↑)** | **RₚLₚ/R (↓)** | **RₚLₚ/L (↓)** |\n|-----------------------|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|:--------------:|\n| Uniform - Identity    | 1.0000 ± 0.000 | 0.3224 ± 0.148 | 0.5572 ± 0.114 | 1.0000 ± 0.000 | 0.3164 ± 0.168 | 0.3162 ± 0.043 | 1.0000 ± 0.000 | 0.5572 ± 0.114 | 0.3322 ± 0.089 | \n| Identity - Identity   | 1.0000 ± 0.000 | 0.3315 ± 0.170 | 0.0971 ± 0.056 | 1.0000 ± 0.000 | 0.3305 ± 0.161 | 0.0039 ± 0.139 | 1.0000 ± 0.000 | 0.0926 ± 0.080 | 0.0042 ± 0.113 |\n| Identity - Uniform    | 0.9994 ± 0.008 | 0.3332 ± 0.160 | 0.0807 ± 0.008 | 0.9903 ± 0.079 | 0.3182 ± 0.145 | 0.0087 ± 0.1125 | 0.9999 ± 0.000 | 0.0870 ± 0.078 | 0.0085 ± 0.113 |\n| **Identity - Ratio** | **1.0000 ± 0.000** | **0.1385 ± 0.232** | **0.0279 ± 0.042** | **1.0000 ± 0.000** | **0.1488 ± 0.132** | **0.2310 ± 0.057** | **1.0000 ± 0.000** |  **0.0373 ± 0.091** | **0.2023 ± 0.097** |\n\n--------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\n> **&nbsp; It remains unclear whether any practical challenges arise in such cases, for example, in the selection of reference points or maintaining modality separation when scaling beyond two modalities.** \n\nWe found LMD maintains Eq. (7) in the main paper (all modality-specific outputs sum exactly to the original prediction) while  preserving the separation property when extending to three modalities. Importantly, we did not observe any instability and used the same reference points selection strategy as in the two-modal case without modification. \nSince this concern was raised by multiple reviewers, we will incorporate these results into the camera-ready version of the paper.\n\n\n## **W2, Q3** :\n> **&nbsp; Figure 2 shows \"perceptual capability\" from the bias term, which is described as representing constant offsets. What exactly does this component capture, residual cross-modal interactions, constant activations, or something else? If it truly represents constant information, does the bias-based prediction remain stable or show minor variations when presented with similar inputs?**\n\nThis inquiry touches on a core component of our method. To summarize, the bias-based prediction refers to the model's perceptual output generated solely from the constant internal values—specifically, the bias terms—without using any sensor input (e.g., RGB values or Radar RCS values). This is tightly aligned with the central philosophy of LMD, which aims to interpret the model perception in a modality specific manner.\n\nThe natural follow-up is : why does this constant bias term exhibit perceptual capability? The perceptual capability of bias term is largely originate from the linearized activation layers. The key lies in how activation layers are handled during linearization. When we linearize these layers, we store information about the activated neurons to reproduce the original model behavior under perturbed inputs. Then, at inference time, instead of using the activation function directly, the model uses the pre-stored information. As a result, the internal constant components (i.e., bias terms) follow these pre-determined activation patterns, which in turn gives rise to perceptual behavior. On the other hand, normalization layers, as discussed in the paper, are responsible only for scaling and thus we hypothesize that they are unlikely to contribute significantly to perceptual expressivity in the bias term.\n\nIn rigorous perspective, as you correctly identified, the bias term captures cross-modal interactions, since the activation status of a neuron at certain location itself is determined largely by the modality-specific information. \n\nAccurately attributing these effects to specific modality contributions is a highly complex, with existing approaches such as Shapley-based methods known to face limitations in such contexts. This observation also relates to the limitation we acknowledged in the paper. We have partially addressed the limitation by integrating our approach with an explanatory model, which offers a complementary perspective. Details can be found in our response to ```Reviewer KNAK```.\n\n\n\n> **&nbsp; While the paper discusses bias components extensively, the semantic meaning of bias remains underexplored**\n\nWe recognize that this can be regarded as a potential weakness, and we plan to elaborate on this point in the camera-ready version by providing a more detailed explanation and discussion.\n\n\n## **W3, Q2** :\n> **&nbsp; The paper doesn’t compare against other post-hoc multimodal interpretability methods, such as SHAP or attention-based attribution techniques in transformer-based models, which are frequently adopted in multimodal fusion settings**\n\nIn response to similar requests from multiple reviewers, we have conducted additional experiments involving SHAP [1] as well as attention-based attribution techniques in transformer-based models. For a comparison with SHAP or LRP, we suggest to check our response to ```Reviewer KNAK``` and ```Reviewer opMu```. \n\n> **&nbsp; Can the proposed method be readily applied to more complex fusion architectures, such as those incorporating cross-modal attention mechanisms — for example, the network used in TransFuser?**\n\nWe have carefully considered the applicability of our method to more complex fusion architectures, such as those incorporating cross-modal attention mechanisms, including the TransFuser. While TransFuser is widely-referenced fusion architecture, we chose to conduct our additional experiments on CRN [2], a more recent model that operates on the nuScenes dataset. Additionally, results on the model —incorporating transformer-based architectures—can be found in our response to ```Reviewer 3nmi```.\n\n----------------------------------------------------------------------------------\n[1] Lundberg, Scott M., and Su-In Lee. \"A unified approach to interpreting model predictions.\" Advances in neural information processing systems 30 (2017).\n\n[2] Kim, Youngseok, et al. \"Crn: Camera radar net for accurate, robust, efficient 3d perception.\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2023."}, {"review_id": "6EhRAjJMuQ", "reviewer": "Reviewer_KNAK", "summary": "The authors propose a post-hoc interpretability method for perception models in autonomous driving, which process multimodal inputs (image, LiDAR or radar data). Their method locally linearizes learned operations to separate modality-specific information and attribute the impact on model outputs.", "questions": "- As mentioned in the weaknesses, can you apply your method to another model and combine it with a feature-based method?\n- Can you briefly discuss how your method relates to methods for mechanistic interpretability?\n- Can you measure non-linear correlation between corrupted and non-corrupted forward passes, e.g. using the xicor metric [4]? This could allow for a more detailed comparison.\n[4] Chatterjee, S., 2019. A new coefficient of correlation. arXiv e-prints, page. arXiv preprint arXiv:1909.10140, 711.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\n- The proposed method splits the processing of a multimodal model and generates predictions using each modality individually. This allows for comparing predictions based on fused features with predictions based on modality-specific features, which is a step toward interpreting the modality-specific contributions.\n- The authors further show that their method disentangles fused features by corrupting one input modality and measuring no impact on the predictions based on the other modality. They perform this experiment for a multimodal model that processes either image and LiDAR or image and radar data. This should be elaborated before publication.\n- The authors ablate different linearization approaches for batch, layer and instance norm layers. They measure the Pearson correlation and MSE between predictions from corrupted and non-corrupted forward passes.\n\nWeaknesses:\n- As a minor issue, while their method seems to be model-agnostic, the authors only evaluate it using one model. Experiments with further models would strengthen the paper. Nevertheless, this issue should at least be discussed in the final contribution.\n- The proposed method is a step toward interpreting the impact of all features per processed modality, but doesn’t quantify or interpret the impact of specific features per modality. Combining the proposed method with a feature-based method like SHAP [1] would significantly strengthen the contribution. \n- As a major shortcoming, the related work subsection on interpretation methods focuses on fairly old methods and doesn’t include recent mechanistic interpretability methods (see [3]), which are also used to analyze motion forecasting models for autonomous driving [4]. \n\n[1] Lundberg, S.M. and Lee, S.I., 2017. A unified approach to interpreting model predictions. Advances in neural information processing systems, 30.\n[2] Bereska, L. and Gavves, S., Mechanistic Interpretability for AI Safety-A Review. Transactions on Machine Learning Research.\n[3] Tas, O.S. and Wagner, R., 2024. Words in Motion: Extracting Interpretable Control Vectors for Motion Transformers. arXiv preprint arXiv:2406.11624.\n\nThe reported strengths make the paper generally interesting and publishable, assuming that the shortcomings will be resolved as outlined above.", "was_revised": true, "final_justification": "Thank you for the rebuttal.\n\nThe additional evidence using an additional model is well appreciated.\nResults using SHAP as well Combination of LMD with SHAP are convincing.\nMore emphasis on a discussion of recent work In particular, recent mechanistic interpretability methods, which are also used to analyze motion forecasting models for autonomous driving would improve the manuscript.\n\nI am ready to raise quality to 3 good now and go to accept.", "rebuttal": "Thank you for the thoughtful constructive feedback and for highlighting the paper’s strengths.\nBelow we address every concern raised and summarize the concrete additions we will include in the camera-ready.\n\n\n## **W1 & Q1** \n> **&nbsp; While the method seems to be model-agnostic, the authors only evaluate it using one model. Experiments with further models would strengthen the paper. Nevertheless, this issue should at least be discussed in the final contribution.**\t\n\nPlease check our responses to ```Reviewer 3nmi```, which contain results on models beyond SimpleBEV, incorporating transformer-based architectures to our method. As you suggested, we will address this issue by explicitly discussing it as part of our contributions.\n\n\n## **W2** \n> **&nbsp; Combining the proposed method with a feature-based method like SHAP would significantly strengthen the contribution. Can you apply your method to another model and combine it with a feature-based method?**\n\n\nWe sincerely thank you for proposing the interesting idea. In response to your suggestion (slightly different from what you suggested), we conducted experiments combining SHAP to our method. Specifically, we treated each modality as a distinct feature in the SHAP framework—for example, in a fusion model incorporating Camera, Radar, and LiDAR, we assumed three input features corresponding to each modality. We computed the full set of Shapley values over all possible coalitions. To estimate each sensor-based prediction, we followed the standard SHAP procedure: we randomly sampled data from the training set to replace the modality of interest, fed it into the model, and subtracted the resulting prediction from that of the original model.\n\nAs noted in [1], SHAP is a local explanation method that assumes feature independence. To evaluate the stability of the SHAP-based explanation under this assumption, we performed the same modality replacement experiment as in our main paper and measured the correlation between the resulting predictions across different modalities. This allowed us to quantify how consistent the SHAP explanation remains in the presence of modality interactions.\n\nAdditionally, we explored an approach that combines LMD with SHAP. Specifically, we first used LMD to decompose the model’s prediction into modality-based components. We then applied SHAP to the bias prediction to redistribute it back across the modality-specific predictions. This approach allows us to refine the attribution by leveraging the strengths of both methods.\n\nWe conducted the modality-replacement experiment using this combined method observing that it better satisfied the assumption of modality-based separability compared to SHAP. Please refer to the table below for the quantitative results.\n\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n**Table A.** Stability Experiments on SHAP and LMD + SHAP (Pearson Correlation)\n| **Modality** | **Metric (↑)** | **SHAP (Corr)** | **LMD + SHAP (Corr)** |\n|--------------|------------|----------------------|--------------------------|\n| **Camera-Radar (2-modal)** | Rₚ/C| 0.6909 ± 0.1119 | 0.9385 ± 0.0412 |\n|              | Cₚ/R | 0.6746 ± 0.0727 | 0.8942 ± 0.0472 |\n| **Camera-LiDAR (2-modal)** | Lₚ/C | 0.7062 ± 0.1172 | 0.9210 ± 0.0422 |\n|              | Cₚ/L      | 0.7205 ± 0.0591 | 0.9095 ± 0.0200 |\n| **Camera-Radar-LiDAR (3-modal)** | RₚLₚ/C | 0.7108 ± 0.1145 | 0.8790 ± 0.0522 |\n|              | CₚLₚ/R      | 0.7066 ± 0.0769 | 0.9574 ± 0.0099 |\n|              |  CₚRₚ/L      | 0.7025 ± 0.0685 | 0.9389 ± 0.0161 |\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n**Table B.** Stability Experiments on SHAP and LMD + SHAP (Mean Squared Error)\n| **Modality** | **Metric (↓)** | **SHAP (MSE)** | **LMD + SHAP (MSE)** |\n|--------------|------------|--------------|------------------|\n| **Camera-Radar (2-modal)** | Rₚ/C | 0.0854 ± 0.0252 | 0.0059 ± 0.0042 |\n|              | Cₚ/R | 0.0191 ± 0.0029 | 0.0136 ± 0.0020 |\n| **Camera-LiDAR (2-modal)** | Lₚ/C | 0.0761 ± 0.0230 | 0.0094 ± 0.0062 |\n|              | Cₚ/L  | 0.0279 ± 0.0038 | 0.0181 ± 0.0028 |\n| **Camera-Radar-LiDAR (3-modal)** | RₚLₚ/C | 0.0651 ± 0.0186 | 0.0375 ± 0.0129 |\n|              | CₚLₚ/R | 0.0073 ± 0.0013 | 0.0031 ± 0.0007 |\n|              | CₚRₚ/L  | 0.0207 ± 0.0031 | 0.0101 ± 0.0017 |\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\n\n## **W3, Q2** \n> **&nbsp; The related work subsection on interpretation methods focuses on fairly old methods and doesn’t include recent mechanistic interpretability methods. Can you briefly discuss how your method relates to methods for mechanistic interpretability?**\t\n\nGiven that this was identified as a major shortcoming, we will provide a more detailed discussion on the connection between our approach and recent advances in mechanistic interpretability.\n\n\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\n\n## **Q3** \n> **&nbsp; Can you measure non-linear correlation between corrupted and non-corrupted forward passes, e.g. using the xicor metric ? This could allow for a more detailed comparison.**\t\n\nWe sincerely thank you for the valuable suggestion. However, due to time constraints during the rebuttal period, we were unable to conduct the proposed experiment.\n\n\n\n[1] Lundberg, Scott M., and Su-In Lee. \"A unified approach to interpreting model predictions.\" Advances in neural information processing systems 30 (2017)."}, {"review_id": "HTDaApBL43", "reviewer": "Reviewer_opMu", "summary": "This paper introduces a method called Layer-Wise Modality Decomposition (LMD) to help understand how different sensors (like cameras, radar, and LiDAR) each contribute to a deep learning model’s prediction in autonomous driving. The method works after the model is trained and does not change the model itself. It breaks down the model’s output layer by layer, showing how much each sensor influenced the result. The authors test this method on real sensor fusion models and show that it can clearly separate the contributions from each sensor, helping make these systems more transparent and interpretable.", "questions": "I am leaning toward the negative side due to the major and minor concerns raised below. \n1. Major question: Can you include or at least discuss comparing LMD with standard interpretability methods such as SHAP, or Grad-CAM in your sensor fusion setting? If these methods are not directly applicable, could you clarify why and possibly provide qualitative or conceptual comparisons?\n2. Minor question: Can you provide a more detailed justification for your treatment of normalization layers (e.g., the use of Identity and Ratio rules)? These rules appear to be heuristic fixes to maintain LRP conservation principles—are there theoretical insights?\n3. (Optional) Can you report on the computational efficiency of your method, such as runtime, memory usage, or scalability to larger models or datasets? This would help assess the practicality of LMD in real-world deployment scenarios.", "limitations": "The paper briefly acknowledges some limitations in the Discussion section, such as the lack of a thorough investigation into the choice of root points and high-order interactions between modalities. However, several important limitations remain insufficiently discussed:  The assumption of modality independence is strong and may not hold in real-world sensor fusion systems, but this is not explicitly addressed. The method’s feasibility for real-time or large-scale deployment is not discussed.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 4, "originality": 3, "strengths_and_weaknesses": "Strengths: (1) The paper presents a novel idea by extending Layer-Wise Relevance Propagation (LRP) and Deep Taylor Decomposition (DTD) to multimodal sensor fusion, enabling layer-wise decomposition by modality—a problem not well addressed in previous work. (2) The proposed method addresses an important gap in interpreting deep learning models used in safety-critical systems like autonomous driving. (3) LMD works post-hoc without modifying the original model, making it practical for real-world systems already in deployment. (4) The paper introduces structured perturbation-based metrics to measure modality separation, and tests the method across both radar-camera and LiDAR-camera setups.\n\nWeaknesses: (1) Major Concern – Please correct me if I’m wrong—I tried my best to understand this point. The LMD method builds upon Deep Taylor Decomposition (DTD) as its core mechanism for relevance propagation. However, as shown in recent rigorous analysis (e.g., Sixt et al., 2022) [1], DTD often reduces to a simple gradient × input formulation under common settings, such as ReLU activations and typical network architectures. This suggests that in practice, DTD (and methods based on it like LMD) may not offer significantly richer or more faithful attributions than simpler gradient-based methods.  (2) Minor Concern –The linearization for normalization layers (BatchNorm, LayerNorm) breaks the relevance-conservation principle that standard LRP relies on. The paper proposes work-arounds, e.g., the identity rule, the ratio rule etc, which seem to restore conservation empirically. However, these fixes are presented as engineering heuristics; the paper does not offer a formal proof.  (3) Major Concern –The experiments lack standard interpretability benchmarks, such as comparisons with SHAP [2] or Grad-CAM [3].\n\n[1] https://keonly.github.io/cards/Layer-wise-relevance-propagation\n[2] Lundberg, S. M., & Lee, S.-I. (2017). A Unified Approach to Interpreting Model Predictions. NeurIPS.\n[3] Selvaraju et al. (2017). Grad-CAM: Visual Explanations from Deep Networks via Gradient-Based Localization. ICCV.", "was_revised": true, "final_justification": "I thank the authors for their additional explanations and new experiments addressing computational and memory complexity, as well as comparisons with SHAP.  The added experiments and clearer conceptual positioning had a positive impact on my evaluation, and I have accordingly raised my score.", "rebuttal": "We sincerely thank you for the careful reading and the constructive feedback. Below we address every point in the order raised and describe the additions that will appear in the revised manuscript.\n\n## **W1** \n> **&nbsp; LMD (based on DTD) may not offer significantly richer or more faithful attributions than simpler gradient-based methods.**\t\n\nAs you correctly pointed out, both LRP and Deep Taylor Decomposition (DTD) can be reduced to an input × gradient formulation, particularly when the root point is chosen with the operating point, also discussed in [1]. \n\nHowever, simple gradient-based methods confront significant limitations when applied to tasks involving high-dimensional inputs and outputs (e.g., perception tasks). In such settings, the computational complexity of computing Jacobians poses challenges applying the methods. \n\nMoreover, as discussed in the main paper, operations such as Layer Normalization present additional challenges: computing gradients through these layers fail to attribute function outputs to input features. This severely hinders interpretability and necessitates alternative strategies that preserve explanatory power while bypassing these limitations. The constraints (e.g., conservation property in LRP) and variants introduced in methods such as LRP and LMD are designed to effectively mitigate these challenges.\n\n\n## **W2, Q2** \n> **&nbsp; The linearization for normalization layers (BatchNorm, LayerNorm) breaks the relevance-conservation principle that standard LRP relies on. The paper proposes work-arounds, e.g., the identity rule, the ratio rule etc, which seem to restore conservation empirically. However, these fixes are presented as engineering heuristics; the paper does not offer a formal proof.**\t\n\nThank you for highlighting the critical aspect. The identity and uniform rules follow directly from Deep Taylor Decomposition (DTD) and from the well-known LRP treatments of renormalization layers. We adopted these rules as LMD variants for comparison experiments since they satisfy our constraints (eq. (7) in our paper). The ratio rule is not a heuristic; it's a closed-form solution to a constrained problem while preserving our assumptions. \n\n\n\n## **W3 & Q1** \n> **&nbsp; The experiments lack standard interpretability benchmarks, such as comparisons with SHAP or Grad-CAM. Could you clarify conceptual comparisons?**\t\n\nIn this reply, we provide a conceptual comparisons between LRP and our proposed LMD method. In LRP, for layers performing element-wise operations such as normalization or activation layers, relevance scores are directly mapped to input neurons, with only the bias splitting rule applied. However, our LMD method adheres closely to the original function behavior by retaining the scale information used in element-wise operations, resulting in disregard of the conservation rule. Additionally, since LRP is a backpropagation-based method, it requires gradient checkpointing, thus implemented with two forward passes and one backward pass. In contrast, LMD requires only one forward pass to store the information for, allowing it to be implemented with a total of two forward passes. Finally, LRP satisfies the conservation rule during the backward pass to ensure that the output relevance score is correctly interpreted. In contrast, LMD follows the behavior of the original function and is focused on modality decomposition. These differences in rules originate from the distinct goals of each method. \n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n**Table A.** Comparison of LRP and LMD method\n| **Method** | **Target**        | **Result**       | **Direction** | **Passes**       | **Element‑Wise Operation** | **Constraint**       |\n|:----------:|:----------------:|:----------------:|:-------------:|:---------------------:|:--------------------------:|:------------------:|\n| **LRP**    | Output Segment   | Input Segment    | Backward      | 2 FWD, 1 BWD | Direct Pass                | Conservation Rule  |\n| **LMD**    | Input Segment    | Out Segment      | Forward       | 2 FWD             | Scaled Pass                | Eq. (7)       |\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\n\n\n* FWD = forward pass, BWD = backward pass.\n\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\nAlso, for a comparison with SHAP and our method, we suggest to check our response to ```Reviewer KNAK```.\n\n\n## **Q3** \n> **&nbsp; Can you report on the computational efficiency of your method, such as runtime, memory usage, or scalability to larger models or datasets? This would help assess the practicality of LMD in real-world deployment scenarios.**\t\n\n\nWe are pleased to provide further details regarding the computational and memory efficiency of our method.\n\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n**Table B.** Computational and memory complexity of LMD, LRP-based and linear-time perturbation methods measured in a single forward pass.\n\n| Method                                   | Passes          | Computational Complexity | Memory Consumption |\n| ---------------------------------------- | ----------------- | ---------------------- | ----------- |\n| **LRP** (gradient checkpointing)         | 2 FWD + 1 BWD | O(1)                   | O(√ Nₗ)     |\n| **Shapley-based** | 2^M FWD            | O(2^M)                  | O(1)        |\n| **LMD**                                  | 2 FWD   | O(1)                   | O(1)        |\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n\n* FWD = forward pass, BWD = backward pass.\nM = number of modalities,\nNₗ = number of layers.\n\nYou can check computational and memory efficiency of our methods from Table B. LRP incurs one backward pass and √ Nₗ memory via checkpointing. Shapley-based methods remain low memory consumption but scale exponentially with multiple modalities. LMD is efficient in terms of both computational complexity and memory consumption: it only requires two forward passes, making it computationally lightweight. Also LMD stores the necessary intermediate information at specific layers and immediately discards it after decomposition while keeping the overall memory usage low. \n\nConsidering the issue was raised by multiple reviewers, We will incorporate it into the main paper to enhance the overall contribution.\n\n-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\n[1] Achtibat, Reduan, et al. \"Attnlrp: attention-aware layer-wise relevance propagation for transformers.\" ICML 2024."}, {"review_id": "Q87zqedY0M", "reviewer": "Reviewer_wjuQ", "summary": "This paper introduces Layer-Wise Modality Decomposition (LMD), a post-hoc interpretability method designed to attribute predictions of multimodal sensor fusion models in autonomous driving to individual input modalities. The paper introduces a method called LMD, designed for interpreting multimodal sensor fusion models in autonomous driving without altering the model architecture. The method is applied to camera-radar and camera-LiDAR fusion scenarios and assessed using structured perturbation-based metrics and visual decomposition approaches.", "questions": "1. The theoretical proof of LMD’s linearization is missing. This paper relies on first-order Taylor expansions to justify LMD’s decomposition, but it never formally proves that this approximation maintains the claimed properties across all layers. If LMD is to be trusted in safety-critical systems, the authors must provide formal guarantees—not just empirical observations. Otherwise, the entire theoretical foundation is not solid.\n2. The proposed LMD should be compared against existing well-known solutions like SHAP, LRP, fANOVA or Grad-CAM. \n3. The term \"model-agnostic\" should be clarified. More details should be provided to demonstrate its importance. In this paper, the proposed solution is only evaluated on camera-radar and camera-LiDAR models, but more experiments should be provided to demonstrate its generality. \n4. The legends of green markers in Figure 2 are missing, and more quantitative validation should be provided.\n5. There is no analysis of LMD’s behavior under adversarial or noisy inputs. Autonomous driving sensors frequently encounter noise, occlusions, and adversarial conditions. If radar data is corrupted, does LMD’s decomposition degrade gracefully, or does it produce nonsensical attributions?\n6. If camera and radar disagree (e.g., one detects an obstacle, the other doesn’t), how does LMD attribute the final prediction? Does it highlight the conflict, or does it silently favor one modality? \n7. The computational overhead of LMD should be discussed. The low latency is important for the safety-critical autonomous system. \n8. The authors point out that root point selection and high-order interactions in activation layers are unresolved, yet how to solve them in LMD is not pending.\n9. How to select the reference point for the Taylor expansion should be explained. \n10. Are the results statistically significant?", "limitations": "The authors do not adequately address the limitations of their work. For example, they do not discuss the computational cost of LMD or how it scales with increasing model size or number of modalities.They do not analyze the sensitivity of LMD to hyperparameter choices or architectural variations. They fail to consider the potential misuse of the method in safety-critical systems where incorrect interpretations could lead to dangerous decisions.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n\n1: The novel model-agnostic approach LMD is proposed to decompose contributions of individual modalities in pretrained fusion models. The research problem in this paper is important for numerous downstream applications.\n\n2. The linearization of activation and normalization layers is mathematically formulated. \n\n3. The proposed solution is broadly evaluated across across camera-radar and camera-LiDAR configurations to demonstrate the generality. \n solution is \n\n\nWeaknesses\n\n1. The paper relies on first-order Taylor expansions to linearize non-linear layers (e.g., activation and normalization layers), but it does not rigorously justify why this approximation is sufficient or how high-order interactions are handled. More details of the theoretical foundation for LMD should be provided.\n\n2. The decomposition process may introduce significant computational costs. The runtime and memory overhead analysis should be provided.\n\n3. The proposed solution is not evaluated under realistic conditions. However, it is important for performance evaluation under noisy or adversarial inputs to demonstrate the efficiency.\n\n4. The presentatoin of this paper should be improved to highlight the key contributions.", "was_revised": true, "final_justification": "The authors have provided more expriments about the model on three-modality setting.\nThe key contribution of this paper is highlighted in the revison.\nI'm willing to update the raking of this paper. However, I am still concerning of the generality of the proposed model for more than three modalities.", "rebuttal": "We sincerely thank you for thorough evaluation and constructive feedback. Your comments provide valuable insights that help us clarify the theoretical foundation of LMD.\n\n## **W1, Q8** \n> **&nbsp; The paper relies on first-order Taylor expansions to linearize non-linear layers (e.g., activation and normalization layers), but it does not rigorously justify why this approximation is sufficient or how high-order interactions are handled. More details of the theoretical foundation for LMD should be provided.**\t\n\nThank you for raising a key question closely related to the core of our work. \nPlease check our responses to ```Reviewer UztF```. \n\n## **Q1** \n> **&nbsp; The theoretical proof of LMD’s linearization is missing.**\n\nWe appreciate the reviewer’s emphasis on formal guarantees. \n\nWe believe the paper includes the theoretical proofs. If you could specify which parts are perceived as missing, we'll clarify the ambiguous points and include the corresponding details in the main paper.\n\n\n## **Q2** \n> **&nbsp; The proposed LMD should be compared against existing well-known solutions like SHAP, LRP, fANOVA or Grad-CAM.**\n\nWe believe this point is crucial for demonstrating the validity and efficiency of our proposed method.\nPlease refer to our responses to ```Reviewer KNAK``` and ```Reviewer opMu```.\n\n\n## **W2, Q7** \n> **&nbsp; The decomposition process may introduce significant computational costs. The runtime and memory overhead analysis should be provided.**\t\n\nWe appreciate the reviewer for highlighting these important concerns. \nPlease check our responses to ```Reviewer opMu```.\n\n\n## **Q3** \n> **&nbsp; More details should be provided to demonstrate its importance. In this paper, the proposed solution is only evaluated on camera-radar and camera-LiDAR models, but more experiments should be provided to demonstrate its generality.**\n\nWe have addressed your concern along with those raised by other reviewers by conducting additional experiments of LMD on 3-modalities fusion and attention-based models. For detailed results and analysis, please refer to our response to ```Reviewer UztF``` and ```Reviewer 3nmi```.\n\n\n## **Q4** \n> **&nbsp; The legends of green markers in Figure 2 are missing.**\t\n\nTo the best of our knowledge, the green markers are correctly included in the figure. If you could clarify which part might have caused the misunderstanding, we would be happy to address it and make the necessary adjustments.\n\n\n## **Q5** \n> **&nbsp; There is no analysis of LMD’s behavior under adversarial or noisy inputs. Autonomous driving sensors frequently encounter noise, occlusions, and adversarial conditions. If radar data is corrupted, does LMD’s decomposition degrade gracefully, or does it produce nonsensical attributions?**\t\n\nWe understand your concern, as this aspect is indeed critical for typical explanation methods. However, our proposed LMD provides an exact description of the pretrained model’s behavior. Therefore, any vulnerability to adversarial or noisy inputs reflects the vulnerability of the pretrained model rather than a weakness of the LMD itself. If you have a different perspective on this point, we would appreciate further clarification and are open to addressing it.\n\n\n## **Q6** \n> **&nbsp; If camera and radar disagree (e.g., one detects an obstacle, the other doesn’t), how does LMD attribute the final prediction? Does it highlight the conflict, or does it silently favor one modality?**\t\n\nFigure.2 in our paper specifically illustrates how LMD interprets the model predictions under such conditions. As with Q5, it is important to note that LMD is not an explanation model but an exact description of the pretrained model, so it does not introduce  biases of favoring a particular modality, which can be a concern in surrogate-model (such as LIME and Kernel-SHAP). If this was not the point you intended to raise, we would greatly appreciate clarification.\n\n\n## **Q9** \n> **&nbsp; How to select the reference point for the Taylor expansion should be explained.**\n\nTo the best of our understanding, the paper does address how the Jacobian is used in relation to the reference point. For activation layers, instead of computing the Jacobian at a specific reference point, we set the diagonal entries of Jacobian to the slope of the line or segment connecting the two operating points to satisfy LMD's constraints. For LayerNorm, computing the Jacobian is not meaningful; therefore, we use a scaling factor as a constant, with the justification provided in the paper. If any aspects of the paper were unclear, we would be happy to take them into consideration accordingly.\n\n\n## **Q10** \n> **&nbsp; Are the results statistically significant?**\n\nThank you for posing a fundamental question regarding our work. The reason why the results statistically significant is that the fact that different modalities are not affected by modality perturbation indicates that we can have strong confidence in the interpretation of modality attribution for the pretrained model's predictions. In other words, the statistically confirmed separation ensures that each modality contribution can be trusted as an isolated interpretation, providing a robust foundation for modality attribution in safety-critical perception systems."}, {"review_id": "lwLkE4sAlB", "reviewer": "Reviewer_3nmi", "summary": "The paper introduces Layer-Wise Modality Decomposition (LMD), a post-hoc interpretability method designed to disentangle modality-specific contributions in pretrained multimodal sensor fusion models for autonomous driving. LMD enables attribution of predictions to individual input modalities like camera and radar or camera and LIDAR across all layers without altering the model architecture.\n\nThe method locally linearises neural network operations to separate modality-specific features and bias, drawing inspiration from techniques like Layer-Wise Relevance Propagation (LRP) and Deep Taylor Decomposition (DTD). LMD’s effectiveness is evaluated through perturbation-based evaluations and visual decompositions by modality, showcasing its ability to clearly distinguish each modality’s role in a complex multimodal model.\n\nExperiments show LMD successfully identifies modality-specific influences, with radar-based predictions responding strongly to radar input changes while camera-based predictions remain stable, and vice versa. While focused on autonomous driving, the paper highlights LMD’s applicability beyond that application, offering a scalable solution for interpreting complex multimodal perception models without imposing constraints on the architecture.", "questions": "- The paper does not discuss limitations or failure cases of LMD. This seems like a crucial missing element. Can you share scenarios where LMD might struggle (e.g., highly correlated modalities, extreme edge cases) and how you plan to address these in future work? \n- Have you considered benchmarking LMD against other post-hoc interpretability methods (e.g. Grad-CAM, SHAP) to further validate its advantages in multimodal settings? This would really help to contextualise the work. \n- The paper claims LMD’s applicability extends beyond autonomous driving (Page 1), and sensor fusion has much broader applicability beyond AV.  However, no specific examples or experiments in other domains are provided. Can you share potential applications in other fields (e.g., medical imaging, robotics) and any preliminary results or plans for such extensions? While it's not critical to address this, this could significantly expand the scope of the paper and contributions.\n\nTo summarise, in general the paper is lacking diversity in experiments performed, particularly in terms of exploring different models beyond SimpleBEV. If this were expanded to include other AV-related models and/or even better models in different domains, this could broaden the appeal and impact of this work, and I would raise my score.", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Clarity: The paper is well-structured, with clear sections that motivate the problem and introduce the proposed Layer-Wise Modality Decomposition (LMD) approach well. Technical concepts are explained with sufficient detail, supported by references to foundational works, and the contributions introduced are contextualised well relative to previous research. While code wasn't available for review, the promise of making code available for the model and experiments would further support the reproducibility, and enable researchers to more easily validate and build upon the method.\n\nExperiments and evaluation: the paper provides validation of their method through the perturbation-based metrics and modality-wise visual decompositions (as seen in the evaluation of camera-radar and camera-LIDAR setting). The perturbation-based metrics and visualizations are well designed to test the method’s core claim of modality separation, and support LMD’s claims. However, comparisons with more baselines and broader modality testing could enhance credibility. In general, the core set of experiments chosen were sensible but quite limited in scope. Broadening this could significantly improve the strength of the paper.\n\nSignificance and Impact: The paper addresses a critical gap in interpretable AI for multimodal perception models, particularly in autonomous driving, where transparency in decision-making is vital for safety and reliability. The model-agnostic nature and applicability to high-capacity architectures make it broadly relevant, with potential impact beyond autonomous driving. On the other hand, the lack of non-driving examples and real-world integration details limits its demonstrated impact. In addition, there are limited modalities explored: the evaluation focuses on camera-radar and camera-LIDAR settings, but the paper does not discuss whether other modality combinations were tested, limiting the generality of conclusions that can be drawn.", "was_revised": true, "final_justification": "Thanks to the authors detailed rebuttals and planned reviewers in response to all the reviewer concerns, I believe they have adequately addressed the concerns raised (including those I raised), and my final score reflects this. In particular, the additional experiments performed by the authors (e.g. LMD to CRN, and more than 2 modalities) were the most important in resolving my outstanding concerns leading to my raised score. It is therefore important that these revisions make it into the final, camera-ready version.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback. We address your concerns as thoroughly as possible in the following reply.\n\n> **&nbsp; Comparisons with more baselines and broader modality testing could enhance credibility**\n\nThank you for suggesting ways to strengthen the credibility of the paper. The results of extending LMD to three modalities can be found in our reply to ```Reviewer UztF```, the experiments combining SHAP with LMD are presented in our response to ```Reviewer KNAK```, and a conceptual comparison between LMD and other interpretability models is provided in our response to ```Reviewer opMU```. \n\nAs you correctly pointed out the limited scope of evaluation, we will include the experiments covered in the rebuttal into the main paper for the camera-ready version.\n\n\n## **Q1** \n> **&nbsp; The paper does not discuss limitations or failure cases of LMD. This seems like a crucial missing element. Can you share scenarios where LMD might struggle (e.g., highly correlated modalities, extreme edge cases) and how you plan to address these in future work?**\t\n\nWe sincerely appreciate that the reviewer’s comments have allowed us to clarify the limitations of LMD and to outline directions for future work.\n\n(1) Failure cases of LMD.\nOne limitation we observed is that while LMD provides an exact modality-wise decomposition for a pretrained model, practical interpretability requires a level of human-eye accessibility. Since LMD decomposes the pretrained model predictions into modality-specific outputs, the resulting maps contain noises which is difficult to interpret. We attribute this to the fact that the fusion model was never trained using single-modality prediction as an objective; hence, the model is not required to produce clean predictions for each modality individually.\n\n(2) Lack of benchmarks.\nTo the best of our knowledge, no existing methods provide comparable interpretability for multimodal fusion outputs in autonomous driving scenarios. This makes it challenging to evaluate LMD against benchmarks. For example, the faithfulness scores [3] which is commonly adopted in domain are not directly applicable to our tasks. As part of our future work, we plan to explore and design rigorous ways to evaluate the faithfulness of LMD in multimodal perception tasks.\n\n\n\n## **Q2** \n> **&nbsp; Have you considered benchmarking LMD against other post-hoc interpretability methods (e.g. Grad-CAM, SHAP) to further validate its advantages in multimodal settings? This would really help to contextualize the work.**\t\n\nIn response to similar requests from multiple reviewers, we performed additional experiments incorporating SHAP [1] and attention-based attribution methods on transformer-based models. For detailed comparisons with SHAP and LRP, please refer to our responses to ```Reviewer KNAK``` and ```Reviewer opMu```. \n\n\n## **Q3** \n> **&nbsp;  The paper is lacking diversity in experiments performed, particularly in terms of exploring different models beyond SimpleBEV. If this were expanded to include other AV-related models and/or even better models in different domains, this could broaden the appeal and impact of this work.**\t\n\nAs many reviewers have pointed out the issue, we conducted additional experiments applying LMD to CRN [2] which is a camera-radar fusion model exploiting attention as a fusion operation. Specifically, the softmax operation in the attention mechanism was treated in the same manner as the activation layers described in the main paper.\n\nWhile we successfully decomposed modality-specific terms in CRN model and we observed that, aside from the constant bias prediction, high-order term arose due to matrix multiplication between input variables. In the camera-ready version, we will include the corresponding qualitative visualizations and a demonstration validating human-eye accessibility.\n\n\n----------------------------------------------------------------------------------\n**Table A.** Comparison (Pearson Correlation) of LMD Variants with CRN [2]\n\n| **Modality**        | **Method**          | **Rₚ/R (↓)**        | **Rₚ/C (↑)**        | **Cₚ/R (↑)**        | **Cₚ/C (↓)**        |\n|---------------------|--------------------|---------------------|---------------------|---------------------|---------------------|\n| **Camera + Radar**  | Uniform - Identity | 0.6123 ± 0.043         | 1.0000 ± 0.000         | 1.0000 ± 0.000         | 0.5912 ± 0.052         |\n|                     | Identity - Identity| 0.2184 ± 0.125         | 1.0000 ± 0.000         | 1.0000 ± 0.000         | 0.3891 ± 0.043         |\n|                     | Identity - Uniform | 0.2572 ± 0.067         | 1.0000 ± 0.000         | 0.9882 ± 0.010         | 0.3827 ± 0.046         |\n|                     | Identity - Ratio   | **0.0552 ± 0.0213**         | **1.0000 ± 0.000**         | **1.0000 ± 0.000**         | **0.1571 ± 0.045**         |\n\n**Table B.** Comparison (Mean Squared Error) of LMD Variants with CRN [2]\n| **Modality**        | **Method**          | **Rₚ/R (↓)**        | **Rₚ/C (↑)**        | **Cₚ/R (↓)**        | **Cₚ/C (↑)**        |\n|---------------------|--------------------|---------------------|---------------------|---------------------|---------------------|\n| **Camera + Radar**  | Uniform - Identity | 11.5812 ± 2.712         | 0.0000 ± 0.000         | 0.0000 ± 0.000         | 39.4135 ± 19.802        |\n|                     | Identity - Identity| 10.5438 ± 1.714         | 0.0000 ± 0.000         | 0.0000 ± 0.000         | 28.8149 ± 18.801        |\n|                     | Identity - Uniform | 9.4676 ± 2.295         | 0.0000 ± 0.000         | 0.0087 ± 0.041         | 27.5582 ± 16.711        |\n|                     | Identity - Ratio   | **8.9732 ± 2.142**         | **0.0000 ± 0.000**         | **0.0000 ± 0.000**         | **54.4890 ± 22.151**        |\n\n\n\n----------------------------------------------------------------------------------\n[1] Lundberg, Scott M., and Su-In Lee. \"A unified approach to interpreting model predictions.\" Advances in neural information processing systems 30 (2017).\n\n[2] Kim, Youngseok, et al. \"Crn: Camera radar net for accurate, robust, efficient 3d perception.\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2023.\n\n[3] Petsiuk, Vitali, Abir Das, and Kate Saenko. \"Rise: Randomized input sampling for explanation of black-box models.\" arXiv preprint arXiv:1806.07421 (2018)."}]

Author rebuttal:

Meta-review: The paper makes a solid contribution to an important problem in autonomous driving safety. Three reviewers explicitly raised their scores after the rebuttal, satisfied with the additional evidence. The comprehensive experiments, theoretical grounding, and practical applicability outweigh the remaining concern about >3 modality scalability. The work establishes a valuable baseline for interpretable multimodal fusion that the community can build upon.
Decision: Accept (poster)