---
title: Layer-Wise Modality Decomposition (LMD)
url: https://www.emergentmind.com/topics/layer-wise-modality-decomposition-lmd
type: topic
---

# Layer-Wise Modality Decomposition (LMD)

Searching arXiv for recent papers explicitly using or closely related to “Layer-Wise Modality Decomposition.”
Layer-Wise Modality Decomposition (LMD) denotes a class of methods that separate multimodal neural representations into modality-attributable components as a function of network depth. In the strictest, explicit sense, the term names a post-hoc interpretability method for pretrained fusion models in autonomous driving that “disentangles modality-specific information across all layers of a pretrained fusion model” and reconstructs each layer as an exact sum of modality-wise components plus a bias or high-order term [2511.00859]. In broader usage, the same phrase is also used informally for architectures and analyses that factorize layer parameters into modality-specific and modality-shared parts, or that quantify layer-wise predictive contributions from different modalities, even when the original papers employ different terminology such as Modality-Adaptive Convolution Decomposition, Layer Matrix Decomposition, or Partial Information Decomposition [2203.01735], [2309.05968], [2602.15580]. This suggests that LMD is best understood as a unifying design and analysis principle: making modality structure explicit at the level of individual layers rather than only at the final embedding or prediction.

## 1. Terminological scope and conceptual variants

The most explicit use of the term appears in “Layer-Wise Modality Decomposition for Interpretable Multimodal Sensor Fusion” [2511.00859], where LMD is defined as a “post-hoc, model-agnostic interpretability method” for multimodal perception models. Its stated goal is to answer, for a given prediction, “how much did each sensor modality (camera / LiDAR / radar) contribute, layer by layer and at the final output?” [2511.00859].

Other papers instantiate closely related ideas without using the exact phrase. In RGB–infrared person re-identification, MID introduces Modality-Adaptive Convolution Decomposition (MACD), which factorizes each decomposed convolutional layer into modality-specific basis layers and a modality-shared coefficient layer; the reconstruction explicitly identifies this as “exactly a layer-wise modality decomposition” in the first three residual blocks of ResNet-50 [2203.01735]. In affective computing, representation decomposition separates aligned image and text representations into one shared low-rank component and two modality-specific sparse components, but the decomposition is applied at a single interface stage between encoder and multimodal LLM rather than repeatedly across network depth [2506.07086]. In multimodal reasoning analysis, Partial Information Decomposition is applied at every Transformer layer to partition predictive information into redundant, vision-unique, language-unique, and synergistic terms, yielding a depth-resolved modality profile [2602.15580]. In audio reasoning distillation, source-wise and layer-wise knowledge distillation distributes modality-specific supervision across token subsets and network depth, which the reconstruction characterizes as “effectively a layer-wise decomposition of modality-specific supervision” [2509.18579].

A further ambiguity arises from “Layer Matrix Decomposition,” also abbreviated LMD, which is not a multimodal method but a layer-wise SVD/eigen-based factorization of trained weight matrices into isometries, metric transforms, and latent manifold maps [2309.05968]. The paper explicitly states that “Layer-Wise Modality Decomposition” does not appear there. A plausible implication is that the acronym LMD is polysemous in current literature, and interpretation depends on whether the decomposition targets modalities, matrices, representations, or predictive information.

## 2. Core formalism of explicit LMD in sensor-fusion networks

In the explicit sensor-fusion formulation, a pretrained fusion network is written as a composition of layer functions
\[
F(x) = f_N \circ f_{N-1} \circ \cdots \circ f_1(x),
\]
with
\[
h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.
\]
For each layer, LMD introduces a modality decomposition
\[
h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},
\]
where \(h_\ell^{(m)}\) is attributed to modality \(m\) and \(h_\ell^{(b)}\) is a bias/high-order component [2511.00859].

The method imposes three properties. First, additivity or conservation:
\[
h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.
\]
Second, functional invariance: there exists a linearized network \(\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1\) such that \(\hat{F}(x) = F(x)\) for the given input \(x\), and each linearized layer distributes over the decomposition input:
\[
\hat{f}_\ell\left(\sum_m h_{\ell-1}^{(m)}\right) = \sum_m \hat{f}_\ell(h_{\ell-1}^{(m)}).
\]
Third, a separation property: perturbing only \(x^{(m)}\) should affect only the corresponding modality-specific sub-network, not the others [2511.00859].

For two modalities such as camera and radar, the paper develops the first fusion-layer decomposition via a first-order Taylor expansion. Re-arranging terms gives
\[
f_1^{(j)}(x^{(c)}, x^{(r)}) = \sum_{m\in\{c,r\}} \sum_i J^{(m)}_{ji} x_i^{(m)} + b_j^{(1)} + \varepsilon,
\]
and the modality-specific layer-1 features are defined as
\[
h_{1,j}^{(c)} = \sum_i J^{(c)}_{ji} x_i^{(c)},\quad
h_{1,j}^{(r)} = \sum_i J^{(r)}_{ji} x_i^{(r)},\quad
h_{1,j}^{(b)} = b_j^{(1)}.
\]
If the remainder vanishes under the chosen linearization, then
\[
h_{1,j} = h_{1,j}^{(c)} + h_{1,j}^{(r)} + h_{1,j}^{(b)}.
\]
For deeper layers approximated as
\[
h_{\ell,j} = \sum_i W_{ji}^{(\ell)} h_{\ell-1,i} + b_j^{(\ell)},
\]
the decomposition propagates by
\[
h_{\ell,j}^{(m)} = \sum_i W_{ji}^{(\ell)} h_{\ell-1,i}^{(m)} \quad (m \neq b),
\]
\[
h_{\ell,j}^{(b)} = b_j^{(\ell)} + \sum_i W_{ji}^{(\ell)} h_{\ell-1,i}^{(b)}.
\]
The final prediction is therefore decomposed as
\[
F(x) = h_N = h_N^{(c)} + h_N^{(r)} + h_N^{(l)} + h_N^{(b)}
\]
for camera, radar, LiDAR, and bias [2511.00859].

This formulation differs from standard saliency in a specific way: LMD does not estimate importance scores over inputs, but constructs modality-specific forward paths through a linearized copy of the original network [2511.00859].

## 3. Linearization, propagation rules, and architectural handling

The explicit LMD method obtains modality separation by linearizing nonlinear operations while preserving the original model’s output on the observed input. The procedure uses two passes: a first pass through the original model to cache layer behavior, and a second pass through a linearized model to propagate modality components [2511.00859].

For elementwise activations, if \(a_j\) is the neuron input and \(o_j = \phi(a_j)\) its output, the cached slope is
\[
c_j = \frac{o_j}{a_j + \varepsilon},
\]
and the linearized activation is
\[
\hat{f}_\ell(h_{\ell-1})_j = c_j \cdot h_{\ell-1,j}.
\]
For ReLU, this is effectively a binary mask. The paper states that this construction reproduces the original layer output at the operating point [2511.00859].

BatchNorm in evaluation mode is treated as an affine mapping plus bias. The affine bias is accumulated into the bias component \(h^{(b)}\), and the method satisfies exact reconstruction when the modality components are summed [2511.00859]. LayerNorm and InstanceNorm are handled differently: LMD fixes the variance from the first pass and applies a ratio rule so that the summed decomposition equals the original LayerNorm output. The paper recommends assigning BatchNorm bias entirely to \(h^{(b)}\) via the identity rule, and using the ratio rule for LayerNorm; empirically this combination performs best on the separation metrics [2511.00859].

The method also extends beyond simple concatenation-based fusion. In SimpleBEV-style architectures, decomposition begins at the first convolution consuming concatenated BEV features. For three modalities, the first fusion layer generalizes to
\[
f_1^{(j)}(x^{(m_1)},\dots,x^{(m_M)}) = \sum_m \sum_i J_{ji}^{(m)} x_i^{(m)} + b_j^{(1)},
\]
after which the same linear propagation rule applies through deeper layers [2511.00859].

Attention-based fusion is more difficult because of bilinear terms such as \(QK^\top\) and softmax. The method assigns same-modality products to the corresponding modality and places cross-modal products into the bias term; softmax is linearized analogously to activations [2511.00859]. This indicates a principled limit of the approach: strongly interaction-driven computations do not admit clean single-modality assignment under this linearization and are absorbed into the bias/high-order component.

A related but architecturally different decomposition appears in MID. There, each decomposed convolution layer is factorized as
\[
\boldsymbol{W}^m = \boldsymbol{\alpha}^m \boldsymbol{\Psi}, \quad m \in \{rgb, ir, mix\},
\]
with modality-specific dictionary bases \(\boldsymbol{\alpha}^{rgb}, \boldsymbol{\alpha}^{ir}, \boldsymbol{\alpha}^{mix}\) and a shared coefficient tensor \(\boldsymbol{\Psi}\) [2203.01735]. The modality-specific bases perform spatial correction, while the shared coefficient performs a shared \(1\times1\) convolution that promotes cross-modality shared semantics [2203.01735]. This is not post-hoc attribution but a layer parameterization that builds modality decomposition directly into the network.

## 4. Information-theoretic and representation-level interpretations

A different line of work instantiates LMD as a decomposition of predictive information rather than activations or weights. In “How Vision Becomes Language,” the layer-wise quantities are a pooled visual representation \(X_V^{(\ell)}\), a pooled language representation \(X_L^{(\ell)}\), and a target variable \(Y\), defined as the logit or probability of the correct answer token [2602.15580]. At each layer, total predictive information is decomposed by Partial Information Decomposition as
\[
I\big(X_V^{(\ell)}, X_L^{(\ell)}; Y\big)
= R^{(\ell)} + U_V^{(\ell)} + U_L^{(\ell)} + S^{(\ell)},
\]
where \(R^{(\ell)}\) is redundancy, \(U_V^{(\ell)}\) vision-unique information, \(U_L^{(\ell)}\) language-unique information, and \(S^{(\ell)}\) synergy [2602.15580].

To make this tractable on LLaVA-scale models, the paper introduces PID Flow, combining PCA, normalizing-flow Gaussianization, and Gaussian PID estimation. Mean-pooled modality summaries are reduced by PCA to retain about 95% variance, Gaussianized with RealNVP flows, and then evaluated with an \(I_{\min}\)-based Gaussian PID [2602.15580]. The operational formulas are
\[
R^{(\ell)} = \min\{I(X_V^{(\ell)};Y), I(X_L^{(\ell)};Y)\},
\]
\[
U_V^{(\ell)} = I(X_V^{(\ell)};Y) - R^{(\ell)},
\]
\[
U_L^{(\ell)} = I(X_L^{(\ell)};Y) - R^{(\ell)},
\]
\[
S^{(\ell)} = I(X_V^{(\ell)}, X_L^{(\ell)}; Y) - I(X_V^{(\ell)};Y) - I(X_L^{(\ell)};Y) + R^{(\ell)}.
\]
This yields a depth-resolved “information state” for each layer [2602.15580].

In affective computing, the analogous object is the aligned CLIP image matrix \(\mathbf{I}\) and text matrix \(\mathbf{T}\), decomposed by joint low-rank matrix recovery into
\[
\mathbf{I} = \mathbf{L} + \mathbf{S}_I, \quad \mathbf{T} = \mathbf{L} + \mathbf{S}_T,
\]
where \(\mathbf{L}\) is a shared low-rank component and \(\mathbf{S}_I,\mathbf{S}_T\) are modality-specific sparse components [2506.07086]. The optimization uses an augmented Lagrangian and ADMM-style updates with singular value thresholding and soft-thresholding, with \(\mu = 10\), \(\lambda = 1\), and 3000 iterations run offline once per sample [2506.07086]. The result is then weighted by a scalar attention mechanism
\[
\mathbf{R} = \alpha_L \mathbf{L} + \alpha_I \mathbf{S}_I + \alpha_T \mathbf{S}_T
\]
and used as a dynamic soft prompt for a multimodal LLM [2506.07086]. Because this decomposition sits between CLIP and the LLM rather than across multiple internal layers, the paper describes it as a stage-wise rather than fully layer-wise modality decomposition [2506.07086].

A plausible synthesis is that explicit LMD methods differ primarily by what they decompose: forward activations [2511.00859], convolution kernels [2203.01735], aligned modality representations [2506.07086], or predictive information [2602.15580].

## 5. Training objectives, supervision patterns, and empirical behavior

The objectives associated with LMD-like methods vary sharply across papers because some are post-hoc analyses and others are train-time architectures.

The explicit sensor-fusion LMD method is post-hoc and does not retrain the model. Its validation therefore relies on structured perturbation-based metrics rather than optimization losses [2511.00859]. For a perturbed modality input, the desired behavior is that the corresponding modality-specific prediction changes strongly while the others remain invariant. Using Pearson correlation coefficient and mean squared error on modality-specific prediction maps, the paper reports that the best-performing configuration—identity rule for BatchNorm and ratio rule for LayerNorm—achieves strong separation. In radar+camera fusion, for example, \(R_{p/R} \approx 0.05\) PCC under radar perturbation and \(R_{p/C} \approx 1.0\) PCC for the camera component, indicating high sensitivity for the perturbed modality and near-invariance for the unperturbed one [2511.00859]. The same configuration yields near-perfect separation in LiDAR+camera and three-modality settings [2511.00859].

MID, by contrast, learns modality decomposition jointly with a re-identification objective. The decomposed convolution network is optimized with three cross-modality center triplet losses and three identification losses:
\[
\begin{aligned}
L_{dcn} = &\; \lambda_{1} L_{ct}^{rgb, ir} + \lambda_{2} L_{ct}^{rgb, mix} + \lambda_{3} L_{ct}^{ir, mix} \\
&+ \lambda_{4} L_{id}^{rgb} + \lambda_{5} L_{id}^{ir} + \lambda_{6} L_{id}^{mix}.
\end{aligned}
\]
The ablation shows that adding MACD to the mixup-only model improves RegDB from \(83.43/79.13\) to \(87.45/84.85\) in rank-1/mAP and SYSU-MM01 from \(56.22/57.12\) to \(60.27/59.40\) [2203.01735]. Another ablation varies the number of decomposed residual blocks \(n_d\): performance peaks at \(n_d = 3\) on both RegDB and SYSU-MM01, while deeper decomposition harms performance [2203.01735]. This supports the interpretation that modality discrepancy is most profitably corrected in early to mid-level layers.

In the audio reasoning framework, the decomposition is enforced through a joint loss
\[
\mathcal{L}_{\rm joint} = \mathcal{L}_{\rm txt} + \alpha_{\rm ac} \mathcal{L}_{\rm ac} + \alpha_{\rm SFT}\mathcal{L}_{\rm SFT},
\]
with \(\alpha_{\rm layer}=0.05\), \(\alpha_{\rm ac}=0.05\), and \(\alpha_{\rm SFT}=0.5\) [2509.18579]. Textual knowledge distillation aligns teacher and student distributions and hidden states on reasoning tokens, while acoustic knowledge distillation aligns hidden states on audio token positions:
\[
\mathcal{L}_{\rm ac} = \sum_{t \in \mathcal{T}_{x}} \sum_{i=1}^{L_S} \mathrm{KD}\big(h^{S0}_{i,t} \,\|\, h^S_{i,t}\big).
\]
On MMAU and IEMOCAP, full layer-wise text KD plus acoustic KD plus SFT improves AQA average to \(73.30\) while recovering SER UA to \(56.03\), compared with \(49.65\) SER UA for layer-wise text KD plus SFT alone [2509.18579]. The paper interprets this as evidence that source-wise decomposition restores acoustic competence while preserving reasoning gains.

In the information-theoretic setting, the empirical behavior is summarized by a “modal transduction” pattern: visual-unique information peaks early and decays, language-unique information rises late and dominates the final prediction, and synergy remains small [2602.15580]. For LLaVA-1.5-7B at the final layer, the average shares are \(R = 1.55\) bits, \(U_L = 18.05\) bits or about \(82.4\%\), \(U_V = 1.40\) bits or about \(6.4\%\), and \(S = 1.28\) bits [2602.15580]. This is not a train-time objective but a quantitative description of how multimodal reasoning unfolds across layers.

## 6. Applications, limitations, and broader significance

The explicit LMD method was developed for autonomous driving perception with camera, radar, and LiDAR fusion, including BEV segmentation models derived from SimpleBEV and attention-based camera–radar fusion networks [2511.00859]. Its main practical use is post-hoc inspection of modality-specific outputs and feature maps without any modification to the original architecture or training process. The paper emphasizes debugging use cases such as over-reliance on a modality, modality-specific failure modes, and sensor redundancy under particular environmental conditions [2511.00859].

MID applies layer-wise decomposition to RGB–infrared person re-identification, where the decomposition is intended to reduce spectral discrepancy and enforce invariant visual semantics [2203.01735]. The affective computing framework uses a single decomposition stage to separate modality agreement from modality contrastness before LLM reasoning on multi-modal aspect-based sentiment analysis, multi-modal emotion analysis, and hateful meme detection [2506.07086]. The audio reasoning framework employs layer-wise modality-decomposed supervision to transfer reasoning ability from text models to an audio LLM while preserving acoustic competence [2509.18579]. The PID-based framework applies LMD as an analysis tool for multimodal Transformers such as LLaVA-1.5-7B and LLaVA-1.6-7B across six GQA reasoning tasks [2602.15580].

Several limitations recur across these variants. In the sensor-fusion setting, the decomposition is exact only at the operating point because it depends on local linearization; the bias term aggregates constant offsets, interaction terms, and attention bilinearities, and therefore has weaker semantic clarity than the modality-specific terms [2511.00859]. In the affective computing setting, the decomposition is not layer-wise across depth, requires heavy offline preprocessing of about 1.3 seconds per image-text pair on CPU for 3000 iterations, and depends on CLIP alignment quality [2506.07086]. In the PID framework, Gaussian PID with \(I_{\min}\) redundancy, PCA, and mean pooling are acknowledged approximations, and the results are currently established only for specific LLaVA variants and GQA tasks [2602.15580]. In audio reasoning distillation, the reliance on textualization and the computational cost of layer-wise KD are explicit limitations [2509.18579].

Taken together, these works support a common interpretation of LMD as a move from monolithic multimodal processing toward depth-resolved modality structure. In some cases this structure is built into parameterization [2203.01735]; in others it is optimized in a decomposition layer [2506.07086], imposed through supervision [2509.18579], inferred through information theory [2602.15580], or extracted post hoc from pretrained networks [2511.00859]. The literature therefore does not yet offer a single canonical LMD formalism. What is consistent across the variants is the claim that modality-specific and modality-shared effects are most informative when exposed at the layer level rather than only at the output.

Source: https://www.emergentmind.com/topics/layer-wise-modality-decomposition-lmd