Layer-Wise Modality Decomposition (LMD)

• Layer-Wise Modality Decomposition (LMD) is a technique that decomposes network layers into modality-specific, shared, and bias components, ensuring additivity and functional invariance.
• It employs linearization methods, such as Taylor expansions, to isolate sensor contributions and propagate modality information through each layer of a pretrained fusion model.
• LMD underpins diverse applications—from autonomous driving to affective computing—while highlighting challenges in handling cross-modal interactions and architectural variations.

Searching arXiv for 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 (Park et al., 2 Nov 2025). 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 (Huang et al., 2022, Shyh-Chang et al., 2023, Wu et al., 17 Feb 2026). 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” (Park et al., 2 Nov 2025), 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?” (Park et al., 2 Nov 2025).

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 (Huang et al., 2022). 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 (Tian et al., 8 Jun 2025). 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 (Wu et al., 17 Feb 2026). 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” (Yang et al., 23 Sep 2025).

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 (Shyh-Chang et al., 2023). 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 (Park et al., 2 Nov 2025).

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: $$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$0 Third, a separation property: perturbing only $$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$1 should affect only the corresponding modality-specific sub-network, not the others (Park et al., 2 Nov 2025).

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

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$2

and the modality-specific layer-1 features are defined as

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$3

If the remainder vanishes under the chosen linearization, then

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$4

For deeper layers approximated as

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$5

the decomposition propagates by

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$6

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$7

The final prediction is therefore decomposed as

$$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$8

for camera, radar, LiDAR, and bias (Park et al., 2 Nov 2025).

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 (Park et al., 2 Nov 2025).

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 (Park et al., 2 Nov 2025).

For elementwise activations, if $$h_0 = x,\quad h_{\ell} = f_\ell(h_{\ell-1}),\quad \ell = 1,\dots,N.$$9 is the neuron input and $$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$0 its output, the cached slope is

$$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$1

and the linearized activation is

$$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$2

For ReLU, this is effectively a binary mask. The paper states that this construction reproduces the original layer output at the operating point (Park et al., 2 Nov 2025).

BatchNorm in evaluation mode is treated as an affine mapping plus bias. The affine bias is accumulated into the bias component $$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$3, and the method satisfies exact reconstruction when the modality components are summed (Park et al., 2 Nov 2025). 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 $$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$4 via the identity rule, and using the ratio rule for LayerNorm; empirically this combination performs best on the separation metrics (Park et al., 2 Nov 2025).

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

$$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$5

after which the same linear propagation rule applies through deeper layers (Park et al., 2 Nov 2025).

Attention-based fusion is more difficult because of bilinear terms such as $$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$6 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 (Park et al., 2 Nov 2025). 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

$$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$7

with modality-specific dictionary bases $$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$8 and a shared coefficient tensor $$h_\ell = \sum_{m \in \{1,\dots,M,b\}} h_\ell^{(m)},$$9 (Huang et al., 2022). The modality-specific bases perform spatial correction, while the shared coefficient performs a shared $h_\ell^{(m)}$0 convolution that promotes cross-modality shared semantics (Huang et al., 2022). 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 $h_\ell^{(m)}$1, a pooled language representation $h_\ell^{(m)}$2, and a target variable $h_\ell^{(m)}$3, defined as the logit or probability of the correct answer token (Wu et al., 17 Feb 2026). At each layer, total predictive information is decomposed by Partial Information Decomposition as

$h_\ell^{(m)}$4

where $h_\ell^{(m)}$5 is redundancy, $h_\ell^{(m)}$6 vision-unique information, $h_\ell^{(m)}$7 language-unique information, and $h_\ell^{(m)}$8 synergy (Wu et al., 17 Feb 2026).

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 $h_\ell^{(m)}$9-based Gaussian PID (Wu et al., 17 Feb 2026). The operational formulas are

$m$0

$m$1

$m$2

$m$3

This yields a depth-resolved “information state” for each layer (Wu et al., 17 Feb 2026).

In affective computing, the analogous object is the aligned CLIP image matrix $m$4 and text matrix $m$5, decomposed by joint low-rank matrix recovery into

$m$6

where $m$7 is a shared low-rank component and $m$8 are modality-specific sparse components (Tian et al., 8 Jun 2025). The optimization uses an augmented Lagrangian and ADMM-style updates with singular value thresholding and soft-thresholding, with $m$9, $h_\ell^{(b)}$0, and 3000 iterations run offline once per sample (Tian et al., 8 Jun 2025). The result is then weighted by a scalar attention mechanism

$h_\ell^{(b)}$1

and used as a dynamic soft prompt for a multimodal LLM (Tian et al., 8 Jun 2025). 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 (Tian et al., 8 Jun 2025).

A plausible synthesis is that explicit LMD methods differ primarily by what they decompose: forward activations (Park et al., 2 Nov 2025), convolution kernels (Huang et al., 2022), aligned modality representations (Tian et al., 8 Jun 2025), or predictive information (Wu et al., 17 Feb 2026).

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 (Park et al., 2 Nov 2025). 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, $h_\ell^{(b)}$2 PCC under radar perturbation and $h_\ell^{(b)}$3 PCC for the camera component, indicating high sensitivity for the perturbed modality and near-invariance for the unperturbed one (Park et al., 2 Nov 2025). The same configuration yields near-perfect separation in LiDAR+camera and three-modality settings (Park et al., 2 Nov 2025).

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: $h_\ell^{(b)}$4 The ablation shows that adding MACD to the mixup-only model improves RegDB from $h_\ell^{(b)}$5 to $h_\ell^{(b)}$6 in rank-1/mAP and SYSU-MM01 from $h_\ell^{(b)}$7 to $h_\ell^{(b)}$8 (Huang et al., 2022). Another ablation varies the number of decomposed residual blocks $h_\ell^{(b)}$9: performance peaks at $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$0 on both RegDB and SYSU-MM01, while deeper decomposition harms performance (Huang et al., 2022). 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

$h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$1

with $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$2, $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$3, and $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$4 (Yang et al., 23 Sep 2025). 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: $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$5 On MMAU and IEMOCAP, full layer-wise text KD plus acoustic KD plus SFT improves AQA average to $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$6 while recovering SER UA to $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$7, compared with $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$8 SER UA for layer-wise text KD plus SFT alone (Yang et al., 23 Sep 2025). 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 (Wu et al., 17 Feb 2026). For LLaVA-1.5-7B at the final layer, the average shares are $h_\ell = \sum_m h_\ell^{(m)} \quad \forall \ell.$9 bits, $\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1$0 bits or about $\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1$1, $\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1$2 bits or about $\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1$3, and $\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1$4 bits (Wu et al., 17 Feb 2026). 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 (Park et al., 2 Nov 2025). 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 (Park et al., 2 Nov 2025).

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 (Huang et al., 2022). 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 (Tian et al., 8 Jun 2025). 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 (Yang et al., 23 Sep 2025). 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 (Wu et al., 17 Feb 2026).

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 (Park et al., 2 Nov 2025). 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 (Tian et al., 8 Jun 2025). In the PID framework, Gaussian PID with $\hat{F} = \hat{f}_N \circ \cdots \circ \hat{f}_1$5 redundancy, PCA, and mean pooling are acknowledged approximations, and the results are currently established only for specific LLaVA variants and GQA tasks (Wu et al., 17 Feb 2026). In audio reasoning distillation, the reliance on textualization and the computational cost of layer-wise KD are explicit limitations (Yang et al., 23 Sep 2025).

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 (Huang et al., 2022); in others it is optimized in a decomposition layer (Tian et al., 8 Jun 2025), imposed through supervision (Yang et al., 23 Sep 2025), inferred through information theory (Wu et al., 17 Feb 2026), or extracted post hoc from pretrained networks (Park et al., 2 Nov 2025). 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.