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Layer-Specific Lipschitz Modulation for Fault-Tolerant Multimodal Representation Learning

Diyar Altinses, Andreas Schwung

Abstract

Modern multimodal systems deployed in industrial and safety-critical environments must remain reliable under partial sensor failures, signal degradation, or cross-modal inconsistencies. This work introduces a mathematically grounded framework for fault-tolerant multimodal representation learning that unifies self-supervised anomaly detection and error correction within a single architecture. Building upon a theoretical analysis of perturbation propagation, we derive Lipschitz- and Jacobian-based criteria that determine whether a neural operator amplifies or attenuates localized faults. Guided by this theory, we propose a two-stage self-supervised training scheme: pre-training a multimodal convolutional autoencoder on clean data to preserve localized anomaly signals in the latent space, and expanding it with a learnable compute block composed of dense layers for correction and contrastive objectives for anomaly identification. Furthermore, we introduce layer-specific Lipschitz modulation and gradient clipping as principled mechanisms to control sensitivity across detection and correction modules. Experimental results on multimodal fault datasets demonstrate that the proposed approach improves both anomaly detection accuracy and reconstruction under sensor corruption. Overall, this framework bridges the gap between analytical robustness guarantees and practical fault-tolerant multimodal learning.

Layer-Specific Lipschitz Modulation for Fault-Tolerant Multimodal Representation Learning

Abstract

Modern multimodal systems deployed in industrial and safety-critical environments must remain reliable under partial sensor failures, signal degradation, or cross-modal inconsistencies. This work introduces a mathematically grounded framework for fault-tolerant multimodal representation learning that unifies self-supervised anomaly detection and error correction within a single architecture. Building upon a theoretical analysis of perturbation propagation, we derive Lipschitz- and Jacobian-based criteria that determine whether a neural operator amplifies or attenuates localized faults. Guided by this theory, we propose a two-stage self-supervised training scheme: pre-training a multimodal convolutional autoencoder on clean data to preserve localized anomaly signals in the latent space, and expanding it with a learnable compute block composed of dense layers for correction and contrastive objectives for anomaly identification. Furthermore, we introduce layer-specific Lipschitz modulation and gradient clipping as principled mechanisms to control sensitivity across detection and correction modules. Experimental results on multimodal fault datasets demonstrate that the proposed approach improves both anomaly detection accuracy and reconstruction under sensor corruption. Overall, this framework bridges the gap between analytical robustness guarantees and practical fault-tolerant multimodal learning.

Paper Structure

This paper contains 27 sections, 7 theorems, 44 equations, 13 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Multimodal Self-Supervised Learning
  4. Fault-tolerant Multimodal Learning
  5. Lipschitz-based Sensitivity Regulation
  6. Propagation of Localized Perturbations
  7. Additive Perturbation Model
  8. Multiplicative perturbations
  9. Comparison of Propagation Dynamics
  10. Training Dynamics and Noise-Dependent Lipschitz Sensitivity
  11. Multimodal alignment-based failure correction
  12. Problem Definition
  13. Pre-trained Multimodal Autoencoder
  14. Self-supervised Compute for Failure Detection and Correction
  15. Layer-Specific Lipschitz Regularization Strategy
  16. ...and 12 more sections

Key Result

Lemma 3.1

Let $W\in\mathbb{R}^{M\times N}$ have independent entries $W_{j,i}$ with $\mathbb{E}[W_{j,i}]=0$ and $\operatorname{Var}[W_{j,i}]=\sigma_W^2$. Let $\delta$ be supported on $S$. Then the expected squared output perturbation satisfies

Figures (13)

  • Figure 1: Overview of the proposed fault-tolerant approach for robotic systems. The architecture includes encoding, correction with contrastive learning and detection modules, and decoding stages to reconstruct the original signal.
  • Figure 2: Operational pipeline for robust industrial deployment. Heterogeneous inputs are encoded into a joint latent space. The representation is analyzed for faults: clean data bypasses the correction block for efficiency, while faulty data is rectified via the Lipschitz-controlled module before executing downstream tasks.
  • Figure 3: Three image modality samples of the three distinct multimodal datasets altinses2025benchmarking.
  • Figure 4: Several randomly applied augmentation techniques on one random camera sample.
  • Figure 5: Visualization of the learned latent feature space using t-SNE. (a) Distribution of clean (purple) versus corrupted (yellow) samples before correction. (b) Distribution of clean (purple) versus corrected (yellow) samples.
  • ...and 8 more figures

Theorems & Definitions (16)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 6 more