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Grounding the Ungrounded: A Spectral-Graph Framework for Quantifying Hallucinations in Multimodal LLMs

Supratik Sarkar, Swagatam Das

TL;DR

The paper introduces a principled spectral-graph diffusion framework to quantify hallucinations in multimodal LLMs by embedding outputs on a multimodal graph Laplacian within an RKHS. It defines a KL-calibrated, smoothed semantic distortion score and an energy-based hallucination measure that decompose across modalities, with time-indexed diffusion and temperature parameters. The core contributions include a multimodal Laplacian, spectral decomposition, Courant–Fischer bounds, and calibration strategies that bound hallucination energy and reveal its modal origins. Empirical validation across COCO, VQAv2, and AudioCaps with multiple inference stacks demonstrates improved detection performance and interpretable energy dynamics, providing a principled basis for evaluation and potential mitigation.

Abstract

Hallucinations in LLMs--especially in multimodal settings--undermine reliability. We present a rigorous information-geometric framework, grounded in diffusion dynamics, to quantify hallucinations in MLLMs where model outputs are embedded via spectral decompositions of multimodal graph Laplacians, and their gaps to a truth manifold define a semantic distortion metric. We derive Courant-Fischer bounds on a temperature-dependent hallucination profile and use RKHS eigenmodes to obtain modality-aware, interpretable measures that track evolution over prompts and time. This reframes hallucination as quantifiable and bounded, providing a principled basis for evaluation and mitigation.

Grounding the Ungrounded: A Spectral-Graph Framework for Quantifying Hallucinations in Multimodal LLMs

TL;DR

The paper introduces a principled spectral-graph diffusion framework to quantify hallucinations in multimodal LLMs by embedding outputs on a multimodal graph Laplacian within an RKHS. It defines a KL-calibrated, smoothed semantic distortion score and an energy-based hallucination measure that decompose across modalities, with time-indexed diffusion and temperature parameters. The core contributions include a multimodal Laplacian, spectral decomposition, Courant–Fischer bounds, and calibration strategies that bound hallucination energy and reveal its modal origins. Empirical validation across COCO, VQAv2, and AudioCaps with multiple inference stacks demonstrates improved detection performance and interpretable energy dynamics, providing a principled basis for evaluation and potential mitigation.

Abstract

Hallucinations in LLMs--especially in multimodal settings--undermine reliability. We present a rigorous information-geometric framework, grounded in diffusion dynamics, to quantify hallucinations in MLLMs where model outputs are embedded via spectral decompositions of multimodal graph Laplacians, and their gaps to a truth manifold define a semantic distortion metric. We derive Courant-Fischer bounds on a temperature-dependent hallucination profile and use RKHS eigenmodes to obtain modality-aware, interpretable measures that track evolution over prompts and time. This reframes hallucination as quantifiable and bounded, providing a principled basis for evaluation and mitigation.

Paper Structure

This paper contains 84 sections, 4 theorems, 103 equations, 3 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Gap.
  3. Our contribution.
  4. Related Work
  5. Preliminaries
  6. Mathematical Foundations
  7. Modeling the LLM outputs
  8. Theoretical Analysis
  9. Semantic Distortion
  10. Extension to Multi-modal Grounding
  11. Formulations to hallucination Energy
  12. Main Results: Proposed Framework
  13. Semantic Graph and Multimodal Laplacian
  14. Spectral Decomposition and Energy Functional
  15. Spectral bounds on hallucination, and time-tecay
  16. ...and 69 more sections

Key Result

Theorem 1

Let a smoothing mass $\varepsilon\in(0,1)$ and a baseline density be fixed, with finite $\rho(x)>0$$\mu$-a.e. and $\int_{\mathcal{X}}\rho(x)\,d\mu(x)=1$; let $K_h(\cdot,\cdot)\in (0, \infty)$ be a $\mu$-Markov kernel (bandwidth $h>0$) and $T_h:L^{1}(\mu)\to L^{1}(\mu)$ be a linear smoother defined f serves as a KL-calibrated smoothed pointwise information gap for tracking hallucinations across pro

Figures (3)

  • Figure 1: Multimodal nested-manifold view of hallucinations. Hollow ellipses denote $\mathcal{X}$, $\mathcal{K}$, $\mathcal{K}_g$.
  • Figure 2: Pipeline for hallucination quantification in MLLMs. For an intuition-building case-study of an image–caption example for an MLLM, see commentsfn:image_caption_example in Appendix \ref{['appendix:notes']}.
  • Figure 3: CF-bounded hallucination energy surfaces (9 panels). Each 3D surface shows $\mathcal{E}_{\mathrm{hall}}^{\mathrm{multi}}$ over temperature $\mathcal{T}_t$ (X) and smoothing mass $\varepsilon$ (Y), clamped between two panel-specific parallel planes marking the CF lower (strictly $>0$) and upper bounds (Z). Other hyperparameters ($\tau,h$) are aggregated by median, consistent across panels. Note: the AudioCaps–BLIP+CLIP+Whisper panel may appear blank if the BLIP vision backbone is intentionally omitted for the audio–text setup; this is expected and documented in our pipeline.

Theorems & Definitions (10)

  • Theorem 1: KL-calibrated smoothed score for hallucination
  • Remark 1
  • Remark 2
  • Lemma 1: Joint measurability of cross inner products
  • Theorem 2: Multimodal energy-based hallucination formalism
  • Corollary 1: Excess-energy hallucination functional
  • proof
  • proof
  • proof
  • proof