Sheaf-Laplacian Obstruction and Projection Hardness for Cross-Modal Compatibility on a Modality-Independent Site

2026-04-08

AI Summary

This study addresses compatibility issues in cross-modal representation alignment by distinguishing two failure mechanisms: the infeasibility of global projection and the lack of global consistency in local projections. Building upon modality-agnostic sample neighborhood structures, the authors propose a cellular-layer inner product space framework that introduces two complementary metrics—projection hardness and layer Laplacian barrier—to reveal the non-transitivity of compatibility. They demonstrate that bridging through intermediate modalities effectively reduces alignment difficulty. Theoretical analysis establishes a bound linking barrier energy to global alignment error and proves that spectral gap governs alignment stability. Under ReLU activation, the feasibility of alignment is rigorously shown to improve with intermediate modalities. The method integrates cellular-layer theory, Lipschitz-constrained projection families, and layered regularized regression, achieving both theoretical rigor and empirical effectiveness.

Abstract

We develop a unified framework for analyzing cross-modal compatibility in learned representations. The core object is a modality-independent neighborhood site on sample indices, equipped with a cellular sheaf of finite-dimensional real inner-product spaces. For a directed modality pair $(a\to b)$, we formalize two complementary incompatibility mechanisms: projection hardness, the minimal complexity within a nested Lipschitz-controlled projection family needed for a single global map to align whitened embeddings; and sheaf-Laplacian obstruction, the minimal spatial variation required by a locally fit field of projection parameters to achieve a target alignment error. The obstruction invariant is implemented via a projection-parameter sheaf whose 0-Laplacian energy exactly matches the smoothness penalty used in sheaf-regularized regression, making the theory directly operational. This separates two distinct failure modes: hardness failure, where no low-complexity global projection exists, and obstruction failure, where local projections exist but cannot be made globally consistent over the semantic neighborhood graph without large parameter variation. We link the sheaf spectral gap to stability of global alignment, derive bounds relating obstruction energy to excess global-map error under mild Lipschitz assumptions, and give explicit constructions showing that compatibility is generally non-transitive. We further define bridging via composed projection families and show, in a concrete ReLU setting, that an intermediate modality can strictly reduce effective hardness even when direct alignment remains infeasible.

Problem

• cross-modal compatibility
• projection hardness
• sheaf-Laplacian obstruction
• global alignment
• modality-independent site

Innovation

• sheaf-Laplacian obstruction
• projection hardness
• cross-modal compatibility
• modality-independent site
• sheaf-regularized regression