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

arXiv: https://arxiv.org/abs/2604.07632 · HTML: https://mlsanity.com/paper/2604.07632/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 31 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Builds on (most cited first, 14 of 14)

- [Learning Transferable Visual Models From Natural Language Supervision](https://mlsanity.com/paper/2103.00020/) (2021)
- [Multimodal Machine Learning: A Survey and Taxonomy](https://mlsanity.com/paper/1705.09406/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [NExT-GPT: Any-to-Any Multimodal LLM](https://mlsanity.com/paper/2309.05519/) (2023)
- [Size-Independent Sample Complexity of Neural Networks](https://mlsanity.com/paper/1712.06541/) (2017)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNs](https://mlsanity.com/paper/2202.04579/) (2022)
- [Revisiting Model Stitching to Compare Neural Representations](https://mlsanity.com/paper/2106.07682/) (2021)
- [Getting aligned on representational alignment](https://mlsanity.com/paper/2310.13018/) (2023)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Universal Neural Machine Translation for Extremely Low Resource Languages](https://mlsanity.com/paper/1802.05368/) (2018)
- [Transfer Learning for Nonparametric Classification: Minimax Rate and Adaptive Classifier](https://mlsanity.com/paper/1906.02903/) (2019)
- [Zero-Resource Translation with Multi-Lingual Neural Machine Translation](https://mlsanity.com/paper/1606.04164/) (2016)
- [Dirac signal processing of higher-order topological signals](https://mlsanity.com/paper/2301.10137/) (2023)

## Built on by (most cited first, 1 of 1)

- [Toward Auto-Research: Mining Falsifiable Research Ideas from Paper Knowledge Graphs with Categorical Structure](https://mlsanity.com/paper/2608.20361/) (2026)

## Abstract (arXiv)

Cross-modal representations vary in how easily they can be aligned, and compatibility is generally non-transitive: two modalities may align through an intermediate modality at lower complexity than through a direct map. We introduce a reference formalism that evaluates all modalities on a fixed neighborhood site and defines two directed invariants. Projection hardness (Ha→ b(varepsilon)) is the minimum complexity within a nested Lipschitz-controlled family required to reach error (varepsilon). For a declared local projection family, sheaf-Laplacian obstruction (Ca→ b(varepsilon)) is the minimum variation of locally fitted projection parameters required to reach the same error. Under identity restrictions, obstruction is the graph Dirichlet energy of a vector-valued parameter field; the sheaf formulation identifies zero energy with successful gluing and extends to edge-dependent transports and heterogeneous parameter spaces. We relate obstruction to the site spectral gap and excess global-map error, and construct ReLU examples showing non-transitive compatibility and a quadratic separation between staged and direct width. Controlled synthetic calibrations recover the predicted hardness separation, cut-induced obstruction scaling, and sensitivity to the fixed site graph.