Level-Crossing Density as a Mesh-Free High-Frequency Auxiliary Loss for Implicit Neural Representations
arXiv:2607.05815v1 Announce Type: new Abstract: The Minkowski functionals of a field's excursion sets -- area, boundary measure, and Euler characteristic -- describe its level-set morphology; the Euler characteristic is the cheapest handle on topology. We derive smooth Monte-Carlo estimators for all three of a continuous neural field, evaluated at scattered points via the co-area formula and Gauss-Bonnet, using only autodiff: no grid, no complex, no persistence. The estimator is accurate to 1-3% against exact topology in 2D and 3D, and costs about 3 ms per iteration where a persistent-homology
The paper addresses a significant challenge in implicit neural representations (INRs) with a novel, computationally efficient method for topological analysis, driven by the increasing complexity and application of such models.
Improved topological understanding and control in INRs can lead to more robust, accurate, and physically plausible models across various fields, from graphics to scientific simulations.
This research provides a mesh-free, autodiff-only approach to assess and potentially control the morphology and topology of neural fields, simplifying complex shape analysis.
- · AI researchers in implicit neural representations
- · 3D graphics and modeling industry
- · Scientific simulation and inverse problems
- · Machine learning hardware developers
- · Traditional mesh-based topological analysis methods
- · Less efficient computational geometry techniques
The ability to accurately and efficiently compute topological features of INRs will enhance their precision and applicability in engineering and design.
This could enable new forms of AI-driven design and optimization where shape and topological constraints are critical for functionality and performance.
Advanced topological understanding in neural fields might contribute to explainable AI and verifiable AI systems by providing deeper insights into generated structures.
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Read at arXiv cs.LG