# Geometry-Aware Graph Construction via Adaptive Spectral Bandwidth Control

arXiv: https://arxiv.org/abs/2609.03306 · HTML: https://mlsanity.com/paper/2609.03306/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 28 references that are in the map.
- First posted on arXiv 3 Sep 2026.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [A Simple Framework for Contrastive Learning of Visual Representations](https://mlsanity.com/paper/2002.05709/) (2020)
- [Emerging Properties in Self-Supervised Vision Transformers](https://mlsanity.com/paper/2104.14294/) (2021)
- [Bootstrap your own latent: A new approach to self-supervised Learning](https://mlsanity.com/paper/2006.07733/) (2020)
- [Improved Baselines with Momentum Contrastive Learning](https://mlsanity.com/paper/2003.04297/) (2020)
- [Barlow Twins: Self-Supervised Learning via Redundancy Reduction](https://mlsanity.com/paper/2103.03230/) (2021)
- [VICReg: Variance-Invariance-Covariance Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2105.04906/) (2021)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [RankMe: Assessing the downstream performance of pretrained self-supervised representations by their rank](https://mlsanity.com/paper/2210.02885/) (2022)
- [Solo-learn: A Library of Self-supervised Methods for Visual Representation Learning](https://mlsanity.com/paper/2108.01775/) (2021)
- [Characterizing Large Language Model Geometry Helps Solve Toxicity Detection and Generation](https://mlsanity.com/paper/2312.01648/) (2023)
- [The Geometry of Self-supervised Learning Models and its Impact on Transfer Learning](https://mlsanity.com/paper/2209.08622/) (2022)
- [T-REGS: Minimum Spanning Tree Regularization for Self-Supervised Learning](https://mlsanity.com/paper/2510.23484/) (2025)
- [IdEst: Assessing Self-Supervised Learning Representations via Intrinsic Dimension](https://mlsanity.com/paper/2606.03338/) (2026)
- [Study of Manifold Geometry using Multiscale Non-Negative Kernel Graphs](https://mlsanity.com/paper/2210.17475/) (2022)

## Abstract (arXiv)

Kernelized graph methods - spectral clustering, diffusion maps, and sparse kernel -regression graphs - that use Gaussian kernels depend on the choice of Gaussian bandwidth sigma, which governs the spectral character of the local kernel operator. When sigma is too small, the kernel overestimates local complexity and treats each sample as an independent direction; when sigma is too large, the kernel collapses multiple directions together, the condition number diverges, and all geometric discrimination is lost. We propose a choice of scale to make the spectral complexity of the kernel consistent with the intrinsic complexity of the underlying manifold. We propose a per-node bandwidth criterion that operationalizes this principle by jointly matching the kernel's effective rank to the local intrinsic dimension estimated via minimum spanning tree, anchoring the search in the manifold-consistent log-log scaling regime. We evaluate SSL embeddings from six encoders on CIFAR-100, showing that adaptive bandwidth consistently improves leave-one-out (LOO) classification and label propagation (LP) accuracy over fixed-bandwidth methods and competing adaptive methods.