# IAN: Iterated Adaptive Neighborhoods for manifold learning and dimensionality estimation

arXiv: https://arxiv.org/abs/2208.09123 · HTML: https://mlsanity.com/paper/2208.09123/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 119 references that are in the map.
- First posted on arXiv 19 Aug 2022.

## Builds on (most cited first, 9 of 9)

- [UMAP: Uniform Manifold Approximation and Projection for Dimension Reduction](https://mlsanity.com/paper/1802.03426/) (2018)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Estimating the intrinsic dimension of datasets by a minimal neighborhood information](https://mlsanity.com/paper/1803.06992/) (2018)
- [Understanding How Dimension Reduction Tools Work: An Empirical Approach to Deciphering t-SNE, UMAP, TriMAP, and PaCMAP for Data Visualization](https://mlsanity.com/paper/2012.04456/) (2020)
- [Visualizing Large-scale and High-dimensional Data](https://mlsanity.com/paper/1602.00370/) (2016)
- [Efficient Algorithms for t-distributed Stochastic Neighborhood Embedding](https://mlsanity.com/paper/1712.09005/) (2017)
- [An Analysis of the t-SNE Algorithm for Data Visualization](https://mlsanity.com/paper/1803.01768/) (2018)
- [t-SNE-CUDA: GPU-Accelerated t-SNE and its Applications to Modern Data](https://mlsanity.com/paper/1807.11824/) (2018)
- [Estimating the sample mean and standard deviation from the sample size, median, range and/or interquartile range](https://mlsanity.com/paper/1407.8038/) (2014)

## Built on by (most cited first, 4 of 4)

- [Intrinsic Dimension Correlation: uncovering nonlinear connections in multimodal representations](https://mlsanity.com/paper/2406.15812/) (2024)
- [Harnessing Data Asymmetry: Manifold Learning in the Finsler World](https://mlsanity.com/paper/2603.11396/) (2026)
- [Manifolds and Modules: How Function Develops in a Neural Foundation Model](https://mlsanity.com/paper/2512.07869/) (2025)
- [Bracketing Uncertainty in Clustering Under the Manifold Hypothesis](https://mlsanity.com/paper/2609.17892/) (2026)

## Abstract (arXiv)

Invoking the manifold assumption in machine learning requires knowledge of the manifold's geometry and dimension, and theory dictates how many samples are required. However, in applications data are limited, sampling may not be uniform, and manifold properties are unknown and (possibly) non-pure; this implies that neighborhoods must adapt to the local structure. We introduce an algorithm for inferring adaptive neighborhoods for data given by a similarity kernel. Starting with a locally-conservative neighborhood (Gabriel) graph, we sparsify it iteratively according to a weighted counterpart. In each step, a linear program yields minimal neighborhoods globally and a volumetric statistic reveals neighbor outliers likely to violate manifold geometry. We apply our adaptive neighborhoods to non-linear dimensionality reduction, geodesic computation and dimension estimation. A comparison against standard algorithms using, e.g., k-nearest neighbors, demonstrates their usefulness. Code for our algorithm will be available at https://github.com/dyballa/IAN