# Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection

arXiv: https://arxiv.org/abs/2106.00827 · HTML: https://mlsanity.com/paper/2106.00827/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 36 references that are in the map.
- First posted on arXiv 1 Jun 2021.

## Builds on (most cited first, 3 of 3)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter](https://mlsanity.com/paper/1910.01108/) (2019)
- [SQuAD: 100,000+ Questions for Machine Comprehension of Text](https://mlsanity.com/paper/1606.05250/) (2016)

## Built on by (most cited first, 6 of 6)

- [Metric Space Magnitude for Evaluating the Diversity of Latent Representations](https://mlsanity.com/paper/2311.16054/) (2023)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [The magnitude of categories of texts enriched by language models](https://mlsanity.com/paper/2501.06662/) (2025)
- [The magnitude vector of images](https://mlsanity.com/paper/2110.15188/) (2021)
- [Approximating Metric Magnitude of Point Sets](https://mlsanity.com/paper/2409.04411/) (2024)
- [Quality-diversity in dissimilarity spaces](https://mlsanity.com/paper/2211.12337/) (2022)

## Abstract (arXiv)

Metric space magnitude, an active field of research in algebraic topology, is a scalar quantity that summarizes the effective number of distinct points that live in a general metric space. The em weighting vector is a closely-related concept that captures, in a nontrivial way, much of the underlying geometry of the original metric space. Recent work has demonstrated that when the metric space is Euclidean, the weighting vector serves as an effective tool for boundary detection. We recast this result and show the weighting vector may be viewed as a solution to a kernelized SVM. As one consequence, we apply this new insight to the task of outlier detection, and we demonstrate performance that is competitive or exceeds performance of state-of-the-art techniques on benchmark data sets. Under mild assumptions, we show the weighting vector, which has computational cost of matrix inversion, can be efficiently approximated in linear time. We show how nearest neighbor methods can approximate solutions to the minimization problems defined by SVMs.