# Metric Space Magnitude for Evaluating the Diversity of Latent Representations

arXiv: https://arxiv.org/abs/2311.16054 · HTML: https://mlsanity.com/paper/2311.16054/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 49 references that are in the map.
- First posted on arXiv 27 Nov 2023.

## Builds on (most cited first, 24 of 24)

- [Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks](https://mlsanity.com/paper/1908.10084/) (2019)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020)
- [GraphRNN: Generating Realistic Graphs with Deep Auto-regressive Models](https://mlsanity.com/paper/1802.08773/) (2018)
- [Assessing Generative Models via Precision and Recall](https://mlsanity.com/paper/1806.00035/) (2018)
- [The Vendi Score: A Diversity Evaluation Metric for Machine Learning](https://mlsanity.com/paper/2210.02410/) (2022)
- [A note on the evaluation of generative models](https://mlsanity.com/paper/1511.01844/) (2015)
- [Evaluation of Text Generation: A Survey](https://mlsanity.com/paper/2006.14799/) (2020)
- [Efficient Graph Generation with Graph Recurrent Attention Networks](https://mlsanity.com/paper/1910.00760/) (2019)
- [Evaluating the Evaluation of Diversity in Natural Language Generation](https://mlsanity.com/paper/2004.02990/) (2020)
- [Scalable Deep Generative Modeling for Sparse Graphs](https://mlsanity.com/paper/2006.15502/) (2020)
- [Geodesic Exponential Kernels: When Curvature and Linearity Conflict](https://mlsanity.com/paper/1411.0296/) (2014)
- [On Evaluation Metrics for Graph Generative Models](https://mlsanity.com/paper/2201.09871/) (2022)
- [Evaluation Metrics for Graph Generative Models: Problems, Pitfalls, and Practical Solutions](https://mlsanity.com/paper/2106.01098/) (2021)
- [Persistent homology detects curvature](https://mlsanity.com/paper/1905.13196/) (2019)
- [Diversity, Density, and Homogeneity: Quantitative Characteristic Metrics for Text Collections](https://mlsanity.com/paper/2003.08529/) (2020)
- [On the effectiveness of persistent homology](https://mlsanity.com/paper/2206.10551/) (2022)
- [Metric Space Magnitude and Generalisation in Neural Networks](https://mlsanity.com/paper/2305.05611/) (2023)
- [Weighting vectors for machine learning: numerical harmonic analysis applied to boundary detection](https://mlsanity.com/paper/2106.00827/) (2021)
- [The magnitude vector of images](https://mlsanity.com/paper/2110.15188/) (2021)
- [Diversity Enhancement via Magnitude](https://mlsanity.com/paper/2201.10037/) (2022)
- [Parallel black-box optimization of expensive high-dimensional multimodal functions via magnitude](https://mlsanity.com/paper/2201.11677/) (2022)

## Built on by (most cited first, 17 of 17)

- [No Metric to Rule Them All: Toward Principled Evaluations of Graph-Learning Datasets](https://mlsanity.com/paper/2502.02379/) (2025)
- [Geometry-Aware Edge Pooling for Graph Neural Networks](https://mlsanity.com/paper/2506.11700/) (2025)
- [From Topology to Retrieval: Decoding Embedding Spaces with Unified Signatures](https://mlsanity.com/paper/2511.22150/) (2025)
- [Approximating Metric Magnitude of Point Sets](https://mlsanity.com/paper/2409.04411/) (2024)
- [Magnitude Distance: A Geometric Measure of Dataset Similarity](https://mlsanity.com/paper/2602.08859/) (2026)
- [Is magnitude 'generically continuous' for finite metric spaces?](https://mlsanity.com/paper/2501.08745/) (2025)
- [Cross-Fusion Distance: A Novel Metric for Measuring Fusion and Separability Between Data Groups in Representation Space](https://mlsanity.com/paper/2601.22036/) (2026)
- [Predict Training Data Quality via Its Geometry in Metric Space](https://mlsanity.com/paper/2510.15970/) (2025)
- [Persistent reachability homology in machine learning applications](https://mlsanity.com/paper/2511.04825/) (2025)
- [Measuring Dataset Diversity from a Geometric Perspective](https://mlsanity.com/paper/2602.09340/) (2026)
- [Local Message-Passing for Discrete Graph Generation](https://mlsanity.com/paper/2603.08825/) (2026)
- [Diversity Curves for Graph Representation Learning](https://mlsanity.com/paper/2605.06466/) (2026)
- [Magnitude-Based Features for Multispecies Spatial Data](https://mlsanity.com/paper/2606.11775/) (2026)
- [Measuring What Persists: Conditioning Mechanisms and a Geometric Framework for AI Agent Identity](https://mlsanity.com/paper/2606.21843/) (2026)
- [Evaluating the Diversity of AI-Generated Content with Diversity Profiles](https://mlsanity.com/paper/2608.17731/) (2026)
- [Generalization in Neural Networks Through the Lens of Magnitude Potential](https://mlsanity.com/paper/2610.01633/) (2026)
- [PLDiv: Measuring Dataset Diversity and Semantic Collapse via Persistence Landscapes](https://mlsanity.com/paper/2610.14031/) (2026)

## Abstract (arXiv)

The magnitude of a metric space is a novel invariant that provides a measure of the 'effective size' of a space across multiple scales, while also capturing numerous geometrical properties, such as curvature, density, or entropy. We develop a family of magnitude-based measures of the intrinsic diversity of latent representations, formalising a novel notion of dissimilarity between magnitude functions of finite metric spaces. Our measures are provably stable under perturbations of the data, can be efficiently calculated, and enable a rigorous multi-scale characterisation and comparison of latent representations. We show their utility and superior performance across different domains and tasks, including (i) the automated estimation of diversity, (ii) the detection of mode collapse, and (iii) the evaluation of generative models for text, image, and graph data.