# A Pseudo-Metric between Probability Distributions based on Depth-Trimmed Regions

arXiv: https://arxiv.org/abs/2103.12711 · HTML: https://mlsanity.com/paper/2103.12711/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 51 of its 161 references that are in the map.
- First posted on arXiv 23 Mar 2021.

## Builds on (most cited first, 30 of 51)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Exploring the Limits of Transfer Learning with a Unified Text-to-Text Transformer](https://mlsanity.com/paper/1910.10683/) (2019)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [BART: Denoising Sequence-to-Sequence Pre-training for Natural Language Generation, Translation, and Comprehension](https://mlsanity.com/paper/1910.13461/) (2019)
- [BERTScore: Evaluating Text Generation with BERT](https://mlsanity.com/paper/1904.09675/) (2019)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Deep contextualized word representations](https://mlsanity.com/paper/1802.05365/) (2018)
- [HuggingFace's Transformers: State-of-the-art Natural Language Processing](https://mlsanity.com/paper/1910.03771/) (2019)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Get To The Point: Summarization with Pointer-Generator Networks](https://mlsanity.com/paper/1704.04368/) (2017)
- [PEGASUS: Pre-training with Extracted Gap-sentences for Abstractive Summarization](https://mlsanity.com/paper/1912.08777/) (2019)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Unified Language Model Pre-training for Natural Language Understanding and Generation](https://mlsanity.com/paper/1905.03197/) (2019)
- [Wasserstein GAN](https://mlsanity.com/paper/1701.07875/) (2017)
- [Text Summarization with Pretrained Encoders](https://mlsanity.com/paper/1908.08345/) (2019)
- [Optimal Transport for Domain Adaptation](https://mlsanity.com/paper/1507.00504/) (2015)
- [MoverScore: Text Generation Evaluating with Contextualized Embeddings and Earth Mover Distance](https://mlsanity.com/paper/1909.02622/) (2019)
- [Advances in Pre-Training Distributed Word Representations](https://mlsanity.com/paper/1712.09405/) (2017)
- [Statistical Aspects of Wasserstein Distances](https://mlsanity.com/paper/1806.05500/) (2018)
- [Stochastic Optimization for Large-scale Optimal Transport](https://mlsanity.com/paper/1605.08527/) (2016)
- [Extractive Summarization as Text Matching](https://mlsanity.com/paper/2004.08795/) (2020)
- [Bottom-Up Abstractive Summarization](https://mlsanity.com/paper/1808.10792/) (2018)
- [Fast Abstractive Summarization with Reinforce-Selected Sentence Rewriting](https://mlsanity.com/paper/1805.11080/) (2018)
- [Re-evaluating Evaluation in Text Summarization](https://mlsanity.com/paper/2010.07100/) (2020)
- [Semantically Conditioned LSTM-based Natural Language Generation for Spoken Dialogue Systems](https://mlsanity.com/paper/1508.01745/) (2015)
- [Ranking Sentences for Extractive Summarization with Reinforcement Learning](https://mlsanity.com/paper/1802.08636/) (2018)
- [Heterogeneous Graph Neural Networks for Extractive Document Summarization](https://mlsanity.com/paper/2004.12393/) (2020)
- [Subspace Robust Wasserstein Distances](https://mlsanity.com/paper/1901.08949/) (2019)

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

- [Of Human Criteria and Automatic Metrics: A Benchmark of the Evaluation of Story Generation](https://mlsanity.com/paper/2208.11646/) (2022)
- [SaulLM-54B & SaulLM-141B: Scaling Up Domain Adaptation for the Legal Domain](https://mlsanity.com/paper/2407.19584/) (2024)
- [What Makes a Good Story and How Can We Measure It? A Comprehensive Survey of Story Evaluation](https://mlsanity.com/paper/2408.14622/) (2024)
- [Unsupervised Layer-wise Score Aggregation for Textual OOD Detection](https://mlsanity.com/paper/2302.09852/) (2023)
- [Learning Disentangled Textual Representations via Statistical Measures of Similarity](https://mlsanity.com/paper/2205.03589/) (2022)
- [Revisiting Instruction Fine-tuned Model Evaluation to Guide Industrial Applications](https://mlsanity.com/paper/2310.14103/) (2023)
- [Functional Anomaly Detection: a Benchmark Study](https://mlsanity.com/paper/2201.05115/) (2022)
- [Enhanced Hallucination Detection in Neural Machine Translation through Simple Detector Aggregation](https://mlsanity.com/paper/2402.13331/) (2024)
- [Hypothesis Transfer Learning with Surrogate Classification Losses: Generalization Bounds through Algorithmic Stability](https://mlsanity.com/paper/2305.19694/) (2023)
- [Beyond Mahalanobis-Based Scores for Textual OOD Detection](https://mlsanity.com/paper/2211.13527/) (2022)
- [Toward Stronger Textual Attack Detectors](https://mlsanity.com/paper/2310.14001/) (2023)
- [PerSEval: Assessing Personalization in Text Summarizers](https://mlsanity.com/paper/2407.00453/) (2024)
- [Repro: An Open-Source Library for Improving the Reproducibility and Usability of Publicly Available Research Code](https://mlsanity.com/paper/2204.13848/) (2022)
- [A Novel Information-Theoretic Objective to Disentangle Representations for Fair Classification](https://mlsanity.com/paper/2310.13990/) (2023)
- [LUNA: A Framework for Language Understanding and Naturalness Assessment](https://mlsanity.com/paper/2401.04522/) (2024)
- [Towards Trustworthy Reranking: A Simple yet Effective Abstention Mechanism](https://mlsanity.com/paper/2402.12997/) (2024)
- [Signature Isolation Forest](https://mlsanity.com/paper/2403.04405/) (2024)

## Abstract (arXiv)

The design of a metric between probability distributions is a longstanding problem motivated by numerous applications in Machine Learning. Focusing on continuous probability distributions on the Euclidean space ℝd, we introduce a novel pseudo-metric between probability distributions by leveraging the extension of univariate quantiles to multivariate spaces. Data depth is a nonparametric statistical tool that measures the centrality of any element x∈ℝd with respect to (w.r.t.) a probability distribution or a data set. It is a natural median-oriented extension of the cumulative distribution function (cdf) to the multivariate case. Thus, its upper-level sets -- the depth-trimmed regions -- give rise to a definition of multivariate quantiles. The new pseudo-metric relies on the average of the Hausdorff distance between the depth-based quantile regions w.r.t. each distribution. Its good behavior w.r.t. major transformation groups, as well as its ability to factor out translations, are depicted. Robustness, an appealing feature of this pseudo-metric, is studied through the finite sample breakdown point. Moreover, we propose an efficient approximation method with linear time complexity w.r.t. the size of the data set and its dimension. The quality of this approximation as well as the performance of the proposed approach are illustrated in numerical experiments.