# Probabilistic Spherical Discriminant Analysis: An Alternative to PLDA for length-normalized embeddings

arXiv: https://arxiv.org/abs/2203.14893 · HTML: https://mlsanity.com/paper/2203.14893/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 19 references that are in the map.
- First posted on arXiv 28 Mar 2022.

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

- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [CosFace: Large Margin Cosine Loss for Deep Face Recognition](https://mlsanity.com/paper/1801.09414/) (2018)
- [Understanding Dimensional Collapse in Contrastive Self-supervised Learning](https://mlsanity.com/paper/2110.09348/) (2021)
- [BUT System Description to VoxCeleb Speaker Recognition Challenge 2019](https://mlsanity.com/paper/1910.12592/) (2019)
- [von Mises-Fisher Mixture Model-based Deep learning: Application to Face Verification](https://mlsanity.com/paper/1706.04264/) (2017)
- [The SpeakIn System for VoxCeleb Speaker Recognition Challange 2021](https://mlsanity.com/paper/2109.01989/) (2021)
- [Analyzing speaker verification embedding extractors and back-ends under language and channel mismatch](https://mlsanity.com/paper/2203.10300/) (2022)
- [Fast variational Bayes for heavy-tailed PLDA applied to i-vectors and x-vectors](https://mlsanity.com/paper/1803.09153/) (2018)
- [Robust Speaker Clustering using Mixtures of von Mises-Fisher Distributions for Naturalistic Audio Streams](https://mlsanity.com/paper/1808.06045/) (2018)

## Built on by (most cited first, 2 of 2)

- [Probabilistic Back-ends for Online Speaker Recognition and Clustering](https://mlsanity.com/paper/2302.09523/) (2023)
- [Toroidal Probabilistic Spherical Discriminant Analysis](https://mlsanity.com/paper/2210.15441/) (2022)

## Abstract (arXiv)

In speaker recognition, where speech segments are mapped to embeddings on the unit hypersphere, two scoring backends are commonly used, namely cosine scoring or PLDA. Both have advantages and disadvantages, depending on the context. Cosine scoring follows naturally from the spherical geometry, but for PLDA the blessing is mixed -- length normalization Gaussianizes the between-speaker distribution, but violates the assumption of a speaker-independent within-speaker distribution. We propose PSDA, an analogue to PLDA that uses Von Mises-Fisher distributions on the hypersphere for both within and between-class distributions. We show how the self-conjugacy of this distribution gives closed-form likelihood-ratio scores, making it a drop-in replacement for PLDA at scoring time. All kinds of trials can be scored, including single-enroll and multi-enroll verification, as well as more complex likelihood-ratios that could be used in clustering and diarization. Learning is done via an EM-algorithm with closed-form updates. We explain the model and present some first experiments.