Provably Accurate and Scalable Linear Classifiers in Hyperbolic Spaces

TL;DR

This paper introduces a unified framework for scalable, provably accurate linear classifiers in hyperbolic spaces, addressing hierarchical data challenges with new algorithms and theoretical guarantees, validated on diverse datasets.

Contribution

The paper presents a novel framework for hyperbolic linear classifiers, including new perceptron and SVM algorithms with performance guarantees and extensions to second-order and online settings.

Findings

Algorithms provably converge with complexity similar to Euclidean methods.

Effective on synthetic and real datasets like CIFAR10 and RNA-seq.

Framework extends to various machine learning problems in hyperbolic spaces.

Abstract

Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform the required learning tasks. For hierarchical data, the space of choice is a hyperbolic space because it guarantees low-distortion embeddings for tree-like structures. The geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. We propose a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. The gist of our approach is to focus on Poincar\'e ball models and formulate the classification problems using tangent space formalisms. Our results include a new hyperbolic perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. Furthermore, we adapt our approach to accommodate second-order perceptrons, where data is preprocessed based on second-order information (correlation) to accelerate convergence, and strategic perceptrons, where potentially manipulated data arrives in an online manner and decisions are made sequentially. The excellent performance of the Poincar\'e second-order and strategic perceptrons shows that the proposed framework can be extended to general machine learning problems in hyperbolic spaces. Our experimental results, pertaining to synthetic, single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet, establish that all algorithms provably converge and have complexity comparable to those of their Euclidean counterparts. Accompanying codes can be found at: https://github.com/thupchnsky/PoincareLinearClassification.