Breaking the Euclidean Barrier: Hyperboloid-Based Biological Sequence Analysis

TL;DR

The paper tackles the challenge that traditional Euclidean embeddings fail to capture hierarchical and tree-like structure in biological sequences. It proposes mapping $k$-mer spectrum representations into hyperboloid space and constructing a kernel matrix via the hyperboloid distance $d(X,Y)=\cosh^{-1}(B(X,Y))$ with $B(X,Y)$ as the Lorentzian inner product, followed by kernel PCA to obtain low-dimensional embeddings for supervised learning. Its core contributions include (i) a novel hyperboloid-based embedding for biological sequences, (ii) theoretical justification ensuring Mercer kernel properties through symmetry and PSD analysis, and (iii) extensive experiments across Spike7k, Human DNA, and Coronavirus Host datasets demonstrating improved classification performance and better inter-class separability. The work demonstrates that hyperbolic geometry can preserve hierarchical information in sequence data, enabling more accurate and structurally informed analyses with potential broad impact in bioinformatics.Overall, the hyperboloid approach provides a principled, kernel-based framework for nonlinear, structurally aware sequence analysis that outperforms several baselines while offering solid theoretical grounding.

Abstract

Genomic sequence analysis plays a crucial role in various scientific and medical domains. Traditional machine-learning approaches often struggle to capture the complex relationships and hierarchical structures of sequence data when working in high-dimensional Euclidean spaces. This limitation hinders accurate sequence classification and similarity measurement. To address these challenges, this research proposes a method to transform the feature representation of biological sequences into the hyperboloid space. By applying a transformation, the sequences are mapped onto the hyperboloid, preserving their inherent structural information. Once the sequences are represented in the hyperboloid space, a kernel matrix is computed based on the hyperboloid features. The kernel matrix captures the pairwise similarities between sequences, enabling more effective analysis of biological sequence relationships. This approach leverages the inner product of the hyperboloid feature vectors to measure the similarity between pairs of sequences. The experimental evaluation of the proposed approach demonstrates its efficacy in capturing important sequence correlations and improving classification accuracy.

Figures (6)

• Figure 1: Hyperboloid isosceles tiling, each triangle has two equal sides and equal angles, we can observe negative curvature and exponential growth.
• Figure 2: Workflow for Hyperboliod Distance-based classification. Each row of the input matrix (a) represents a sequence vector computed using $k$-mers spectrum. The sequence is first converted into kernel matrix using hyperboloid distance (b) and (c), then the kernel matrix is processed through Kernel-PCA (d) and used in the classification algorithms.
• Figure 3: t-SNE plots for Hyperboloid Model using Spike7k, Human DNA, and Coronavirus Host datasets.
• Figure 4: Heatmap for classes in Coronavirus Host dataset.
• Figure 5: Heatmap for classes in Human DNA dataset.

Theorems & Definitions (1)

• Remark 1