Principal Nested Cones

2026-04-21

AI Summary

This work addresses the challenge of nonlinear dimensionality reduction for high-dimensional conical data, which requires simultaneous preservation of both scale and shape structures—a goal typically neglected by existing methods that prioritize one over the other. The authors propose the Principal Nested Cones (PNC) framework, which, for the first time, unifies the modeling of scale–shape interactions in nonlinear dimensionality reduction. PNC achieves this by progressively projecting data onto a sequence of nested hypercones and leverages PCA for computationally efficient approximation. Evaluated on both synthetic and real-world datasets—including morphometric, developmental, and molecular data—the method effectively uncovers nonlinear scale–shape coupling, substantially improves representation and reconstruction accuracy, and yields interpretable low-dimensional embeddings, demonstrating scalability to ultra-high-dimensional settings.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Search and Optimization: Non-convex Optimization
• Computer Vision: Other Foundations of Computer Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Security and Privacy: Large-scale security measurements

Abstract

In many applications, the data lie on a type of cone, where there is a distinction between an overall scale variable and the remaining scale-free structure. For example, the joint size and shape of objects are points on a cone, where size represents scale, and shape is the scale-free structure. Dimension reduction is central in such applications, as shape data are often high-dimensional. Interactions between shape and size are widespread and of significant interest in real-world applications. However, most existing methods either lack a single notion of size or focus solely on shape, effectively removing size information. We propose Principal Nested Cones (PNC), a nonlinear dimension reduction framework that preserves both shape and size. PNC represents data through a sequence of nested hypercones and progressively projects observations onto lower-dimensional cone spaces. The resulting PNC scores provide low-dimensional representations that jointly capture size-shape variation in an interpretable manner. To enable scalable computation in ultra-high-dimensional settings, we develop a fast approximation combining PCA-based transformation with standard PNC. Simulation studies and real data applications demonstrate that PNC captures nonlinear size-shape structure, improves representation and reconstruction, and yields interpretable insights across morphometric, developmental, and molecular datasets.

Problem

• cone
• shape
• size
• dimension reduction
• scale-free

Innovation

• Principal Nested Cones
• nonlinear dimension reduction
• shape-size interaction
• hypercones
• scalable approximation