Principal Nested Cones

TL;DR

Principal Nested Cones (PNC) is a novel nonlinear dimension reduction method designed to simultaneously preserve and interpret both size and shape information in cone-structured data, applicable to high-dimensional morphometric and molecular datasets.

Contribution

PNC introduces a sequence of nested hypercones for joint size-shape data reduction, with a scalable approximation for ultra-high-dimensional applications.

Findings

PNC effectively captures nonlinear size-shape structures in simulations.

PNC improves data representation and reconstruction in real datasets.

PNC provides interpretable insights in morphometric, developmental, and molecular studies.

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.