Metric Oja Depth, New Statistical Tool for Estimating the Most Central Objects

The Oja depth (simplicial volume depth) is one of the classical statistical techniques for measuring the central tendency of data in multivariate space. Despite the widespread emergence of object data like images, texts, matrices or graphs, a well-developed and suitable version of Oja depth for object data is lacking. To address this shortcoming, a novel measure of statistical depth, the metric Oja depth applicable to any object data, is proposed. Two competing strategies are used for optimizing metric depth functions, i.e., finding the deepest objects with respect to them. The performance of the metric Oja depth is compared with three other depth functions (half-space, lens, and spatial) in diverse data scenarios. Keywords: Object Data, Metric Oja depth, Statistical depth, Optimization, Metric statistics

💡 Research Summary

The paper addresses a pressing gap in modern statistical analysis: the lack of a robust, theoretically sound depth function that can be applied to non‑Euclidean object data such as images, texts, graphs, and matrices. While the classical Oja depth (also known as simplicial volume depth) is well‑established for multivariate Euclidean data, it does not extend naturally to arbitrary metric spaces where many contemporary data types reside. To fill this void, the authors propose the Metric Oja Depth (MOJ), a novel statistical depth measure defined for any metric space ((\mathcal X,d)).

Conceptual Foundations
The authors begin by reviewing existing metric depth functions—metric half‑space depth (MHD), metric lens depth (MLD), and metric spatial depth (MSD)—highlighting their reliance on pairwise distance comparisons and their desirable properties such as invariance to distance‑preserving transformations and robustness. However, each of these depths captures a different geometric intuition, and none directly mirrors the volume‑based intuition underlying Oja depth.

Definition of Metric Oja Depth
For a fixed point (x\in\mathcal X) and three independent random objects (X_1,X_2,X_3\sim P), the authors define an event (L(x_1,x_2,x_3)) that holds when the triangle inequality becomes an equality, i.e., (x_2) lies exactly “between” (x_1) and (x_3). They then construct a 3×3 matrix (B_3(x,X_1,X_2,X_3)) whose entries are half‑differences of squared distances, reminiscent of the Cayley‑Menger determinant used to compute simplex volumes in Euclidean space. The depth is finally defined as

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