[Paper Review] A Correlation-Based Distance

Jean-Luc Falcone, Paul Albuquerque|ArXiv.org|Feb 27, 2004

Neural Networks and Applications4 citations

TL;DR

This paper proposes a correlation-based distance metric on R^D that measures dissimilarity between data points based on their standardized, centered components, mapping them to a hypersphere. The distance is defined as d(x,y) = sqrt(1 - [corr(x,y)]^2), which equals sin(α) where α is the angle between the normalized vectors; the center of mass is derived via eigendecomposition of a covariance-like matrix under spherical constraints.

ABSTRACT

In this short technical report, we define on the sample space R^D a distance between data points which depends on their correlation. We also derive an expression for the center of mass of a set of points with respect to this distance.

Motivation & Objective

• To define a novel distance metric on R^D that reflects correlation structure between data points.
• To address the limitation of standard Euclidean distance in capturing relational patterns in high-dimensional data.
• To derive the center of mass of a set of points under this correlation-based distance on the hypersphere.
• To ensure the distance metric satisfies key properties such as symmetry and non-negativity, though it is degenerate on R^D.
• To establish a non-degenerate metric in the projective space P^D by leveraging directional equivalence of vectors.

Proposed method

• Transform each data point x ∈ R^D ∕ Diag to its standardized, centered form x* = (x - x̄)/σ_x, mapping it to the hypersphere S^D(√D).
• Define the correlation-based distance as d(x,y) = √(1 - [corr(x,y)]^2) = √(1 - (x*·y*)²/D²), which equals sin(α), the sine of the angle between x* and y*.
• Use Lagrange multipliers to minimize the average squared distance F(g) = (1/N)Σ[d(g,x_j)]² under the constraint that g lies on the hypersphere S^D(√D).
• Derive the optimization condition as M g = λ g, where M is a symmetric D×D matrix formed from the outer products of the data points.
• Solve for the center of mass g as the eigenvector of M corresponding to the largest eigenvalue, normalized to lie on the hypersphere.
• Recognize that the distance is degenerate on R^D but becomes non-degenerate on the projective space P^D due to directional equivalence of vectors.

Experimental results

Research questions

• RQ1 How can a distance metric be defined on R^D that reflects correlation between data points rather than raw Euclidean differences?
• RQ2 What are the metric properties (symmetry, non-negativity, triangle inequality) of the proposed correlation-based distance?
• RQ3 How can the center of mass of a set of points be computed under this new distance metric?
• RQ4 In what space does the proposed distance become non-degenerate, and why?
• RQ5 What is the geometric and algebraic structure of the optimization problem for the center of mass under this metric?

Key findings

• The proposed distance d(x,y) = √(1 - [corr(x,y)]²) satisfies symmetry and non-negativity, and d(x,y) = 0 if and only if x* = ±y*, indicating perfect correlation.
• The distance is equivalent to sin(α), where α is the angle between the standardized vectors x* and y*, confirming its geometric interpretation.
• The triangle inequality holds due to the sine function's subadditivity under angle addition, verified via rotation into a common plane.
• The center of mass g is found as the eigenvector of the matrix M = (1/(ND)) Σ_j x_j x_j^T corresponding to the largest eigenvalue, normalized to lie on the hypersphere.
• The distance is degenerate on R^D ∕ Diag because d(x,y) = 0 does not imply x = y, but becomes non-degenerate on the projective space P^D.
• The optimization for the center of mass reduces to an eigenvalue problem, making the method computationally tractable for high-dimensional data.