proximalmid

Proximalmid is a term used in discussions of optimization and data representation to denote a central representative point obtained through proximal-regularized averaging. The name combines proximal, referencing proximal operators and regularization, with mid, hinting at a central or midpoint notion. The concept is not standardized and appears in a small subset of theoretical literature and illustrative examples.

Definition (informal). Given a finite collection of points in a Hilbert space, proxmid is defined as the unique point that minimizes a proximal-regularized objective balancing fidelity to the inputs with a smoothness or compactness preference. Different authors formalize the objective in distinct ways, but the core ideas are convexity, a unique minimizer, and computability via proximal-operator–style iterations.

Properties. Proximalmid tends to behave as a robust center: it lies near the data mean but attains less sensitivity to outliers than a simple average, depending on the chosen regularization. By adjusting the regularization strength, its position can interpolate between the mean and more centralized points. It is related to classical centers such as the geometric median and barycenter, yet it is distinguished by its proximal regularization framework.

Applications and landscape. Potential uses include robust clustering, multi-criteria decision analysis, and image processing where a stable central representative is desirable under regularization. Ongoing work seeks formal existence and uniqueness conditions, practical algorithms for large-scale problems, and comparisons with standard centers.