Formal derivation of perceptron freedom from high-dimensional geometric properties

Establish a rigorous mathematical derivation showing that perceptron freedom—namely, the near-universal availability of linear separations by a single hyperplane in high-dimensional spaces—follows directly from four specific geometric properties of high-dimensional spaces: concentration of measure, quasi-orthogonality of random vectors, exponential capacity for packing nearly orthogonal directions, and regularity of data manifolds.

Background

Throughout the paper, perceptron freedom is described as the regime in which high dimensionality yields an abundance of separating hyperplanes, making linear separability generically available (drawing on Cover’s theorem and related high-dimensional phenomena). The introduction and Sections 3.1–3.2 identify four properties—concentration of measure, quasi-orthogonality, exponential capacity, and manifold regularity—as key geometric enablers.

The conclusion explicitly states that a formal derivation of perceptron freedom from these four properties remains an open question, motivating a rigorous link between qualitative geometric intuition and a complete mathematical proof.