HubHSP Graph: Effective Data Sampling for Pivot-Based Representation Strategies

Abstract

Given a finite dataset in a metric space, we investigate the definition of a representative sample. Such a definition is important in data analysis strategies to seed algorithms (such as $k$-means) and for pivot-based data indexing techniques. We discuss the geometrical and statistical facets of such a definition.

We propose the Hubness Half Space Partitioning (HubHSP) strategy as an effective sampling heuristic that combines both geometric and statistical constraints. We show that the HubHSP sampling strategy is sound and stable in non-uniform high-dimensional regimes and compares favorably with classical sampling techniques.

Annexes

HubHSP Projection. The HSP selects its neighbors based on increasing distance after discarding half-planes. Since the neighbors selected by the HubHSP can occur in random order of their distance values from the central point $x_i$, it is critical to consider them as projected over a common sphere centered at $x_i$.

The most canonical choice is the sphere $C_i$ including the first neighbor $x_l$ of $x_i$. Note $\rho _i=d(x_l,x_i)$ its radius (the distance between $x_i$ and its closest neighbor), then a point $x_j$ is projected as $\tilde{x}_j$ onto $C_i$ by:

$$\tilde{x}_j={{\,\textrm{Proj}\,}}_{C_i}(x_j)=\mathop {\textrm{argmin}}_{x\in C_i}d(x,x_j) = x_i+\rho _i\frac{x_j-x_i}{d(x_j,x_i)}$$

Main Mathematical Symbols