The Delaunay Density Diagnostic

TL;DR

A general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry is introduced and can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed datasets of input–output pairs.

Abstract

Accurate approximation of a real-valued function depends on two aspects of the available data: the density of inputs within the domain of interest and the variation of the outputs over that domain. There are few methods for assessing whether the density of inputs is \textit{sufficient} to identify the relevant variations in outputs -- i.e., the ``geometric scale'' of the function -- despite the fact that sampling density is closely tied to the success or failure of an approximation method. In this paper, we introduce a general purpose, computational approach to detecting the geometric scale of real-valued functions over a fixed domain using a deterministic interpolation technique from computational geometry. The algorithm is intended to work on scalar data in moderate dimensions (2-10). Our algorithm is based on the observation that a sequence of piecewise linear interpolants will converge to a continuous function at a quadratic rate (in $L^2$ norm) if and only if the data are sampled densely enough to distinguish the feature from noise (assuming sufficiently regular sampling). We present numerical experiments demonstrating how our method can identify feature scale, estimate uncertainty in feature scale, and assess the sampling density for fixed (i.e., static) datasets of input-output pairs. We include analytical results in support of our numerical findings and have released lightweight code that can be adapted for use in a variety of data science settings.

Figures (8)

• Figure 1: An arbitrary collection of points $\mathcal{D}$ in $\mathbb R^2$ (left) has a unique triangular mesh (right, blue lines), called the Delaunay triangulation. Every triangle in the Delaunay mesh satisfies the "empty ball criterion": the open circumball whose boundary passes through the vertices of the triangle does not contain any points from $\mathcal{D}$. The boundaries of the circumballs for the Delaunay triangulation in the figure above are shown as grey circles. For $d > 2$, these properties generalize to meshes of $d$-simplices and associated $d$-dimensional circumballs.
• Figure 2: We validate \ref{['alg:DD_msd']} and \ref{['alg:DD_grad-msd']} by assessing the computed rate over a range of average sample spacings, for $f$ shown in the top row. The mean rate (black dot series) shows the average of the computed rate over 100 trials with different random initial seeds. The inter-quartile (between $25^\text{th}$ and $75^\text{th}$ percentiles) and inter-decile (between $10^\text{th}$ and $90^\text{th}$ percentiles) ranges are shown in dark blue and light blue bands, respectively. (a) Pure noise is consistently detected as having the "noisy features" rate (0 for MSD, $-1$ for grad-MSD). (b) Fine scale features are only recoverable if average sample spacing is small enough.
• Figure 3: Additional experiments as in \ref{['fig:validation1']}. (a) Insufficiently sampled fine scale features are detected as noise; the large scale quadratic feature is recoverable with larger average sample spacing. (b) Smooth variation (at the scale of inquiry) is consistently detected as having the "recoverable features" rate (2 for MSD, 1 for grad-MSD).
• Figure 4: Left: A basic visualization of the Ackley function on $[-4,4]^2$. Right: Legend for the plots in \ref{['fig:ackley2']}.
• Figure 5: For the Ackley function on $\mathbb R^2$---see \ref{['fig:ackley1']}---we examine the effect of changing the upsampling rate $b$ on the rates computed by \ref{['alg:DD_msd']} and \ref{['alg:DD_grad-msd']}. Using the values of $b$ indicated, we confirm that a larger $b$ value corresponds to a larger step size (in the horizontal axis) and a smaller variation in the computed rate, as evidenced by the narrower inter-quartile and inter-decile ranges as $b$ increases. For larger average sample spacings, the small number of samples is the cause of the increased variation. In each case, the mean rate has the same trend, reflecting the multi-scale nature of the Ackley function.

Theorems & Definitions (4)

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