An Empirical Study of Moment Estimators for Quantile Approximation

We empirically evaluate lightweight moment estimators for the single-pass quantile approximation problem, including maximum entropy methods and orthogonal series with Fourier, Cosine, Legendre, Chebyshev and Hermite basis functions. We show how to apply stable summation formulas to offset numerical precision issues for higher-order moments, leading to reliable single-pass moment estimators up to order 15. Additionally, we provide an algorithm for GPU-accelerated quantile approximation based on parallel tree reduction. Experiments evaluate the accuracy and runtime of moment estimators against the state-of-the-art KLL quantile estimator on 14,072 real-world datasets drawn from the OpenML database. Our analysis highlights the effectiveness of variants of moment-based quantile approximation for highly space efficient summaries: their average performance using as few as five sample moments can approach the performance of a KLL sketch containing 500 elements. Experiments also illustrate the difficulty of applying the method reliably and showcases which moment-based approximations can be expected to fail or perform poorly.

KLL±approximate quantile sketches over dynamic datasets

Recently the long standing problem of optimal construction of quantile sketches was resolved byKarnin,Lang, andLiberty using the KLL sketch (FOCS 2016). The algorithm for KLL is restricted to online insert operations and no delete operations. For many real-world applications, it is necessary to support delete operations. When the data set is updated dynamically, i.e., when data elements are inserted and deleted, the quantile sketch should reflect the changes. In this paper, we proposeKLL±, the first quantile approximation algorithm to operate in thebounded deletionmodel to account for both inserts and deletes in a given data stream. KLL±extends the functionality of KLL sketches to support arbitrary updates with small space overhead. The space bound for KLL±is [EQUATION], where ∈ and δ are constants that determine precision and failure probability, and α bounds the number of deletions with respect to insert operations. The experimental evaluation of KLL±highlights that with minimal space overhead, KLL±achieves comparable accuracy in quantile approximation to KLL.

QUANTILE APPROXIMATION FOR ROBUST STATISTICAL ESTIMATION AND k-ENCLOSING PROBLEMS

Given a set P of n points in Rd, a fundamental problem in computational geometry is concerned with finding the smallest shape of some type that encloses all the points of P. Well-known instances of this problem include finding the smallest enclosing box, minimum volume ball, and minimum volume annulus. In this paper we consider the following variant: Given a set of n points in Rd, find the smallest shape in question that contains at least k points or a certain quantile of the data. This type of problem is known as a k-enclosing problem. We present a simple algorithmic framework for computing quantile approximations for the minimum strip, ellipsoid, and annulus containing a given quantile of the points. The algorithms run in O(n log n) time.