Convex Distance Function Research Articles

Discrete Piecewise Monotonic Approximation by a Strictly Convex Distance Function

Theory and algorithms are presented for the following smoothing problem. We are given n measurements of a real-valued function that have been altered by random errors caused by the deriving process. For a given integer k, some efficient algorithms are developed that approximate the data by minimizing the sum of strictly convex functions of the errors in such a way that the approximated values are made up of at most k monotonic sections. If $k = 1$, then the problem can be solved by a special strictly convex programming calculation. If $k > 1$, then there are $O({n^k})$ possible choices of the monotonic sections, so that it is impossible to test each one separately. A characterization theorem is derived that allows dynamic programming to be used for dividing the data into optimal disjoint sections of adjacent data, where each section requires a single monotonic calculation. It is remarkable that the theorem reduces the work for a global minimum to $O(n)$ monotonic calculations to subranges of data and $O(k{s^2})$ computer operations, where $s - 2$ is the number of sign changes in the sequence of the first divided differences of the data. Further, certain monotonicity properties of the extrema of best approximations with k and $k - 1$, and with k and $k - 2$ monotonic sections make the calculation quite efficient. A Fortran 77 program has been written and some numerical results illustrate the performance of the smoothing technique in a variety of data sets.

Computing minimum length paths of a given homotopy class

In this paper, we show that the universal covering space of a surface can be used to unify previous results on computing paths in a simple polygon. We optimize a given path among obstacles in the plane under the Euclidean and link metrics and under polygonal convex distance functions. Besides revealing connections between the minimum paths under these three distance functions, the framework provided by the universal cover leads to simplified linear-time algorithms for shortest path trees, for minimum-link paths in simple polygons, and for paths restricted to c given orientations.

SIMPLE ALGORITHMS FOR ENUMERATING INTERPOINT DISTANCES AND FINDING k NEAREST NEIGHBORS

We present an O(n log n+k log k) time and O(n+k) space algorithm which takes as input a set of n points in the plane and enumerates the k smallest distances between pairs of points in nondecreasing order. We also present an O(n log n+kn log k) solution to the problem of finding the k nearest neighbors for each of n points. Both algorithms are conceptually very simple, are easy to implement, and are based on a common data structure: the Delaunay triangulation. Variants of the algorithms work for any convex distance function metric.

An algorithm for isotonic regression with arbitrary convex distance function

In the present paper we consider the isotonic regression problem with an arbitrary convex distance function d(·), and the main purpose being to present an algorithm for obtaining all isotonic regression under this reasonable assumption on d(·). Further, we consider a piece-wise linear distance function d(·) of the type d( t)= C −| t| for t<0 and d( t)= C +| t| for t⩾0 and get an isotonic pth fractile regression by choosing p=C + (C −+C +)

An analysis of scatter decomposition

A formal analysis of a powerful mapping technique known as scatter decomposition is provided. Scatter decomposition divides an irregular computational domain into a large number of equally sized pieces and distributes them modularly among processors. A probabilistic model of workload in one dimension is used to formally explain why and when scatter decomposition works. The first result is that if a correlation in workload is a convex function of distance, then scattering a more finely decomposed domain yields a lower average processor workload variance. The second result shows that if the workload process is a stationary Gaussian and the correlation function decreases linearly in distance until becoming zero and then remain zero, scattering a more finely decomposed domain yields a lower expected maximum processor workload. It is shown that if the correlation function decreases linearly across the entire domain, then among all mappings that assign an equal number of domain pieces to each processor, scatter decomposition minimizes the average processor workload variance. The dependence of these results on the assumption of decreasing correlation is illustrated with situations where a coarser granularity actually achieves better load balance.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Tori whose covering spaces have convex distance functions

Tori whose covering spaces have convex distance functions

Polar analogues of two theorems by Minkowski

Since Minkowski's time, much progress has been made in the geometry of numbers, even as far as the geometry of numbers of convex bodies is concerned. But, surprisingly, one rather obvious interpretation of classical theorems in this theory has so far escaped notice.Minkowski's basic theorem establishes an upper estimate for the smallest positive value of a convex distance function F(x) on the lattice of all points x with integral coordinates. By contrast, we shall establish a lower estimate for F(x) at all the real points X on a suitable hyperplanewith integral coefficients u1, …, un not all zero. We arrive at this estimate by means of applying to Minkowski's Theorem the classical concept of polarity relative to the unit hypersphereThis concept of polarity allows generally to associate with known theorems on point lattices analogous theorems on what we call hyperplane lattices. These new theorems, although implicit in the old ones, seem to have some interest and perhaps further work on hyperplane lattices may lead to useful results.In the first sections of this note a number of notations and results from the classical theory will be collected. The later sections deal then with the consequences of polarity.