8,261 research outputs found
The Metric Nearness Problem
Metric nearness refers to the problem of optimally restoring metric properties to
distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric
data can be important in various settings, for example, in clustering, classification, metric-based
indexing, query processing, and graph theoretic approximation algorithms. This paper formulates
and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a “nearest” set
of distances that satisfy the properties of a metric—principally the triangle inequality. For solving
this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative
projection method. An intriguing aspect of the metric nearness problem is that a special case turns
out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and
develops a new algorithm for the latter problem using a primal-dual method. Applications to graph
clustering are provided as an illustration. We include experiments that demonstrate the computational
superiority of triangle fixing over general purpose convex programming software. Finally, we
conclude by suggesting various useful extensions and generalizations to metric nearness
A Better Alternative to Piecewise Linear Time Series Segmentation
Time series are difficult to monitor, summarize and predict. Segmentation
organizes time series into few intervals having uniform characteristics
(flatness, linearity, modality, monotonicity and so on). For scalability, we
require fast linear time algorithms. The popular piecewise linear model can
determine where the data goes up or down and at what rate. Unfortunately, when
the data does not follow a linear model, the computation of the local slope
creates overfitting. We propose an adaptive time series model where the
polynomial degree of each interval vary (constant, linear and so on). Given a
number of regressors, the cost of each interval is its polynomial degree:
constant intervals cost 1 regressor, linear intervals cost 2 regressors, and so
on. Our goal is to minimize the Euclidean (l_2) error for a given model
complexity. Experimentally, we investigate the model where intervals can be
either constant or linear. Over synthetic random walks, historical stock market
prices, and electrocardiograms, the adaptive model provides a more accurate
segmentation than the piecewise linear model without increasing the
cross-validation error or the running time, while providing a richer vocabulary
to applications. Implementation issues, such as numerical stability and
real-world performance, are discussed.Comment: to appear in SIAM Data Mining 200
A cell-based smoothed finite element method for kinematic limit analysis
This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged
- …