70,735 research outputs found
Two curve Chebyshev approximation and its application to signal clustering
In this paper we extend a number of important results of the classical
Chebyshev approximation theory to the case of simultaneous approximation of two
or more functions. The need for this extension is application driven, since
such kind of problems appears in the area of curve (signal) clustering. In this
paper we propose a new efficient algorithm for signal clustering and develop a
procedure that allows one to reuse the results obtained at the previous
iteration without recomputing the cluster centres from scratch. This approach
is based on the extension of the classical de la Vallee-Poussin's procedure
originally developed for polynomial approximation. In this paper, we also
develop necessary and sufficient optimality conditions for two curve Chebyshev
approximation, that is our core tool for curve clustering. These results are
based on application of nonsmooth convex analysis
Drawing Area-Proportional Euler Diagrams Representing Up To Three Sets
Area-proportional Euler diagrams representing three sets are commonly used to visualize the results of medical experiments, business data, and information from other applications where statistical results are best shown using interlinking curves. Currently, there is no tool that will reliably visualize exact area-proportional diagrams for up to three sets. Limited success, in terms of diagram accuracy, has been achieved for a small number of cases, such as Venn-2 and Venn-3 where all intersections between the sets must be represented. Euler diagrams do not have to include all intersections and so permit the visualization of cases where some intersections have a zero value. This paper describes a general, implemented, method for visualizing all 40 Euler-3 diagrams in an area-proportional manner. We provide techniques for generating the curves with circles and convex polygons, analyze the drawability of data with these shapes, and give a mechanism for deciding whether such data can be drawn with circles. For the cases where non-convex curves are necessary, our method draws an appropriate diagram using non-convex polygons. Thus, we are now always able to automatically visualize data for up to three sets
Adaptive Momentum for Neural Network Optimization
In this thesis, we develop a novel and efficient algorithm for optimizing neural networks inspired by a recently proposed geodesic optimization algorithm. Our algorithm, which we call Stochastic Geodesic Optimization (SGeO), utilizes an adaptive coefficient on top of Polyaks Heavy Ball method effectively controlling the amount of weight put on the previous update to the parameters based on the change of direction in the optimization path. Experimental results on strongly convex functions with Lipschitz gradients and deep Autoencoder benchmarks show that SGeO reaches lower errors than established first-order methods and competes well with lower or similar errors to a recent second-order method called K-FAC (Kronecker-Factored Approximate Curvature). We also incorporate Nesterov style lookahead gradient into our algorithm (SGeO-N) and observe notable improvements. We believe that our research will open up new directions for high-dimensional neural network optimization where combining the efficiency of first-order methods and the effectiveness of second-order methods proves a promising avenue to explore
Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition
We provide efficient constant factor approximation algorithms for the
problems of finding a hierarchical clustering of a point set in any metric
space, minimizing the sum of minimimum spanning tree lengths within each
cluster, and in the hyperbolic or Euclidean planes, minimizing the sum of
cluster perimeters. Our algorithms for the hyperbolic and Euclidean planes can
also be used to provide a pants decomposition, that is, a set of disjoint
simple closed curves partitioning the plane minus the input points into subsets
with exactly three boundary components, with approximately minimum total
length. In the Euclidean case, these curves are squares; in the hyperbolic
case, they combine our Euclidean square pants decomposition with our tree
clustering method for general metric spaces.Comment: 22 pages, 14 figures. This version replaces the proof of what is now
Lemma 5.2, as the previous proof was erroneou
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