Topic
Metric (mathematics)
About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.
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TL;DR: A one parameter family of variable metric updates is developed by considering a fundamental decomposition of the Hessian that underlies Variable Metric Algorithms and considers particular choices of the parameter.
Abstract: We develop a one parameter family of variable metric updates by considering a fundamental decomposition of the Hessian that underlies Variable Metric Algorithms. The relationship with other Variable Metric Updates is discussed. Considerations based on the condition of the Hessian inverse approximation indicate particular choices of the parameter and these are discussed in the second half of this paper.
333 citations
01 Jan 1982
TL;DR: The problem of allocating area to modules at the highest level of a top-down decomposition is treated and a theorem of Schoenberg is applied to obtain a good embedding of the module space into the plane.
Abstract: The problem of allocating area to modules at the highest level of a top-down decomposition is treated in this paper. A theorem of Schoenberg is applied to obtain a good embedding of the module space into the plane. The dutch metric is introduced to transform netlist information - if available - into a distance matrix. This metric is flexible enough to enable the user to steer the design in an interactive environment, and rigorous enough to yield results satisfying optimality criterions. The embedding is used to derive the topology of the floorplan in the form of the structure tree of a slicing structure. To store the partial structure tree during the construction a concise and convenient data structure, the shorthand tree, is introduced. For any aspect ratio of the chip a minimum area floorplan can be generated. The paper also shows how wiring space predictions can be incorporated, how varying degrees of module flexibility can be accounted for, and how fixing bonding pad macros affects the procedure.
333 citations
TL;DR: A chi-squared distance analysis is used to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations and the class conditional probabilities are smoother in the modified neighborhoods, whereby better classification performance can be achieved.
Abstract: Nearest-neighbor classification assumes locally constant class conditional probabilities. This assumption becomes invalid in high dimensions with finite samples due to the curse of dimensionality. Severe bias can be introduced under these conditions when using the nearest-neighbor rule. We propose a locally adaptive nearest-neighbor classification method to try to minimize bias. We use a chi-squared distance analysis to compute a flexible metric for producing neighborhoods that are highly adaptive to query locations. Neighborhoods are elongated along less relevant feature dimensions and constricted along most influential ones. As a result, the class conditional probabilities are smoother in the modified neighborhoods, whereby better classification performance can be achieved. The efficacy of our method is validated and compared against other techniques using both simulated and real-world data.
332 citations