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欧几里德投影(Euclidean projection)

时间:2014-05-16 20:08:59      阅读:1697      评论:0      收藏:0      [点我收藏+]

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Euclidean projection on a set

An Euclidean projection of a point bubuko.com,布布扣 on a set bubuko.com,布布扣 is a point that achieves the smallest Euclidean distance from bubuko.com,布布扣 to the set. That is, it is any solution to the optimization problem

bubuko.com,布布扣

When the set bubuko.com,布布扣 is convex, there is a unique solution to the above problem. In particular, the projection on an affine subspace is unique.

Example: assume that bubuko.com,布布扣 is the hyperplane

bubuko.com,布布扣

The projection problem reads as a linearly constrained least-squares problem, of particularly simple form:

bubuko.com,布布扣

The projection of bubuko.com,布布扣 on bubuko.com,布布扣 turns out to be aligned with the coefficient vector bubuko.com,布布扣. Indeed, components of bubuko.com,布布扣 orthogonal to bubuko.com,布布扣 don‘t appear in the constraint, and only increase the objective value. Setting bubuko.com,布布扣 in the equation defining the hyperplane and solving for the scalar bubuko.com,布布扣 we obtain bubuko.com,布布扣, so that the projection is bubuko.com,布布扣.

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凸集(Convex Set):

Definitions

A subset bubuko.com,布布扣 of bubuko.com,布布扣 is said to be convex if and only if it contains the line segment between any two points in it:

bubuko.com,布布扣

Subspaces and affine sets, such as lines, planes and higher-dimensional ‘‘flat’’ sets, are obviously convex, as they contain the entire line passing through any two points, not just the line segment. That is, there is no restriction on the scalar bubuko.com,布布扣 anymore in the above condition.

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当且仅当bubuko.com,布布扣上的子集bubuko.com,布布扣包含任意二点直接的线段满足:

bubuko.com,布布扣
很明显,子空间和仿射集合,诸如直线,平面以及更高维度的水平集均是凸集。

Examples:

A set is said to be a convex cone if it is convex, and has the property that if bubuko.com,布布扣, then bubuko.com,布布扣 for every bubuko.com,布布扣.

欧几里德投影(Euclidean projection),布布扣,bubuko.com

欧几里德投影(Euclidean projection)

标签:style   class   c   http   int   a   

原文地址:http://www.cnblogs.com/sdnfever/p/3725348.html

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