Nearest Neighbors Clarification Rule(Alternative Approaches) --Han47 10:34, 10 March 2008 (EDT)
Alternative Approach
find invariant coordination $ \varphi : \Re ^k \rightarrow \Re ^n $ --Han47 10:41, 10 March 2008 (EDT) such that $ \varphi (x) = \varphi (\bar x) $ for all $ x, \bar x $ which are related by a rotation & translation
Do not trivialize!
e.g.) $ \varphi (x) =0 $ gives us invariant coordinate but lose separation
Want $ \varphi (x) = \varphi (\bar x) $ $ \Leftrightarrow x, \bar x $ are related by a rotation and translation
Example $ p=(p_1,p_2,\cdots, p_N) \in \Re ^{3 \times N} $ $ \varphi $ maps representation position of taps on body onto $ (d_{12},d_{13},d_{14},\cdots , d_{N-1, N} ) $
where $ d_{ij} $= Euclidean distance between $ p_i $ and $ p_j $
Can reconstruct up to a rotation and translation
Warning: Euclidean distance in invariant coordination space has nothing to do with Euclidean distance or proanstes distance in initial feature space
Nearest Neighbor in $ \Re ^2 $ yields tessalation (tiling of floor with 2D shapes such that 1) no holes and 2) cover all of $ \Re ^2 $)
Shape of cells depends on metric chosen
E.g., if feature vectors are such that vectors related by a rotation belong to same class $ \rightarrow $ metric should be chosen so that files are rotationally symmetric.
