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<math>P_d(e \mid \vec{x})=\int p_d(e \mid \vec{x}, \vec{x}_d)p_d(\vec{x'}_d \mid \vec{x})</math> | <math>P_d(e \mid \vec{x})=\int p_d(e \mid \vec{x}, \vec{x}_d)p_d(\vec{x'}_d \mid \vec{x})</math> | ||
| + | but <math>\lim _{d \rightarrow \infty } p_d (\vec{x'}_d \mid \vec{x})=\delta {\vec{x'}-\vec{x}}</math> | ||
| + | |||
| + | because probability that sample falls into region R centered at <math>\vec{x}</math> is <math>P_{R}=\int _R p(\vec{x' _d})d \vec{x'}_d</math>. | ||
| + | |||
| + | So, if <math>p(\vec{x}) \neq 0</math> (true almost everywhere), then probability that all samples fall outside R is <math>\lim _{d \rightarrow \infty} | ||
| + | {(1-P_{R})}^d =0</math> | ||
| + | |||
| + | So, <math>\lim _{d \rightarrow \infty} \vec{x}_d = \vec{x}</math> and <math>p_d (\vec{x'}_d \mid \vec{x})=\delta (\vec{x'}_d -\vec{x})+\epsilon _d (\vec{x})</math>where <math>\lim _{d \rightarrow \infty } \epsilon _d (x)=0</math> | ||
Revision as of 11:33, 10 March 2008
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.
Instead of working with (x,y) rotationally invariant, work with $ z=\sqrt{x^2 + y^2} $
How good is Nearest Neighbor rule? 1) training error is zero: don't care 2) test error? Want to Bayes error rate
- Nearest Neighbor error rate
recall Probability of error (error rate) on test data is $ P(e)=\int p(e \mid \vec{x}) p(\vec{x}) d\vec{x} $
Let $ P_d(e) $ be the error rate when d training samples are used.
Let $ P=\lim_{d \rightarrow \infty } P_{d}(e) $
Claim limit error rate $ P=\int (1-\sum _{i=1} ^{c}p^2 (\omega _i \mid \vec{x}))p(x)dx $
Proof of claim: Given observation $ \vec{x} $, denode by $ \vec{x}_d $ the nearest neighbor of $ \vec{x} $ among $ \{\vec{x}_1,\vec{x}_2, \cdots , \vec{x}_d \} $
$ P_d(e \mid \vec{x})=\int p_d(e \mid \vec{x}, \vec{x}_d)p_d(\vec{x'}_d \mid \vec{x}) $ but $ \lim _{d \rightarrow \infty } p_d (\vec{x'}_d \mid \vec{x})=\delta {\vec{x'}-\vec{x}} $
because probability that sample falls into region R centered at $ \vec{x} $ is $ P_{R}=\int _R p(\vec{x' _d})d \vec{x'}_d $.
So, if $ p(\vec{x}) \neq 0 $ (true almost everywhere), then probability that all samples fall outside R is $ \lim _{d \rightarrow \infty} {(1-P_{R})}^d =0 $
So, $ \lim _{d \rightarrow \infty} \vec{x}_d = \vec{x} $ and $ p_d (\vec{x'}_d \mid \vec{x})=\delta (\vec{x'}_d -\vec{x})+\epsilon _d (\vec{x}) $where $ \lim _{d \rightarrow \infty } \epsilon _d (x)=0 $
