Editing Learning vector quantization (section)

=== LVQ3 ===
[[File:Apollonian_circles.svg|thumb|Some Apollonian circles. Every blue circle intersects every red circle at a right angle. Every red circle passes through the two points ''{{mvar|C, D}}'', and every blue circle separates the two points.]]
Initialize several code vectors per label. Iterate until convergence criteria is reached.

# Sample a datum <math>x_i</math>, and find out two code vectors <math>w_j, w_k</math> closest to it.
# Let <math>d_j := \|x_i - w_j\|, d_k := \|x_i - w_k\|</math>.
# If <math>\min \left(\frac{d_j}{d_k}, \frac{d_k}{d_j}\right)>s </math>, where <math>s=\frac{1-w}{1+w}</math>, then
#* If <math>w_j</math> and <math>x_i</math> have the same class, and <math>w_k</math> and <math>x_i</math> have different classes, then <math>w_j \leftarrow w_j + \alpha_t(x_i - w_j)</math> and <math>w_k \leftarrow w_k - \alpha_t(x_i - w_k)</math>.
#* If <math>w_k</math> and <math>x_i</math> have the same class, and <math>w_j</math> and <math>x_i</math> have different classes, then <math>w_j \leftarrow w_j - \alpha_t(x_i - w_j)</math> and <math>w_k \leftarrow w_k + \alpha_t(x_i - w_k)</math>.
#* If <math>w_j</math> and <math>w_k</math> and <math>x_i</math> have the same class, then <math>w_j \leftarrow w_j - \epsilon\alpha_t(x_i - w_j)</math> and <math>w_k \leftarrow w_k + \epsilon\alpha_t(x_i - w_k)</math>.
#* If <math>w_k</math> and <math>x_i</math> have different classes, and <math>w_j</math> and <math>x_i</math> have different classes, then the original paper simply does not explain what happens in this case, but presumably nothing happens in this case.
# Otherwise, skip.
Note that condition <math>\min \left(\frac{d_j}{d_k}, \frac{d_k}{d_j}\right)>s </math>, where <math>s=\frac{1-w}{1+w}</math>, precisely means that the point <math>x_i</math> falls between two [[Apollonian circles|Apollonian spheres]].