Proof of bias term in nearest centroid classifier

S

Sauraj

New member
Joined
Jul 6, 2019
Messages
39
Hello, I dont even know how to start the proof

New Text Document.jpg


This is true but only if I know that w=µ2−µ1∣∣µ2−µ1∣∣\displaystyle w = \frac{ µ_2 - µ_1 }{ ||µ_2-µ_1|| } if β=wTµ1+µ22\displaystyle \beta = w^T \frac {µ_1 + µ_2} {2} from here (picture above in link) so that in my case w should be:

w=w∆−wo∣∣w∆−wo∣∣=w∆−wo(w∆−wo)(w∆−wo)T\displaystyle w = \frac {w_∆ - w_o} {||w_∆ - w_o||} = \frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T}

if I insert everything in wTx−β\displaystyle w^Tx-\beta:

(w∆−wo(w∆−wo)(w∆−wo)T)Tx−(w∆−wo(w∆−wo)(w∆−wo)T)Two+w∆2=\displaystyle (\frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T})^Tx-(\frac {w_∆ - w_o} {(w_∆ - w_o)(w_∆ - w_o)^T})^T \frac {w_o + w_∆} {2} =

((w∆−wo)T(w∆−wo)T(w∆−wo))x−((w∆−wo)T(w∆−wo)T(w∆−wo))wo+w∆2\displaystyle (\frac {(w_∆ - w_o)^T} {(w_∆ - w_o)^T(w_∆ - w_o)})x-(\frac {(w_∆ - w_o)^T} {(w_∆ - w_o)^T(w_∆ - w_o)}) \frac {w_o + w_∆} {2} and now I have the expected beta

But how to calculate this if I dont know what w is, I only have β=wTwo+w∆2\displaystyle \beta = w^T \frac {w_o + w_∆} {2} and w=wo−w∆\displaystyle w = w_o - w_∆ from the formula above
 
You must log in or register to reply here.
Top