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So the margin is: $$\frac{2}{||\mathbf{w}||}$$
TODO: please interpret this for my monkey brain because right now it's just math in my head
+> ok so here is what I seem to understand so far the SVM itself is a classifer based on the nearest points, and at the classifier the linear formula is zero, and on one side it moves in the + class and on the other side ot the - class (ok sounds fair enough so far) -> and that means the the "w" vector always points in the direction of the boundary.
+
we wanted to maximise the margin so:$$\max_{\mathbf{w},w_0}\frac{2}{||\mathbf{w}||}$$
which is the same as minimising: $$\min_{\mathbf{w},w_0}\frac{1}{2}||\mathbf{w}||^2$$
![[Pasted image 20251105165950.png]]
$$1-\max_c p_c$$
Entropy:
-$$-\sum_c p_c\log(p_c)$$
+$$-\sum_c p_c\log_2(p_c)$$
Gini Index:
$$\sum_c p_c(1-p_c)$$
- Constriant: $w^Tw=1\rightarrow||w||^2=1$
- requires w (direction of projection) to be unit vector (length is 1); solves scaling problem and guarantees unique solution
-## zero-mean data
+## zero-mean (and unit variance?) data
- recall the definition of variance ![[Pasted image 20251106004843.png]]
- for PCA shift your data to zero-mean
- for each feature, calcualte the mean, subtract the mean from each data point in that feature
- our objective is to max. variance $$\max_{||w||^2=1}\text{var}(w^TX)$$
- assuming zero-mean data: $$\text{var}(w^TX)=\frac{1}{n}(w^TX)(w^TX)^T=\frac{1}{n}w^TXX^Tw=w^TMw$$
## lagrange multiplier
-- introduce a language multiplier $\lambda$ to incorporate the constraint into our optimization $$L(w,\lambda)=w^TMw-\lambda(w^Tw-1)$$
+- introduce a lagrange multiplier $\lambda$ to incorporate the constraint into our optimization $$L(w,\lambda)=w^TMw-\lambda(w^Tw-1)$$
![[Pasted image 20251106005249.png]]
- it penalizes any deviation from the constraint
- if the constraint is violated (i.e. $w^Tw\neq 1$)it will either increase or decrease the value of the Lagrangian
projects / research-obsidian.git / commitdiff