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Machine learning specialization - Coggle Diagram

- - - - The K-means algorithm is a method to automatically cluster similar data points together.
      - K-means is an iterative procedure that
        * Starts by guessing the initial centroids, and then * Refines this guess by * Repeatedly assigning examples to their closest centroids, and then * Recomputing the centroids based on the assignments.
      - ``` python # Initialize centroids # K is the number of clusters centroids = kMeans_init_centroids(X, K) for iter in range(iterations): # Cluster assignment step: # Assign each data point to the closest centroid. # idx[i] corresponds to the index of the centroid # assigned to example i idx = find_closest_centroids(X, centroids) # Move centroid step: # Compute means based on centroid assignments centroids = compute_centroids(X, idx, K) ```
  - - - The collaborative filtering algorithm in the setting of movie
        recommendations considers a set of $n$-dimensional parameter vectors
        $\mathbf{x}^{(0)},...,\mathbf{x}^{(n_m-1)}$, $\mathbf{w}^{(0)},...,\mathbf{w}^{(n_u-1)}$ and $b^{(0)},...,b^{(n_u-1)}$, where the
        model predicts the rating for movie $i$ by user $j$ as
        $y^{(i,j)} = \mathbf{w}^{(j)}\cdot \mathbf{x}^{(i)} + b^{(j)}$ . Given a dataset that consists of
        a set of ratings produced by some users on some movies, you wish to
        learn the parameter vectors $\mathbf{x}^{(0)},...,\mathbf{x}^{(n_m-1)},
        \mathbf{w}^{(0)},...,\mathbf{w}^{(n_u-1)}$ and $b^{(0)},...,b^{(n_u-1)}$ that produce the best fit (minimizes
        the squared error).
      - ```The collaborative filtering cost function is given by
      $$J({\mathbf{x}^{(0)},...,\mathbf{x}^{(n_m-1)},\mathbf{w}^{(0)},b^{(0)},...,\mathbf{w}^{(n_u-1)},b^{(nu-1)}})= \left[ \frac{1}{2}\sum{(i,j):r(i,j)=1}(\mathbf{w}^{(j)} \cdot \mathbf{x}^{(i)} + b^{(j)} - y^{(i,j)})^2 \right]
      
      \underbrace{\left[
      \frac{\lambda}{2}
      \sum_{j=0}^{nu-1}\sum{k=0}^{n-1}(\mathbf{w}^{(j)}_k)^2
      
      \frac{\lambda}{2}\sum_{i=0}^{nm-1}\sum{k=0}^{n-1}(\mathbf{x}_k^{(i)})^2
      \right]}_{regularization}
      \tag{1}$$
      The first summation in (1) is "for all $i$, $j$ where $r(i,j)$ equals $1$" and could be written:
      
      $$ = \left[ \frac{1}{2}\sum_{j=0}^{n_u-1} \sum_{i=0}^{n_m-1}r(i,j)*(\mathbf{w}^{(j)} \cdot \mathbf{x}^{(i)} + b^{(j)} - y^{(i,j)})^2 \right] +\text{regularization} $$
```
- python code
  (np.sum((((np.matmul(X_r,W_r.T) + b_r -Y_r)2)* R_r/2)) + (np.sum(X_r2) + np.sum(Wr*2)) lambda/2)