Logistic regression: iterative algorithm The algorithm alternates between the following four steps untol convergence 1. Estimate s; = o(a¹ x); i = 1,.., Nsamples 2. Evaluate the error e = s, -y, where y, are the true labels 3. Evaluate the gradient g = x₁(s; - y₁) 4. Update a → a-yG using gradient descent with a step size g Here, x; are the vectorized digits of dimension 784 x 1. X; are vectors of length 785 x 1, obtained by adding a 1 to the end. Note that the above operations can be computed efficiently in the matrix form as 1. Estimate s = o(a¹ X), where s is a 1x matrix. 2. Evaluate the error e = s-y 3. Evaluate the gradient X(s - y) 4. Update a → a-yG using gradient descent with a step size g Complete the code below Nsamples, Nfeatures = X_train.shape Nclasses = 2 a = np.random.randn (Nfeatures+1,Nclasses-1) Xtilde = np.concatenate((X_train,np.ones((Nsamples,1))),axis=1). T gamma = 1e-1 for iter in range(1500): z = np.dot(a.T,Xtilde) y_pred sigmoid(z) = error = y_train - y_pred.T gradient = -np.dot (xtilde, error)/Nsamples a = a gamma*gradient if(np.mod (iter, 100)==0): print("Error = ",np.sum(error**2)) fig, ax plt.subplots (1,2) = ax[0].plot(s[:,0:200].T) ax[0].plot(y_train[0:200]) ax[0].set_title('True and predicted labels") ax[1].plot(error) ax[1].set_title('Prediction Errors') plt.show() plt.imshow(np.reshape(a[:-1], (28,28))) plt.title("weights")

Continue your investigations into Pythagorean triples and/or Euler Bricks. Possible topics include: • Connections between Fibonacci numbers and Pythagorean triples. The number 5 is both a Pythagorean hypoteneuse and Fibonacci number. Do other numbers share these properties? • Pythagorean quadruples (x, y, z, w) where x² + y² + z² = w². ● Let x + y + z be the perimeter of the triple (x, y, z). In 1900, Derrick Lehmer proved that if N(p) is the number of primitive Pythagorean triples of perimeter ≤p, then N (p) log 2 P πT² as p→∞.

How Kruskal's Algorithm Works?

Can someone explain to me the concept behind Karnaugh maps? Please use simplest detail possible since I'm a beginner.