# Math Worksheet

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Figure 1: Just a directed graph 1. Consider the directed graph G, shown in Figure 1 (a) (5 points) Write down the adjacency matrix, A, of G. (b) (5 points) Is the matrix A irreducible? Why or why not? (c) (5 points) Write down the transition matrix P associated to G. (d) (5 points) Write pseudocode for the PageRank algorithm. UID: 2. (10 points) Show that the function f (x) = xn is convex for all even and positive n. 3. Consider the Sufficient Lemma we proved when studying gradient descent. Lemma 1 (Sufficient Descent). Suppose that 1. f (x) is L-Lipschitz differentiable. 2. xk+1 = xk − α∇f (xk ). 3. α = 1/L. Then f (xk+1 ) ≤ f (xk ) − 1 k∇f (xk )k22 L In this problem you will answer some questions about the proof of this Lemma. Proof. f (xk+1 ) ≤ f (xk ) + ∇f (xk )> (xk+1 − xk ) + L kxk+1 − xk k22 2 α2 L k∇f (xk )k22 = f (xk ) − αk∇f (xk )k22 + 2   αL = f (xk ) − α 1 − k∇f (xk )k22 2 (1) (2) (a) (5 points) From which Assumption (1,2 or 3) do we obtain equation (1) ? (b) (5 points) Explain how we get from (1) to (2). (c) (5 points) How large can we take α while still guaranteeing descent (i.e. f (xk+1 ) < f (xk )) ? UID: 4. (10 points) Solve the following linear program graphically. That is, sketch the feasible region, identify the corners and evaluate which one yields the minimum value of the cost function. minimize 2x1 − 3x2 subject to: − x1 + x2 ≤ 10 4x1 + x2 ≤ 8 x1 ≥ 0 and x2 ≥ 0 5. All three parts of this question refer to the constrained optimization problem minimize f (x) 3 x∈X ⊂R (3) Where the constraint set X is defined as X = {x ∈ R3 : x21 = 1 and x2 − x3 = 4} (a) (5 points) What is a penalty function, Q(x) for this constraint set. (b) (5 points) Explain the idea of the penalty method for solving (3). (c) (5 points) Suppose it is absolutely critical we find an approximate solution to (3) that lies within the constraint set X . Is it a good idea to use the Penalty method? Why or why not? UID: 6. (15 points) Consider the dishonest casino problem modeled by the Hidden Markov Model (HMM) given below. Suppose we observe the sequence of emissions Y 6 = HT T T HT . 0.6 0.75 0.4 fair Loaded 0.5 0.5 Heads Tails 0.25 0.3 0.7 Heads Tails Use Viterbi’s algorithm to find the most probable sequence of hidden states. (Here, the hidden states are of course “F” and “L”). Assume an initial distribution of P[x1 = F ] = 0.5 and P[x1 = L] = 0.5. Your answer should include the dynamic programming table you construct as part of the algorithm. 7. The logistic regression model is fθ (x) = σ(θ > x +b) where σ represents the sigmoid function and θ, b are parameters to be learned. (a) (5 points) Why do we use the sigmoid function here? Specifically, explain what purpose it is serving, and why it is a better choice than the step function H(z) =  1 if z ≥ 0 . 0 if z < 0 (b) (5 points) How do we choose good values for the parameters θ and b?  > (c) (5 points) Suppose x ∈ R2 , and θ = −1 4 while b = 7. Determine an equation for the classification boundary.
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Mathematical Methods of Data Theory

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1. a)
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A graph is irreducible if and only if it is a connected graph. This implies that for any two
vertices 1, 2, there exists a path whose endpoints are connected. The graph above is a
connected graph making the matrix irreducible.
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procedure PageRank(G, iteration)
for all p in the graph do opg[p] 0 do
dp ⁡ 𝑥1 = −10. This leads to the creation of two
points, (-10, 0) and (0, 10).
𝑥2 ≤ 𝑥1 + 10
G...

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