1.
a. Let 𝐾, 𝐿 be two kernels (operating on the same space) and let 𝛼, 𝛽 be two positive scalars.
Prove that 𝛼𝐾 + 𝛽𝐿 is a kernel.
b. Provide (two different) examples of non-zero kernels 𝐾, 𝐿 (operating on the same space), so that:
i. 𝐾 − 𝐿 is a kernel. ii. 𝐾 − 𝐿 is not a kernel.
Prove your answers.
2. Use Lagrange Multipliers to find the maximum and minimum values of the function subject to the given constraints:
Function: 𝑓(𝑥, 𝑦, 𝑧) = 𝑥0 + 𝑦0 + 𝑧0. Constraint: 𝑔(𝑥, 𝑦, 𝑧) = 4233 + 6533 + 6733 = 1, where 𝛼 > 𝛽 > 0
3. Let 𝑋 = ℝ=. Let
𝐶 = 𝐻 = {ℎ(𝑎, 𝑏, 𝑐) = {(𝑥, 𝑦, 𝑧) 𝑎, |𝑦| ≤ 𝑏, |𝑧| . 𝑎, 𝑏, 𝑐 ∈ ℝL} the
set of all origin centered boxes. Describe a polynomial sample complexity algorithm 𝐿 that learns 𝐶 using 𝐻. State the time complexity and the sample complexity of your suggested algorithm. Prove all your steps.
