1.
a. Let ๐พ, ๐ฟ be two kernels (operating on the same space) and let ๐ผ, ๐ฝ be two positive scalars.
Prove that ๐ผ๐พ + ๐ฝ๐ฟ is a kernel.
b. Provide (two different) examples of non-zero kernels ๐พ, ๐ฟ (operating on the same space), so that:
i. ๐พ − ๐ฟ is a kernel. ii. ๐พ − ๐ฟ 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: ๐(๐ฅ, ๐ฆ, ๐ง) = ๐ฅ0 + ๐ฆ0 + ๐ง0. Constraint: ๐(๐ฅ, ๐ฆ, ๐ง) = 4233 + 6533 + 6733 = 1, where ๐ผ > ๐ฝ > 0
3. Let ๐ = โ=. Let
๐ถ = ๐ป = {โ(๐, ๐, ๐) = {(๐ฅ, ๐ฆ, ๐ง) ๐, |๐ฆ| ≤ ๐, |๐ง| . ๐, ๐, ๐ ∈ โL} the
set of all origin centered boxes. Describe a polynomial sample complexity algorithm ๐ฟ that learns ๐ถ using ๐ป. State the time complexity and the sample complexity of your suggested algorithm. Prove all your steps.
