COMS 4771 Machine Learning (**add semester here**) Problem Set #1
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Problem 1
Examples of blackboard and calligraphic letters: Rd ⊃ Sd−1, C ⊂ B. Examples of bold-faced letters (perhaps suitable for matrix and vectors):
L(x, λ) = f (x) − ⟨λ, Ax − b⟩. (1) Example of a custom-defined math operator:
var(X) = EX2 − (EX)2.
Example of references: the Lagrangian is given in Eq. (1), and Theorem 1 is interesting.
Example of adaptively-sized parentheses:
n 1/n n 1/n n
xi + yi ≤ (xi +yi)
i=1 i=1 i=1 Example of aligned equations:
Pr(X =1|Y =1)= Pr(X =1 ∧ Y =1) Pr(Y = 1)
1/n
.
= Pr(Y =1|X =1)·Pr(X =1). (2) Pr(Y = 1)
Usual expression for Bayes’ rule
Example of a theorem:
Theorem 1 (Euclid). There are infinitely many primes.
Euclid’s proof. There is at least one prime, namely 2. Now pick any finite list of primes p1, p2, . . . , pn. It suffices to show that there is another prime not on the list. Let p := ni=1 pi + 1, which is not any of the primes on the list. If p is prime, then we’re done. So suppose instead that p is not prime. Then there is prime q which divides p. If q is one of the primes on the list, then it would divide p − ni=1 pi = 1, which is impossible. Therefore q is not one of the n primes in the list, so we’re done.
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COMS 4771 Machine Learning (**add semester here**) Here is a centered table.
ABCD
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