TheoryofDeepLearning.2022

13
Examples of Theorems, Proofs, Algorithms, Tables, Figures
In this chapter, Zhao provide examples of many things, like Theorems,
Lemmas, Algorithms, Tables, and Figures. If anyone has
question, feel free to contact Zhao directly.
13.1 Example of Theorems and Lemmas
We provide some examples
Theorem 13.1.1 (d-dimension sparse Fourier transform). There is an
algorithm (procedure FourierSparseRecovery in Algorithm 2) that
runs in ??? times and outputs ??? such that ???.
Note that, usually, if we provide the algorithm of the Theorem/Lemma.
Theorem should try to ref the corresponding Algorithm.
For the name of Theorem/Lemma/Corollary ..., let us only capitalize
the first word,
Lemma 13.1.2 (Upper bound on the gradient).
Theorem 13.1.3 (Main result).
13.2 Example of Long Equation Proofs
We can rewrite ‖Ax ′ − b‖ 2 2
in the following sense,
‖Ax ′ − b‖ 2 2 = ‖Ax′ − Ax ∗ + AA † b − b‖ 2 2
= ‖Ax ∗ − Ax ′ ‖ 2 2 + ‖Ax∗ − b‖ 2 2
= ‖Ax ∗ − Ax ′ ‖ 2 2 + OPT2
where the first step follows from x ∗ = A † b, the second step follows
from Pythagorean Theorem, and the last step follows from OPT :=
‖Ax ∗ − b‖ 2 .