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\lhead{CS 592--ATK, SPRING 2022}
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\title{Homework 5}
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\item {\bfseries Fourier Transformation Matrix.} (20 points)
We shall provide an alternate mechanism to construct the Fourier transformation matrix.
Recall that, for functions $\zo^n\to\bbR$, we defined the basis functions as follows.
For all $S,x\in\zo^n$, we defined
$$ \chi_S(x) \defeq (-1)^{S_1\cdot x_1 + S_2\cdot x_2 + \dotsi + S_n\cdot x_n}$$
Given this definition of the Fourier basis functions, the definition of the Fourier transformation matrix $\cF_n \in \frac1N \{+1,-1\}^{N\times N}$, where $N=2^n$, is as follows.
We shall use row indices $i\in\{0,1,\dotsc,N-1\}$ and $j\in\{0,1,\dotsc,N-1\}$ and define
$$ \left(\cF\right)_{i,j} \defeq \frac1N\chi_j(i)$$
Now, we begin the new definition using {\em matrix tensor product}.
Let $A\in\bbR^{a\times b}$ and $B\in \bbR^{a'\times b'}$ be two matrices.
We define the {\em block matrix} $C=A\otimes B$ as follows.
For $i\in\{1,\dotsc,a\}$ and $b\in\{1,\dotsc,b\}$
$$ C_{i,j} \defeq a_{i,j}B$$
{\bfseries Base case.} Define
$$\cG_1 \defeq \frac12\left[\begin{matrix} 1 &1 \\1&-1\end{matrix}\right]$$
{\bfseries Recursive construction.} Define, for $n>1$, $\cG_n \defeq \cG_{1}\otimes\cG_{n-1}$.
Prove, by induction, that $\cF_n = \cG_n$.
\noindent{\bfseries Solution.}\newline
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\item {\bfseries Smoothed Function Property.} (20 points)
Let $f\colon\zo^n\to\bbR$ be a function.
Let $L_p(f)$ be the norm defined as follows
$$ L_p(f) \defeq \left(\frac1N \sum_{x\in\zo^n} \abs{f(x)}^p\right)^{1/p}$$
For any $\rho\in[0,1]$, prove that $L_p(T_\rho(f)) \leq L_p(f)$.
Equality holds if and only if $f$ is a constant function, or $\rho=1$.
\noindent{\bfseries Solution.}\newline
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\item {\bfseries Most Random functions are Small Biased.} (20 points)
Let $f\colon\zo^n\to\{+1,-1\}$ be a boolean function.
Suppose we consider a {\em random} boolean function such that, for every $x\in\zo^n$, we assign $f(x)$ independently and uniformly at random from the set $\{+1,-1\}$.
Recall that a function $f$ is small biased if $\abs{\bias_f(S)}\leq\eps$ for all $0\neq S\in\zo^n$.
Formally state and prove a concentration result that proves: ``a random boolean function is small-biased with very high probability.''
\noindent{\bfseries Solution.}\newline
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\item {\bfseries Differential Operator.} (20 points)
We shall consider functions $\zo^n\to\bbR$.
Let us introduce a notation.
Given $x\in\zo^n$, we represent $x|_{i=1}$ as the bit-string identical to $x$ except that its $i$-th coordinate is fixed to 1.
Similarly, $x|_{i=0}$ is the bit-string that is identical to $x$ except that its $i$-th coordinate is fixed to $0$.
Let $D_i(f)$ be the function $\zo^n\to\bbR$ defined as follows
$$ D_i(f)(x) = f(x|_{i=1})-f(x|_{i=0})$$
Express $\widehat{D_i(f)}$ as a function of $\hf$.
\noindent{\bfseries Solution.}\newline
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\item {\bfseries Flats are Small-biased Distribution.} (20 points)
We shall consider function $\bbZ_p\to\bbC$ in this problem.
Define $\omega = \exp(2\pi \imath /p)$.
Recall that we defined, for $S\in\bbZ_p$, as follows
$$ \bias_f(S) = \sum_{x\in\bbZ_p} f(x)\omega^{S\cdot x}$$
Let $\X$ be a uniform distribution over the set $\{0,1,\dotsc,t-1\}$, for some integer $t