\documentclass[11pt]{article} \input{headers05.tex} \lhead{CS 592--ATK, SPRING 2022} \rhead{Name: Type your name here} %%% <-- REPLACE Hemanta K. Maji WITH YOUR NAME HERE \usepackage[strict]{changepage} \newcommand{\nextoddpage}{\checkoddpage\ifoddpage{\ \newpage\ \newpage}\else{\ \newpage}\fi} \begin{document} \title{Homework 5} \date{} \maketitle \thispagestyle{fancy} \pagestyle{fancy} \begin{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \ %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \ %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \ %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% PROBLEM 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 \ %%% <-- ERASE THIS LINE AND WRITE YOUR SOLUTION HERE \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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