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\noindent {\large {\bf Analysis of Boolean Functions}}  {\hfill {\bf CMU 18-859S / 21-801A, Fall 2012}}

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{\sc Problem Set 3} \\
{\bf Due: Monday, Oct.\ 1, beginning of class} \\
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\noindent {\bf Homework policy}: Please work on the homework by yourself; it isn't intended to be too difficult.  Questions about the homework or other course material can be asked on Piazza.\\

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\begin{enumerate}
    \item Let $p \in [1, \infty)$.  Given $f \btR$ we define its \emph{$p$-norm} to be $\|f\|_p = \E_{\bx}[|f(\bx)|^p]^{1/p}$ and its \emph{Fourier $p$-norm} to be $\snorm{f}_p = (\sum_{S} |\wh{f}(S)|^p)^{1/p}$.  We also define $\|f\|_\infty = \max_x \{|f(x)|\}$ and $\snorm{f}_\infty = \max_S \{|\wh{f}(S)|\}$.
        \begin{enumerate}
            \item For $1 \leq p \leq q \leq \infty$, show that $\|f\|_p \leq \|f\|_q$.  (Hint: one solution uses the fact that the function $t \mapsto t^{q/p}$ is convex on $[0, \infty)$.)  Conversely, show $\snorm{f}_p \geq \snorm{f}_q$.
            \item Show that $\snorm{f}_\infty \leq \|f\|_1$ and $\|f\|_\infty \leq \snorm{f}_1$.
        \end{enumerate}
    \item In this exercise you are asked to prove some fancily-named properties of the noise operator $\T_\rho$.
          \begin{enumerate}
             \item Show that $\T_\rho$ is ``positivity-preserving'' for all $\rho \in [-1,1]$, meaning $f \geq 0 \Rightarrow \T_\rho f \geq 0$.

                 Show also that it is ``positivity-improving'' for all $\rho \in (-1,1)$, meaning $f \geq 0, f \not \equiv 0 \Rightarrow \T_\rho f > 0$.
             \item Show the ``semigroup property'': $\T_{\rho_1} \circ \T_{\rho_2} = \T_{\rho_1 \rho_2}$ for all $\rho_1, \rho_2, \in [0,1]$.

                 (If you like, prove it even for $\rho_1, \rho_2 \in [-1,1]$.)
             \item Show that $\T_\rho$ is a ``contraction on $L^p$'' for all $p \geq 1$ and $\rho \in [-1,1]$; i.e., $\|\T_\rho f\|_p \leq \|f\|_p$.
          \end{enumerate}
    \item For functions $f \co \F_2^n \to \R$, sometimes it is more natural to index the Fourier coefficients not by subsets $S \subseteq [n]$ but by elements $\gamma \in \F_2^n$; here we identify a subset with its indicator vector.  In this case we would write the Fourier expansion as
        \[
            f = \sum_{\gamma \in \F_2^n} \wh{f}(S) \chi_\gamma, \qquad \text{where } \chi_\gamma(x) = (-1)^{\gamma \cdot x}
        \]
        and $\gamma \cdot x$ is the dot-product of $\gamma$ and $x$ in the vector space $\F_2^n$.  Note that for $\beta, \gamma \in \F_2^n$ we have $\chi_{\beta} \chi_{\gamma} = \chi_{\beta + \gamma}$.
        \begin{enumerate}
            \item Let $H$ be a vector subspace of $\F_2^n$.  Let $H^\perp$ be its ``perpendicular subspace''; i.e., $H^\perp = \{\gamma \in \F_2^n : \gamma \cdot x = 0 \text{ for all } x \in H\}$. Show that the indicator function $1_H \co \F_2^n \to \{0,1\}$ of~$H$ has the Fourier expansion $1_H = \sum_{\gamma \in H^\perp} 2^{-k} \chi_\gamma$, where $k = \dim(H^\perp)$. (Remark: $k = n - \dim(H)$ is sometimes denoted $\codim(H)$.)
            \item Given the subspace $H$ and also $y \in \F_2^n$, the set $H + y = \{h + y : h \in H\}$ is called an ``affine subspace'' of~$\F_2^n$.  Show that the indicator function $1_{H+y} \co \F_2^n \to \{0,1\}$ of this affine subspace has the Fourier expansion $1_{H+y} = \sum_{\gamma \in H^\perp} 2^{-k} \chi_\gamma(y) \chi_\gamma$, where again $k = \dim(H^\perp)$.
        \end{enumerate}
    \item Suppose the Fourier spectrum of $f \btR$ is $\eps_1$-concentrated on $\calF$ and that $g \btR$ satisfies $\|f - g\|_2^2 \leq \eps_2$.  Show that the Fourier spectrum of $g$ is~$2(\eps_1+\eps_2)$-concentrated on~$\calF$.
    \item Given $s \in \N^+$, let $\calC$ be the class of all functions $f \btb$ expressible as $f(x) = g(h_1(x), \dots, h_s(x))$, where $h_1, \dots, h_s \btb$ are weighted majority functions and $g \co \bits^s \to \bits$ is any function. Show that $\calC$ is learnable from random examples to error~$\eps$ in time~$n^{O(s^2/\eps^2)}$.  You may use Peres's Theorem, that $\NS_\delta[h] \leq 2\sqrt{\delta}$ for all $\delta \in [0,\frac12]$ and all weighted majorities~$h$. (Hint: how can you bound $\NS_\delta[f]$?)
    \item \begin{enumerate}
              \item Let $k \in \N^+$ and let $\calC = \{f \btb \mid \deg(f) \leq k\}$. (In particular, $\calC$ contains all functions computable by depth-$k$ decision trees.)  Show that $\calC$ is learnable from random examples with error~$0$ in time $n^k \cdot \poly(n, 2^k)$.  You may use the following ``Degree/Granularity Fact'': for every $f \in \calC$ and every $S \subseteq [n]$, the Fourier coefficient $\wh{f}(S)$ is an integer multiple of $2^{1-k}$.
              \item (Bonus.) Prove the Degree/Granularity Fact.
          \end{enumerate}
\end{enumerate}

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