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Level $k$ bounds in Analysis of Boolean functions

In Ryan O'Donnell's book Analysis of Boolean functions, following Corollary 9.25 the following appears: If $f\colon \{-1,1\}^n \to \{0,1\}$, and we have $\mathbb{E}[f] = \alpha$, then for any integer ...
199 views

Dual to hypercontractive inequality

Recall the hypercontractive inequality: Let $\rho = \sqrt{\frac{p-1}{q-1}}$, then $||T_\rho(f)||_q \leq ||f||_p$ In https://www.cs.cmu.edu/~odonnell/papers/analysis-survey.pdf it is stated that the ...
254 views

How tight is the XOR lemma?

The XOR lemma states that if you have a distribution $D$ on $\{0,1\}^n$, and all the Fourier coefficients of $2^n D$ are small, then it is close in $L_1$ to the uniform distribution. Specifically, ...
143 views

Sensitivity and Low-Degree Approximation under Non-Uniform Distribution

I am searching for generalizations of analysis of Boolean functions when the input strings are distributed according to a general non-uniform distribution, possibly with arbitrary dependencies between ...
217 views

217 views

527 views

Beigel-Tarui transformation of ACC cricuits

I am reading the appendix about ACC lower bounds for NEXP in Arora and Barak's Computational Complexity book. http://www.cs.princeton.edu/theory/uploads/Compbook/accnexp.pdf One of the key lemmas is a ...
66 views

Estimating the average contiguous sequence length of a balanced boolean function

Given a balanced Boolean function $f:\{0,1\}^n\mapsto\{0,1\}$, is there a sampling strategy that needs less than $O(2^n)$ samples to estimate the average length of contiguous sequences of 1's? ...
179 views

Natural Proofs and methods for polylog depth circuit lower bounds

I have a question about the following question and its answer. Status on circuit lower bounds for polylog-bounded depth circuits In the above question, it is asked about methods to prove lower bounds ...
92 views

Bound for the spectral norm of a boolean function [duplicate]

As we know that the upper bound for the spectral norm of a Boolean function on $n$ variables is $2^{n/2}$. Is it possible to improve this bound..? Can somebody provide me an example of a Boolean ...
822 views

Is the the spectral norm of a Boolean function bounded by the degree of its Fourier expansion?

Let $f: \{-1,1\}^n \rightarrow \{-1,1\}$ be a Boolean function. The Fourier expansion of $f$ is $$f(T) = \sum_{S \subseteq [n]} \widehat{f}(S)\ \chi_S(T)$$ where $\widehat{f}(S)$ are real numbers ...
787 views