Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [fourier-analysis]

The tag has no usage guidance.

3
votes
1answer
89 views

Which (almost) balanced Boolean function has smallest “total” influence

The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f : \{-1,1\}^n \xrightarrow{} \{-1,1\}$ $$ \max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot \...
3
votes
1answer
143 views

Lower bound on the support size of an $\epsilon$-biased distribution

Let $D$ be an $\epsilon$-biased distribution we want to show that $$\text{Supp}(D)\geq \Omega\bigg(\frac{n}{\epsilon^2\log(\frac{1}{\epsilon})}\bigg)$$ I know that there are some proofs for this but I ...
6
votes
1answer
116 views

Average-case analogue of Small-bias Spaces

Recall that an $\epsilon$-biased space is a set $S \subset \{0,1\}^n$ such that for every non-zero linear test $\alpha \in \{0,1\}^n \setminus \{0\}^n$, the expected bias $$| \mathbb{E}_{x \in S} [ (-...
9
votes
1answer
177 views

Has there been any progress in tightening the exponent in the result that polylog independence fools $AC_0$?

Braverman showed that distributions which are $(log \frac{m}{\epsilon})^{O(d^2)}$-wise independent $\epsilon$-fool depth $d$ $AC^0$ circuits of size $m$ by "gluing together" the Smolensky ...
1
vote
0answers
59 views

Relative Two-Function Hypercontractive Inequality $\langle T_{\rho\sigma} f,g \rangle \le \langle T_{\sigma} f,g \rangle^q$

The hypercontractive theorem (or Bonami Beckner inequality) says (Ryan O'Donnell): This of course also directly gives the 'intermediate' inequality $\|T_{\rho\sigma}f\|_q \le \|T_\sigma f\|_{1+(q-1)\...
3
votes
0answers
40 views

$p$-biased two-function hypercontractivity

The Hypercontractivity theorem (or Bonami Beckner inequality) is a very useful tool. Unfortunately, it isn't easy to carry over to other spaces than the uniform boolean cube. In Ryan O'Donnel's ...
2
votes
1answer
181 views

Fourier decomposition in terms of another basis

Given a Boolean function $f:\{-1,1\}^n\rightarrow \{-1,1\}$, it is well know that the Fourier decomposition of $f$ can be written as $f(x)=\sum_{S\subseteq \{1,\ldots,n\}} \widehat{f}(S) \prod_{i\in S}...
4
votes
0answers
110 views

Relationship between sparsity and rank of a boolean function

I have the following question when I was going through the proof of the following theorem. Theorem. For XOR function $f \circ XOR$, $rank(M_{f \circ XOR}) = ||\hat f ||_0$ where $M_{f \circ XOR}$ is ...
4
votes
1answer
141 views

Basic property of boolean functions with restrictions

For $f:\{\pm1\}^n\to\mathbb{R}$, $I\subset\{1,\dots,n\}$ and $x\in\{\pm1\}^{\{1,\dots,n\}\setminus I}$ we define $f_I[x]:\{\pm1\}^I\to\mathbb{R}$ by $f_I[x](y)=f(x,y)$. (We denote by ($x,y$) the ...
6
votes
0answers
118 views

Are there generalizations for Chow's theorem?

The Chow's theorem as it stands holds only for a single linear threshold gate. That these gates are uniquely determined by their first $n+1$ Fourier coefficients. Are there other circuits for which ...
10
votes
1answer
217 views

Is this “subgroup packing” polytope integral?

Let $\Gamma$ be a finite abelian group, and let $P$ be the polytope in $\mathbb{R}^\Gamma$ defined to be the points $x$ satisfying the following inequalities: $$\begin{array}{cl} \sum_{g\in G} x_g \...
14
votes
0answers
288 views

Fourier spectrum of the parity of two monotone Boolean functions

This is a question that I've been pondering, on and off, for a while, and unsuccessfully. I'd be very interested in any insight regarding this conjecture. (Or rather, these conjectures.) Recall that, ...
5
votes
0answers
121 views

About counting the “total size” of non-zero Fourier coefficients of a Boolean function

Given a $f: \{-1,1\}^n \rightarrow \mathbb{R}$, I want to compute this quantity, $\sum_{ \hat{f}(S) \neq 0, S \subseteq 2^{[n]}} \vert S \vert $ i.e the sum of the sizes of the subsets of $[n]$ ...
-1
votes
1answer
119 views

Sparsity of a Boolean function and its Fourier depth [closed]

For a function $f : \{-1,1\}^n \rightarrow \mathbb{R}$ one can ask for its $l_0$ norm in the indicator basis i.e the number of vertices on which the function is non-zero. Does this sparsity parameter ...
8
votes
0answers
100 views

Learning function with a few low-order Fourier coefficients, from uniformly random samples

Let $f:\{-1,+1\}^n \to \{-1,+1\}$ be a boolean function where all of the energy of the Fourier transform of $f$ is concentrated in a small number of low-order coefficients, say $k$ coefficients each ...
7
votes
1answer
215 views

Is a monotone boolean function monotone as a multilinear polynomial?

Let $f:\{-1,1\}^n \rightarrow \{0,1\}$ be a monotonically increasing boolean function. That is, $f(x) \leq f(y)$, if coordinate-wise $x \leq y$. Now, let $F: [-1,1]^n \rightarrow \mathbb{R}$ be $f$ ...
1
vote
0answers
69 views

Fourier expansion of boolean functions and affine subspaces

Let $f:\mathbb{F}^n_2\rightarrow \{0,1\}$ be a function constant on an affine subspace $V$ of co-dimension $t$. Assume that that $V$ is a linear subspace, by replacing $f(x)$ with $f(x+v)$ for some $v ...
5
votes
0answers
98 views

Is there a simple argument for this Hemi-Icosahedron Boolean function?

This is problem 1(e) from Homework 1 of the course about Analysis of Boolean functions at CMU in 2012 as well as problem 1.1(n) on p.34 of Ryan O'Donnell's Analysis of Boolean Functions. Compute ...
1
vote
0answers
30 views

Are there uses for a Fourier transform of length $n^m$ with elements of maximum size $n$?

In essence, I'm trying to get a better feel for when there is a use for FFT with small coefficients, compared to the length, assuming that we get a better runtime. I've been toying with an idea for a ...
2
votes
0answers
111 views

What would faster Fourier Transform(FFT?) and/or multiplication algorithms imply?

There are many problems which have implications on P vs. NP and other complexity classes. Supposing that we're interested in Fourier transforms and/or multiplication algorithms, do faster Fourier ...
3
votes
0answers
91 views

On FFT and trigonometric matrix eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$ B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots ...
0
votes
1answer
81 views

Question about discarding the second register in the standard approach of hidden subgroup algorithm

My questions: What does discarding the second register mean for the standard approach of hidden subgroup algorithm? Why does discarding let the first register end up in a mixed state? My ...
-3
votes
1answer
135 views

Dimension of the Fourier transform for $S_5$ [closed]

My question: What is the dimension of the Fourier transform for $S_5$? My effort: The dimensions of the seven irreps of $S_5$ are $1,1,4,4,5,5,6$. According to the notes of Andrew Childs, the ...
2
votes
0answers
150 views

Is the nonnegativeness of a polynomial hard for $\mathsf{NP}_\mathbb{R}$?

It is clear that the following problem is in $\mathsf{NP}_\mathbb{R}$. Input: a list $P$ of triplets $(a,s,t)$ where $s$ and $t$ are nonnegative integers. Output: is there an $x\in \mathbb{R}$ such ...
13
votes
0answers
279 views

Sign patterns for Fourier coefficients of Boolean functions

Given a sequence of real numbers $(a_i)$, the sign-pattern sequence $(s_i)$ is defined by $s_i = +$ if $a_i \geq 0$ and $s_i = -$ otherwise. For a boolean function $f: \{0,1\}^n \to \{0,1\}$, ...
4
votes
1answer
254 views

$\ell_1$ norm of Fourier coefficients vector for the hypercube

Let $G$ be the normzlied hypercube graph on $2^d$. It is a Cayley graph and it is well known that its eigenvalues are given by $\lambda_r = 1-2\frac{|r|}{d}$ for every $r \in \{0,1\}^d$. Given a ...
4
votes
1answer
89 views

Is being fooled by limited independence preserved by products?

Question. Let $f,g : \{\pm 1\}^n \to \{\pm 1\}$ be $\varepsilon$-fooled by $k$-wise independence -- i.e. for any $k$-wise independent random variable $X$, $\left|\mathbb{E}[f(X)] - \mathbb{E}[f(U)]\...
5
votes
0answers
159 views

Fourier Analysis on checking whether there exists a vector in hypercube orthogonal to a set of vectors?

I know virtually noting about Fourier Analysis and I'd like to know whether it's worth to learn this topic for my problem. My problem is: Given vectors $h_1,\cdots,h_k\in\{+1,-1\}^n$ where $k < n$,...
10
votes
0answers
153 views

Variance of bounded functions with rapidly decaying Fourier coefficients

I have the following conjecture about bounded functions on the hypercube. Any help resolving it (proof, counterexample, some ideas) is much appreciated. Conjecture. Let $f : \{ -1, +1 \}^n \to [-1, +...
3
votes
0answers
142 views

An identity about the Majority function?

Let $f:\{-1,1\}^n \rightarrow \{-1,1\}$ be any boolean function. Let $Maj_n$ represent the majority function. Let $\langle f,g \rangle = E[f(x)g(x)]$ and $\mathcal{I}(f) = E_x[\# i, s.t. f(x)\neq f(x \...
0
votes
0answers
124 views

Determining if a function is constant or not using period finding

Consider an arbitrary boolean function $$f: {\lbrace 0,1 \rbrace}^n \rightarrow \lbrace 0,1 \rbrace$$ which we write as: $$f(x_1, x_2 ... x_n) $$ where each $x_i$ is a boolean variable We note ...
0
votes
0answers
122 views

Computing the period of a function using a quantum computer

Consider a blackbox function $$f(x): Z \rightarrow \lbrace 0,1 \rbrace $$ Which inputs an integer and outputs 0 or 1 with bit complexity n. If the period P of this function satisfies $$P \in O(2^{...
1
vote
0answers
90 views

Period of a Multivariable Function using Quantum Computing

consider a function $$f(x_1,x_2...x_n)$$ I am told it is possible to compute the period of the function as a vector $$<l_1,l_2...l_n>$$ where each l denotes the period of the function for ...
2
votes
0answers
55 views

Is there a relation between $l_p$-norms of functions with same Fourier spectra but w.r.t different measures on the Hamming cube?

Informally, I want to ask if two functions $f$ and $g$ on the Hamming cube have the same Fourier spectra but w.r.t different measure and basis, then is $||f||_p$ related to $||g||_p$? (Where each $p$-...
2
votes
0answers
102 views

Geometry on a space of polynomial functions

I am considering some geometric concepts in a space of functions.But I am not sure the concept I consider is already defined in some references. Let $P_{1},...,P_{N}:\mathbb{F}_{2}^{n}\rightarrow \...
7
votes
0answers
176 views

An alternative proof of hypercontractivity of the Becker-Bonami operator

The Becker-Bonami operator $T_p$ on a function $f(x) : \{0,1\}^d \to \mathbb{R}$ is defined as follows. Let $\nu(x)$ be a perturbation of $x$ so that every bit remains the same with probability $p$ or ...
0
votes
0answers
82 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 ...
6
votes
2answers
613 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 ...
16
votes
2answers
391 views

An extension of the noise operator

In a problem I am currently working on, an extension of the noise operator arises naturally, and I was curious whether there has been prior work. First let me revise the basic noise operator $T_{\...
11
votes
1answer
295 views

Upperbound on the degree of a boolean function in terms of its sensitivity

A very interesting open problem in the study of complexity measures of Boolean function is the so called sensitivity vs block sensitivity conjecture. For background on sensitivity versus block ...
16
votes
2answers
746 views

Robustness of splitting a junta

We say that a Boolean function $f: \{0,1\}^n \to \{0,1\}$ is a $k$-junta if $f$ has at most $k$ influencing variables. Let $f: \{0,1\}^n \to \{0,1\}$ be a $2k$-junta. Denote the variables of $f$ by $...
18
votes
2answers
836 views

Are all the functions whose fourier weight is concentrated on the small sized sets computed by AC0 circuits?

Are all the functions whose fourier weight is concentrated on the small sized sets(or terms with low degree) computed by $\mathsf{AC}^0$ circuits ?
42
votes
12answers
3k views

Applications of representation theory of the symmetric group

Inspired by this question and in particular the final paragraph of Or's answer, I have the following question: Do you know of any applications of the representation theory of the symmetric group in ...
58
votes
3answers
6k views

Why does Fourier analysis of Boolean functions “work”?

Over the years I have gotten used to seeing many TCS theorems proved using discrete Fourier analysis. The Walsh-Fourier (Hadamard) transform is useful in virtually every subfield of TCS, including ...
13
votes
1answer
462 views

Can one prove $\mathsf{PARITY} \notin \mathsf{AC}^0$ using Linial-Mansour-Nisan theorem and the knowledge of fourier spectrum of $\mathsf{PARITY}$?

Result 1: Linial-Mansour-Nisan theorem says that the fourier weight of the functions computed by the $\mathsf{AC}^0$ circuits is concentrated on the subsets of small size with high probability. ...
12
votes
1answer
652 views

The entropy of a convolution over the hypercube

Say we have a function $f:\mathbb{Z}_2^n \to \mathbb{R}$, such that $\sum _{x\in \mathbb{Z}_2^n} f(x)^2 = 1$ (so we can think of $\{ f(x)^2\} _{x\in \mathbb{Z}_2^n}$ as a distribution). It is natural ...
5
votes
0answers
125 views

Truncated convolution algorithm

Suppose I have $n+1$-dimensional vectors $a$ and $b$. For each $i = 0, \dots, n$, I also have integers $0 \leq j_0(i) \leq j_1(i) \leq i$. I want to compute the following variant of a convolution: $$ ...
19
votes
2answers
835 views

Linearly independent Fourier coefficients

A basic property of vector spaces is that a vector space $V \subseteq \mathbb{F}_2^n$ of dimension $n-d$ can be characterized by $d$ linearly independent linear constraints - that is, there exist $d$ ...
14
votes
1answer
515 views

Best query complexity of Goldreich-Levin / Kushilevitz-Mansour learning algorithm

What is the best known query complexity of Goldreich-Levin learning algorithm? Lecture notes from Luca Trevisan's blog, Lemma 3, states it as $O(1/\epsilon^4 n \log n)$. Is this the best known in ...
3
votes
1answer
239 views

Nearly Optimal Sparse Walsh-Fourier Tranform

In the recent paper: Nearly Optimal Sparse Fourier Transform[Haitham Hassanieh, Piotr Indyk, Dina Katabi, Eric Price], the authors show an $O(k \log n)$-time algorithm for the problem of computing the ...