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Questions tagged [fourier-analysis]

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Distance between Fourier distributions of independent random Boolean functions

For a boolean function $f: \{-1,+1\}^n \to \{-1,+1\}$, the squared Fourier coefficients $\{\hat{f}(S)^2\}_{S \subseteq \{0,1\}^n}$ form a probability distribution. I want to know what the total ...
helloworld's user avatar
4 votes
0 answers
52 views

"Inverting" the fourier spectrum representation of a boolean function to recover a circuit representaiton

Given a boolean circuit, or an equivalent boolean expression, we can compute its fourier spectrum to yield a real-valued (multilinear) polynomial representation. What about the other way around? ...
Thomas Shrimpton's user avatar
0 votes
0 answers
45 views

Is it possible to estimate the positive outcomes of a boolean function using an optimized version of Goldreich-Levin?

Let $\mathcal{X} = \{-1,1\}^n$ and $h: \mathcal{X} \to \{-1,1\}$, h can be expanded in the basis of monomials for the uniform distribution, or also can have a distribution free expansion (Gram-Schmidt ...
rivana's user avatar
  • 65
1 vote
0 answers
46 views

what are some Lower bound for finding large fourier coefficients of boolean function (above a threshold)?

Is there some known lower bounds for estimating large fourier coefficients of boolean functions? And were there any comparison of tightness with the upper bound of Goldreich Levin algorithm?
rivana's user avatar
  • 65
2 votes
0 answers
46 views

Does Goldreich-Levin algorithm for finding large Fourier coefficients have time complexity upper bound = sample complexity upper bound?

I'm currently working on finding better bounds for Goldreich-Levin algorithm for estimating large Fourier coefficients of a boolean function. I was surprised seeing that the upper bounds for time ...
rivana's user avatar
  • 65
1 vote
0 answers
52 views

Is there an efficient Goldreich-Levin algorithm that generalizes to agnostic PAC setting?

Goldreich Levin algorithm is an algorithm that based on some assumption (boundness on Fourier coefficients) outputs the indices for most significant Fourier coefficients of a boolean function, however ...
rivana's user avatar
  • 65
1 vote
0 answers
47 views

Compute Fourier coefficients from Single Fourier coefficient and initial vector?

I have some vector $\vec v\in\mathbb{Z}_q^n$, and would like to obtain $n$ vectors $\vec f_0,\dots, \vec f_{n-1}$ where $\vec f_i = (\mathcal{F}(\vec v)_i,0,\dots,0)$, i.e. each vector is a single ...
Mark Schultz-Wu's user avatar
4 votes
1 answer
179 views

What is the time complexity of fermionic Fourier transform?

Suppose $N = 2^L$ and we are interested in performing the following transformation a $\mapsto$ a_hat on arrays of $N$ complex ...
fiktor's user avatar
  • 141
2 votes
0 answers
61 views

Learning boolean functions with input-ouput examples and side-information

The Kushilevitz-Mansour, "low-degree", and Goldreich-Levin algorithms aim to learn a function $f: \{0,1\}^n \rightarrow \{0,1\}$ from a sufficiently large set of input-output examples $(x_i, ...
Tom Shrimpton's user avatar
1 vote
0 answers
40 views

Is there an efficient way to compute the (top entries of) the parity spectrum of a function on the Boolean cube?

Suppose I have the Boolaen cube $B=\{-1, 1\}^n$, and let $F$ be the $2^n$ dimensional space of real-valued functions on it. Then the parity functions form a basis for $F$: $$b_S(x)=\prod_{i\in S}x_i,\...
Jack M's user avatar
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4 votes
0 answers
87 views

Harmonic analysis of sequences of Boolean functions (i.e. of words in $(\{0,1\}^n)^*$)

Is there any research on harmonic analysis of sequences of Boolean functions, which represent the application of a Boolean function on a word in $(\{0,1\}^n)^*$? I'm looking for any reference on this, ...
Shaull's user avatar
  • 5,646
5 votes
1 answer
447 views

A boolean function $f: \{0,1\}^n \rightarrow \{0,1\}$ is chosen at random from all $2^{2^n}$ such $f$. What do the Fourier coefficients look like?

As in the title. I'm not sure where to start here. My guess is that in expectation at least a constant fraction are non zero, and as a result there would exist some "large" coefs. and some "small" ...
zfkmz's user avatar
  • 187
6 votes
1 answer
438 views

Is the basis of parity functions the only orthonormal basis for Boolean functions?

Is there another orthonormal basis of functions for Boolean functions? Or, more specifically, besides the parity functions, is there another explicit function (which is common and has a name) that can ...
tigercub97's user avatar
5 votes
2 answers
369 views

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 ...
Andy's user avatar
  • 245
5 votes
0 answers
117 views

Majority function stability under deletion and addition of entries

It is well known that the majority function is stable under random flipping of bits. That is, if $v$ is a random binary vector, and then we re-sample each bit of $v$ with probability $\delta$ and get $...
Bartolinio's user avatar
2 votes
0 answers
150 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 ...
Michael Hahn's user avatar
4 votes
1 answer
166 views

Given a subset of of the hypercube and an affine transform of it, find the affine map

This is a follow up to this resolved question. Suppose we are given a set of bitvectors $A\subseteq\mathbb{F}_2^d$ and an invertible affine transformed copy of it $$B=\{Mx + s\mid x\in A\}$$ for some ...
boinkboink's user avatar
7 votes
2 answers
358 views

Given a subset of the hypercube and a copy translated by s, find s

Problem: Suppose we are given an $n$ element subset $A\subseteq\{0,1\}^d$ of the $d$ dimensional hypercube and a translated copy $B= A+s$ by some secret $s\in\{0,1\}^d$. Find $s$ as fast as possible ...
boinkboink's user avatar
1 vote
0 answers
66 views

relations between the degrees of a boolean function and its absolute function

Given a boolean function $f:\{0,1\}^n\rightarrow\mathbb{R}$ of degree $d$, is there any upper bound in terms of $d$ on the degree of the function $|f|$, where $|f|(x)=|f(x)|$. Here the degree of $f$ ...
user07001129's user avatar
1 vote
0 answers
91 views

Agnostic query learning of decision trees

Gopalan, Kalai, Klivans gave an algorithm https://dl.acm.org/citation.cfm?id=1374376.1374451 for agnostically learning decision trees $h:\{0,1\}^n\to\{0,1\}$ under the uniform distribution given ...
Aryeh's user avatar
  • 10.6k
4 votes
1 answer
299 views

Which (almost) balanced Boolean function has smallest "total" influence

The well known Kahn–Kalai–Linial (KKL) Theorem says that for any Boolean function $f\colon \{-1,1\}^n \xrightarrow{} \{-1,1\}$ $$ \max_{i \in [n]} \{\mathbf{Inf}_i[f] \} \geq \mathop{\bf Var}[f] \cdot ...
user52879's user avatar
3 votes
1 answer
258 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 ...
user621824's user avatar
6 votes
1 answer
134 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} [ (-...
BharatRam's user avatar
  • 393
9 votes
1 answer
249 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 ...
Samuel Schlesinger's user avatar
1 vote
0 answers
76 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)\...
Thomas Ahle's user avatar
2 votes
0 answers
91 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 ...
Thomas Ahle's user avatar
2 votes
1 answer
240 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}...
Annonymous's user avatar
4 votes
0 answers
210 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 ...
withhighprob's user avatar
3 votes
1 answer
224 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 ...
Nathan's user avatar
  • 179
7 votes
0 answers
154 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 ...
gradstudent's user avatar
  • 1,453
10 votes
1 answer
257 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 \...
Andrew Morgan's user avatar
14 votes
0 answers
372 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, ...
Clement C.'s user avatar
  • 4,471
4 votes
0 answers
164 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 $[n]$ ...
gradstudent's user avatar
  • 1,453
-1 votes
1 answer
295 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 ...
gradstudent's user avatar
  • 1,453
7 votes
0 answers
124 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 ...
D.W.'s user avatar
  • 12.1k
8 votes
1 answer
314 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 $f$ ...
user43879's user avatar
1 vote
0 answers
156 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 ...
Marco's user avatar
  • 31
7 votes
0 answers
244 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 the ...
Chill2Macht's user avatar
1 vote
0 answers
35 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 ...
Matt Groff's user avatar
  • 2,100
2 votes
0 answers
133 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 ...
Matt Groff's user avatar
  • 2,100
2 votes
0 answers
147 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 ...
DVD's user avatar
  • 71
0 votes
1 answer
121 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 ...
Omar Shehab's user avatar
-3 votes
1 answer
144 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 ...
Omar Shehab's user avatar
2 votes
0 answers
152 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 ...
user34585's user avatar
14 votes
0 answers
330 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 $f: \{0,1\}^n \to \{0,1\}$, ...
arnab's user avatar
  • 7,000
4 votes
1 answer
357 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 ...
Dean's user avatar
  • 225
4 votes
1 answer
107 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)]\...
Thomas's user avatar
  • 2,803
5 votes
0 answers
165 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 $k < n$,...
Federico Magallanez's user avatar
11 votes
0 answers
174 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, +...
Thomas's user avatar
  • 2,803
3 votes
0 answers
195 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 \...
NAg's user avatar
  • 666