Primer on Fourier Analysis

Dana MoshkovitzPrinceton University and

The Institute for Advanced Study

Fourier Analysis in Theoretical Computer Science

Fourier Analysis in Theoretical Computer Science (Unofficial List)

“The Fourier Magic”

“something that looks scary to analyze”

“bunch of (in)equalities”

Fourier Analysis

Today: Explain the “Fourier Magic”

What is it? Why is it useful?

What does it do?When to use it?

What do we know about it?

It’s Just a Different Way to Look at Functions

It’s Changing Basis

• Background: Real/complex functions form vector space

• Idea: Represent functions in Fourier basis, which is the basis of the shift operators (representation by frequency).

• Advantage: Convolution (complicated “global” operation on functions) becomes simple (“local”) in Fourier basis

• Generality: Here will only consider the Boolean case – very-special case

Fourier Basis (Boolean Cube Case)

• Boolean cube: additive group Z2n

• Space of functions: Z2n.

– Inner product space where f,g=Ex[f(x)g(x)].

• Characters: (x+y)=(x)(y)

Foundations

• Claim (Characterization): The characters are the eigenvectors of the shift operators Ssf(x)→ f(x+s).

• Corollary (Basis): The characters form an orthonormal basis.

• Claim (Explicit): The characters are the functions S(x) = (-1)iSxi for S[n].

Fourier Transform = Polynomial Expansion

• Fourier coefficients: f^(S) = f,S.

• Note: f^()=Ex[f(x)]

• Polynomial expansion: substitute yi=(-1)xi

f(y1,…,yn) = Sµ[n]f^(S)i2Syi

• Fourier transform: f f^

The Fourier Spectrum

|S|level

Degree-k Polynomial

|S|

0

k

k-Junta

|S|

0

k

Orthonormal BasesParseval Identity (generalized

Pythagorean Thm): For any f,

S(f^(S))2 = Ex[ (f (x))2]

So, for Boolean f:{±1}n→{±1}, we have:

x(f^(x))2 = 1

In general, for any f,g, f,g = 2nf^,g^

Convolution

Convolution:(f*g)(x) = Ey[f(y)g(x-y)]

ExampleWeighted average:(f*w)(0) = Ey[f(y)w(y)]

Convolution in Fourier Basis

Claim: For any f,g, (f*g)^ f^·g^

Proof: By expanding according to definition.

Things You Can Do with Convolution

Parts of The Spectrum• Variance: Varx[f(x)] = Ex[f(x)2] - Ex[f(x)]2 = S≠; f^(S)2

• Influence of i’th variable: Infi(f) = Px[f(x)≠f(xei)] = S3i f^(S)2

Smoothening f

• Perturbation: x»±y : for each i, – yi = xi with probability 1-±

– yi = 1-xi otherwise

• T±f(x) = Ex»±y[f(y)]

• Convolution: T±f f*P(noise=µ)

• Fourier: (T±f)^ (1-2±)|S|·f^

Smoothed Function is Close to Low Degree!

Tail: Part of |T±f|22 on levels ¸ k is:

· (1-2±)2k |f|22· e-c±k

Hence, weight on levels ¸ C · 1/ · log 1/

Hypercontractivity

Theorem (Bonami, Gross): For f, for ± · √(p-1)/(q-1),

|T±f|q · |f|p

Roughly, and incorrectly ;-): “T±f much [in a “tougher” norm] smoother than f”

Noise Sensitivity and Stability

• Noise Sensitivity: NS±(f) = Px»±y (f(x)f(y))

• Correlation: NS±(f) = 2(E[f]-f,T±f)

• Stability: Set := 1/2-/2S½(f) = f,T±f

• Fourier: S±(f) = f^, |S|f^

= §S |S| f^(S)2

Thresholds Are Stablest and Hardness of Approximation

• What is it? Isoperimetric inequality on noise stability [MOO05].

• Applications to hardness of approximation (e.g., Max-Cut [KKMO04]).

• Derived from “Invariance Principle” (extended Central Limit Theorem), used by the [R08] extension of [KKMO04].

Thresholds Are Stablest

Theorem [MOO’05]: Fix 0<<1. For balanced f (i.e., E[f]=0) where Infi(f)≤ for all i,

Sρ(f) ≤ 2/π · arcsin ρ + O( (loglog 1/²)/log1/²)

≈ noise stability of threshold functions t(x)=sign(∑aixi), ∑ai

2=1

More Material

• There are excellent courses on Fourier Analysis

available on the homepages of: Irit Dinur and

Ehud Friedgut, Guy Kindler, Subhash Khot,

Elchanan Mossel, Ryan O’Donnell, Oded Regev.