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Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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Page 1: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Primer on Fourier Analysis

Dana MoshkovitzPrinceton University and

The Institute for Advanced Study

Page 2: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Fourier Analysis in Theoretical Computer Science

Page 3: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Fourier Analysis in Theoretical Computer Science (Unofficial List)

Page 4: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

“The Fourier Magic”

“something that looks scary to analyze”

“bunch of (in)equalities”

Fourier Analysis

Page 5: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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?

Page 6: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

It’s Just a Different Way to Look at Functions

Page 7: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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

Page 8: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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)

Page 9: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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].

Page 10: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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^

Page 11: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

The Fourier Spectrum

|S|level

Page 12: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Degree-k Polynomial

|S|

0

k

Page 13: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

k-Junta

|S|

0

k

Page 14: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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^

Page 15: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Convolution

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

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

Page 16: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Convolution in Fourier Basis

Claim: For any f,g, (f*g)^ f^·g^

Proof: By expanding according to definition.

Page 17: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

Things You Can Do with Convolution

Page 18: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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

Page 19: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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^

Page 20: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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/

Page 21: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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”

Page 22: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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

Page 23: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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].

Page 24: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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

Page 25: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study

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.