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![Page 1: Primer on Fourier Analysis Dana Moshkovitz Princeton University and The Institute for Advanced Study.](https://reader033.fdocuments.us/reader033/viewer/2022051517/56649ea75503460f94ba9a87/html5/thumbnails/1.jpg)
Primer on Fourier Analysis
Dana MoshkovitzPrinceton University and
The Institute for Advanced Study
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Fourier Analysis in Theoretical Computer Science
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Fourier Analysis in Theoretical Computer Science (Unofficial List)
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“The Fourier Magic”
“something that looks scary to analyze”
“bunch of (in)equalities”
Fourier Analysis
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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?
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It’s Just a Different Way to Look at Functions
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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
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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)
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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].
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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^
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The Fourier Spectrum
|S|level
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Degree-k Polynomial
|S|
0
k
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k-Junta
|S|
0
k
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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^
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Convolution
Convolution:(f*g)(x) = Ey[f(y)g(x-y)]
ExampleWeighted average:(f*w)(0) = Ey[f(y)w(y)]
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Convolution in Fourier Basis
Claim: For any f,g, (f*g)^ f^·g^
Proof: By expanding according to definition.
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Things You Can Do with Convolution
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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
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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^
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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/
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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”
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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
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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].
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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
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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.