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

Fourier Analysis in Theoretical Computer Science

Fourier Analysis in Theoretical Computer Science (Unofficial List) Polynomials multiplication (FFT) List Decoding [AGS03] Collective Coin Flipping [BL,KKL] Analysis of expansion/sampling (e.g., [MR06]) Computational Learning [KM] Linearity testing [BLR] Analysis of threshold phenomena Hardness of Approximation (dictator testing) [H97] Voting/social choice schemes … Quantum Computing

“something that looks scary to analyze” “bunch of (in)equalities” “The Fourier Magic” Fourier Analysis “something that looks scary to analyze” “bunch of (in)equalities”

Today: Explain the “Fourier Magic” Why is it useful? What is it? 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)iSxi for S[n]. Shift operators occur “everywhere” [try to think of examples; some will be given in the sequel]. In particular, convolution involves shifting. Analysis in the eigenvectors basis seems “natural”. Notice: in the Boolean case, the characters correspond to the linear function <s,x> for the binary vectors s. This is what is used for “linearity testing”. [the (-1) is for addition <-->multiplication]

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 level n n-1 … n/2 |S| … 1

Degree-k Polynomial n n-1 … n/2 |S| … 1 k

k-Junta n n-1 … n/2 |S| … 1 k-junta is a degenerate case – instead of considering n vars, we could have considered k k

Orthonormal Bases Parseval 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 = 2nf^,g^

Convolution Convolution: (f*g)(x) = Ey[f(y)g(x-y)] Example Weighted 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: … n/2 … 1 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(xei)] = S3i f^(S)2 Intuition: Variance = the non-constant part of the function Influence of i = the part of the spectrum that depends on i

Smoothening f Perturbation: x»±y : for each i, T±f(x) = Ex»±y[f(y)] 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 NS±(f) = Px»±y (f(x)f(y)) Correlation: NS±(f) = 2(E[f]-f,T±f) Stability: Set  := 1/2-/2 S½(f) = f,T±f Fourier: S±(f) = f^, |S|f^ = §S |S| f^(S)2 NS±(f) = 2f^() – 2S(1-2±)|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]. Isoperimetry = Largest area for specified boundary/[Here:]smallest boundary for given area. For more applications – look at [MOO]. The application go far beyond hardness of approximation.

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), ∑ai2=1 Balanced is [somewhat] necessary: constant function is stable! Can “play” with that – consider somewhat non-constant function. In hardness of approximation – usually easy to ensure the function is balanced. Also, commonly in hardness constructions we consider negative rho. - All influences are small = no variable determines the function to a large extent Can “play” with that – first remove influential variables. But in hardness construction this is exactly what we exploit: stable balanced functions have an influential variable!

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.