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

“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 Z 2 n Space of functions: Z 2 n . – Inner product space where  f,g  =E x [f(x)g(x)]. Characters:  (x+y)=  (x)  (y)

Foundations Claim (Characterization): The characters are the eigenvectors of the shift operators S s f(x)→ f(x+s). Corollary (Basis): The characters form an orthonormal basis. Claim (Explicit): The characters are the functions  S (x) = (-1)  i  S x i for S  [n].

Fourier Transform = Polynomial Expansion Fourier coefficients: f ^ (S) =  f,  S . Note: f ^ (  )=E x [f(x)] Polynomial expansion: substitute y i =(-1) x i f(y 1,…,y n ) =  S µ [n] f ^ (S)  i 2 S y i Fourier transform: f  f ^

The Fourier Spectrum n n-1 … n/2 … 1 0 |S| level

Degree-k Polynomial n n-1 … n/2 … 1 0 |S| 0 k

k-Junta n n-1 … n/2 … 1 0 |S| 0 k

Orthonormal Bases Parseval Identity (generalized Pythagorean Thm): For any f,  S (f ^ (S)) 2 = E x [ (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  = 2 n  f ^,g ^ 

Convolution Convolution: (f*g)(x) = E y [f(y)g(x-y)] Example Weighted average: (f*w)(0) = E y [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: Var x [f(x)] = E x [f(x) 2 ] - E x [f(x)] 2 =  S ≠ ; f ^ (S) 2 Influence of i’th variable: Inf i (f) = P x [f(x)≠f(x  e i )] =  S 3 i f ^ (S) 2 n n-1 … n/2 … 1 0

Smoothening f Perturbation: x » ± y : for each i, – y i = x i with probability 1- ± – y i = 1-x i otherwise T ± f(x) = E x » ± 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| 2 2 on levels ¸ k is: · (1-2 ± ) 2k |f| 2 2 · 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) = P x » ± y (f(x)  f(y)) Correlation: NS ± (f) = 2  (E[f]-  f,T ± f  ) Stability: Set  := 1/2-  /2 S ½ (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 Inf i (f)≤  for all i, S ρ (f) ≤ 2/π · arcsin ρ + O( (loglog 1/ ² )/log1/ ² ) ≈ noise stability of threshold functions t(x)=sign(∑a i x i ), ∑a i 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.