Discrete Fourier Analysis

2022 May 17

[1] O'Donnell, Ryan. Analysis of boolean functions. Cambridge University Press, 2014.

[2] Filmus, Yuval. "Analysis of Boolean functions." (2016).

[3] De Wolf, Ronald. "A brief introduction to Fourier analysis on the Boolean cube." Theory of Computing (2008): 1-20.

[3] is short, very concise, neat and contains pointers to other materials.

Fourier basis are orthonormal

Fourier transform is a linear map. 

Fourier basis are the only linear functions. Linearity tests are essentially testing how close a function is to a Fourier basis. The measure for this closeness or distance can be statistical distance, inner product, 2-norm, etc. 

For a function, the set of heavy Fourier coefficients is small. (\epsilon then 1/(\epsilon^2))

Convolution is bilinear

XOR lemma: the bias of the XOR of independent distributions over {0,1} is the product of individual biases, if tested with character S=1. (Always 1 if tested with S=0)

Min-entropy source gives small collision probability

Min-entropy extractor: Given a min-entropy source  X (or in essence, any source with small collision probability) and a small-bias distribution Y, then the statistical distance between (X\xor Y) and uniform is small. 

Leftover hash lemma (LHL): first, we establish that a single deterministic hash function cannot extract random bits from all high min-entropy sources. A simple thought experiment is to fix a hash function, we can always construct a high min-entropy source that makes this function output a constant. Then we can use a family of hash functions, or more precisely, universal hash functions. LHL states that this family of universal hash functions can extract randomness from all min-entropy sources. 

Noise operator is a linear map.

Norm has monotonicity, meaning that for 1<=p<q, p-norm is no greater than q-norm. 

Norm has contractivity, meaning that for p>=1, p-norm on the smoothened (run through noise operator) version of a function is no greater than the p-norm on the original function.

Norm has hypercontractivity, meaning that for 1<=p<q, q-norm on the smoothened version of a function is no greater than the p-norm on the original function. Here the noise operator has a specific noise parameter. [Theorem 4.1 in [3]]

KKL theorem as a special case for hypercontractivity, with p=1+\delta, q=2. 

Friedgut: if a Boolean function has total influence \t, then it's close to a 2^{O(\t)}-junta Boolean function (meaning that this function only depends on 2^{O(\t)} of the coordinates/variables). The proof is to construct the junta function. The bound is tight because for address/tribe function, this many variables are needed. 

Bourgain: for a balanced Boolean function (equal number of inputs resulting in output 0 and 1), the weight on higher degree Fourier coefficients cannot decay too fast unless it is close to a junta function.  

Goldreich-Levin hardcore predicate: we can construct PRG from OWF. Proof by contradiction: if the construction is not secure, then we can invert the OWF with Goldreich-Levin algorithm. 

Overall, Fourier analysis tries to capture the properties of the original function with properties of Fourier coefficients. The applications, as also mentioned in [3], include property testing, PCP, explaining threshold phenomena via analysis of influences and juntas, an alternative proof for Arrow's theorem, verification for ML algorithms (t-sparse Boolean functions), etc.