Learning Significant Fourier Coefficients over Finite Abelian Groups

Fourier transform is among the most widely used tools in computer science. Computing the Fourier transform of a signal of length N may be done in time Θ(N logN) using the Fast Fourier Transform (FFT) algorithm. This time bound clearly cannot be improved below Θ(N), because the output itself is of length N . Nonetheless, it turns out that in many applications it suffices to find only the significant Fourier coefficients, i.e., Fourier coefficients occupying, say, at least 1% of the energy of the signal. This motivates the problem discussed in this entry: the problem of efficiently finding and approximating the significant Fourier coefficients of a given signal (SFT, in short). A naive solution for SFT is to first compute the entire Fourier transform of the given signal and then to output only the significant Fourier coefficients; thus yielding no complexity improvement over algorithms computing the entire Fourier transform. In contrast, SFT can be solved far more efficiently in running time Θ̃(logN) and while reading at most Θ̃(logN) out of the N signal’s entries [2]. This fast algorithm for SFT opens the way to applications taken from diverse areas including computational learning, error correcting codes, cryptography and algorithms. We now formally define the SFT problem, restricting our attention to discrete signals. We use functional notation where a signal is a function f : G → C over a finite abelian group G, its energy is ‖f‖2 def = 1 |G| ∑ x∈G f(x) 2, and its maximal amplitude is ‖f‖∞ def = max {|f(x)| |x ∈ G}.1 For ease of presentation we assume without loss of generality that G = ZN1 × ZN2 × . . .× ZNk for N1, . . . , Nk ∈ Z+ (i.e., positive integers), and ZN is the additive group of integers modulo N . The Fourier transform of f is the function f̂ : G→ C defined for each α = (α1, . . . , αk) ∈ G by