TY - GEN

T1 - Deterministic sparse fourier approximation via fooling arithmetic progressions

AU - Akavia, Adi

PY - 2010

Y1 - 2010

N2 - A significant Fourier transform (SFT) algorithm, given a threshold τ and oracle access to a function f, outputs (the frequencies and approximate values of) all the τ-significant Fourier coefficients of f, i.e., the Fourier coefficients whose magnitude exceeds τ|| f ||2 2. In this paper we present the first deterministic SFT algorithm for functions f over ℤN which is: (1) Local, i.e., its running time is polynomial in logN, 1/τ and L1(f̂) (the L1 norm of f's Fourier transform). (2) Robust to random noise. This strictly extends the class of compressible/Fourier sparse functions over ℤN efficiently handled by prior deterministic algorithms. As a corollary we obtain deterministic and robust algorithms for sparse Fourier approximation, compressed sensing and sketching. As a central tool, we prove that there are: 1. Explicit sets A of size poly((ln N)d, 1/ε) with ε-discrepancy in all rank d Bohr sets in ℤN. This extends the Razborov-Szemeredi-Wigderson result on ε-discrepancy in arithmetic progressions to Bohr sets, which are their higher rank analogue. 2. Explicit sets AP of size poly(lnN, 1/ε) that ε-approximate the uniform distribution over a given arithmetic progression P in ℤN, in the sense that |Exε A χ(x) ∼ ExεP χ(x)| < ε for all linear tests χ in ℤN. This extends results on small biased sets, which are sets approximating the uniform distribution over the entire domain, to sets approximating uniform distributions over (arbitrary size) arithmetic progressions. These results may be of independent interest.

AB - A significant Fourier transform (SFT) algorithm, given a threshold τ and oracle access to a function f, outputs (the frequencies and approximate values of) all the τ-significant Fourier coefficients of f, i.e., the Fourier coefficients whose magnitude exceeds τ|| f ||2 2. In this paper we present the first deterministic SFT algorithm for functions f over ℤN which is: (1) Local, i.e., its running time is polynomial in logN, 1/τ and L1(f̂) (the L1 norm of f's Fourier transform). (2) Robust to random noise. This strictly extends the class of compressible/Fourier sparse functions over ℤN efficiently handled by prior deterministic algorithms. As a corollary we obtain deterministic and robust algorithms for sparse Fourier approximation, compressed sensing and sketching. As a central tool, we prove that there are: 1. Explicit sets A of size poly((ln N)d, 1/ε) with ε-discrepancy in all rank d Bohr sets in ℤN. This extends the Razborov-Szemeredi-Wigderson result on ε-discrepancy in arithmetic progressions to Bohr sets, which are their higher rank analogue. 2. Explicit sets AP of size poly(lnN, 1/ε) that ε-approximate the uniform distribution over a given arithmetic progression P in ℤN, in the sense that |Exε A χ(x) ∼ ExεP χ(x)| < ε for all linear tests χ in ℤN. This extends results on small biased sets, which are sets approximating the uniform distribution over the entire domain, to sets approximating uniform distributions over (arbitrary size) arithmetic progressions. These results may be of independent interest.

UR - http://www.scopus.com/inward/record.url?scp=84894738603&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84894738603

SN - 9780982252925

T3 - COLT 2010 - The 23rd Conference on Learning Theory

SP - 381

EP - 393

BT - COLT 2010 - The 23rd Conference on Learning Theory

T2 - 23rd Conference on Learning Theory, COLT 2010

Y2 - 27 June 2010 through 29 June 2010

ER -