Fast Fourier Sparsity Testing over the Boolean Hypercube

Grigory Yaroslavtsevhttp://grigory.us

Joint with Andrew Arnold (Waterloo), Arturs Backurs (MIT), Eric Blais (Waterloo) and Krzysztof Onak (IBM).

Fourier Analysis

• Notation switch:

• Functions as vectors form a vector space:

• Inner product on functions = “correlation”:

Fourier Analysis

• For let character • Fact: Every function can be uniquely

represented as a multilinear polynomial

• Fourier coefficient of on • Parseval’s Thm: For any

PAC-style learning

• PAC-learning under uniform distribution: for a class of functions , given access to and find a hypothesis such that

• Query model :– , for any

• Fourier analysis helps because of sparsity in Fourier spectrum– Low-degree concentration– Concentration on a small number of Fourier

coefficients

Fourier Analysis and LearningDef (Fourier Concentration): Fourier spectrum of is -concentrated on a collection of subsets 𝔽 if:Sparse Fourier Transform [Goldreich-Levin/Kushilevitz-

Mansour]: Class which is -concentrated on M sets can be PAC-learned with queries:dist

Testing Sparsity in

-close : dist(,k-sparse)

⇒

k-sparse

NO

Property Tester

-close

Accept with probability

Reject with probability

⇒

⇒Don’t care

⇒

Tolerant Property Tester

Accept with probability

Reject with probability

⇒

⇒Don’t care

NO

-close-close

k-sparse

Previous work under Hamming

• Testing sparsity of Boolean functions under Hamming distance– [Gopalan,O’Donnell,Servedio,Shiplka,Wimmer’11

• Non-tolerant test• Complexity • Reduction to testing under • Lower bound

– [Yoshida, Wimmer’13] • Tolerant test• Complexity

• Our results give a tolerant test with almost quadratic improvement on [GOSSW’11]

Pairwise Independent Hashing

∀𝑺 ,𝑻 Pr [h (𝑺 )=𝑎 , h (𝑻 )=𝑏 ]=Pr [h (𝑺)=𝑎 ] Pr [h (𝑻 )=𝑏 ]

Pairwise Fourier Hashing [FGKP’09]

= Cosets of a random linear subspace of

= Projection of on the coset

Energy =||𝒇 ||2

2=∑

𝑏||𝒇 𝑏||2

2

Testing k-sparsity [GOSSW’11]

# ¿𝑂 (𝑘2 )⇒• Fact: random samples from suffice to estimate

up to with prob. • Algorithm: Estimate all projections up to with

probability • Complexity: , only non-tolerant

Our Algorithm

• Take # cosets B • Let be a random sample from • For a coset let median(), where • Output

• Complexity: • Fact: The “median estimators” suffice to estimate

all up to

Analysis

• Two main challenges– Top-k coefficients may collide– Noise from non top-k coefficients

• Take # cosets B • Let be a random sample from • For a coset let median(), where • Output

Analysis: Large coefficients

Lemma: Fix If all coefficients are then for buckets the weight in buckets with collisions

Proof:• # coefficients • Pr[coefficient collides] • By Markov w.p. the colliding weight

Analysis: Small coefficients

Lemma: Fix If all coefficients are then for buckets the weight in any subset of size is – “Light buckets” with weight contribute – “Heavy buckets” contribute : • Weighted # collisions

• Each in a “heavy bucket” contributes to • Overall:

Analysis: Putting it together

Lemma: If the previous two lemmas hold then the -error of the algorithm is at most • instead of because of error in singleton

heavy coefficients• Crude bound because of pairwise

independence + Cauchy-Schwarz

If B = and -error

Other results + Open Problems

• Our result: non-tolerant test– Using BLR-test to check linearity of projections

• Lower bound of [GOSSW’11] is • Extensions to other domains – Sparse FFT on the line [Hassanieh,

Indyk,Katabi,Price’12]– Sparse FFT in d dimensions [Indyk, Kapralov’14]

• Other properties that can be tested in – Monotonicity, Lipschitzness, convexity [Berman,

Raskhodnikova, Y. ‘14]