Testing Fourier dimensionality and sparsity Ryan ODonnell Carnegie

Testing Fourier dimensionality and sparsity Ryan O’Donnell Carnegie Mellon University joint with Parikshit Gopalan Rocco Servedio Amir Shpilka Karl Wimmer UW & Microsoft SVC Columbia Technion CMU & Duquesne

Outline 1. Testing boolean functions overview 2. Statement & proof sketch of Result 2 3. Statements of other results

Property Testing Characterizingtestability fairly well-understood for & graphs [AS 05 a, AS 05 b, AFNS 06, AT 08, …] codes [KS 07, KS 08, …]. But wide open for boolean functions.

Testable boolean function properties poly(s/ϵ) q’s

Characterization? ? Fischer’s survey [Fis 01] suggests Fourier may be key. This paper: Having concise Fourier representation is testable.

Fourier analysis of boolean functions

Fourier analysis of boolean functions

Fourier analysis of boolean functions There are 2 n linear functions, Every uniquely expressible as a real #

Fourier sparsity Def: f is s-sparse ⇔ Eg: # of nonzero is ≤ s Linear functions are 1 -sparse. k-juntas are 2 k-sparse.

Result #2 Thm: “Is s-sparse? ” is ϵ-testable with poly(s/ϵ) nonadaptive queries.

Physical space Fourier space

Fourier space

Hashing Fourier coefficients idea s 2 buckets

Hashing Fourier coefficients idea s 2 buckets

λ: subpace of codimension d

affine subpace of λ: codimension d b∈

Hashing Fourier coefficients idea 2 d = s 2 buckets (aff. subsps. ) indexed by F 2 Birthday Paradox? OK, by pairwise independence.

Projection functions

Projection functions

Projection functions b

Physical to Fourier link b ex:

Physical to Fourier link b ex: Pbf(x) = avg. of ±f on 2 d = s 2 strings rel’d to x

The Test 1. Hash to a random 2 d = s 2 buckets b. 2. For each b, 3. ϵ’-test if Pbf(x) ≡ 0, 4. Let B = {b : Pbf wasn’t ≡ 0} 5. Say NO if |B| > s for ϵ’ = poly(ϵ/s)

The Test 1. Hash to a random 2 d = s 2 buckets b. 2. For each b, 3. ϵ’-test if Pbf(x) ≡ 0, 4. Let B = {b : Pbf wasn’t ≡ 0} 5. Say NO if |B| > s 6. Say YES for ϵ’ = poly(ϵ/s)

The Test 1. Hash to a random 2 d = s 2 buckets b. 2. For each b, 3. ϵ’-test if Pbf(x) ≡ 0, for ϵ’ = poly(ϵ/s) 4. Let B = {b : Pbf wasn’t ≡ 0} 5. Say NO if |B| > s 6. Say YES poly(s/ϵ) nonadapt. queries

The Test Recall that for b ∈ B we “should” have Pbf = Analysis easier if you also do: 6. For each b ∈ B, 7. Test |Pbf| constant 8. Test sgn(Pbf) is linear, using [BLR 90]

The Test Recall that for b ∈ B we “should” have Pbf = Analysis easier if you also do: 6. For each b ∈ B, 7. Test |Pbf| constant 8. Test sgn(Pbf) is linear, using [BLR 90]

Key for analysis Granularity Theorem: If is s-sparse, then for all λ. (hence 0 or > 1/s)

Other results Def: f is k-dimensional if { } lies in a k-dimensional subspace. ⇔ f is a “k-junta of parities” Result 1: “Is f k-dimensional? ” is ϵ-testable with 2 O(k)/ϵ nonadaptive queries.

Other results Result 4: Exact, properlearning algorithm for s-sparse functions under unif. (Easy consequence of Granularity Theorem. )

Other results Result 5: “Every subclassof k-dimensional functions is testable with 2 O(k)/ϵ nonadaptive queries. ” (Uses “Testing by implicit learning” [DLMORSW 07])