Gowers Norm, Function Limits, and Parameter Estimation Yuichi Yoshida National Institute of Informatics, and Preferred Infrastructure, Inc. January 12, 2016 Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 1 / 20
Affine-invariant Parameter Definition A parameter π maps a function f : F n 2 → { 0 , 1 } to a value in [0 , 1]. Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 2 / 20
Affine-invariant Parameter Definition A parameter π maps a function f : F n 2 → { 0 , 1 } to a value in [0 , 1]. Definition A parameter π is affine-invariant if π ( f ) = π ( f ◦ A ) for any bijective affine transformation A : F n 2 → F n 2 . E.g. • # of ones divided by 2 n . • Minimum Hamming distance to a linear function / 2 n . • Spectral norm (= the sum of absolute Fourier coefficients) / 2 n . Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 2 / 20
Parameter Estimation Definition An algorithm is an estimator for a parameter π if, given • n ∈ N , • a query access to f : F n 2 → { 0 , 1 } , and • an error parameter ǫ > 0, it approximates π ( f ) to within ǫ with probability at least 2 / 3. Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 3 / 20
Parameter Estimation Definition An algorithm is an estimator for a parameter π if, given • n ∈ N , • a query access to f : F n 2 → { 0 , 1 } , and • an error parameter ǫ > 0, it approximates π ( f ) to within ǫ with probability at least 2 / 3. Definition π is constant-query estimable if there is an estimator with query complexity that depends only on ǫ (and not on n ). Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 3 / 20
Oblivious Estimator Definition A (constant-query) oblivious estimator f 0 1 • Samples a random affine 1 1 f | H subspace H of dimension h ( ǫ ). H • Determines its output 0 1 based only on the restriction f | H . 0 0 Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 4 / 20
Oblivious Estimator Definition A (constant-query) oblivious estimator f 0 1 • Samples a random affine 1 1 f | H subspace H of dimension h ( ǫ ). H • Determines its output 0 1 based only on the restriction f | H . 0 0 • Avoid “unnatural” parameters such as π ( f ) = n mod 2. • For natural parameters, a constant-query estimator implies an oblivious constant-query estimator. Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 4 / 20
Main Result Theorem (Informal) An affine-invariant parameter π is (obliviously) constant-query estimable � For any function sequence ( f i : F i 2 → { 0 , 1 } ) i ∈ N that “converges” in a certain metric, the sequence π ( f i ) converges. Related work: • A similar characterization for (dense) graphs [LS06]. Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 5 / 20
Applications: Property testing Definition f : F n 2 → { 0 , 1 } is ǫ -far from P if, 2 | f ( x ) � = g ( x ) } / 2 n ≥ ǫ. g ∈P # { x ∈ F n d P ( f ) := min Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 6 / 20
Applications: Property testing Definition f : F n 2 → { 0 , 1 } is ǫ -far from P if, 2 | f ( x ) � = g ( x ) } / 2 n ≥ ǫ. g ∈P # { x ∈ F n d P ( f ) := min A tester for a property P : All functions Given P Accept w.p. 2/3 • n ∈ N , • a query access to f : F n 2 → { 0 , 1 } , and • an error parameter ǫ > 0, ε -far Reject w.p. 2/3 Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 6 / 20
Property Testing: Characterization Corollary (Informal) An affine-invariant property P is constant-query testable � For any function sequence ( f i : F i 2 → { 0 , 1 } ) i ∈ N that “converges” in a certain metric, the sequence d P ( f i ) converges. Simplified a previous characterization [Yos14], which involves many quantifiers and objects with seven parameters (regularity-instances). Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 7 / 20
Property Testing: Specific Properties Corollary (Informal) Suppose that a property P satisfies: • Any f ∈ P is of the form f ( x ) = Γ( P 1 ( x ) , . . . , P c ( x ) , Q 1 ( x ) , . . . , Q c ′ ( x )) , where P i ’s are low-degree polynomials, Q i ’s are products of linear functions, c + c ′ = O (1) , Γ : F c + c ′ → { 0 , 1 } . 2 • (A minor condition) Then, P is obliviously constant-query testable. Includes low-degree polynomials and having small spectral norm. Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 8 / 20
“Convergence” in a Certain Metric “For any function sequence ( f i : F n i 2 → { 0 , 1 } ) that converges in a certain metric, the sequence π ( f i ) converges.” We have two issues: • Metric? Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 9 / 20
“Convergence” in a Certain Metric “For any function sequence ( f i : F n i 2 → { 0 , 1 } ) that converges in a certain metric, the sequence π ( f i ) converges.” We have two issues: • Metric? ⇒ Gowers norm Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 9 / 20
“Convergence” in a Certain Metric “For any function sequence ( f i : F n i 2 → { 0 , 1 } ) that converges in a certain metric, the sequence π ( f i ) converges.” We have two issues: • Metric? ⇒ Gowers norm • Convergence of functions on different domains? Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 9 / 20
“Convergence” in a Certain Metric “For any function sequence ( f i : F n i 2 → { 0 , 1 } ) that converges in a certain metric, the sequence π ( f i ) converges.” We have two issues: • Metric? ⇒ Gowers norm • Convergence of functions on different domains? ⇒ Non-standard analysis Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 9 / 20
Gowers Norm Definition Let f : F n 2 → R . The Gowers norm of order d for f is 1 / 2 d � � � f � U d := f ( x + y i ) . E x , y 1 ,..., y d I ⊆{ 1 ,..., d } i ∈ I • � · � U d measures correlation with “polynomials” of degree < d . Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 10 / 20
A Metric for Functions on an Identical Domain µ f , h : distribution of f restricted to an affine subspace of dimension h . Fact � f − g ◦ A � U d is small (for large d) ⇒ µ f , h ≈ µ g , h . Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 11 / 20
A Metric for Functions on an Identical Domain µ f , h : distribution of f restricted to an affine subspace of dimension h . Fact � f − g ◦ A � U d is small (for large d) ⇒ µ f , h ≈ µ g , h . Define υ d ( f , g ) := A :affine bijection � f − g ◦ A � U d min Fact υ d ( f , g ) is small ⇔ µ f , h ≈ µ g , h . Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 11 / 20
A Metric for Functions on an Identical Domain Observation Constant-query estimability ⇔ small υ d ( f , g ) implies π ( f ) ≈ π ( g ) . Proof sketch. π is constant-query estimable. ⇔ If f and g are indistinguishable by a constant-query estimator (i.e., µ f , h ≈ µ g , h ), then π ( f ) ≈ π ( g ). ⇔ Small υ d ( f , g ) implies π ( f ) ≈ π ( g ). Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 12 / 20
Convergence of a Function Sequence If υ d were a metric defined over functions on different domains, then “small υ d ( f , g ) implies π ( f ) ≈ π ( g ) ” can be rephrased as “If a function sequence ( f i : F i 2 → { 0 , 1 } ) i ∈ N converges in the υ d -metric, then π ( f i ) converges.” Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 13 / 20
Convergence of a Function Sequence If υ d were a metric defined over functions on different domains, then “small υ d ( f , g ) implies π ( f ) ≈ π ( g ) ” can be rephrased as “If a function sequence ( f i : F i 2 → { 0 , 1 } ) i ∈ N converges in the υ d -metric, then π ( f i ) converges.” To make this statement meaningful, we extend υ d using non-standard analysis . Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 13 / 20
Brief Introduction to Non-standard Analysis Non-standard analysis allows us to syntactically define a limit of any sequence (even if there’s no metric). Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 14 / 20
Brief Introduction to Non-standard Analysis • ω : a “nice” family of subsets of N . ( non-principal ultrafilter ) Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 15 / 20
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