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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

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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

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Affine-invariant Parameter

Definition A parameter π maps a function f : Fn

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

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Affine-invariant Parameter

Definition A parameter π maps a function f : Fn

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 : Fn

2 → Fn 2.

E.g.

  • # of ones divided by 2n.
  • Minimum Hamming distance to a linear function / 2n.
  • Spectral norm (= the sum of absolute Fourier coefficients) / 2n.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 2 / 20

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Parameter Estimation

Definition An algorithm is an estimator for a parameter π if, given

  • n ∈ N,
  • a query access to f : Fn

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

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Parameter Estimation

Definition An algorithm is an estimator for a parameter π if, given

  • n ∈ N,
  • a query access to f : Fn

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

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Oblivious Estimator

Definition

1 1 1 1

H f|H f

A (constant-query) oblivious estimator

  • Samples a random affine

subspace H of dimension h(ǫ).

  • Determines its output

based only on the restriction f |H.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 4 / 20

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Oblivious Estimator

Definition

1 1 1 1

H f|H f

A (constant-query) oblivious estimator

  • Samples a random affine

subspace H of dimension h(ǫ).

  • Determines its output

based only on the restriction f |H.

  • Avoid “unnatural” parameters such as π(f ) = n mod 2.
  • For natural parameters, a constant-query estimator implies an
  • blivious constant-query estimator.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 4 / 20

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Main Result

Theorem (Informal) An affine-invariant parameter π is (obliviously) constant-query estimable

  • For any function sequence (fi : Fi

2 → {0, 1})i∈N that “converges” in a

certain metric, the sequence π(fi) 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

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Applications: Property testing

Definition f : Fn

2 → {0, 1} is ǫ-far from P if,

dP(f ) := min

g∈P #{x ∈ Fn 2 | f (x) = g(x)}/2n ≥ ǫ.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 6 / 20

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Applications: Property testing

Definition f : Fn

2 → {0, 1} is ǫ-far from P if,

dP(f ) := min

g∈P #{x ∈ Fn 2 | f (x) = g(x)}/2n ≥ ǫ.

Accept w.p. 2/3 Reject w.p. 2/3 P ε-far All functions

A tester for a property P: Given

  • n ∈ N,
  • a query access

to f : Fn

2 → {0, 1}, and

  • an error parameter ǫ > 0,

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 6 / 20

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Property Testing: Characterization

Corollary (Informal) An affine-invariant property P is constant-query testable

  • For any function sequence (fi : Fi

2 → {0, 1})i∈N that “converges” in a

certain metric, the sequence dP(fi) 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

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Property Testing: Specific Properties

Corollary (Informal) Suppose that a property P satisfies:

  • Any f ∈ P is of the form

f (x) = Γ(P1(x), . . . , Pc(x), Q1(x), . . . , Qc′(x)), where Pi’s are low-degree polynomials, Qi’s are products of linear functions, c + c′ = O(1), Γ : Fc+c′

2

→ {0, 1}.

  • (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

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“Convergence” in a Certain Metric

“For any function sequence (fi : Fni

2 → {0, 1}) that

converges in a certain metric, the sequence π(fi) converges.” We have two issues:

  • Metric?

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 9 / 20

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“Convergence” in a Certain Metric

“For any function sequence (fi : Fni

2 → {0, 1}) that

converges in a certain metric, the sequence π(fi) 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

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“Convergence” in a Certain Metric

“For any function sequence (fi : Fni

2 → {0, 1}) that

converges in a certain metric, the sequence π(fi) 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

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“Convergence” in a Certain Metric

“For any function sequence (fi : Fni

2 → {0, 1}) that

converges in a certain metric, the sequence π(fi) 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

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Gowers Norm

Definition Let f : Fn

2 → R. The Gowers norm of order d for f is

f Ud :=   E

x,y1,...,yd

  • I⊆{1,...,d}

f (x +

  • i∈I

yi)  

1/2d

.

  • · Ud measures correlation with “polynomials” of degree < d.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 10 / 20

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A Metric for Functions on an Identical Domain

µf ,h: distribution of f restricted to an affine subspace of dimension h. Fact f − g ◦ AUd is small (for large d) ⇒ µf ,h ≈ µg,h.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 11 / 20

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A Metric for Functions on an Identical Domain

µf ,h: distribution of f restricted to an affine subspace of dimension h. Fact f − g ◦ AUd is small (for large d) ⇒ µf ,h ≈ µg,h. Define υd(f , g) := min

A:affine bijection f − g ◦ AUd

Fact υd(f , g) is small ⇔ µf ,h ≈ µg,h.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 11 / 20

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A Metric for Functions on an Identical Domain

Observation Constant-query estimability ⇔ small υd(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., µf ,h ≈ µg,h), then π(f ) ≈ π(g). ⇔ Small υd(f , g) implies π(f ) ≈ π(g).

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 12 / 20

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Convergence of a Function Sequence

If υd were a metric defined over functions on different domains, then “small υd(f , g) implies π(f ) ≈ π(g)” can be rephrased as “If a function sequence (fi : Fi

2 → {0, 1})i∈N converges in

the υd-metric, then π(fi) converges.”

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 13 / 20

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Convergence of a Function Sequence

If υd were a metric defined over functions on different domains, then “small υd(f , g) implies π(f ) ≈ π(g)” can be rephrased as “If a function sequence (fi : Fi

2 → {0, 1})i∈N converges in

the υd-metric, then π(fi) converges.” To make this statement meaningful, we extend υd using non-standard analysis.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 13 / 20

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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

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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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Brief Introduction to Non-standard Analysis

  • ω: a “nice” family of subsets of N. (non-principal ultrafilter)
  • Introduce an equivalence relation ∼ on number sequences,

where (ai)i∈N ∼ (bi)i∈N iff {i ∈ N | ai = bi} ∈ ω.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 15 / 20

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Brief Introduction to Non-standard Analysis

  • ω: a “nice” family of subsets of N. (non-principal ultrafilter)
  • Introduce an equivalence relation ∼ on number sequences,

where (ai)i∈N ∼ (bi)i∈N iff {i ∈ N | ai = bi} ∈ ω.

  • The ultralimit of a sequence (ai)i∈N, denoted by lim

i→ω ai, is the

equivalence class it belongs to.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 15 / 20

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Brief Introduction to Non-standard Analysis

  • ω: a “nice” family of subsets of N. (non-principal ultrafilter)
  • Introduce an equivalence relation ∼ on number sequences,

where (ai)i∈N ∼ (bi)i∈N iff {i ∈ N | ai = bi} ∈ ω.

  • The ultralimit of a sequence (ai)i∈N, denoted by lim

i→ω ai, is the

equivalence class it belongs to.

  • Most operations can be naturally lifted to ultralimits.
  • E.g. lim

i→ω ai + lim i→ω bi = lim i→ω(ai + bi).

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 15 / 20

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Brief Introduction to Non-standard Analysis

  • ω: a “nice” family of subsets of N. (non-principal ultrafilter)
  • Introduce an equivalence relation ∼ on number sequences,

where (ai)i∈N ∼ (bi)i∈N iff {i ∈ N | ai = bi} ∈ ω.

  • The ultralimit of a sequence (ai)i∈N, denoted by lim

i→ω ai, is the

equivalence class it belongs to.

  • Most operations can be naturally lifted to ultralimits.
  • E.g. lim

i→ω ai + lim i→ω bi = lim i→ω(ai + bi).

  • A first order sentence φ is true in the ultralimit world ⇔ φ is

true for ω-many i’s. ( Lo´ s’ theorem)

  • E.g. lim

i→ω ai + lim i→ω bi = lim i→ω ci ⇔ {i | ai + bi = ci} ∈ ω .

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 15 / 20

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υd-Metric over Function Limits

The function limit f of a function sequence (fi : Fi

2 → {0, 1}) is

defined as f(lim

i→ω xi) = lim i→ω fi(xi).

(Formally, we take the standard part)

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 16 / 20

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υd-Metric over Function Limits

The function limit f of a function sequence (fi : Fi

2 → {0, 1}) is

defined as f(lim

i→ω xi) = lim i→ω fi(xi).

(Formally, we take the standard part)

Definition For two function limits f, g, we define υd(f, g) := inf

A f − g ◦ AUd,

where A is over ultralimits of sequences of affine bijections.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 16 / 20

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Non-standard Analysis

Definition For a function f : Fn

2 → {0, 1}, let ∗f = the function limit of the sequence (f ◦ Ai)i∈N,

where Ai : Fi

2 → Fn 2 is an arbitrary full-rank affine transformation.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 17 / 20

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Non-standard Analysis

Definition For a function f : Fn

2 → {0, 1}, let ∗f = the function limit of the sequence (f ◦ Ai)i∈N,

where Ai : Fi

2 → Fn 2 is an arbitrary full-rank affine transformation.

Definition (fi) is υd-convergent if the sequence (∗fi) converges in the υd-metric. The choice of Ai’s is not important when discussing υd-convergence.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 17 / 20

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Main Result

Using the same idea as the identical domain case, we obtain: Theorem An affine-invariant parameter π is (obliviously) constant-query estimable

  • If a function sequence (fi : Fi

2 → {0, 1})i∈N is υd-convergent for any

d ∈ N, then the sequence π(fi) converges. Proof ingredients:

  • Tools from higher order Fourier analysis: non-classical

polynomials, decomposition theorem.

  • Another notion of convergence.

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 18 / 20

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Summary and Open Problems

  • Defined υd-metric over function limits and obtained a concise

characterization of constant-query estimable affine-invariant parameters.

  • F2 can be generalized to Fp for any prime p, and for any prime

power using recent techniques [BL15, BB15].

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 19 / 20

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Summary and Open Problems

  • Defined υd-metric over function limits and obtained a concise

characterization of constant-query estimable affine-invariant parameters.

  • F2 can be generalized to Fp for any prime p, and for any prime

power using recent techniques [BL15, BB15].

  • Can we use our characterization to show other specific

parameters are constant-query estimable?

  • Can we characterize properties that are constant-query testable

with one-sided error using function limits?

Yuichi Yoshida (NII and PFI) Gowers Norm, Function Limits, and Parameter Estimation January 12, 2016 19 / 20