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Linear point-tensor evaluations are Boolean quadratics

Lax342547.QuadraticDegree · concepts/Lax342547/QuadraticDegree.lean · lax-342547

proven

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    Natural Language Statement

    Lemma

    Actual outer-product tensors evaluated by an arbitrary linear functional have Boolean degree at most two; the checked nonvanishing bound exactifies error below one quarter.

    Concept map
    16 concepts; 7 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 3 statements. Each proof establishes one of them relative to its assumptions.

    1 linear_outer_product_degree proven

    2 quadratic_error_exactification proven

    3 quadratic_function_degree proven

    Lean source view on GitHub

    1import Lax342547.BooleanWeight
    2import Mathlib.Data.Matrix.Basis
    3
    4/-!
    5---
    6title: Linear point-tensor evaluations are Boolean quadratics
    7type: lemma
    8---
    9Actual outer-product tensors evaluated by an arbitrary linear functional have Boolean degree at most two; the checked nonvanishing bound exactifies error below one quarter.
    10-/
    11
    12namespace Lax342547.QuadraticDegree
    13
    14open Lax342547.MomentSpace Lax342547.SelectorInterpolation
    15open scoped BigOperators
    16
    17axiom quadratic_function_degree {b : ℕ} {I : Type} [Fintype I]
    18 (A : Matrix I I Binary) (z : I → (Fin b → Binary) → Binary)
    19 (hz : ∀ i, z i ∈ selectorSpan b 1) :
    20 (fun x => ∑ i, ∑ j, A i j*z i x*z j x) ∈ selectorSpan b 2
    21
    22axiom linear_outer_product_degree {b : ℕ} {I : Type} [Fintype I]
    23 (φ : Module.Dual Binary (Matrix I I Binary)) (z : I → (Fin b → Binary) → Binary)
    24 (hz : ∀ i, z i ∈ selectorSpan b 1) :
    25 (fun x => φ (Matrix.of (fun i j => z i x*z j x))) ∈ selectorSpan b 2
    26
    27axiom quadratic_error_exactification {b : ℕ} (f : (Fin b → Binary) → Binary)
    28 (hf : f ∈ selectorSpan b 2)
    29 (herr : Lax342547.RetainedImages.cellMass (fun _ : Fin b → Binary => 1/(2 : ℝ)^b)
    30 (fun x => f x ≠ 0) < 1/4) : f = 0
    31
    32end Lax342547.QuadraticDegree
    33
    Show ProofShow ProofShow Proof
    Builds on
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    From Mathlib

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