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ea76aac | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Computability.MinUncut.Estimates.Away
namespace OAI
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter
open scoped Topology NNReal
open scoped BigOperators
open MeasureTheory ProbabilityTheory Polynomial Filter
open scoped BigOperators Topology
open MeasureTheory ProbabilityTheory WithLp
open scoped BigOperators RealInnerProductSpace
open scoped BigOperators
namespace MinUncut.RowNoise
open MeasureTheory ProbabilityTheory BinaryFourier GaussianHermite
open scoped BigOperators
variable {R W ι : Type*} [Fintype R] [DecidableEq R] [Fintype W] [DecidableEq W]
[AddCommGroup W] [Module F₂ W] [Fintype ι] [DecidableEq ι]
omit [AddCommGroup W] [Module F₂ W] in
lemma selectedProject_gaussian_local (J : Finset (Finset R × (ι → ℕ)))
(u : (R → W) → (ι → ℝ) → ℝ) (x : ι) (F : Finset ι)
(hJ : ∀ j ∈ J, ∀ y ∈ F, y≠x → j.2 y=0) (B : R → W)
(c C : ι → ℝ) (hC : ∀ y ∉ F, c y=C y) :
selectedProject J u B c=selectedProject J u B (Function.update C x (c x)) := by
unfold selectedProject
apply Finset.sum_congr rfl
intro j hj
simp only [maskHermite_eq,psi_localized j.2 x F (hJ j hj) c C hC]
omit [AddCommGroup W] [Module F₂ W] in
lemma selectedProject_gaussian_memLp (J : Finset (Finset R × (ι → ℕ)))
(u : (R → W) → (ι → ℝ) → ℝ) (x : ι) (B : R → W) (C : ι → ℝ) :
MemLp (fun t => selectedProject J u B (Function.update C x t)) 2 γ := by
unfold selectedProject
apply memLp_finsetSum
intro j _
simp only [maskHermite_eq]
exact (memLp_psi_update j.2 x C).mul_const _
omit [AddCommGroup W] [Module F₂ W] in
lemma selectedProject_gaussian_mean (J : Finset (Finset R × (ι → ℕ)))
(u : (R → W) → (ι → ℝ) → ℝ) (x : ι) (hJ : ∀ j ∈ J, 0<j.2 x)
(B : R → W) (C : ι → ℝ) :
(∫ t, selectedProject J u B (Function.update C x t) ∂γ)=0 := by
unfold selectedProject
rw [integral_finsetSum]
· apply Finset.sum_eq_zero
intro j hj
simp only [maskHermite_eq,integral_mul_const,integral_psi_update j.2 x C (hJ j hj),zero_mul]
· intro j _
simp only [maskHermite_eq]
exact ((memLp_psi_update j.2 x C).mul_const _).integrable (by norm_num)
end MinUncut.RowNoise
namespace MinUncut.Inner
open MeasureTheory ProbabilityTheory BinaryFourier GaussianHermite RowNoise
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A]
variable {m n : ℕ}
def freeGaussianPoints (x : Point m n) {k : ℕ} (H : Subbox.PointedBox x k) :
Finset (Point m n) := Finset.univ.filter (Subbox.InBox H)
lemma retained_zero_positive (x : Point m n) {k : ℕ} (H : Subbox.PointedBox x k)
(J : Finset (Finset (Row m n) × (Point m n → ℕ)))
(hJ : ∀ j ∈ J, j.1.card < (m-(∑ u, j.2 u)).choose 2)
(j : Finset (Row m n) × (Point m n → ℕ)) (hj : j∈retainedClass x H none J) :
0<j.2 x := by
have ht := Finset.mem_filter.mp hj
have hc := Finset.mem_filter.mp ht.1
exact (maskClass_none x j.2 j.1 (hJ j hc.1)).mp hc.2
lemma retained_gaussian_local (x : Point m n) {k : ℕ} (H : Subbox.PointedBox x k)
(a : Option (PairClass m)) (J : Finset (Finset (Row m n) × (Point m n → ℕ)))
(j : Finset (Row m n) × (Point m n → ℕ)) (hj : j∈retainedClass x H a J)
(y : Point m n) (hy : y∈freeGaussianPoints x H) (hne : y≠x) : j.2 y=0 := by
apply Nat.eq_zero_of_not_pos
intro hp
exact (Finset.mem_filter.mp hj).2 (Or.inl ⟨y, by simpa [hermiteSupport] using hp,
hne,(Finset.mem_filter.mp hy).2⟩)
def gaussianSlice (f : FoldedProof A) (σ η : ℝ) (x : Point m n)
{k : ℕ} (H : Subbox.PointedBox x k)
(J : Finset (Finset (Row m n) × (Point m n → ℕ)))
(B : FaceArray A m n) (C : Point m n → ℝ) (t : ℝ) : ℝ :=
selectedProject (retainedClass x H none J) (fun B c => gradient f B σ η c x)
B (Function.update C x t)
lemma retained_gaussian_slice (f : FoldedProof A) (σ η : ℝ) (x : Point m n)
{k : ℕ} (H : Subbox.PointedBox x k)
(J : Finset (Finset (Row m n) × (Point m n → ℕ)))
(B : FaceArray A m n) (c C : Point m n → ℝ)
(hC : ∀ y ∉ freeGaussianPoints x H, c y=C y) :
selectedProject (retainedClass x H none J) (fun B c => gradient f B σ η c x) B c =
gaussianSlice f σ η x H J B C (c x) :=
selectedProject_gaussian_local _ _ x _ (retained_gaussian_local x H none J) B c C hC
lemma gaussianSlice_memLp (f : FoldedProof A) (σ η : ℝ) (x : Point m n)
{k : ℕ} (H : Subbox.PointedBox x k)
(J : Finset (Finset (Row m n) × (Point m n → ℕ)))
(B : FaceArray A m n) (C : Point m n → ℝ) :
MemLp (gaussianSlice f σ η x H J B C) 2 γ :=
selectedProject_gaussian_memLp _ _ x B C
lemma gaussianSlice_mean (f : FoldedProof A) (σ η : ℝ) (x : Point m n)
{k : ℕ} (H : Subbox.PointedBox x k)
(J : Finset (Finset (Row m n) × (Point m n → ℕ)))
(hJ : ∀ j ∈ J, j.1.card < (m-(∑ u, j.2 u)).choose 2)
(B : FaceArray A m n) (C : Point m n → ℝ) :
(∫ t, gaussianSlice f σ η x H J B C t ∂γ)=0 :=
selectedProject_gaussian_mean _ _ x (retained_zero_positive x H J hJ) B C
end MinUncut.Inner
end
end OAI
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