| import Mathlib |
| import OAI.Computability.MinUncut.Analysis.SelectedProjectGaussianLocal |
| import OAI.Computability.MinUncut.Analysis.CenteredPointCorrelationPower |
| import OAI.Computability.MinUncut.Estimates.SelectedProjectSliceContinuous |
|
|
| 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.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} |
|
|
| lemma gaussianSlice_continuous (f : FoldedProof A) (σ η : ℝ) (x : Point m n) |
| {k : ℕ} (Y : Subbox.PointedBox x k) |
| (J : Finset (Finset (Row m n) × (Point m n → ℕ))) |
| (B : FaceArray A m n) (C : Point m n → ℝ) : |
| Continuous (gaussianSlice f σ η x Y J B C) := |
| selectedProject_slice_continuous _ _ _ _ _ |
|
|
| theorem processed_zero_correlation_power (f : FoldedProof A) (σ η : ℝ) |
| (hm : 0 < m) (hn : 0 < n) {k : ℕ} (Y : ∀ x : Point m n, Subbox.PointedBox x k) |
| (J : Finset (Finset (Row m n) × (Point m n → ℕ))) |
| (B : FaceArray A m n) (C : Point m n → ℝ) {A₀ : ℝ} (hA : 0≤A₀) : |
| (∫ c, faceCorrelation (pointField (fun x => Slice.centeredClip γ A₀ |
| (gaussianSlice f σ η x (Y x) J B C)) c) ∂γpi (Point m n))^(2^m) ≤ |
| (2*A₀)^(2^m)*((m:ℝ)/n) := by |
| apply centered_point_correlation_power γ _ |
| (fun x => Slice.centeredClip_continuous γ A₀ _ (gaussianSlice_continuous f σ η x (Y x) J B C)) |
| (by positivity) |
| (fun x t => Slice.centeredClip_abs_le hA (gaussianSlice_memLp f σ η x (Y x) J B C) t) |
| (fun x => Slice.centeredClip_mean_zero hA (gaussianSlice_memLp f σ η x (Y x) J B C)) hm hn |
|
|
| lemma processed_zero_slice_loss (f : FoldedProof A) (σ η : ℝ) (x : Point m n) |
| {k : ℕ} (Y : 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) {A₀ : ℝ} (hA : 0 < A₀) : |
| (∫ t, |gaussianSlice f σ η x Y J B C t- |
| Slice.centeredClip γ A₀ (gaussianSlice f σ η x Y J B C) t| ∂γ) ≤ |
| 2*(∫ t, (gaussianSlice f σ η x Y J B C t)^2 ∂γ)/A₀ := by |
| exact Slice.centeredClip_loss hA (gaussianSlice_memLp f σ η x Y J B C) |
| (gaussianSlice_mean f σ η x Y J hJ B C) |
| end MinUncut.Inner |
|
|
| end |
|
|
| end OAI |
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|
