| import Mathlib |
| import OAI.Probability.SKRatio.Matrices.WeightedSquares |
| import OAI.Probability.SKRatio.Matrices.CompactMatrixMasks |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set Real |
| namespace SKRatio.Planted |
| open SKRatioClock.Regression MatrixNet |
|
|
| theorem weighted_square_operator_rare {X : Type*} [PseudoMetricSpace X] [CompactSpace X] |
| (β : ℝ) (G : X × X → ℝ) (hG : Continuous G) {u : ℝ} (hu : 0<u) : |
| ExponentiallyRare (law β) (fun n => {g | ∃ x : Fin n → X, |
| u < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (Matrix.of fun i j => squareError β g i j*G (x i,x j))‖}) := by |
| obtain ⟨C,hC,hrow⟩ := square_rows_bounded β |
| obtain ⟨δ,hδ,hmask⟩ := continuous_kernel_insertion G hG |
| (show 0≤C+β^2 by positivity) hu |
| apply (hrow.union (squareError_rare β hδ)).mono |
| refine Filter.Eventually.of_forall ?_ |
| rintro n g ⟨x,hx⟩ |
| by_contra hn |
| have hr : ∀ i, ∑ j, |squareError β g i j| ≤ C+β^2 := by |
| intro i |
| exact (squareError_absolute_row β g i).trans (add_le_add_left |
| (le_of_not_gt (fun hi => hn (Or.inl ⟨i,hi⟩))) _) |
| have hc : ∀ j, ∑ i, |squareError β g i j| ≤ C+β^2 := by |
| intro j |
| simpa only [squareError_symmetric β g] using hr j |
| have hd : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (squareError β g)‖ ≤ δ := |
| le_of_not_gt (fun h => hn (Or.inr h)) |
| exact (not_lt_of_ge (hmask (squareError β g) hr hc hd x)) hx |
|
|
| end SKRatio.Planted |
|
|
| end |
| end |
|
|
| end OAI |
|
|
