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7d9f027 | 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 | 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
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