| import Mathlib |
| import OAI.Probability.SKGap.Matrix.LogDetIntegral |
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|
| namespace OAI |
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|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory Real Set Filter |
| open scoped BigOperators Matrix.Norms.Frobenius NNReal Topology |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| |
| lemma regularizedLogDet_lipschitzWith [Nonempty ι] {γ : ℝ} (hγ : 0 < γ) : |
| LipschitzWith ⟨1/sqrt ((Fintype.card ι:ℝ)*γ),by positivity⟩ |
| (LogDet.regularizedLogDet γ : Matrix ι ι ℝ → ℝ) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro M N |
| change |LogDet.regularizedLogDet γ M-LogDet.regularizedLogDet γ N| ≤ |
| (1/sqrt ((Fintype.card ι:ℝ)*γ))*‖M-N‖ |
| exact LogDet.regularizedLogDet_lipschitz hγ M N |
|
|
| lemma logPath_regularized_tail [Nonempty ι] {j γ u : ℝ} (hj : 0 < j) (hγ : 0 < γ) |
| (hu : 0 ≤ u) {a : ι → ℝ} (ha : ∀ i, 0 ≤ a i) (ha1 : ∀ i, a i ≤ 1) : |
| let F := fun g => LogDet.regularizedLogDet γ (logPath ((j/(Fintype.card ι:ℝ))*∑ i,a i) a |
| (goeMatrix (j/(Fintype.card ι:ℝ)) g) 1) |
| let μ := Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1) |
| μ.real {g | u ≤ |F g-∫ x,F x ∂μ|} ≤ |
| 2*Real.exp (-γ*(Fintype.card ι:ℝ)^2*u^2/(π^2*j)) := by |
| have hn : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos |
| have hc := (regularizedLogDet_lipschitzWith hγ).comp |
| ((logPath_one_lipschitz (j := j) ha ha1).comp (goeMatrix_L2_lipschitz (j/(Fintype.card ι:ℝ)))) |
| let L : ℝ≥0 := ⟨(1/sqrt ((Fintype.card ι:ℝ)*γ))*sqrt (2*(j/(Fintype.card ι:ℝ))),by positivity⟩ |
| have hc' : LipschitzWith L (fun x : EuclideanSpace ℝ (MatrixCoordinates ι) => |
| LogDet.regularizedLogDet γ (logPath ((j/(Fintype.card ι:ℝ))*∑ i,a i) a |
| (goeMatrix (j/(Fintype.card ι:ℝ)) x.ofLp) 1)) := by |
| exact hc.weaken (by |
| change (1/sqrt ((Fintype.card ι:ℝ)*γ))*(1*sqrt (2*(j/(Fintype.card ι:ℝ)))) ≤ |
| (1/sqrt ((Fintype.card ι:ℝ)*γ))*sqrt (2*(j/(Fintype.card ι:ℝ))) |
| rw [_root_.one_mul]) |
| have hL : 0 < L := NNReal.coe_pos.mp (by |
| change 0 < (1/sqrt ((Fintype.card ι:ℝ)*γ))*sqrt (2*(j/(Fintype.card ι:ℝ))) |
| exact mul_pos (one_div_pos.mpr (sqrt_pos.mpr (mul_pos hn hγ))) |
| (sqrt_pos.mpr (mul_pos (by norm_num) (div_pos hj hn)))) |
| have h := gaussianProduct_tail_lipschitz |
| (f := fun g => LogDet.regularizedLogDet γ (logPath ((j/(Fintype.card ι:ℝ))*∑ i,a i) a |
| (goeMatrix (j/(Fintype.card ι:ℝ)) g) 1)) hc' hL hu |
| apply h.trans_eq |
| congr 2 |
| change -2*u^2/(π^2*((1/sqrt ((Fintype.card ι:ℝ)*γ))*sqrt (2*(j/(Fintype.card ι:ℝ))))^2)=_ |
| rw [mul_pow,_root_.one_div_pow,sq_sqrt (by positivity : 0 ≤ (Fintype.card ι:ℝ)*γ), |
| sq_sqrt (by positivity : 0 ≤ 2*(j/(Fintype.card ι:ℝ)))] |
| field_simp |
|
|
| lemma regularizedLogDet_transpose {γ : ℝ} (hγ : γ ≠ 0) (M : Matrix ι ι ℝ) : |
| LogDet.regularizedLogDet γ Mᵀ=LogDet.regularizedLogDet γ M := by |
| have e₁ : γ • (1+(γ⁻¹ • M)*Mᵀ)=M*Mᵀ+γ • 1 := by |
| rw [smul_add,smul_mul_assoc,smul_smul,mul_inv_cancel₀ hγ,one_smul] |
| ac_rfl |
| have e₂ : γ • (1+Mᵀ*(γ⁻¹ • M))=Mᵀ*M+γ • 1 := by |
| rw [smul_add,mul_smul_comm,smul_smul,mul_inv_cancel₀ hγ,one_smul] |
| ac_rfl |
| unfold LogDet.regularizedLogDet LogDet.regularizedGram |
| rw [transpose_transpose,← e₁,← e₂,det_smul,det_smul,det_one_add_mul_comm] |
| end SKGap |
| end |
| end |
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| end OAI |
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