| import Mathlib |
| import OAI.Probability.SKGap.Matrix.LogDetIntegral |
| import OAI.Probability.SKGap.Stability.LogDetMean |
| import OAI.Probability.SKGap.Localization.BoundedIntervalFubini |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory Real Set Filter |
| open RealComplex |
| open scoped BigOperators Matrix.Norms.Frobenius SchwartzMap Topology |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
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| lemma rare_logdet_coefficients {n j γ : ℝ} (hn : 0 < n) (hj : 0 ≤ j) (hγ : 0 < γ) : |
| sqrt n*(1+j)/sqrt (n*γ)=(1+j)/sqrt γ ∧ |
| n^2*sqrt (2*j/n)/sqrt (n*γ)=n*sqrt (2*j)/sqrt γ := by |
| have hs : 0 < sqrt n := sqrt_pos.mpr hn |
| have hg : 0 < sqrt γ := sqrt_pos.mpr hγ |
| rw [sqrt_mul hn.le,sqrt_div (by positivity : 0 ≤ 2*j)] |
| constructor |
| · field_simp |
| · field_simp |
| rw [sq_sqrt hn.le] |
| ring |
|
|
| def logdetMeanError (j R B C rate γ : ℝ) : ℝ := |
| 2*(2*j+1)*C+(|log γ|+(1+j)/sqrt γ+2*((2*j+R)*B))*(3/rate)+ |
| (sqrt (2*j)/sqrt γ)*(8*sqrt 3/rate^2) |
|
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| lemma logdetMeanError_nonneg {j R B C rate γ : ℝ} (hj : 0 ≤ j) (hR : 0 ≤ R) |
| (hB : 0 ≤ B) (hC : 0 ≤ C) (hrate : 0 < rate) (hγ : 0 < γ) : |
| 0 ≤ logdetMeanError j R B C rate γ := by unfold logdetMeanError; positivity |
|
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| lemma logPath_regularized_expected [Nonempty ι] (f : 𝓢(ℝ,ℂ)) {R lo hi j C γ rate : ℝ} |
| (hR : 0 ≤ R) (hlo : 0 < lo) (hf : ∀ x ∈ Icc lo hi, f x=(x:ℂ)⁻¹) |
| (hj : 0 ≤ j) (hC : 0 ≤ C) (hγ : 0 < γ) (hrate : 0 < rate) |
| {a : ι → ℝ} (ha : ∀ i, 0 ≤ a i) (ha1 : ∀ i, a i ≤ 1) |
| (hK : ∀ z ∈ Icc (0:ℝ) 1, ∀ i, |pathExpected f R hR j a z i-1| ≤ C/(Fintype.card ι:ℝ)) |
| (hWK : ∀ z ∈ Icc (0:ℝ) 1, ∀ i, |
| |(∫ g, (realProject R hR (goeMatrix (j/(Fintype.card ι:ℝ)) g)* |
| pathK f R hR j a z (goeMatrix (j/(Fintype.card ι:ℝ)) g)) i i |
| ∂Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1))- |
| z*((j/(Fintype.card ι:ℝ))*∑ b,a b)| ≤ C/(Fintype.card ι:ℝ)) |
| (hp : (Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1)).real |
| {g | goeMatrix (j/(Fintype.card ι:ℝ)) g ∉ logPathGood R lo hi ((j/(Fintype.card ι:ℝ))*∑ b,a b) a} ≤ |
| 3*Real.exp (-rate*(Fintype.card ι:ℝ))) : |
| (∫ g, LogDet.regularizedLogDet γ (logPath ((j/(Fintype.card ι:ℝ))*∑ b,a b) a |
| (goeMatrix (j/(Fintype.card ι:ℝ)) g) 1) |
| ∂Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1)) ≤ |
| j*((∑ i,a i)/(Fintype.card ι:ℝ))^2+γ*(pathBound f R j 1)^2+ |
| logdetMeanError j R (pathBound f R j 1) C rate γ/(Fintype.card ι:ℝ) := by |
| let μ := Measure.pi (fun _ : MatrixCoordinates ι => gaussianReal 0 1) |
| let r := j/(Fintype.card ι:ℝ) |
| let q := r*∑ i,a i |
| let B := pathBound f R j 1 |
| let D := (2*j+R)*B |
| let T := fun p : ℝ × (MatrixCoordinates ι → ℝ) => logPathCutoffDerivative f R hR j a (goeMatrix r p.2) p.1 |
| let I := fun g => ∫ z in (0:ℝ)..1, T (z,g) |
| let F := fun g => LogDet.regularizedLogDet γ (logPath q a (goeMatrix r g) 1) |
| let s := {g | goeMatrix r g ∉ logPathGood R lo hi q a} |
| have hn : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos |
| have hn1 : (1:ℝ) ≤ Fintype.card ι := by exact_mod_cast Fintype.card_pos |
| have hB : 0 ≤ B := (path_constants_nonneg f hR hj (by norm_num : (0:ℝ)≤1)).1 |
| have hD : 0 ≤ D := by dsimp [D]; positivity |
| have ht : Continuous T := (continuous_logPathCutoffDerivative f hR j a).comp |
| (continuous_fst.prodMk ((goeMatrix_pi_lipschitz r).continuous.comp continuous_snd)) |
| have htb (z : ℝ) (hz : z ∈ Icc (0:ℝ) 1) (g) : |T (z,g)| ≤ D := |
| logPathCutoffDerivative_bound f hR hj ha ha1 hz _ |
| have htm (z : ℝ) (hz : z ∈ Icc (0:ℝ) 1) : |
| |(∫ g, T (z,g) ∂μ)-z*(j*((∑ i,a i)/(Fintype.card ι:ℝ))^2)| ≤ (2*j+1)*C/(Fintype.card ι:ℝ) := by |
| simpa only [mul_assoc] using logPathCutoffDerivative_mean_error f hR hj hC ha ha1 hz (hK z hz) (hWK z hz) |
| obtain ⟨hI,hIm⟩ := bounded_interval_mean (μ := μ) ht htb htm |
| have hIb (g) : |I g| ≤ D := by |
| have hh := intervalIntegral.norm_integral_le_of_norm_le_const (a := (0:ℝ)) (b := 1) |
| (f := fun z => T (z,g)) (C := D) (fun z hz => by |
| have hz' : z ∈ Ioc (0:ℝ) 1 := by simpa only [uIoc_of_le (by norm_num : (0:ℝ)≤1)] using hz |
| simpa only [Real.norm_eq_abs] using htb z ⟨hz'.1.le,hz'.2⟩ g) |
| simpa only [I,Real.norm_eq_abs,sub_zero,abs_one,mul_one] using hh |
| have hs : MeasurableSet s := ((logPathGood_isClosed R lo hi q a).preimage (goeMatrix_pi_lipschitz r).continuous).measurableSet.compl |
| have hF : Integrable F μ := logPath_regularized_integrable hγ ha ha1 |
| have hgood (g) (hg : g ∉ s) : F g ≤ 2*I g+γ*B^2 := by |
| have hg' : goeMatrix r g ∈ logPathGood R lo hi q a := by simpa only [s,mem_ofPred_eq,not_not] using hg |
| have hsymm : (goeMatrix r g)ᵀ=goeMatrix r g := goeMatrix_transpose r g |
| have hev := logdetPath_integral f hlo hf hR j ha (goeMatrix r g) hsymm hg' |
| have hu := pathK_agrees_at f hlo hf hR j ha (by norm_num : (0:ℝ)≤1) (goeMatrix r g) hsymm hg'.1 |
| (hg'.2 1 (by constructor <;> norm_num)).1 (hg'.2 1 (by constructor <;> norm_num)).2 |
| have hb : opNorm (logPath q a (goeMatrix r g) 1)⁻¹ ≤ B := by |
| rw [← hu.2] |
| exact (pathK_bounds f hR hj (by norm_num : (0:ℝ)≤1) ha ha1 (by constructor <;> norm_num)).1 _ |
| have hh := LogDet.regularizedLogDet_le_logdet_opNorm hγ.le _ hu.1 hb |
| change F g ≤ 2*I g+γ*B^2 |
| dsimp only [I,T] |
| rw [hev] |
| exact hh.trans_eq (by ring) |
| have hex := expectation_upper_good hs hF hI (mul_nonneg hγ.le (sq_nonneg B)) hIb hgood |
| have hrare := logPath_regularized_rare hj hγ ha ha1 hs |
| obtain ⟨hc₁,hc₂⟩ := rare_logdet_coefficients hn hj hγ |
| rw [hc₁,hc₂] at hrare |
| obtain ⟨hp₁,_,hp₃⟩ := exponential_tail_absorptions hrate hn1 hp |
| have hsmall : (|log γ|+(1+j)/sqrt γ+2*D)*μ.real s+ |
| (sqrt (2*j)/sqrt γ)*((Fintype.card ι:ℝ)*sqrt (μ.real s)) ≤ |
| (|log γ|+(1+j)/sqrt γ+2*D)*((3/rate)/(Fintype.card ι:ℝ))+ |
| (sqrt (2*j)/sqrt γ)*((8*sqrt 3/rate^2)/(Fintype.card ι:ℝ)) := by |
| exact add_le_add (mul_le_mul_of_nonneg_left hp₁ (by positivity)) |
| (mul_le_mul_of_nonneg_left hp₃ (by positivity)) |
| have hib := (abs_le.mp hIm).2 |
| dsimp only [F,I,T,q,r,μ,B,D,logdetMeanError] at hex hrare hib hsmall ⊢ |
| simp only [div_eq_mul_inv] at hex hrare hib hsmall ⊢ |
| nlinarith only [hex,hrare,hib,hsmall] |
| end SKGap |
| end |
| end |
|
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| end OAI |
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