| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.PairGauge |
|
|
| namespace OAI |
|
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| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory |
| open scoped Classical NNReal |
|
|
| variable {d : ℕ} |
|
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| lemma probabilityFlow_growth {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (x : Point d) {r T : ℝ} (hr : 0≤r) (hr1 : r≤1) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT : 0≤T) (hT1 : T<1) (z : Point d) : |
| ‖fullProbabilityFlow hF x hr hl T z‖≤circuitGrowthConstant*(circuitD F x+‖z‖) := by |
| have hf := probabilityFlow_initial_displacement hF x hr hl z ⟨hT,hT1.le⟩ |
| have hn := norm_le_norm_sub_add (fullProbabilityFlow hF x hr hl T z) z |
| rw [field_zero_eq_gradient] at hf |
| have hD : ‖gradient F x‖≤circuitD F x := le_add_of_nonneg_left (Real.sqrt_nonneg _) |
| have hd := sqrt_le_circuitD F x |
| have hc := circuitD_nonneg F x |
| have h₁ := mul_le_mul_of_nonneg_right hr1 (norm_nonneg (gradient F x)) |
| have h₂ := mul_le_mul_of_nonneg_right (show (lam:ℝ)*r^2≤1 by linarith) |
| (show 0≤Real.pi*Real.sqrt d+‖z‖ by positivity) |
| have hp := mul_le_mul_of_nonneg_left hd Real.pi_pos.le |
| have hb : r*‖gradient F x‖+((lam:ℝ)*r^2)*(Real.pi*Real.sqrt d+‖z‖)≤ |
| (Real.pi+1)*(circuitD F x+‖z‖) := by |
| nlinarith [mul_nonneg Real.pi_pos.le (norm_nonneg z)] |
| have hh := mul_le_mul_of_nonneg_left hb terminalTransportConstant_pos.le |
| have hz : ‖z‖≤(Real.pi+1)*(circuitD F x+‖z‖) := by |
| nlinarith [mul_nonneg Real.pi_pos.le (norm_nonneg z),mul_nonneg Real.pi_pos.le hc] |
| have hc' : terminalTransportConstant+1≤8*(terminalTransportConstant+1) := by |
| linarith [terminalTransportConstant_pos] |
| have hh' := mul_le_mul_of_nonneg_right hc' (show 0≤(Real.pi+1)*(circuitD F x+‖z‖) by positivity) |
| dsimp [circuitGrowthConstant] |
| nlinarith |
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|
| lemma interpolationLaw_norm_sq {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (x : Point d) {r T : ℝ} (hr : 0≤r) (hr1 : r≤1) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT : 0≤T) (hT1 : T<1) : |
| Integrable (fun z : Point d => ‖z‖^2) (interpolationLaw F x r T) ∧ |
| (∫z : Point d,‖z‖^2 ∂interpolationLaw F x r T)≤4*circuitGrowthConstant^2*(circuitD F x)^2 := by |
| have hm := (fullProbabilityFlow_lipschitz hF x hr hl ⟨hT,hT1.le⟩).continuous.measurable |
| have hg := gaussian_growth_sq_moment (fullProbabilityFlow_lipschitz hF x hr hl ⟨hT,hT1.le⟩).continuous |
| circuitGrowthConstant_pos.le (circuitD_nonneg F x) |
| (probabilityFlow_growth hF x hr hr1 hl hT hT1) |
| rw [←fullProbabilityFlow_interpolation_closed hF x hr hl hT hT1] |
| refine ⟨(integrable_map_measure (by fun_prop) hm.aemeasurable).mpr hg.1,?_⟩ |
| rw [integral_map hm.aemeasurable (by fun_prop)] |
| apply hg.2.trans |
| have hs := pow_le_pow_left₀ (Real.sqrt_nonneg (d:ℝ)) (sqrt_le_circuitD F x) 2 |
| rw [Real.sq_sqrt (Nat.cast_nonneg d)] at hs |
| nlinarith [mul_le_mul_of_nonneg_left hs (sq_nonneg circuitGrowthConstant)] |
|
|
| lemma pairGauge_sq_bound {C : ℝ} (hC : 0≤C) (z : Point d × Point d) : |
| (pairGauge C z)^2≤4*(1+C)^2+4*‖z.1‖^2+4*‖z.2‖^2 := by |
| have hp : ‖z‖≤‖z.1‖+‖z.2‖ := by |
| change max ‖z.1‖ ‖z.2‖≤_ |
| exact max_le (le_add_of_nonneg_right (norm_nonneg _)) (le_add_of_nonneg_left (norm_nonneg _)) |
| have hq : 0≤pairGauge C z := by unfold pairGauge; positivity |
| have hb : pairGauge C z≤1+C+‖z.1‖+‖z.2‖ := by dsimp [pairGauge]; linarith |
| have hh := pow_le_pow_left₀ hq hb 2 |
| nlinarith [sq_nonneg (1+C-‖z.1‖-‖z.2‖),sq_nonneg (‖z.1‖-‖z.2‖)] |
|
|
| lemma pairGauge_moment (μ : Measure (Point d)) [IsProbabilityMeasure μ] |
| (hi : Integrable (fun z : Point d => ‖z‖^2) μ) {C : ℝ} (hC : 0≤C) : |
| Integrable (fun z => (pairGauge C z)^2) (μ.prod (stdGaussian (Point d))) ∧ |
| (∫z,(pairGauge C z)^2 ∂μ.prod (stdGaussian (Point d)))≤ |
| 4*(1+C)^2+4*(∫z,‖z‖^2 ∂μ)+4*d := by |
| have hj : Integrable (fun z : Point d => ‖z‖^2) (stdGaussian (Point d)) := by |
| simpa only [id_eq] using (IsGaussian.memLp_two_id (μ:=stdGaussian (Point d))).norm.integrable_sq |
| have hdom : Integrable (fun z : Point d × Point d => 4*(1+C)^2+4*‖z.1‖^2+4*‖z.2‖^2) |
| (μ.prod (stdGaussian (Point d))) := ((integrable_const (4*(1+C)^2)).add ((hi.comp_fst (stdGaussian (Point d))).const_mul 4)).add |
| ((hj.comp_snd μ).const_mul 4) |
| have hgi : Integrable (fun z => (pairGauge C z)^2) (μ.prod (stdGaussian (Point d))) := by |
| apply hdom.mono' ((pairGauge_measurable C).pow_const 2).aestronglyMeasurable |
| exact Filter.Eventually.of_forall (fun z => by simpa only [Real.norm_eq_abs,abs_sq,Pi.add_apply] using pairGauge_sq_bound hC z) |
| refine ⟨hgi,?_⟩ |
| apply (integral_mono hgi hdom (pairGauge_sq_bound hC)).trans_eq |
| have hf : Integrable (fun z : Point d × Point d => 4*‖z.1‖^2) (μ.prod (stdGaussian (Point d))) := (hi.comp_fst _).const_mul _ |
| have hs : Integrable (fun z : Point d × Point d => 4*‖z.2‖^2) (μ.prod (stdGaussian (Point d))) := (hj.comp_snd _).const_mul _ |
| have hl : Integrable (fun z : Point d × Point d => 4*(1+C)^2+4*‖z.1‖^2) (μ.prod (stdGaussian (Point d))) := (integrable_const _).add hf |
| rw [integral_add hl hs,integral_add (integrable_const _) hf,integral_const,probReal_univ,one_smul, |
| integral_const_mul,integral_const_mul] |
| rw [integral_fun_fst (fun z : Point d => ‖z‖^2),integral_fun_snd (fun z : Point d => ‖z‖^2)] |
| simp only [probReal_univ,one_smul,stdGaussian_sq_norm] |
|
|
| lemma pairGauge_interpolation_moment {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| (x : Point d) {r T : ℝ} (hr : 0≤r) (hr1 : r≤1) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT : 0≤T) (hT1 : T<1) (hd : 1≤d) : |
| Integrable (fun z => (pairGauge (circuitD F x) z)^2) |
| ((interpolationLaw F x r T).prod (stdGaussian (Point d))) ∧ |
| (∫z,(pairGauge (circuitD F x) z)^2 ∂(interpolationLaw F x r T).prod (stdGaussian (Point d)))≤ |
| (20+16*circuitGrowthConstant^2)*(circuitD F x)^2 := by |
| let := interpolationLaw_probability hF x hr (by linarith only [hl]) T |
| have hm := interpolationLaw_norm_sq hF x hr hr1 hl hT hT1 |
| have hg := pairGauge_moment _ hm.1 (circuitD_nonneg F x) |
| refine ⟨hg.1,hg.2.trans ?_⟩ |
| have hs : (1:ℝ)≤Real.sqrt d := by |
| exact (by norm_num : (1:ℝ)=Real.sqrt 1).trans_le (Real.sqrt_le_sqrt (by exact_mod_cast hd)) |
| have hD := hs.trans (sqrt_le_circuitD F x) |
| have hd2 := pow_le_pow_left₀ (Real.sqrt_nonneg (d:ℝ)) (sqrt_le_circuitD F x) 2 |
| rw [Real.sq_sqrt (Nat.cast_nonneg d)] at hd2 |
| have hs2 := pow_le_pow_left₀ (show 0≤1+circuitD F x by positivity [circuitD_nonneg F x]) |
| (show 1+circuitD F x≤2*circuitD F x by linarith) 2 |
| nlinarith [hm.2] |
|
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| end LogConcaveSampling |
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| end |
|
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| end |
|
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| end OAI |
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