| import Mathlib |
| import OAI.Probability.LogConcave.OraclePrograms.CircuitD |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open MeasureTheory ProbabilityTheory |
| open scoped Classical NNReal |
|
|
| variable {d : ℕ} |
|
|
| def anchorGauge (C : ℝ) (z : Point d) : ℝ := 1+C+‖z‖ |
|
|
| lemma anchorGauge_measurable (C : ℝ) : Measurable (anchorGauge (d:=d) C) := by |
| unfold anchorGauge; fun_prop |
| lemma one_le_anchorGauge {C : ℝ} (hC : 0≤C) (z : Point d) : 1≤anchorGauge C z := by |
| unfold anchorGauge; linarith [norm_nonneg z] |
| lemma anchorGauge_change (C : ℝ) (x y : Point d) : |
| anchorGauge C y≤anchorGauge C x+dist x y := by |
| have hh := norm_le_norm_add_norm_sub x y |
| dsimp [anchorGauge] |
| rw [dist_eq_norm] |
| linarith |
| lemma anchorGauge_gaussian_integrable (C : ℝ) : |
| Integrable (fun z : Point d => (anchorGauge C z)^2) (stdGaussian (Point d)) := by |
| have h := (memLp_const (1+C) (p:=2) (μ:=stdGaussian (Point d))).add |
| (IsGaussian.memLp_two_id (μ:=stdGaussian (Point d))).norm |
| exact h.integrable_sq |
| lemma anchorGauge_gaussian_moment (C : ℝ) : |
| (∫z : Point d,(anchorGauge C z)^2 ∂stdGaussian (Point d))≤2*(1+C)^2+2*d := by |
| have hi2 : 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 hi : Integrable (fun z : Point d => 2*(1+C)^2+2*‖z‖^2) (stdGaussian (Point d)) := |
| (integrable_const _).add (hi2.const_mul _) |
| apply (integral_mono (anchorGauge_gaussian_integrable C) hi (fun z => ?_)).trans_eq |
| (show (∫z : Point d,2*(1+C)^2+2*‖z‖^2 ∂stdGaussian (Point d))=2*(1+C)^2+2*d by |
| rw [integral_add (integrable_const _) (hi2.const_mul _), |
| integral_const,probReal_univ,one_smul,integral_const_mul,stdGaussian_sq_norm]) |
| dsimp [anchorGauge] |
| nlinarith [sq_nonneg (1+C-‖z‖)] |
| lemma anchorGauge_circuitD_moment (hd : 1≤d) (F : Point d → ℝ) (x : Point d) : |
| (∫z : Point d,(anchorGauge (circuitD F x) z)^2 ∂stdGaussian (Point d))≤10*(circuitD F x)^2 := by |
| apply (anchorGauge_gaussian_moment _).trans |
| have hs := Real.sq_sqrt (Nat.cast_nonneg d : (0:ℝ)≤d) |
| have hdim : (1:ℝ)≤d := by exact_mod_cast hd |
| have hD := sqrt_le_circuitD F x |
| have h0 := circuitD_nonneg F x |
| have hr0 := Real.sqrt_nonneg (d:ℝ) |
| have hr1 : 1≤Real.sqrt (d:ℝ) := by nlinarith |
| have h1 : 1≤circuitD F x := hr1.trans hD |
| have hsq := pow_le_pow_left₀ hr0 hD 2 |
| nlinarith [sq_nonneg (circuitD F x-1)] |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
