| import Mathlib |
| import OAI.Probability.LogConcave.JetEstimates.Growth |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
|
|
| namespace LogConcaveSampling |
| open MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| lemma integral_mul_sq_le {X : Type*} [MeasurableSpace X] {μ : Measure X} |
| {f g : X → ℝ} (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| (∫ x, f x*g x ∂μ)^2 ≤ (∫ x, (f x)^2 ∂μ)*(∫ x, (g x)^2 ∂μ) := by |
| have hi (f g : X → ℝ) (hf : MemLp f 2 μ) (hg : MemLp g 2 μ) : |
| inner ℝ (hf.toLp f) (hg.toLp g) = ∫ x, f x*g x ∂μ := by |
| rw [L2.inner_def] |
| apply integral_congr_ae |
| filter_upwards [hf.coeFn_toLp,hg.coeFn_toLp] with x hx hy |
| simp [hx,hy,mul_comm] |
| have hh := sq_le_sq₀ (norm_nonneg (inner ℝ (hf.toLp f) (hg.toLp g))) |
| (mul_nonneg (norm_nonneg (hf.toLp f)) (norm_nonneg (hg.toLp g))) |>.mpr |
| (norm_inner_le_norm (𝕜 := ℝ) (hf.toLp f) (hg.toLp g)) |
| rw [Real.norm_eq_abs,sq_abs,mul_pow,← real_inner_self_eq_norm_sq, |
| ← real_inner_self_eq_norm_sq,hi,hi,hi] at hh |
| simpa only [pow_two] using hh |
|
|
| namespace Appell |
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] |
| variable {ι ϑ κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype ϑ] [DecidableEq ϑ] |
| [Fintype κ] [DecidableEq κ] |
|
|
| def crossMoment (μ : Measure E) (b : OrthonormalBasis κ ℝ E) |
| (a : ι → κ) (c : ϑ → κ) : ℝ := |
| ∫ x, slotPolynomial μ b a Finset.univ x * slotPolynomial μ b c Finset.univ x ∂μ |
|
|
| omit [DecidableEq κ] in |
| lemma sum_crossMoment (μ : Measure E) [IsProbabilityMeasure μ] (hμ : HasMoments μ) |
| (b : OrthonormalBasis κ ℝ E) (T : (ι → κ) → ℝ) (U : (ϑ → κ) → ℝ) : |
| (∑ a, ∑ c, T a * crossMoment μ b a c * U c) = |
| ∫ x, tensorPolynomial μ b T x * tensorPolynomial μ b U x ∂μ := by |
| have hh (a : ι → κ) (c : ϑ → κ) : Integrable |
| (fun x => T a * slotPolynomial μ b a Finset.univ x * |
| (U c * slotPolynomial μ b c Finset.univ x)) μ := |
| ((((slotPolynomial_growth μ b a _).mul (slotPolynomial_growth μ b c _)).integrable |
| ((slotPolynomial_smooth μ b a _).continuous.mul |
| (slotPolynomial_smooth μ b c _).continuous).aestronglyMeasurable hμ).const_mul |
| (T a*U c)).congr (Filter.Eventually.of_forall (fun x => by ring)) |
| simp only [tensorPolynomial,Finset.sum_mul] |
| simp only [Finset.mul_sum] |
| rw [integral_finsetSum _ (fun a ha => integrable_finsetSum _ (fun c hc => hh a c))] |
| simp_rw [integral_finsetSum _ (fun c hc => hh _ c)] |
| apply Finset.sum_congr rfl |
| intro a ha |
| apply Finset.sum_congr rfl |
| intro c hc |
| rw [show (fun x => T a * slotPolynomial μ b a Finset.univ x * |
| (U c * slotPolynomial μ b c Finset.univ x)) = |
| (fun x => (T a*U c) * (slotPolynomial μ b a Finset.univ x * |
| slotPolynomial μ b c Finset.univ x)) by funext x; ring,integral_const_mul] |
| simp only [crossMoment] |
| ring |
|
|
| theorem crossMoment_sq_bound (μ : Measure E) [IsProbabilityMeasure μ] (hμ : HasMoments μ) |
| {β : ℝ} (hβ : 0 ≤ β) (hP : HasPoincare μ β) (b : OrthonormalBasis κ ℝ E) |
| (T : (ι → κ) → ℝ) (U : (ϑ → κ) → ℝ) : |
| (∑ a, ∑ c, T a * crossMoment μ b a c * U c)^2 ≤ |
| (β^(Fintype.card ι)*((Fintype.card ι).factorial:ℝ)^2) * |
| (β^(Fintype.card ϑ)*((Fintype.card ϑ).factorial:ℝ)^2) * |
| (∑ a, (T a)^2)*(∑ c, (U c)^2) := by |
| rw [sum_crossMoment μ hμ] |
| have hh := (integral_mul_sq_le (tensorPolynomial_memLp μ hμ b T) |
| (tensorPolynomial_memLp μ hμ b U)).trans |
| (mul_le_mul (tensorPolynomial_L2 μ hμ hβ hP b T) |
| (tensorPolynomial_L2 μ hμ hβ hP b U) |
| (integral_nonneg (fun _ => sq_nonneg _)) (by positivity)) |
| exact hh.trans_eq (by ring) |
| end Appell |
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
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