openAI-math-dataset / lean /OAI /Probability /LogConcave /Complexity /KernelActionNormalizedBudget.lean
| import Mathlib | |
| import OAI.Probability.LogConcave.Sampling.ProbabilityNormalizedTerm | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Set MeasureTheory Quadrature | |
| open scoped Classical BigOperators NNReal | |
| lemma circuitD_dimension {d : ℕ} (F : Point d → ℝ) (x : Point d) : | |
| (d:ℝ)≤(circuitD F x)^2 := by | |
| have h := pow_le_pow_left₀ (Real.sqrt_nonneg (d:ℝ)) (sqrt_le_circuitD F x) 2 | |
| simpa only [Real.sq_sqrt (Nat.cast_nonneg d)] using h | |
| lemma harmonicNormalizedTerm_mono (n : ℕ) {R z Z : ℝ} (hR : 0≤R) (hz : 0≤z) (hzZ : z≤Z) : | |
| harmonicNormalizedTerm n R z≤harmonicNormalizedTerm n R Z := by | |
| unfold harmonicNormalizedTerm | |
| rw [abs_of_nonneg hz,abs_of_nonneg (hz.trans hzZ)] | |
| gcongr | |
| lemma kernelNormalizedTerm_mono (d k n N : ℕ) (lam Ap Ah Lp : ℝ≥0) | |
| (r R h : ℝ) {z Z : ℝ} (hR : 0≤R) (hz : 0≤z) (hzZ : z≤Z) : | |
| kernelNormalizedTerm d k n N lam Ap Ah Lp r R h z≤ | |
| kernelNormalizedTerm d k n N lam Ap Ah Lp r R h Z := by | |
| dsimp only [kernelNormalizedTerm] | |
| gcongr | |
| exact harmonicNormalizedTerm_mono n hR hz hzZ | |
| lemma weighted_uniform_upper {ι : Type*} [Fintype ι] (a f : ι → ℝ) {B : ℝ} | |
| (ha : ∀i,0≤a i) (hf : ∀i,f i≤B) : | |
| (∑i,a i*f i)≤(∑i,a i)*B := by | |
| rw [Finset.sum_mul] | |
| exact Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hf i) (ha i)) | |
| def kernelActionNormalizedBudget (d k n m N : ℕ) (lam Ap Ah Lp : ℝ≥0) | |
| (r R h ψ C J : ℝ) : ℝ := | |
| let S := (∑i,|derivativeWeight (angleNodes m) i|)/ψ | |
| let V := (numericalContractionCoefficient Ap Ah Lp+1)*numericalErrorCoefficient C J | |
| let B := kernelNormalizedTerm d k n N lam Ap Ah Lp r R h (ψ*m) | |
| 600*(V*S*B)^2+2*S^2*((ψ*m)^(m+1)/(m.factorial:ℝ))^2* | |
| (R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1) | |
| lemma kernelActionNormalizedBudget_nonneg (d k n m N : ℕ) (lam Ap Ah Lp : ℝ≥0) | |
| (r R h ψ C J : ℝ) (hR : 0≤R) : 0≤kernelActionNormalizedBudget d k n m N lam Ap Ah Lp r R h ψ C J := by | |
| unfold kernelActionNormalizedBudget | |
| positivity [harmonicCorrectionBudget_nonneg (m+1)] | |
| theorem kernelActionRmsBudget_normalized {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} | |
| (hF : Primitive F lam) (x : Point d) (Ap Ah Lp : ℝ≥0) {r R ρ h ψ C J : ℝ} | |
| (hr : 0≤r) (hr1 : r≤1) (hR : 0<R) (hh : 0≤h) (hψ : 0<ψ) (hC : 0≤C) (hJ : 0≤J) | |
| (hl : (lam:ℝ)*r^2≤1/2) (hL : (lam:ℝ)*r≤1) (hρ0 : 0≤ρ) (hρ1 : ρ<1) (k n m N : ℕ) : | |
| kernelActionRmsBudget F x lam Ap Ah Lp r R ρ h ψ C J k n m N ≤ | |
| (((lam:ℝ)*r^2)*circuitD F x)^2* | |
| kernelActionNormalizedBudget d k n m N lam Ap Ah Lp r R h ψ C J := by | |
| let l := (lam:ℝ)*r^2 | |
| let D := circuitD F x | |
| let S := (∑i,|derivativeWeight (angleNodes m) i|)/ψ | |
| let V := (numericalContractionCoefficient Ap Ah Lp+1)*numericalErrorCoefficient C J | |
| let B := kernelNormalizedTerm d k n N lam Ap Ah Lp r R h (ψ*m) | |
| have hl0 : 0≤l := by positivity | |
| have hD : 0≤D := circuitD_nonneg F x | |
| have hV : 0≤V := by dsimp [V]; positivity [numericalContractionCoefficient_pos Ap Ah Lp, | |
| numericalErrorCoefficient_pos C J] | |
| have hS : 0≤S := by dsimp [S]; positivity | |
| have hw : (∑i,|derivativeWeight (angleNodes m) i/ψ|)=S := by | |
| simp only [abs_div,abs_of_pos hψ,←Finset.sum_div,S] | |
| have hb (i : Fin (m+1)) : kernelValueRmsBudget F x lam Ap Ah Lp r R ρ h | |
| (ψ*angleNodes m i) C J k n N≤300*(V*l*B*D)^2 := by | |
| have hi0 : 0≤angleNodes m i := by dsimp [angleNodes]; positivity | |
| have hi1 : angleNodes m i≤(m:ℝ) := by dsimp [angleNodes]; exact_mod_cast Nat.le_of_lt_succ i.2 | |
| have hz : 0≤ψ*angleNodes m i := mul_nonneg hψ.le hi0 | |
| apply (kernelValueRmsBudget_normalized hF x Ap Ah Lp hr hr1 hR hh hz hC hJ hl hL hρ0 hρ1 k n N).trans | |
| have hmono := kernelNormalizedTerm_mono d k n N lam Ap Ah Lp r R h hR.le hz | |
| (mul_le_mul_of_nonneg_left hi1 hψ.le) | |
| have hbase : 0≤kernelNormalizedTerm d k n N lam Ap Ah Lp r R h (ψ*angleNodes m i) := by | |
| unfold kernelNormalizedTerm | |
| positivity [probabilityNormalizedTerm_nonneg d k n hh, | |
| harmonicNormalizedTerm_nonneg (z:=ψ*angleNodes m i) n hR.le, | |
| numericalContractionCoefficient_pos Ap Ah Lp] | |
| change 300*(V*l*_*D)^2≤_ | |
| gcongr | |
| have hsum := weighted_uniform_upper (fun i => |derivativeWeight (angleNodes m) i/ψ|) | |
| (fun i => kernelValueRmsBudget F x lam Ap Ah Lp r R ρ h (ψ*angleNodes m i) C J k n N) | |
| (fun i => abs_nonneg _) hb | |
| rw [hw] at hsum | |
| have hnum := mul_le_mul_of_nonneg_left hsum (show 0≤2*S by positivity) | |
| have hdim := circuitD_dimension F x | |
| have hc : 0≤harmonicCorrectionBudget (m+1) := harmonicCorrectionBudget_nonneg (m+1) | |
| have hang : 2*(S^2*(((ψ*m)^(m+1)/(m.factorial:ℝ))^2* | |
| ((d*l^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1))))≤ | |
| 2*(S^2*(((ψ*m)^(m+1)/(m.factorial:ℝ))^2* | |
| ((D^2*l^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)))) := by | |
| gcongr | |
| unfold kernelActionRmsBudget | |
| rw [hw] | |
| calc | |
| _ ≤ 2*S*(S*(300*(V*l*B*D)^2))+2*(S^2*(((ψ*m)^(m+1)/(m.factorial:ℝ))^2* | |
| ((D^2*l^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1)))) := by | |
| apply add_le_add | |
| · simpa only [mul_assoc] using hnum | |
| · exact hang | |
| _ = _ := by dsimp [kernelActionNormalizedBudget,V,B,S,l,D]; ring | |
| end LogConcaveSampling | |
| end | |
| end | |
| end OAI | |