| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.MeanMaterialVector |
| import OAI.Probability.LogConcave.Numerics.LogMeshLength |
| import OAI.Probability.LogConcave.Sampling.Jet |
| import OAI.Probability.LogConcave.Numerics.BasisDerivative |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory Quadrature |
| open scoped Classical BigOperators NNReal |
|
|
| def terminalDerivativeBudget (n d : ℕ) (lam : ℝ≥0) (r : ℝ) : ℝ := |
| (r*((lam:ℝ)*r)*Real.exp (Real.pi^2/4))* |
| ((TensorSum.terminalQ n).val.momentBudget d 0 2).toReal |
|
|
| lemma terminalDerivativeBudget_nonneg (n d : ℕ) (lam : ℝ≥0) (r : ℝ) : |
| 0≤terminalDerivativeBudget n d lam r := by |
| unfold terminalDerivativeBudget |
| have h : 0≤r*((lam:ℝ)*r) := by nlinarith [sq_nonneg r,lam.coe_nonneg] |
| positivity |
|
|
| variable {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) |
|
|
| include hlam in |
| lemma terminalActionChain_sq_uniform (hd : 1≤d) (n : ℕ) {a b : ℝ} |
| (ha : 0≤a) (hb : b≤T) (t : ℝ) (ht : t∈Icc a b) : |
| let μ := (interpolationLaw F x r T).prod (stdGaussian (Point d)) |
| let B := (terminalDerivativeBudget n d lam r* |
| ((Real.sqrt (1-b^2))⁻¹)^(2*n))^2*d |
| Integrable (fun p => ‖terminalActionChain hF x hr hl hT0 hT1 (n+1) (t,p)‖^2) μ ∧ |
| (∫p,‖terminalActionChain hF x hr hl hT0 hT1 (n+1) (t,p)‖^2 ∂μ)≤B := by |
| dsimp only |
| have hh := terminalActionChain_sq hF x hr hlam hl hT0 hT1 hd n |
| ⟨t,ha.trans ht.1,ht.2.trans hb⟩ |
| refine ⟨hh.1,hh.2.trans ?_⟩ |
| apply mul_le_mul_of_nonneg_right _ (Nat.cast_nonneg _) |
| apply pow_le_pow_left₀ (by positivity) |
| apply mul_le_mul_of_nonneg_left _ (terminalDerivativeBudget_nonneg n d lam r) |
| exact pow_le_pow_left₀ (by positivity) |
| (inverse_sqrt_time_mono (ha.trans ht.1) ht.2 (hb.trans_lt hT1)) _ |
|
|
| lemma terminal_cell_budget_cancel (n : ℕ) {a b h : ℝ} (ha : 0≤a) |
| (hab : a≤b) (hb : b<1) (hh : 0≤h) (hc : b-a≤h*(1-b)) : |
| ((b-a)^(n+1))^2*((terminalDerivativeBudget (n+1) d lam r* |
| ((Real.sqrt (1-b^2))⁻¹)^(2*(n+1)))^2*d)≤ |
| (terminalDerivativeBudget (n+1) d lam r*h^(n+1))^2*d := by |
| have he := cell_singularity_cancel ha hab le_rfl hb hh hc n |
| have hA := terminalDerivativeBudget_nonneg (n+1) d lam r |
| have hs : 0≤((Real.sqrt (1-b^2))⁻¹)^(2*(n+1)) := by positivity |
| have he' : (b-a)^(n+1)*((Real.sqrt (1-b^2))⁻¹)^(2*(n+1))≤h^(n+1) := by |
| simpa only [Nat.mul_add,Nat.mul_one] using he |
| calc |
| _ = ((terminalDerivativeBudget (n+1) d lam r)* |
| ((b-a)^(n+1)*((Real.sqrt (1-b^2))⁻¹)^(2*(n+1))))^2*d := by ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (pow_le_pow_left₀ (mul_nonneg hA (mul_nonneg (pow_nonneg (sub_nonneg.mpr hab) _) hs)) |
| (mul_le_mul_of_nonneg_left he' hA) 2) (Nat.cast_nonneg _) |
|
|
| include hlam in |
|
|
| theorem terminal_derivative_cell_rms {I : Type*} [Fintype I] [DecidableEq I] |
| (u : I → ℝ) (hu : Function.Injective u) (hun : ∀i,u i∈Icc (0:ℝ) 1) |
| (n : ℕ) (hn : n+2≤Fintype.card I) (hd : 1≤d) |
| {a b h t : ℝ} (ha : 0≤a) (hab : a<b) (hb : b≤T) |
| (hh : 0≤h) (hc : b-a≤h*(1-b)) (ht : t∈Icc (0:ℝ) 1) : |
| let J := terminalActionChain hF x hr hl hT0 hT1 |
| let μ := (interpolationLaw F x r T).prod (stdGaussian (Point d)) |
| let err := fun z => J 1 (a+(b-a)*t,z)-(b-a)⁻¹ • |
| derivativeInterpolation u (fun i => J 0 (a+(b-a)*u i,z)) t |
| Integrable (fun z => ‖err z‖^2) μ ∧ |
| (∫z,‖err z‖^2 ∂μ)≤2*(1+(∑i,|basisDerivative u i t|)^2)* |
| (terminalDerivativeBudget (n+1) d lam r*h^(n+1))^2*d := by |
| dsimp only |
| let := interpolationLaw_probability hF x hr.le (by linarith) T |
| let J := terminalActionChain hF x hr hl hT0 hT1 |
| have hm (s : ℝ) (hs : s∈Icc a (a+(b-a))) := |
| terminalActionChain_sq_uniform hF x hr hlam hl hT0 hT1 hd (n+1) ha hb s |
| (by simpa only [add_sub_cancel] using hs) |
| have hv := derivative_cell_error_rms u hu hun n hn J |
| (fun k => (terminalActionChain_smooth hF x hr hl hT0 hT1 k).continuous) |
| (terminalActionChain_hasDerivAt hF x hr hl hT0 hT1) |
| (sub_pos.mpr hab) ht (by positivity) |
| (fun s hs => (hm s hs).1) (fun s hs => (hm s hs).2) |
| refine ⟨hv.1,hv.2.trans ?_⟩ |
| have he := terminal_cell_budget_cancel (d:=d) (lam:=lam) (r:=r) n ha hab.le |
| (hb.trans_lt hT1) hh hc |
| convert mul_le_mul_of_nonneg_left he (by positivity : 0≤2*(1+(∑i,|basisDerivative u i t|)^2)) using 1 <;> ring |
|
|
| include hlam in |
|
|
| theorem terminal_hermite_cell_rms {I : Type*} [Fintype I] [DecidableEq I] |
| (u : I → ℝ) (hu : Function.Injective u) (i₀ : I) (hu0 : u i₀=0) |
| (hun : ∀i,u i∈Icc (0:ℝ) 1) (n : ℕ) (hn : n+1≤Fintype.card I) (hd : 1≤d) |
| {b h : ℝ} (hb0 : 0<b) (hb : b≤T) (hh : 0≤h) (hc : b≤h*(1-b)) : |
| let J := terminalActionChain hF x hr hl hT0 hT1 |
| let μ := (interpolationLaw F x r T).prod (stdGaussian (Point d)) |
| let err := fun z => (∫t in (0:ℝ)..1,t⁻¹ • J 1 (b*t,z))- |
| ∑i,(zeroHermiteWeight u i₀ i/b) • J 0 (b*u i,z) |
| Integrable (fun z => ‖err z‖^2) μ ∧ |
| (∫z,‖err z‖^2 ∂μ)≤2*(1+(∑i,|zeroHermiteWeight u i₀ i|)^2)* |
| (terminalDerivativeBudget (n+1) d lam r*h^(n+1))^2*d := by |
| dsimp only |
| let := interpolationLaw_probability hF x hr.le (by linarith) T |
| let J := terminalActionChain hF x hr hl hT0 hT1 |
| have hm (s : ℝ) (hs : s∈Icc (0:ℝ) b) := |
| terminalActionChain_sq_uniform hF x hr hlam hl hT0 hT1 hd (n+1) le_rfl hb s hs |
| have hv := zeroHermite_derivative_error_rms u hu i₀ hu0 hun n hn J |
| (fun k => (terminalActionChain_smooth hF x hr hl hT0 hT1 k).continuous) |
| (terminalActionChain_hasDerivAt hF x hr hl hT0 hT1) |
| (terminalActionChain_first_zero hF x hr hlam hl hT0 hT1) |
| hb0 (by positivity) (fun s hs => (hm s hs).1) (fun s hs => (hm s hs).2) |
| refine ⟨hv.1,hv.2.trans ?_⟩ |
| have he := terminal_cell_budget_cancel (d:=d) (lam:=lam) (r:=r) n |
| (a:=0) le_rfl hb0.le (hb.trans_lt hT1) hh (by simpa only [sub_zero] using hc) |
| simp only [sub_zero] at he |
| convert mul_le_mul_of_nonneg_left he (by positivity : 0≤2*(1+(∑i,|zeroHermiteWeight u i₀ i|)^2)) using 1 <;> ring |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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