| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.ScaledMeanJet |
| import OAI.Probability.LogConcave.JetEstimates.GadjScalar |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Filter |
| open scoped Topology Classical BigOperators NNReal RealInnerProductSpace |
|
|
| def jointU {d : ℕ} {κ : Type*} (F : Point d → ℝ) (x : Point d) (r L : ℝ) |
| (v : κ → Point d) (l : List κ) (i : Fin d) (p : ℝ × Point d) : ℝ := |
| conditionalU F x r p.1 p.2 (EuclideanSpace.basisFun (Fin d) ℝ i) L v l |
|
|
| lemma jointU_eventually_scaled {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) |
| {κ : Type*} (v : κ → Point d) (l : List κ) (i : Fin d) (y : Point d) : |
| jointU F x r ((lam:ℝ)*r) v l i=ᶠ[nhds (ρ,y)]scaledMeanJet F x r v l i := by |
| have hn : ∀ᶠp : ℝ × Point d in nhds (ρ,y),p.1∈Set.Ioo (0:ℝ) 1 := |
| continuousAt_fst.eventually (Ioo_mem_nhds hρ0 hρ1) |
| filter_upwards [hn] with p hp |
| exact (scaledMeanJet_eq_U hF x hr hlam hl hp.1 hp.2 v l i p.2).symm |
|
|
| lemma jointU_smooth_pos {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) |
| {κ : Type*} (v : κ → Point d) (l : List κ) (i : Fin d) (y : Point d) : |
| ContDiffAt ℝ (⊤:ℕ∞) (jointU F x r ((lam:ℝ)*r) v l i) (ρ,y) := by |
| apply ContDiffAt.congr_of_eventuallyEq _ (jointU_eventually_scaled hF x hr hlam hl hρ0 hρ1 v l i y) |
| unfold scaledMeanJet |
| exact ((contDiffAt_fst.pow _).inv (pow_ne_zero _ hρ0.ne')).mul |
| (JetCalculus.smooth_jet_at (jointMean_smooth_at hF x hr.le hl (by nlinarith) i) _ l) |
|
|
| lemma split_power_cancel {κ : Type*} [DecidableEq κ] (l : List κ) (s : Finset κ) |
| {ρ : ℝ} (hρ : ρ≠0) : |
| (ρ^l.length)⁻¹*ρ^(l.filter (fun j => j∈s)).length* |
| ρ^((l.filter (fun j => j∉s)).length+1)=ρ := by |
| have hl : l.length=(l.filter (fun j => j∈s)).length+(l.filter (fun j => j∉s)).length := by |
| simpa only [←decide_not] using l.length_eq_length_filter_add (fun j => decide (j∈s)) |
| rw [hl,pow_add,pow_succ] |
| field_simp |
|
|
| theorem material_conditionalU_pos {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0<r) |
| (hlam : 0<lam) (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0<ρ) (hρ1 : ρ<1) |
| {κ : Type*} [DecidableEq κ] (v : κ → Point d) (l : List κ) (hlN : l.Nodup) |
| (i : Fin d) (y : Point d) : |
| JetCalculus.mdir (1,0) jointSpace r (jointMean F x r) |
| (jointU F x r ((lam:ℝ)*r) v l i) (ρ,y)= |
| JetCalculus.gadj (JetCalculus.spaceBasis d) (fun k z => jointScore F x r k (ρ,z)) |
| (fun k z => jointU F x r ((lam:ℝ)*r) |
| (Sum.elim v (fun _ : Unit => JetCalculus.spaceBasis d k)) (Sum.inr ()::l.map Sum.inl) i (ρ,z)) y+ |
| 2*r*ρ*∑s∈l.toFinset.powerset.erase ∅,∑k, |
| jointU F x r ((lam:ℝ)*r) v (l.filter (fun j => j∈s)) k (ρ,y)* |
| jointU F x r ((lam:ℝ)*r) (Sum.elim v (fun _ : Unit => JetCalculus.spaceBasis d k)) |
| (Sum.inr ()::(l.filter (fun j => j∉s)).map Sum.inl) i (ρ,y) := by |
| have he := (JetCalculus.mdir_eventuallyEq (1,0) jointSpace r (M:=jointMean F x r) (fun _ => EventuallyEq.rfl) |
| (jointU_eventually_scaled hF x hr hlam hl hρ0 hρ1 v l i y)).eq_of_nhds |
| rw [he,material_scaledMeanJet hF x hr hlam hl hρ0 hρ1 v l hlN] |
| have hU (k : Fin d) := conditionalU_smooth_pos hF x hr hlam hl hρ0 hρ1 |
| (Sum.elim v (fun _ : Unit => JetCalculus.spaceBasis d k)) (Sum.inr ()::l.map Sum.inl) |
| (EuclideanSpace.basisFun (Fin d) ℝ i) |
| have hdir := dir_mean_jet_eq_scaled_U hF x hr hlam hl hρ0.le hρ1 v l i |
| simp_rw [hdir] |
| rw [JetCalculus.gadj_scalar _ _ (fun k => (hU k).differentiable (by simp))] |
| dsimp only |
| have hscale : (ρ^(l.length+1))⁻¹*ρ^(l.length+1)=1 := inv_mul_cancel₀ (pow_ne_zero _ hρ0.ne') |
| rw [←mul_assoc,hscale,one_mul] |
| congr 1 |
| rw [Finset.mul_sum,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro s _ |
| rw [Finset.mul_sum,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro k _ |
| rw [mean_jet_eq_scaled_U hF x hr hlam hl hρ0.le hρ1, |
| dir_mean_jet_eq_scaled_U hF x hr hlam hl hρ0.le hρ1] |
| dsimp only |
| have hs := split_power_cancel l s hρ0.ne' |
| unfold jointU |
| calc |
| _ = 2*r*((ρ^l.length)⁻¹*ρ^(l.filter (fun j => j∈s)).length* |
| ρ^((l.filter (fun j => j∉s)).length+1))* |
| (conditionalU F x r ρ y (EuclideanSpace.basisFun (Fin d) ℝ k) ((lam:ℝ)*r) v |
| (l.filter (fun j => j∈s))*conditionalU F x r ρ y (EuclideanSpace.basisFun (Fin d) ℝ i) |
| ((lam:ℝ)*r) (Sum.elim v (fun _ : Unit => JetCalculus.spaceBasis d k)) |
| (Sum.inr ()::(l.filter (fun j => j∉s)).map Sum.inl)) := by ring |
| _ = _ := by rw [hs] |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
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| end |
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| end OAI |
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