| import Mathlib |
| import OAI.Probability.LogConcave.Complexity.KernelAnalyticNormalizedBudget |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators NNReal |
|
|
| namespace MeanTree |
| variable {X : Type*} [MeasurableSpace X] {d : ℕ} |
|
|
| def forward (r T h : ℝ) (n N : ℕ) (x y : MeanTree X d) |
| (i : ProbabilityNode T h n) : MeanTree X d := |
| picard (probabilityWeight T h n) |
| (fun j E => scale (-r) (conditional r (probabilityNodeTime T h n j) x E)) |
| (fun _ => y) N i |
|
|
| lemma eval_forward {F : Point d → ℝ} {lam : ℝ≥0} (hF : Primitive F lam) |
| {r T h : ℝ} {n N : ℕ} (hn : 0<n) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (x y : MeanTree X d) (z : X) (i : ProbabilityNode T h n) : |
| eval F (forward r T h n N x y i) z= |
| probabilityPicard F (eval F x z) r T h n N (eval F y z) i := by |
| unfold forward probabilityPicard |
| have ha (j : ProbabilityNode T h n) : 0<1-(probabilityNodeTime T h n j)^2 := by |
| have hj := probabilityNodeTime_mem hT0 hT1 hh hn j |
| have ht := hj.2.trans_lt hT1 |
| nlinarith [hj.1] |
| have he := eval_picard F (probabilityWeight T h n) |
| (fun j E => scale (-r) (conditional r (probabilityNodeTime T h n j) x E)) |
| (fun _ => y) (fun z j => probabilityMeanVelocity F (eval F x z) r (probabilityNodeTime T h n j)) |
| (fun j E z => by simp only [eval_scale,eval_conditional hF r _ x E z (ha j),probabilityMeanVelocity]) N z i |
| simpa only [FinitePicard.nodes_eq_iterate] using he |
|
|
| lemma depth_forward (r T h : ℝ) (n N : ℕ) (x y : MeanTree X d) |
| (i : ProbabilityNode T h n) {b : ℕ} (hx : depth x≤b) (hy : depth y≤b) : |
| depth (forward r T h n N x y i)≤b+N := by |
| apply (depth_picard _ _ _ (D:=1) (fun _ => hy) ?_ N i).trans (by omega) |
| intro j E |
| rw [depth_scale] |
| exact depth_conditional_le _ _ _ _ (hx.trans (le_max_left _ _)) (le_max_right _ _) |
| end MeanTree |
|
|
| def sampleMeanCircuit {d : ℕ} (F : Point d → ℝ) (x : Point d) (r T h : ℝ) |
| (n N : ℕ) (e : ProbabilityNode T h n) (z : Point d) : Point d := |
| T⁻¹ • probabilityPicard F x r T h n N z e |
|
|
| def samplingEnvelope (d k n N : ℕ) (C q h : ℝ) : ℝ := |
| 4*(2*C*dimensionLog d^k*(2*h)^(n+1)+circuitGrowthConstant*q^N)^2 |
|
|
| theorem sampleMeanCircuit_rms (n : ℕ) (hn : 0<n) : |
| ∃A : ℝ≥0,∃C : ℝ,0≤C ∧ ∃k : ℕ,∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, |
| ∀hF : Primitive F lam,∀(x : Point d) {r : ℝ},∀hr : 0<r, |
| r≤1 → 0<lam → ∀hl : (lam:ℝ)*r^2≤1/2,(lam:ℝ)*r≤1 → 1≤d → |
| ∀{T h : ℝ},1/2≤T → T<1 → 0<h → h≤Real.log 2 → |
| A*probabilityMeanLipschitz lam r≤1/2 → ∀N : ℕ,∀e : ProbabilityNode T h n, |
| Integrable (fun z => ‖sampleMeanCircuit F x r T h n N e z- |
| T⁻¹ • fullProbabilityFlow hF x hr.le hl (probabilityNodeTime T h n e) z‖^2) |
| (stdGaussian (Point d)) ∧ |
| (∫z,‖sampleMeanCircuit F x r T h n N e z- |
| T⁻¹ • fullProbabilityFlow hF x hr.le hl (probabilityNodeTime T h n e) z‖^2 ∂stdGaussian (Point d))≤ |
| (circuitD F x)^2*samplingEnvelope d k n N C (A*probabilityMeanLipschitz lam r) h := by |
| obtain ⟨A,C,hC,k,h⟩ := probabilityPicard_forward_rms n hn |
| refine ⟨A,C,hC,k,?_⟩ |
| intro d F lam hF x r hr hr1 hlam hl hL hd T hmesh hT hT1 hh hhsmall hq N e |
| have he := h hF x hr hlam hl hd (by linarith) hT1 hh hhsmall hq N e |
| have hT0 : 0<T := by linarith |
| have hTin : 0≤T⁻¹ := inv_nonneg.mpr hT0.le |
| have hTi : T⁻¹≤2 := (inv_le_comm₀ hT0 (by norm_num)).mpr (by simpa using hT) |
| have hi := probabilityInitialRms_growth F x lam hr.le hL |
| have hi' : probabilityInitialRms F x lam r≤circuitGrowthConstant*circuitD F x := by |
| apply hi.trans |
| nlinarith [mul_le_mul_of_nonneg_left hr1 |
| (mul_nonneg circuitGrowthConstant_pos.le (circuitD_nonneg F x))] |
| have hD := sqrt_le_circuitD F x |
| have hD0 := circuitD_nonneg F x |
| have hlo : 0≤(lam:ℝ)*r^2 := by positivity |
| have hlog := dimensionLog_nonneg d |
| let q : ℝ := A*probabilityMeanLipschitz lam r |
| let B : ℝ := 2*C*dimensionLog d^k*(2*hmesh)^(n+1)+circuitGrowthConstant*q^N |
| have heq : (fun z => ‖sampleMeanCircuit F x r T hmesh n N e z- |
| T⁻¹ • fullProbabilityFlow hF x hr.le hl (probabilityNodeTime T hmesh n e) z‖^2)= |
| (fun z => (T⁻¹)^2*‖probabilityPicard F x r T hmesh n N z e- |
| fullProbabilityFlow hF x hr.le hl (probabilityNodeTime T hmesh n e) z‖^2) := by |
| funext z |
| simp only [sampleMeanCircuit,←smul_sub,norm_smul,Real.norm_eq_abs,mul_pow,sq_abs] |
| rw [heq] |
| refine ⟨he.1.const_mul _,?_⟩ |
| rw [integral_const_mul] |
| have hb : 2*C*((lam:ℝ)*r^2)*Real.sqrt d*(1+Real.log ((d:ℝ)+1))^k*(2*hmesh)^(n+1)+ |
| q^N*probabilityInitialRms F x lam r≤circuitD F x*B := by |
| have hls : (lam:ℝ)*r^2*Real.sqrt d≤circuitD F x := by |
| exact (mul_le_of_le_one_left (Real.sqrt_nonneg _) (by linarith)).trans hD |
| have h₁ := mul_le_mul_of_nonneg_left hls |
| (show 0≤2*C*dimensionLog d^k*(2*hmesh)^(n+1) by positivity) |
| have h₂ := mul_le_mul_of_nonneg_left hi' (show 0≤q^N by dsimp [q]; positivity) |
| dsimp [B,dimensionLog] at * |
| nlinarith |
| have hp := pow_le_pow_left₀ (show 0≤2*C*((lam:ℝ)*r^2)*Real.sqrt d*(1+Real.log ((d:ℝ)+1))^k*(2*hmesh)^(n+1)+ |
| q^N*probabilityInitialRms F x lam r by |
| change 0≤2*C*((lam:ℝ)*r^2)*Real.sqrt d*dimensionLog d^k*(2*hmesh)^(n+1)+q^N*probabilityInitialRms F x lam r |
| positivity [probabilityInitialRms_nonneg F x lam r]) hb 2 |
| have hti := pow_le_pow_left₀ hTin hTi 2 |
| calc |
| _ ≤ (T⁻¹)^2*(circuitD F x*B)^2 := mul_le_mul_of_nonneg_left (he.2.trans hp) (sq_nonneg _) |
| _ ≤ 4*(circuitD F x*B)^2 := by nlinarith [mul_le_mul_of_nonneg_right hti (sq_nonneg (circuitD F x*B))] |
| _ = _ := by dsimp [samplingEnvelope,B,q]; ring |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.MeanTree |
| open Set MeasureTheory Quadrature FinitePicard |
| open scoped Classical BigOperators |
|
|
| variable {X : Type*} [MeasurableSpace X] {d : ℕ} |
|
|
| def oriented (Y G : X → Point d) (θ : ℝ) : X → Point d := |
| fun z => Real.cos θ • Y z+Real.sin θ • G z |
|
|
| omit [MeasurableSpace X] in |
| @[simp] lemma oriented_zero (Y G : X → Point d) : oriented Y G 0=Y := by |
| funext z; simp [oriented] |
|
|
| def DirectCenter (r T v : ℝ) (x Y G : X → Point d) (r' : ℝ) (b : X → Point d) : Prop := |
| ∃ρ∈Icc (0:ℝ) T,∃θ,|θ|≤v ∧ r'=r*Real.sqrt (1-ρ^2) ∧ |
| b=fun z => x z+(r*ρ) • oriented Y G θ z |
|
|
| lemma conditional_centers_direct {r T v ρ θ : ℝ} {x Y G : X → Point d} |
| (xE yE : MeanTree X d) (hx : base xE=x) (hy : base yE=oriented Y G θ) |
| (hρ : ρ∈Icc (0:ℝ) T) (hθ : |θ|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) : |
| centers (DirectCenter r T v x Y G) (conditional r ρ xE yE) := by |
| apply centers_conditional _ _ _ _ _ _ hxc hyc |
| exact ⟨ρ,hρ,θ,hθ,rfl,by rw [hx,hy]⟩ |
|
|
| lemma probability_centers_direct {r T h v θ : ℝ} {n N : ℕ} |
| (hn : 0<n) (hT : 0<T) (hT1 : T<1) (hh : 0<h) |
| {x Y G : X → Point d} (xE yE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=oriented Y G θ) (hθ : |θ|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (s i : ProbabilityNode T h n) : |
| base (probability r T h n N s xE yE i)=base yE ∧ |
| centers (DirectCenter r T v x Y G) (probability r T h n N s xE yE i) := by |
| apply picard_shape _ _ _ _ (fun _ => hyc) |
| intro j k E he hc |
| refine ⟨by simp,?_⟩ |
| apply (centers_scale _ _ _).2 |
| exact conditional_centers_direct xE E hx (he.trans hy) |
| (probabilityNodeTime_mem hT hT1 hh hn j) hθ hxc hc |
|
|
| lemma forward_centers_direct {r T h v θ : ℝ} {n N : ℕ} |
| (hn : 0<n) (hT : 0<T) (hT1 : T<1) (hh : 0<h) |
| {x Y G : X → Point d} (xE yE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=oriented Y G θ) (hθ : |θ|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (i : ProbabilityNode T h n) : |
| base (forward r T h n N xE yE i)=base yE ∧ |
| centers (DirectCenter r T v x Y G) (forward r T h n N xE yE i) := by |
| apply picard_shape _ _ _ _ (fun _ => hyc) |
| intro j k E he hc |
| refine ⟨by simp,?_⟩ |
| apply (centers_scale _ _ _).2 |
| exact conditional_centers_direct xE E hx (he.trans hy) |
| (probabilityNodeTime_mem hT hT1 hh hn j) hθ hxc hc |
|
|
| lemma harmonic_centers_direct {r T v ρ θ : ℝ} {n N : ℕ} (hn : 0<n) |
| {x Y G : X → Point d} (xE yE gE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=Y) (hg : base gE=G) |
| (hρ : ρ∈Icc (0:ℝ) T) (hθ : |θ|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (hgc : centers (DirectCenter r T v x Y G) gE) (i : Fin (n+1)) : |
| base (harmonic r ρ θ n N xE yE gE i)=oriented Y G (θ*probabilityNodes n i) ∧ |
| centers (DirectCenter r T v x Y G) (harmonic r ρ θ n N xE yE gE i) := by |
| have ha : ∀j : Fin (n+1), base (add (scale (Real.cos (θ*probabilityNodes n j)) yE) |
| (scale (Real.sin (θ*probabilityNodes n j)) gE))=oriented Y G (θ*probabilityNodes n j) := by |
| intro j; simp only [base_add,base_scale,hy,hg]; rfl |
| have hp := picard_shape (DirectCenter r T v x Y G) (harmonicWeight n θ) |
| (fun _ E => scale ρ (scale (-r) (conditional r ρ xE E))) |
| (fun j => add (scale (Real.cos (θ*probabilityNodes n j)) yE) |
| (scale (Real.sin (θ*probabilityNodes n j)) gE)) |
| (fun _ => (centers_add _ _ _).2 ⟨(centers_scale _ _ _).2 hyc,(centers_scale _ _ _).2 hgc⟩) ?_ N i |
| · exact ⟨hp.1.trans (ha i),hp.2⟩ |
| · intro j k E he hc |
| refine ⟨by simp,?_⟩ |
| apply (centers_scale _ _ _).2 |
| apply (centers_scale _ _ _).2 |
| apply conditional_centers_direct xE E hx (he.trans (ha k)) hρ _ hxc hc |
| rw [abs_mul,abs_of_nonneg (probabilityNodes_mem hn k).1] |
| exact (mul_le_mul_of_nonneg_left (probabilityNodes_mem hn k).2 (abs_nonneg θ)).trans (by simpa using hθ) |
|
|
| lemma kernelCorrection_centers_direct {r T h v θ : ℝ} {n N : ℕ} |
| (hn : 0<n) (hT : 0<T) (hT1 : T<1) (hh : 0<h) |
| {x Y G : X → Point d} (xE yE gE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=Y) (hg : base gE=G) |
| (hθ : |θ|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (hgc : centers (DirectCenter r T v x Y G) gE) (s e : ProbabilityNode T h n) : |
| base (kernelCorrection r T h θ n N s e xE yE gE)=(fun _ => 0) ∧ |
| centers (DirectCenter r T v x Y G) (kernelCorrection r T h θ n N s e xE yE gE) := by |
| have hv : 0≤v := (abs_nonneg _).trans hθ |
| have hb := probability_centers_direct (N:=N) hn hT hT1 hh xE yE hx |
| (by simpa using hy : base yE=oriented Y G 0) (by simpa using hv) hxc hyc e s |
| have hw := harmonic_centers_direct (N:=N) hn xE (probability r T h n N e xE yE s) gE hx |
| (hb.1.trans hy) hg (probabilityNodeTime_mem hT hT1 hh hn s) hθ hxc hb.2 hgc (Fin.last n) |
| have vlast : probabilityNodes n (Fin.last n)=1 := by |
| simp [probabilityNodes,hn.ne'] |
| rw [vlast,mul_one] at hw |
| constructor |
| · funext z |
| simp only [kernelCorrection,base_sumFamily,base_scale,base_conditional,smul_zero,Finset.sum_const_zero] |
| · unfold kernelCorrection |
| apply centers_sumFamily |
| intro j |
| apply (centers_scale _ _ _).2 |
| apply (centers_scale _ _ _).2 |
| have hp := probability_centers_direct (N:=N) hn hT hT1 hh xE |
| (harmonic r (probabilityNodeTime T h n s) θ n N xE (probability r T h n N e xE yE s) gE (Fin.last n)) |
| hx hw.1 hθ hxc hw.2 s j |
| exact conditional_centers_direct _ _ hx (hp.1.trans hw.1) |
| (probabilityNodeTime_mem hT hT1 hh hn j) hθ hxc hp.2 |
| |
| lemma kernelQuadrature_centers_direct {r T h ψ v : ℝ} {n m N : ℕ} |
| (hn : 0<n) (hT : 0<T) (hT1 : T<1) (hh : 0<h) |
| {x Y G : X → Point d} (xE yE gE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=Y) (hg : base gE=G) |
| (hψ : ∀j : Fin (m+1),|ψ*angleNodes m j|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (hgc : centers (DirectCenter r T v x Y G) gE) |
| (e : ProbabilityNode T h n) (w : ProbabilityNode T h n → ℝ) : |
| base (kernelQuadrature r T h ψ n m N e w xE yE gE)=(fun _ => 0) ∧ |
| centers (DirectCenter r T v x Y G) (kernelQuadrature r T h ψ n m N e w xE yE gE) := by |
| have hk (i : ProbabilityNode T h n) (j : Fin (m+1)) := |
| kernelCorrection_centers_direct (N:=N) hn hT hT1 hh xE yE gE hx hy hg (hψ j) hxc hyc hgc i e |
| constructor |
| · funext z |
| simp only [kernelQuadrature,kernelAction,base_sumFamily,base_scale,(hk _ _).1,smul_zero,Finset.sum_const_zero] |
| · apply centers_sumFamily |
| intro i |
| apply (centers_scale _ _ _).2 |
| apply centers_sumFamily |
| intro j |
| exact (centers_scale _ _ _).2 (hk i j).2 |
| end LogConcaveSampling.MeanTree |
| |
| end |
| |
| end |
| |
| section |
| |
| noncomputable section |
| namespace LogConcaveSampling.MeanTree |
| open Set MeasureTheory Quadrature |
| open scoped Classical BigOperators |
| |
| variable {X : Type*} [MeasurableSpace X] {d : ℕ} |
| |
| lemma pairPicard_shape {I : Type*} [Fintype I] (P : ℝ → (X → Point d) → Prop) |
| (w : I → I → ℝ) (φ : MeanTree X d × MeanTree X d → MeanTree X d × MeanTree X d) |
| (a : MeanTree X d × MeanTree X d) |
| (ha : centers P a.1 ∧ centers P a.2) |
| (hφ : ∀p,base p.1=base a.1 → base p.2=base a.2 → centers P p.1 → centers P p.2 → |
| base (φ p).1=(fun _ => 0) ∧ base (φ p).2=(fun _ => 0) ∧ |
| centers P (φ p).1 ∧ centers P (φ p).2) |
| (N : ℕ) (i : I) : |
| base (pairPicard w φ a N i).1=base a.1 ∧ base (pairPicard w φ a N i).2=base a.2 ∧ |
| centers P (pairPicard w φ a N i).1 ∧ centers P (pairPicard w φ a N i).2 := by |
| induction N generalizing i with |
| | zero => exact ⟨rfl,rfl,ha⟩ |
| | succ N ih => |
| have hp (j : I) := hφ (pairPicard w φ a N j) (ih j).1 (ih j).2.1 (ih j).2.2.1 (ih j).2.2.2 |
| refine ⟨?_,?_,?_,?_⟩ |
| · funext z |
| simp only [pairPicard,base_add,base_sumFamily,base_scale,(hp _).1,smul_zero,Finset.sum_const_zero,add_zero] |
| · funext z |
| simp only [pairPicard,base_add,base_sumFamily,base_scale,(hp _).2.1,smul_zero,Finset.sum_const_zero,add_zero] |
| · exact (centers_add _ _ _).2 ⟨ha.1,centers_sumFamily _ _ (fun j => (centers_scale _ _ _).2 (hp j).2.2.1)⟩ |
| · exact (centers_add _ _ _).2 ⟨ha.2,centers_sumFamily _ _ (fun j => (centers_scale _ _ _).2 (hp j).2.2.2)⟩ |
| |
| lemma velocity_centers_direct {r T h ψ v s : ℝ} {n m N : ℕ} |
| (hT : 0<T) (hT1 : T<1) (hh : 0<h) (hv : 0≤v) |
| {x Y G : X → Point d} (xE yE gE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=Y) (hg : base gE=G) |
| (hψ : ∀j : Fin (m+1),|ψ*angleNodes m j|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (hgc : centers (DirectCenter r T v x Y G) gE) |
| (e : ProbabilityNode T h (n+1)) : |
| base (velocity r T h ψ s n m N e xE (yE,gE)).1=(fun _ => 0) ∧ |
| base (velocity r T h ψ s n m N e xE (yE,gE)).2=(fun _ => 0) ∧ |
| centers (DirectCenter r T v x Y G) (velocity r T h ψ s n m N e xE (yE,gE)).1 ∧ |
| centers (DirectCenter r T v x Y G) (velocity r T h ψ s n m N e xE (yE,gE)).2 := by |
| have hk := kernelQuadrature_centers_direct (N:=N) (Nat.succ_pos n) hT hT1 hh xE yE gE hx hy hg hψ hxc hyc hgc e (terminalQuadratureWeight T h n) |
| have hm : centers (DirectCenter r T v x Y G) (mean r xE) := by |
| apply centers_mean _ _ _ _ hxc |
| refine ⟨0,⟨by norm_num,hT.le⟩,0,by simpa using hv,?_,?_⟩ |
| · simp |
| · simpa only [mul_zero,zero_smul,add_zero] using hx |
| have hc := conditional_centers_direct xE yE hx (by simpa using hy : base yE=oriented Y G 0) |
| ⟨hT.le,le_rfl⟩ (by simpa using hv : |(0:ℝ)|≤v) hxc hyc |
| refine ⟨?_,?_,?_,?_⟩ |
| · funext z; simp only [velocity,base_scale,hk.1,smul_zero] |
| · funext z; simp only [velocity,base_scale,base_add,base_conditional,base_mean,smul_zero,add_zero] |
| · exact (centers_scale _ _ _).2 hk.2 |
| · exact (centers_scale _ _ _).2 ((centers_add _ _ _).2 ⟨hc,(centers_scale _ _ _).2 hm⟩) |
| |
| lemma centeringStates_centers_direct {r T h ψ v s : ℝ} {n m N nc Nc : ℕ} |
| (hT : 0<T) (hT1 : T<1) (hh : 0<h) (hv : 0≤v) |
| {x Y G : X → Point d} (xE yE gE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=Y) (hg : base gE=G) |
| (hψ : ∀j : Fin (m+1),|ψ*angleNodes m j|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (hgc : centers (DirectCenter r T v x Y G) gE) |
| (e : ProbabilityNode T h (n+1)) (i : Fin (nc+1)) : |
| base (centeringStates r T h ψ s n m N nc Nc e xE yE gE i).1=Y ∧ |
| base (centeringStates r T h ψ s n m N nc Nc e xE yE gE i).2=G ∧ |
| centers (DirectCenter r T v x Y G) (centeringStates r T h ψ s n m N nc Nc e xE yE gE i).1 ∧ |
| centers (DirectCenter r T v x Y G) (centeringStates r T h ψ s n m N nc Nc e xE yE gE i).2 := by |
| have hp := pairPicard_shape (DirectCenter r T v x Y G) (centeringWeight nc) |
| (velocity r T h ψ s n m N e xE) (yE,gE) ⟨hyc,hgc⟩ |
| (fun p hp hq hpc hqc => velocity_centers_direct hT hT1 hh hv xE p.1 p.2 hx |
| (hp.trans hy) (hq.trans hg) hψ hxc hpc hqc e) Nc i |
| exact ⟨hp.1.trans hy,hp.2.1.trans hg,hp.2.2⟩ |
| |
| lemma meanCircuit_centers_direct {r T h ψ v s : ℝ} {n m N nc Nc : ℕ} |
| (hT : 0<T) (hT1 : T<1) (hh : 0<h) (hv : 0≤v) |
| {x Y G : X → Point d} (xE yE gE : MeanTree X d) |
| (hx : base xE=x) (hy : base yE=Y) (hg : base gE=G) |
| (hψ : ∀j : Fin (m+1),|ψ*angleNodes m j|≤v) |
| (hxc : centers (DirectCenter r T v x Y G) xE) |
| (hyc : centers (DirectCenter r T v x Y G) yE) |
| (hgc : centers (DirectCenter r T v x Y G) gE) |
| (e : ProbabilityNode T h (n+1)) : |
| base (meanCircuit r T h ψ s n m N nc Nc e xE yE gE)=(fun z => s • G z) ∧ |
| centers (DirectCenter r T v x Y G) (meanCircuit r T h ψ s n m N nc Nc e xE yE gE) := by |
| have hp (i : Fin (nc+1)) := centeringStates_centers_direct (N:=N) (Nc:=Nc) (s:=s) hT hT1 hh hv xE yE gE hx hy hg hψ hxc hyc hgc e i |
| constructor |
| · funext z |
| simp only [meanCircuit,base_add,base_scale,base_sumFamily,base_conditional,hg, |
| smul_zero,Finset.sum_const_zero,add_zero] |
| · apply (centers_add _ _ _).2 |
| refine ⟨(centers_scale _ _ _).2 hgc,centers_sumFamily _ _ ?_⟩ |
| intro j |
| apply (centers_scale _ _ _).2 |
| exact conditional_centers_direct xE _ hx (by simpa using (hp j).1 : base _=oriented Y G 0) |
| ⟨hT.le,le_rfl⟩ (by simpa using hv : |(0:ℝ)|≤v) hxc (hp j).2.2.1 |
| end LogConcaveSampling.MeanTree |
| |
| end |
| |
| end |
| |
| end OAI |
| |
