| import Mathlib |
| import OAI.Probability.LogConcave.TensorGraphs.RankEquiv |
| import OAI.Probability.LogConcave.JetEstimates.AllSplitBound |
|
|
| 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 |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| namespace LogConcaveSampling.GraphSchedule |
| open scoped Classical |
|
|
| variable {V U : Type*} [Fintype V] [Fintype U] |
| (key : V → ℕ) (src dst : U → V) |
| local instance edgeDecEqT (q : ℕ) : DecidableEq (Edge (U:=U) q) := Classical.decEq _ |
| local instance finDecEqT (d : ℕ) : DecidableEq (Fin d) := Classical.decEq _ |
|
|
| noncomputable def incidentMatrix (hkey : Function.Injective key) (hne : ∀u,src u≠dst u) |
| (first last v : V) (q d : ℕ) |
| (T : ({t : Edge (U:=U) q |
| (Frontier.Outgoing (emit key src dst first last q) (key v) → Fin d) → |
| (Frontier.Incoming (absorb key src dst first last q) (key v) → Fin d) → ℝ := |
| fun o i => T (fun t => Sum.elim i o (incidentSplit key src dst hkey hne first last v q t)) |
|
|
| omit [Fintype U] in |
| lemma incidentMatrix_global (hkey : Function.Injective key) (hne : ∀u,src u≠dst u) |
| (first last v : V) (q d : ℕ) |
| (T : ({t : Edge (U:=U) q |
| (c : Edge (U:=U) q → Fin d) : |
| incidentMatrix key src dst hkey hne first last v q d T |
| (fun t => c t.val) (fun t => c t.val)=T (fun t => c t.val) := by |
| unfold incidentMatrix |
| congr 1 |
| funext t |
| dsimp only [incidentSplit,Equiv.coe_fn_mk] |
| split_ifs <;> rfl |
|
|
| lemma incidentMatrix_bound (hkey : Function.Injective key) (hne : ∀u,src u≠dst u) |
| (first last v : V) (q d : ℕ) |
| (T : ({t : Edge (U:=U) q |
| (M : ℝ) (hT : TensorEnergy.AllSplitBound T M) |
| [Nonempty (Frontier.Incoming (absorb key src dst first last q) (key v))] |
| [Nonempty (Frontier.Outgoing (emit key src dst first last q) (key v))] : |
| TensorEnergy.Bound (incidentMatrix key src dst hkey hne first last v q d T) (M^2) := |
| hT _ _ (incidentSplit key src dst hkey hne first last v q) |
| end LogConcaveSampling.GraphSchedule |
| namespace LogConcaveSampling.GraphSchedule |
| open scoped BigOperators Classical |
| open TensorEnergy |
|
|
| variable {V U : Type*} [Fintype V] [Fintype U] [Nonempty V] |
| local instance edgeDecEqTG (q : ℕ) : DecidableEq (Edge (U:=U) q) := Classical.decEq _ |
| local instance finDecEqTG (d : ℕ) : DecidableEq (Fin d) := Classical.decEq _ |
|
|
| theorem tensor_graph_bound (e : V ≃ Fin (Fintype.card V)) (src dst : U → V) |
| (hne : ∀u,src u≠dst u) (first last : V) (q d : ℕ) |
| (hsplit : ∀v, |
| Nonempty (Frontier.Incoming (absorb (fun v => (e v).val) src dst first last q) (e v).val) ∧ |
| Nonempty (Frontier.Outgoing (emit (fun v => (e v).val) src dst first last q) (e v).val)) |
| (T : ∀v,({t : Edge (U:=U) q |
| (M : V → ℝ) (hM : ∀v,0≤M v) (hT : ∀v,AllSplitBound (T v) (M v)) : |
| |∑c : U ⊕ Fin q → Fin d, ∏v,T v (fun t => duplicate c t.val)|≤ |
| (d:ℝ)^q*(∏v,M v) := by |
| let key : V → ℕ := fun v => (e v).val |
| have hkey : Function.Injective key := Fin.val_injective.comp e.injective |
| let D : ℕ → Type _ := fun k => |
| (Frontier.Outgoing (emit key src dst first last q) k → Fin d) → |
| (Frontier.Incoming (absorb key src dst first last q) k → Fin d) → ℝ |
| let a : ∀v,D (e v).val := fun v => |
| incidentMatrix key src dst hkey hne first last v q d (T v) |
| let A : ∀k,D k := rankExtend e (fun _ _ _ => 0) a |
| have hAt (v : V) : A (e v).val=a v := rankExtend_at e _ a v |
| have hA (v : V) : Bound (A (e v).val) ((M v)^2) := by |
| rw [hAt] |
| let := (hsplit v).1 |
| let := (hsplit v).2 |
| exact incidentMatrix_bound key src dst hkey hne first last v q d (T v) (M v) (hT v) |
| have hb := ranked_coordinate_bound e src dst hne first last q d A M hM hA |
| have heq (v : V) (c : U ⊕ Fin q → Fin d) : |
| A (e v).val (fun t => duplicate c t.val) (fun t => duplicate c t.val)= |
| T v (fun t => duplicate c t.val) := by |
| rw [hAt] |
| exact incidentMatrix_global key src dst hkey hne first last v q d (T v) (duplicate c) |
| convert hb using 1 |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro c _ |
| apply Finset.prod_congr rfl |
| intro v _ |
| exact (heq v c).symm |
| end LogConcaveSampling.GraphSchedule |
| namespace LogConcaveSampling.GraphSchedule |
| open MeasureTheory TensorEnergy |
| open scoped BigOperators Classical ENNReal |
|
|
| variable {Ω P H U : Type*} [MeasurableSpace Ω] [Fintype P] [Fintype H] [Fintype U] |
| [Nonempty P] |
| local instance edgeDecEqTGI (q : ℕ) : DecidableEq (Edge (U:=U) q) := Classical.decEq _ |
| local instance finDecEqTGI (d : ℕ) : DecidableEq (Fin d) := Classical.decEq _ |
|
|
| theorem tensor_graph_integral_bound (μ : Measure Ω) |
| (e : P ⊕ H ≃ Fin (Fintype.card (P ⊕ H))) (src dst : U → P ⊕ H) |
| (hne : ∀u,src u≠dst u) (first last : P ⊕ H) (q d : ℕ) |
| (hsplit : ∀v, |
| Nonempty (Frontier.Incoming (absorb (fun v => (e v).val) src dst first last q) (e v).val) ∧ |
| Nonempty (Frontier.Outgoing (emit (fun v => (e v).val) src dst first last q) (e v).val)) |
| (T : Ω → ∀v,({t : Edge (U:=U) q |
| (M : P ⊕ H → Ω → ℝ) (hM : ∀v x,0≤M v x) |
| (hT : ∀v x,AllSplitBound (T x v) (M v x)) |
| (B : P → ℝ≥0∞) (C : H → ℝ≥0∞) |
| (hMp : ∀i,AEMeasurable (fun x => ENNReal.ofReal (M (Sum.inl i) x)) μ) |
| (hp : ∀i,(∫⁻x,ENNReal.ofReal (M (Sum.inl i) x)^(Fintype.card P) ∂μ)≤B i^(Fintype.card P)) |
| (hh : ∀h x,ENNReal.ofReal (M (Sum.inr h) x)≤C h) (hC : ∀h,C h≠⊤) : |
| (∫⁻x,ENNReal.ofReal |∑c : U ⊕ Fin q → Fin d, |
| ∏v,T x v (fun t => duplicate c t.val)| ∂μ)≤ |
| (d:ℝ≥0∞)^q*((∏i,B i)*(∏h,C h)) := by |
| have hpw (x : Ω) : ENNReal.ofReal |∑c : U ⊕ Fin q → Fin d, |
| ∏v,T x v (fun t => duplicate c t.val)|≤ |
| (d:ℝ≥0∞)^q*((∏i,ENNReal.ofReal (M (Sum.inl i) x))* |
| (∏h,ENNReal.ofReal (M (Sum.inr h) x))) := by |
| have hb := ENNReal.ofReal_le_ofReal (tensor_graph_bound e src dst hne first last q d |
| hsplit (T x) (fun v => M v x) (fun v => hM v x) (fun v => hT v x)) |
| rw [ENNReal.ofReal_mul (by positivity),ENNReal.ofReal_pow (by positivity), |
| ENNReal.ofReal_natCast,ENNReal.ofReal_prod_of_nonneg (fun v _ => hM v x), |
| Fintype.prod_sum_type] at hb |
| exact hb |
| calc |
| _ ≤ ∫⁻x,(d:ℝ≥0∞)^q*((∏i,ENNReal.ofReal (M (Sum.inl i) x))* |
| (∏h,ENNReal.ofReal (M (Sum.inr h) x))) ∂μ := lintegral_mono hpw |
| _ = (d:ℝ≥0∞)^q*(∫⁻x,(∏i,ENNReal.ofReal (M (Sum.inl i) x))* |
| (∏h,ENNReal.ofReal (M (Sum.inr h) x)) ∂μ) := |
| lintegral_const_mul' _ _ (by finiteness) |
| _ ≤ _ := mul_le_mul' le_rfl |
| (NetworkMoments.lintegral_primary_auxiliary_le hMp hp hh hC) |
| end LogConcaveSampling.GraphSchedule |
| namespace LogConcaveSampling.AdjointRemoval |
| open MeasureTheory GraphSchedule TensorEnergy |
| open scoped BigOperators Classical ENNReal |
|
|
| variable {n : ℕ} {S : State (Fin (n+1))} |
| local instance edgeDecEqAT (β : Fin n → ℕ) (q : ℕ) : |
| DecidableEq (Edge (U:=InternalEdge β) q) := Classical.decEq _ |
| local instance cutDecEqAT (β : Fin n → ℕ) (q : ℕ) : |
| DecidableEq (InternalEdge β ⊕ Fin q) := |
| @instDecidableEqSum _ _ (Classical.decEq _) (Classical.decEq _) |
|
|
| theorem actual_tensor_contraction_bound |
| (hS : S∈expand (Fintype.card (Fin (n+1))) initial) |
| (β : Fin n → ℕ) (hβ : ∀k,0<β k) (q d : ℕ) (hq : 0<q) |
| (T : ∀v : TargetVertex S, |
| ({t : Edge (U:=InternalEdge β) q |
| IncidentAt (V:=TargetVertex S) (edgeSource (S:=S) β) (edgeTarget (S:=S) hS β) |
| (Sum.inl 0) (Sum.inl (Fin.last n)) v q t} → Fin d) → ℝ) |
| (M : TargetVertex S → ℝ) (hM : ∀v,0≤M v) (hT : ∀v,AllSplitBound (T v) (M v)) : |
| |∑c : InternalEdge β ⊕ Fin q → Fin d, ∏v,T v (fun t => duplicate c t.val)|≤ |
| (d:ℝ)^q*(∏v,M v) := by |
| obtain ⟨e,he⟩ := actual_ranked_schedule hS β hβ |
| exact tensor_graph_bound e (edgeSource (S:=S) β) (edgeTarget (S:=S) hS β) |
| (internal_source_ne hS β) (Sum.inl 0) (Sum.inl (Fin.last n)) q d (he q hq) T M hM hT |
|
|
| theorem actual_tensor_integral_bound {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) |
| (hS : S∈expand (Fintype.card (Fin (n+1))) initial) |
| (β : Fin n → ℕ) (hβ : ∀k,0<β k) (q d : ℕ) (hq : 0<q) |
| (T : Ω → ∀v : TargetVertex S, |
| ({t : Edge (U:=InternalEdge β) q |
| IncidentAt (V:=TargetVertex S) (edgeSource (S:=S) β) (edgeTarget (S:=S) hS β) |
| (Sum.inl 0) (Sum.inl (Fin.last n)) v q t} → Fin d) → ℝ) |
| (M : TargetVertex S → Ω → ℝ) (hM : ∀v x,0≤M v x) |
| (hT : ∀v x,AllSplitBound (T x v) (M v x)) |
| (B : Fin (n+1) → ℝ≥0∞) (C : S.hessians → ℝ≥0∞) |
| (hMp : ∀i,AEMeasurable (fun x => ENNReal.ofReal (M (Sum.inl i) x)) μ) |
| (hp : ∀i,(∫⁻x,ENNReal.ofReal (M (Sum.inl i) x)^(n+1) ∂μ)≤B i^(n+1)) |
| (hh : ∀h x,ENNReal.ofReal (M (Sum.inr h) x)≤C h) (hC : ∀h,C h≠⊤) : |
| (∫⁻x,ENNReal.ofReal |∑c : InternalEdge β ⊕ Fin q → Fin d, |
| ∏v,T x v (fun t => duplicate c t.val)| ∂μ)≤ |
| (d:ℝ≥0∞)^q*((∏i,B i)*(∏h,C h)) := by |
| obtain ⟨e,he⟩ := actual_ranked_schedule hS β hβ |
| exact tensor_graph_integral_bound μ e (edgeSource (S:=S) β) (edgeTarget (S:=S) hS β) |
| (internal_source_ne hS β) (Sum.inl 0) (Sum.inl (Fin.last n)) q d (he q hq) T M hM hT |
| B C hMp (by simpa only [Fintype.card_fin] using hp) hh hC |
| end LogConcaveSampling.AdjointRemoval |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
