| import OAI.Combinatorics.ContingencyTables.Sampling.AllSmallFibres |
| import OAI.Combinatorics.ContingencyTables.Sampling.UnitGraphEnergy |
|
|
| namespace OAI |
|
|
| /-! |
|
|
| Every retained completion then has weight one. The same local graph is |
| sampled by accepting every valid proposal; no reference row is chosen. |
| -/ |
|
|
| namespace ContingencyTables.SmallGraphProfiles |
|
|
| open scoped BigOperators Classical |
| open CompletionCounts CommonBasesFPRAS FiniteExposureVariance |
|
|
| noncomputable section |
| universe u |
| variable {I J : Type u} [Fintype I] [Fintype J] [LinearOrder I] [LinearOrder J] |
| variable (r : I → ℕ) (c : J → ℕ) [Nonempty (PaperStates r c)] |
| local notation "d" => dimensionAllowance (I := I) (J := J) |
| local instance : DecidableEq (PaperStates r c) := Classical.decEq _ |
|
|
| local notation "Adj" => physicalAdjacent r c (d^20) (d^12) (stateCapacity r c) (stateCapacity_small r c) |
|
|
| def unitSmallChain : FiniteChain (PaperStates r c) := |
| MetropolisChain.chain (fun _ => 1) (fun _ => by norm_num) Adj |
| (physicalAdjacent_symmetric r c _ _ (stateCapacity r c) (stateCapacity_small r c)) |
| (smallProposal d) ((smallProposal_bounds d (by unfold dimensionAllowance; omega)).1.le) |
| (5*d^2+1) |
| (physicalAdjacent_degree_allowance r c _ _ (stateCapacity r c) (stateCapacity_small r c)) |
| (smallProposal_degree d (by unfold dimensionAllowance; omega)) |
|
|
| lemma unitSmallChain_pi (z : PaperStates r c) : |
| (unitSmallChain r c).π z=1/(Fintype.card (PaperStates r c):ℝ) := by |
| simp only [unitSmallChain,MetropolisChain.chain,FiniteChain.ofMoves, |
| MetropolisChain.normalizer,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] |
|
|
| lemma unitSmallChain_energy (H : PaperStates r c → ℝ) : |
| (unitSmallChain r c).energy H= |
| (smallProposal d/(Fintype.card (PaperStates r c):ℝ))* |
| ((∑ x,∑ y,if Adj x y then (H x-H y)^2 else 0)/2) := by |
| exact MetropolisChain.unit_chain_energy Adj |
| (physicalAdjacent_symmetric r c _ _ (stateCapacity r c) (stateCapacity_small r c)) |
| (smallProposal d) ((smallProposal_bounds d (by unfold dimensionAllowance; omega)).1.le) |
| (5*d^2+1) |
| (physicalAdjacent_degree_allowance r c _ _ (stateCapacity r c) (stateCapacity_small r c)) |
| (smallProposal_degree d (by unfold dimensionAllowance; omega)) H |
|
|
| omit [Nonempty (PaperStates r c)] in |
| lemma stateGraphEnergy_allSmall (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true) |
| (H : PaperStates r c → ℝ) : |
| stateGraphEnergy r c H=(∑ x,∑ y,if Adj x y then (H x-H y)^2 else 0)/2 := by |
| have he := fullEnergy_physical_eq r c (d^20) (d^12) (stateCapacity r c) (stateCapacity_small r c) H |
| change stateGraphEnergy r c H=(∑ x,∑ y,if Adj x y then |
| min (stateWeight r c x) (stateWeight r c y)*(H x-H y)^2 else 0)/2 at he |
| simpa only [stateWeight_allSmall r c hsmall,min_self,one_mul] using he |
|
|
| omit [LinearOrder I] [LinearOrder J] [Nonempty (PaperStates r c)] in |
| lemma allowance_fourteen [Nonempty I] [Nonempty J] : 14 ≤ d := by |
| have hi := Fintype.card_pos (α := I) |
| have hj := Fintype.card_pos (α := J) |
| unfold dimensionAllowance |
| nlinarith |
|
|
| lemma unitSmallChain_pi_weight |
| (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true) (z : PaperStates r c) : |
| (unitSmallChain r c).π z=stateWeight r c z/(∑ y,stateWeight r c y) := by |
| rw [unitSmallChain_pi] |
| simp only [stateWeight_allSmall r c hsmall,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] |
|
|
| lemma unitSmallChain_energy_lower |
| (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true) |
| (H : PaperStates r c → ℝ) : |
| (smallProposal d)/(2*(∑ z,stateWeight r c z))*stateGraphEnergy r c H ≤ |
| (unitSmallChain r c).energy H := by |
| have hsum : (∑ z,stateWeight r c z)=(Fintype.card (PaperStates r c):ℝ) := by |
| simp only [stateWeight_allSmall r c hsmall,Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one] |
| have hi := unitSmallChain_energy r c H |
| rw [←stateGraphEnergy_allSmall r c hsmall H] at hi |
| rw [hsum] |
| calc |
| _ = ((smallProposal d/(Fintype.card (PaperStates r c):ℝ))*stateGraphEnergy r c H)/2 := by ring |
| _ = (unitSmallChain r c).energy H/2 := by rw [hi] |
| _ ≤ (unitSmallChain r c).energy H := by linarith [(unitSmallChain r c).energy_nonneg H] |
|
|
| /-- The full graph bound includes the empty large rectangle. -/ |
| theorem unitSmallChain_poincare [Nonempty I] [Nonempty J] |
| (hsmall : ∀ i j,FirstPaperPhysicalMarginal.paperSmall r c i j=true) |
| (H : PaperStates r c → ℝ) : |
| (unitSmallChain r c).variance H ≤ (d:ℝ)^160*(unitSmallChain r c).energy H := by |
| let K : ℝ := 770*(d:ℝ)^4*((d^20:ℕ):ℝ)^6 |
| have hβ := (smallProposal_bounds d (by unfold dimensionAllowance; omega)).1 |
| have hv : variance (stateWeight r c) H ≤ K*stateGraphEnergy r c H := |
| firstPaper_physical_variance r c H |
| have hp := chain_poincare_of_weighted_graph (unitSmallChain r c) (stateWeight r c) H |
| (stateWeight_positive r c) (unitSmallChain_pi_weight r c hsmall) |
| (stateGraphEnergy r c H) K (smallProposal d) (by dsimp [K]; positivity) hβ hv |
| (unitSmallChain_energy_lower r c hsmall H) |
| exact hp.trans (mul_le_mul_of_nonneg_right |
| (smallProposal_poincare_budget d allowance_fourteen) ((unitSmallChain r c).energy_nonneg H)) |
|
|
| end |
| end ContingencyTables.SmallGraphProfiles |
|
|
| end OAI |
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|
