| import OAI.Combinatorics.PerfectMatching.WeightedSpectrum |
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| namespace OAI |
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| universe u_1 u_2 |
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| section |
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| noncomputable section |
| open scoped BigOperators Topology |
| open Matrix Filter |
| namespace MatchingEntropy |
| variable {V : Type u_1} {E : Type u_2} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
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| theorem separableDefect_attains_max (G : LooplessGraph V E) |
| (heven : Even (Fintype.card V)) (hPM : Nonempty G.Matching) : |
| ∃ x ∈ G.polytope, ∀ y ∈ G.polytope, |
| separableLower y-maxMatchingEntropy G y ≤ separableLower x-maxMatchingEntropy G x := by |
| have hs : UpperSemicontinuousOn (fun y => separableLower y-maxMatchingEntropy G y) |
| G.polytope := by |
| simpa only [sub_eq_add_neg, Pi.neg_apply] using |
| continuous_separableLower.continuousOn.upperSemicontinuousOn.add |
| (maxMatchingEntropy_lowerSemicontinuousOn G heven).neg |
| have hn : G.polytope.Nonempty := by |
| obtain ⟨M⟩ := hPM |
| exact ⟨G.indicator M,subset_convexHull ℝ _ (Set.mem_range_self M)⟩ |
| exact UpperSemicontinuousOn.exists_isMaxOn hn G.compact_polytope hs |
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| theorem matching_covariance_rank_face (G : LooplessGraph V E) {x : E → ℝ} |
| (heven : Even (Fintype.card V)) {p : G.Matching → ℝ} |
| (hp : p ∈ G.feasibleLaws x) (hopt : entropy p=maxMatchingEntropy G x) : |
| Fintype.card {e // 0 < x e} ≤ G.faceCodimension x+ |
| (covariance p (matchingNormalized G x)).rank := by |
| let W := weightedDirectionSpace p (matchingNormalized G x) |
| let L : G.faceSpace x →ₗ[ℝ] W := |
| ((normalizeLinear x).comp (G.faceSpace x).subtype).codRestrict W |
| (fun u => normalized_face_mem G heven hp hopt u.property) |
| have hi : Function.Injective L := by |
| intro u v huv |
| apply Subtype.ext |
| funext e |
| have hh := congrFun (congrArg Subtype.val huv) e |
| change u.val e/Real.sqrt (x e.val)=v.val e/Real.sqrt (x e.val) at hh |
| exact (div_left_inj' (ne_of_gt (Real.sqrt_pos.mpr e.property))).mp hh |
| have hdim := LinearMap.finrank_le_finrank_of_injective hi |
| have hq := (G.faceSpace x).finrank_quotient_add_finrank |
| rw [covariance_rank hp.1.1] |
| have hq' : G.faceCodimension x+Module.finrank ℝ (G.faceSpace x)=Fintype.card {e // 0 < x e} := by |
| simpa only [LooplessGraph.faceCodimension,Module.finrank_pi,Module.finrank_self,Finset.sum_const, |
| Finset.card_univ,smul_eq_mul,mul_one] using hq |
| change Module.finrank ℝ (G.faceSpace x) ≤ Module.finrank ℝ W at hdim |
| dsimp only [W] at hdim |
| omega |
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| theorem faceCodimension_strict (G : LooplessGraph V E) {x : E → ℝ} |
| (heven : Even (Fintype.card V)) (hn : 0 < Fintype.card V) (hx : x ∈ G.polytope) : |
| G.faceCodimension x < 2*Fintype.card V := by |
| by_cases hn4 : 4 ≤ Fintype.card V |
| · have hb := LooplessGraph.faceCodimension_strong_aux (Fintype.card V) G rfl heven hn4 x |
| (G.polytope_subset_constraintPolytope hx) |
| omega |
| · have hnt : ¬∃ S : Finset V, Odd S.card ∧ 3≤S.card ∧ 3≤Sᶜ.card ∧ G.cutMass x S=1 := by |
| rintro ⟨S,_,hlo,hhi,_⟩ |
| have hh := Finset.card_compl_add_card S |
| omega |
| have hb := G.faceCodimension_no_tight heven hnt |
| omega |
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| omit [Fintype V] [DecidableEq V] in |
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| theorem matching_separable_covariance (G : LooplessGraph V E) {x : E → ℝ} |
| {p : G.Matching → ℝ} (hp : p ∈ G.feasibleLaws x) |
| (hopt : entropy p=maxMatchingEntropy G x) |
| (hmax : ∀ y ∈ G.polytope, separableLower y-maxMatchingEntropy G y ≤ |
| separableLower x-maxMatchingEntropy G x) |
| (u : {e // 0 < x e} → ℝ) : |
| u ⬝ᵥ (covariance p (matchingNormalized G x) *ᵥ u) ≤ |
| ∑ e, (1-3*x e.val/(1-x e.val))*((covariance p (matchingNormalized G x) *ᵥ u) e)^2 := by |
| let w := matchingScoreWeights x u |
| let c := ∑ e, w e*x e |
| let a := linearScore G w c |
| let v := G.mean (fun M => p M*a M) |
| have hx : x ∈ G.polytope := (G.mem_polytope_iff x).mpr ⟨p,hp⟩ |
| have hcenter : ∑ M, p M*a M=0 := by |
| dsimp only [a,c] |
| rw [linearScore_expectation,hp.2,hp.1.2,mul_one,sub_self] |
| have hh := matching_separable_second_variation G hp hopt hmax w c hcenter |
| have hzero : ∀ e, ¬0 < x e → v e=0 := by |
| intro e he |
| have he0 : x e=0 := le_antisymm (le_of_not_gt he) (G.coordinate_mem_Icc hx e).1 |
| exact mean_score_support G hp.1 a e (by rw [hp.2,he0]) |
| have hnorm : (∑ e, (v e)^2/x e) = |
| ∑ e, ((covariance p (matchingNormalized G x) *ᵥ u) e)^2 := by |
| rw [sum_positive_support x _ (fun e he => by rw [hzero e he]; simp)] |
| apply Finset.sum_congr rfl |
| intro e _ |
| rw [matchingNormalized_mulVec G hp,div_pow,Real.sq_sqrt e.property.le] |
| have hbar : (∑ e, (v e)^2/(1-x e)) = |
| ∑ e, x e.val/(1-x e.val)*((covariance p (matchingNormalized G x) *ᵥ u) e)^2 := by |
| rw [sum_positive_support x _ (fun e he => by rw [hzero e he]; simp)] |
| apply Finset.sum_congr rfl |
| intro e _ |
| rw [matchingNormalized_mulVec G hp,div_pow,Real.sq_sqrt e.property.le] |
| change (v e.val)^2/(1-x e.val)=x e.val/(1-x e.val)*((v e.val)^2/x e.val) |
| field_simp [ne_of_gt e.property] |
| have hquad : (∑ M, p M*(a M)^2)=u ⬝ᵥ (covariance p (matchingNormalized G x) *ᵥ u) := by |
| rw [covariance_quadratic hp.1.1] |
| apply Finset.sum_congr rfl |
| intro M _ |
| rw [← matchingNormalized_score G x u M] |
| change -(∑ e, (v e)^2/x e)+3*(∑ e, (v e)^2/(1-x e))+(∑ M,p M*(a M)^2) ≤ 0 at hh |
| rw [hnorm,hbar,hquad] at hh |
| have heq : (∑ e, (1-3*x e.val/(1-x e.val))*((covariance p (matchingNormalized G x) *ᵥ u) e)^2) = |
| (∑ e, ((covariance p (matchingNormalized G x) *ᵥ u) e)^2)- |
| 3*(∑ e, x e.val/(1-x e.val)*((covariance p (matchingNormalized G x) *ᵥ u) e)^2) := by |
| simp_rw [sub_mul,one_mul,Finset.sum_sub_distrib,Finset.mul_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro e _ |
| ring |
| rw [heq] |
| linarith |
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| omit [Fintype V] [DecidableEq V] in |
| theorem matching_covariance_diagonal (G : LooplessGraph V E) {x : E → ℝ} |
| {p : G.Matching → ℝ} (hp : p ∈ G.feasibleLaws x) (e : {e // 0<x e}) : |
| covariance p (matchingNormalized G x) e e=1-x e.val := by |
| rw [covariance_entry hp.1.1] |
| have hh := normalizedCentered_variance hp.1.2 |
| (fun M (e : {e // 0<x e}) => G.indicator M e.val) |
| (by intro M e; simp only [LooplessGraph.indicator]; split <;> norm_num) |
| (fun e => x e.val) (fun e => congrFun hp.2 e.val) (fun e => e.property) e |
| convert hh using 1 |
| apply Finset.sum_congr rfl |
| intro M _ |
| change p M*matchingNormalized G x M e*matchingNormalized G x M e= |
| p M*(matchingNormalized G x M e)^2 |
| ring |
| |
| theorem separable_max_has_forced (G : LooplessGraph V E) {x : E → ℝ} |
| (heven : Even (Fintype.card V)) (hn : 0 < Fintype.card V) (hx : x ∈ G.polytope) |
| (hmax : ∀ y ∈ G.polytope, separableLower y-maxMatchingEntropy G y ≤ |
| separableLower x-maxMatchingEntropy G x) : ∃ e, x e=1 := by |
| by_contra hforced |
| push Not at hforced |
| obtain ⟨p,hp,hopt⟩ := maxMatchingEntropy_attained G hx |
| have hsecond := covariance_weighted_rank (covariance_posSemidef p (matchingNormalized G x)) |
| (fun e => 1-3*x e.val/(1-x e.val)) (matching_separable_covariance G hp hopt hmax) |
| have htrace : (∑ e : {e // 0<x e}, (1-3*x e.val/(1-x e.val))* |
| covariance p (matchingNormalized G x) e e) = Fintype.card {e // 0<x e}-2*(Fintype.card V:ℝ) := by |
| have he (e : {e // 0<x e}) : (1-3*x e.val/(1-x e.val))* |
| covariance p (matchingNormalized G x) e e=1-4*x e.val := by |
| rw [matching_covariance_diagonal G hp] |
| have hh : 1-x e.val ≠ 0 := sub_ne_zero.mpr (Ne.symm (hforced e.val)) |
| field_simp |
| ring |
| simp_rw [he] |
| rw [Finset.sum_sub_distrib,← Finset.mul_sum] |
| obtain ⟨r,hr⟩ := heven |
| have hv : Fintype.card V=2*r := by omega |
| have hs : (∑ e : {e // 0<x e}, x e.val)=(r:ℝ) := by |
| rw [← G.marginal_sum hv hx] |
| exact (sum_positive_support x x (fun e he => |
| le_antisymm (le_of_not_gt he) (G.coordinate_mem_Icc hx e).1)).symm |
| rw [hs,hv] |
| simp only [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,mul_one,Nat.cast_mul,Nat.cast_ofNat] |
| ring |
| rw [htrace] at hsecond |
| have hrank := matching_covariance_rank_face G heven hp hopt |
| have hstrict := faceCodimension_strict G heven hn hx |
| have hrankR : (Fintype.card {e // 0<x e}:ℝ) ≤ G.faceCodimension x+ |
| (covariance p (matchingNormalized G x)).rank := by exact_mod_cast hrank |
| have hstrictR : (G.faceCodimension x:ℝ)<2*(Fintype.card V:ℝ) := by exact_mod_cast hstrict |
| linarith |
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| end MatchingEntropy |
| end |
| end |
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| end OAI |
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