| import OAI.Combinatorics.MatchingEntropy.Continuity |
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| namespace OAI |
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| noncomputable section |
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| open scoped BigOperators Topology |
| open Filter Matrix |
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| universe uV uE uI uJ uA uB uC uW uF uD uι |
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| namespace MatchingEntropyBounds |
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| variable {V : Type uV} {E : Type uE} [Fintype V] [Fintype E] [DecidableEq V] [DecidableEq E] |
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| def matchingNormalized (G : LooplessGraph V E) (x : E → ℝ) : |
| G.Matching → {e |
| normalizedCentered (fun M e => G.indicator M e.val) (fun e => x e.val) |
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| omit [Fintype V] [DecidableEq V] in |
| theorem matchingNormalized_average (G : LooplessGraph V E) {x y : E → ℝ} |
| {q : G.Matching → ℝ} (hq : q ∈ G.feasibleLaws y) (e : {e |
| (∑ M, q M*matchingNormalized G x M e)=(y e.val-x e.val)/Real.sqrt (x e.val) := by |
| simp only [matchingNormalized,normalizedCentered,← mul_div_assoc,mul_sub, |
| ← Finset.sum_div,Finset.sum_sub_distrib,← Finset.sum_mul,hq.1.2,one_mul] |
| rw [show (∑ M, q M*G.indicator M e.val)=y e.val from congrFun hq.2 e.val] |
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| omit [Fintype V] [DecidableEq V] in |
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| theorem midpoint_law_support (G : LooplessGraph V E) {x v : E → ℝ} {a : ℝ} |
| {p q r : G.Matching → ℝ} (hp : p ∈ G.feasibleLaws x) |
| (hopt : entropy p=maxMatchingEntropy G x) |
| (hq : q ∈ G.feasibleLaws (x+a • v)) |
| (hr : r ∈ G.feasibleLaws (x+(-a) • v)) : |
| ∀ M, 0 < q M → 0 < p M := by |
| let w : G.Matching → ℝ := (1/2:ℝ) • q+(1/2:ℝ) • r |
| have hw : w ∈ G.feasibleLaws x := by |
| refine ⟨convex_probabilitySimplex _ hq.1 hr.1 (by norm_num) (by norm_num) (by norm_num),?_⟩ |
| change G.meanLinear ((1/2:ℝ) • q+(1/2:ℝ) • r) = _ |
| rw [map_add,map_smul,map_smul] |
| change (1/2:ℝ) • G.mean q+(1/2:ℝ) • G.mean r = _ |
| rw [hq.2,hr.2] |
| ext e |
| simp only [Pi.add_apply,Pi.smul_apply,smul_eq_mul] |
| ring |
| intro M hM |
| apply matching_optimizer_support G hp hw hopt |
| change 0 < (1/2:ℝ)*q M+(1/2:ℝ)*r M |
| have hh := hr.1.1 M |
| linarith |
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| theorem normalized_face_mem (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) |
| {u : {e |
| (fun e => u e/Real.sqrt (x e.val)) ∈ weightedDirectionSpace p (matchingNormalized G x) := by |
| have hx : x ∈ G.polytope := (G.mem_polytope_iff x).mpr ⟨p,hp⟩ |
| have hev := G.faceSpace_eventually_line (G.polytope_subset_constraintPolytope hx) hu |
| have hneg : ∀ᶠ t : ℝ in 𝓝 0, x+(-t) • LooplessGraph.positiveExtension x u ∈ G.constraintPolytope := by |
| have hc : Tendsto (fun t : ℝ => -t) (𝓝 0) (𝓝 0) := by simpa using continuous_neg.tendsto (0:ℝ) |
| exact hc.eventually hev |
| have hh : ∀ᶠ t : ℝ in 𝓝[>] 0, |
| (x+t • LooplessGraph.positiveExtension x u ∈ G.constraintPolytope ∧ |
| x+(-t) • LooplessGraph.positiveExtension x u ∈ G.constraintPolytope) ∧ 0 < t := |
| ((hev.and hneg).filter_mono nhdsWithin_le_nhds).and eventually_mem_nhdsWithin |
| obtain ⟨a,⟨ha,hna⟩,ha0⟩ := hh.exists |
| rw [← G.polytope_eq_constraints heven] at ha hna |
| obtain ⟨q,hq⟩ := (G.mem_polytope_iff _).mp ha |
| obtain ⟨r,hr⟩ := (G.mem_polytope_iff _).mp hna |
| have hsupp := midpoint_law_support G hp hopt hq hr |
| have hsum : (∑ M, q M • matchingNormalized G x M) ∈ |
| weightedDirectionSpace p (matchingNormalized G x) := by |
| apply Submodule.sum_mem |
| intro M _ |
| by_cases hM : 0 < p M |
| · exact Submodule.smul_mem _ _ (Submodule.subset_span ⟨⟨M,hM⟩,rfl⟩) |
| · have hq0 : q M=0 := le_antisymm (le_of_not_gt fun hqM => hM (hsupp M hqM)) (hq.1.1 M) |
| simp only [hq0,zero_smul] |
| exact Submodule.zero_mem _ |
| have hin := Submodule.smul_mem _ a⁻¹ hsum |
| have he : a⁻¹ • (∑ M, q M • matchingNormalized G x M) = |
| (fun e => u e/Real.sqrt (x e.val)) := by |
| ext e |
| simp only [Pi.smul_apply,Finset.sum_apply,smul_eq_mul] |
| rw [matchingNormalized_average G hq] |
| simp only [Pi.add_apply,Pi.smul_apply,smul_eq_mul,LooplessGraph.positiveExtension_pos] |
| field_simp |
| ring |
| rwa [he] at hin |
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| def normalizeLinear (x : E → ℝ) : |
| ({e |
| toFun := fun u e => u e/Real.sqrt (x e.val) |
| map_add' := by intro u v; ext e; exact add_div _ _ _ |
| map_smul' := by intro a u; ext e; exact mul_div_assoc _ _ _ |
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| omit [DecidableEq E] in |
| theorem sum_positive_support (x f : E → ℝ) (hf : ∀ e, ¬ 0 < x e → f e=0) : |
| (∑ e, f e) = ∑ e : {e |
| classical |
| have hz : (∑ e : {e |
| Finset.sum_eq_zero fun e _ => hf e.val e.property |
| have hh := Fintype.sum_subtype_add_sum_subtype (fun e => 0 < x e) f |
| rw [hz,add_zero] at hh |
| exact hh.symm |
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| def matchingScoreWeights (x : E → ℝ) (u : {e |
| LooplessGraph.positiveExtension x (fun e => u e/Real.sqrt (x e.val)) |
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| omit [Fintype V] [DecidableEq V] in |
| theorem matchingNormalized_score (G : LooplessGraph V E) (x : E → ℝ) |
| (u : {e |
| linearScore G (matchingScoreWeights x u) (∑ e, matchingScoreWeights x u e*x e) M = |
| matchingNormalized G x M ⬝ᵥ u := by |
| classical |
| unfold linearScore |
| rw [← Finset.sum_sub_distrib] |
| rw [sum_positive_support x _ (fun e he => by simp [matchingScoreWeights,LooplessGraph.positiveExtension,he])] |
| rw [dotProduct] |
| apply Finset.sum_congr rfl |
| intro e _ |
| simp only [matchingScoreWeights,LooplessGraph.positiveExtension_pos,matchingNormalized,normalizedCentered] |
| ring |
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| omit [Fintype V] [DecidableEq V] in |
| theorem matchingNormalized_mulVec (G : LooplessGraph V E) {x : E → ℝ} |
| {p : G.Matching → ℝ} (hp : p ∈ G.feasibleLaws x) |
| (u : {e |
| (covariance p (matchingNormalized G x) *ᵥ u) e = |
| G.mean (fun M => p M * linearScore G (matchingScoreWeights x u) |
| (∑ e, matchingScoreWeights x u e*x e) M) e.val / Real.sqrt (x e.val) := by |
| let a := linearScore G (matchingScoreWeights x u) (∑ e, matchingScoreWeights x u e*x e) |
| have hcenter : ∑ M, p M*a M=0 := by |
| dsimp only [a] |
| rw [linearScore_expectation,hp.2,hp.1.2,mul_one,sub_self] |
| calc |
| _ = ∑ M, (p M*a M*G.indicator M e.val-x e.val*(p M*a M))/Real.sqrt (x e.val) := by |
| rw [covariance_mulVec hp.1.1] |
| apply Finset.sum_congr rfl |
| intro M _ |
| rw [← matchingNormalized_score G x u M] |
| change p M*((G.indicator M e.val-x e.val)/Real.sqrt (x e.val))*a M = _ |
| ring |
| _ = _ := by |
| rw [← Finset.sum_div,Finset.sum_sub_distrib,← Finset.mul_sum,hcenter,mul_zero,sub_zero] |
| rfl |
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| end MatchingEntropyBounds |
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| end |
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| end OAI |
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