| import OAI.Analysis.Brenier.FixedMass |
|
|
| namespace OAI |
|
|
| /-! Compactness and uniqueness imply continuous dependence of minimizers. |
| No differentiability or inverse-function theorem is used. -/ |
|
|
| noncomputable section |
| open Set Filter |
| open scoped Topology |
|
|
| namespace Problem358.CompactArgmin |
|
|
| variable {P C I : Type*} [TopologicalSpace P] [TopologicalSpace C] |
|
|
| /-- A compactly constrained family of minimizers converges to the unique minimizer |
| of the limiting continuous objective. The feasible set may be noncompact. -/ |
| theorem tendsto_of_unique_minimizer {l : Filter I} {p : I → P} {p₀ : P} |
| {c : I → C} {c₀ : C} {S K : Set C} {F : P × C → ℝ} |
| (hK : IsCompact K) (hS : IsClosed S) (hF : Continuous F) |
| (hp : Tendsto p l (𝓝 p₀)) (hbounded : ∀ᶠ i in l, c i ∈ K) |
| (hfeasible : ∀ᶠ i in l, c i ∈ S) |
| (hmin : ∀ᶠ i in l, ∀ z ∈ S, F (p i, c i) ≤ F (p i, z)) |
| (hunique : ∀ z ∈ S, IsMinOn (fun w => F (p₀, w)) S z → z = c₀) : |
| Tendsto c l (𝓝 c₀) := by |
| apply hK.tendsto_nhds_of_unique_mapClusterPt hbounded |
| intro z _ hz |
| obtain ⟨U, hUl, hUz⟩ := mapClusterPt_iff_ultrafilter.mp hz |
| have hzS : z ∈ S := hS.mem_of_tendsto hUz (hUl hfeasible) |
| apply hunique z hzS |
| intro w hw |
| have hleft : Tendsto (fun i => F (p i, c i)) (U : Filter I) (𝓝 (F (p₀, z))) := |
| hF.continuousAt.tendsto.comp ((hp.mono_left hUl).prodMk_nhds hUz) |
| have hright : Tendsto (fun i => F (p i, w)) (U : Filter I) (𝓝 (F (p₀, w))) := |
| hF.continuousAt.tendsto.comp ((hp.mono_left hUl).prodMk_nhds tendsto_const_nhds) |
| apply le_of_tendsto_of_tendsto hleft hright |
| filter_upwards [hUl hmin] with i hi |
| exact hi w hw |
|
|
| /-- Local compact bounds and uniqueness give continuity of an arbitrary selected minimizer. -/ |
| theorem continuousAt_unique_minimizer {c : P → C} {p : P} |
| {S K : Set C} {F : P × C → ℝ} |
| (hK : IsCompact K) (hS : IsClosed S) (hF : Continuous F) |
| (hbounded : ∀ᶠ q in 𝓝 p, c q ∈ K) |
| (hfeasible : ∀ᶠ q in 𝓝 p, c q ∈ S) |
| (hmin : ∀ᶠ q in 𝓝 p, ∀ z ∈ S, F (q, c q) ≤ F (q, z)) |
| (hunique : ∀ z ∈ S, IsMinOn (fun w => F (p, w)) S z → z = c p) : |
| ContinuousAt c p := |
| tendsto_of_unique_minimizer hK hS hF tendsto_id hbounded hfeasible hmin hunique |
|
|
| end Problem358.CompactArgmin |
|
|
| end |
|
|
| noncomputable section |
| open Set MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| namespace Problem358.MaxPotential |
|
|
| variable {ι : Type*} [Fintype ι] [Nonempty ι] |
|
|
| /-- Finite affine maxima depend jointly continuously on sites, intercepts, and source points. -/ |
| theorem continuous_joint_potential {d : ℕ} : |
| Continuous (fun p : ((ι → E d) × (ι → ℝ)) × E d => |
| potential p.1.1 p.1.2 p.2) := by |
| have h : Continuous (Finset.univ.sup' Finset.univ_nonempty |
| (fun i (p : ((ι → E d) × (ι → ℝ)) × E d) => ⟪p.2, p.1.1 i⟫ + p.1.2 i)) := by |
| apply Continuous.finset_sup' |
| intro i _ |
| fun_prop |
| convert h using 1 |
| funext p |
| simp only [potential, Finset.sup'_apply] |
| |
| /-- The integrated finite affine objective is jointly continuous in all sites and intercepts. -/ |
| theorem continuous_uniform_integral_potential {d : ℕ} {K : Set (E d)} |
| (hK : IsCompact K) (hne : (interior K).Nonempty) : |
| Continuous (fun p : (ι → E d) × (ι → ℝ) => |
| ∫ x, potential p.1 p.2 x ∂uniformMeasure K) := by |
| let := UniformSource.probability hK hne |
| have h := continuous_parametric_integral_of_continuous |
| (μ := uniformMeasure K) |
| (f := fun (p : (ι → E d) × (ι → ℝ)) (x : E d) => potential p.1 p.2 x) |
| (continuous_joint_potential (ι := ι) (d := d)) hK |
| rwa [Measure.restrict_eq_self_of_ae_mem (UniformSource.ae_mem hK.measurableSet)] at h |
| |
| end Problem358.MaxPotential |
| |
| end |
| |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped RealInnerProductSpace BigOperators Topology |
| |
| namespace Problem358.MaxPotential |
| |
| variable {ι : Type*} [Fintype ι] [Nonempty ι] |
| |
| /-- Any normalized minimizing intercept selection depends continuously on the sites |
| at every injective configuration. This needs neither moving-cell derivatives nor IFT. -/ |
| theorem continuousAt_uniform_intercept_selection {d : ℕ} {K : Set (E d)} |
| (hK : IsCompact K) (hconv : Convex ℝ K) (hne : (interior K).Nonempty) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) |
| (c : (ι → E d) → ι → ℝ) |
| (hc0 : ∀ y, ∑ i, m i * c y i = 0) |
| (hcmin : ∀ y (a : ι → ℝ), (∑ i, m i * a i) = 0 → |
| (∫ x, potential y (c y) x ∂uniformMeasure K) ≤ ∫ x, potential y a x ∂uniformMeasure K) |
| (y : ι → E d) (hy : Function.Injective y) : ContinuousAt c y := by |
| let := UniformSource.probability hK hne |
| let L : NNReal := ⟨‖y‖ + 1, by positivity⟩ |
| let B : ℝ := 2 * (L : ℝ) * ∫ x, ‖x‖ ∂uniformMeasure K |
| let S : Set (ι → ℝ) := {a | ∑ i, m i * a i = 0} |
| let J : Set (ι → ℝ) := Set.Icc (fun i => -B / m i) (fun _ => B) |
| have hS : IsClosed S := isClosed_eq (by fun_prop) continuous_const |
| have hJ : IsCompact J := isCompact_Icc |
| have hnorm : ∀ᶠ z : ι → E d in 𝓝 y, ‖z‖ < (L : ℝ) := by |
| exact continuous_norm.continuousAt.eventually_lt continuousAt_const (by change ‖y‖ < ‖y‖ + 1; linarith) |
| have hbounded : ∀ᶠ z in 𝓝 y, c z ∈ J := by |
| filter_upwards [hnorm] with z hz |
| apply minimizer_mem_box (uniformMeasure K) z (c z) m L |
| (fun i => (norm_le_pi_norm z i).trans hz.le) hm hsum (hc0 z) |
| (UniformSource.integrable_norm hK) |
| exact hcmin z 0 (by simp) |
| apply CompactArgmin.continuousAt_unique_minimizer hJ hS |
| (continuous_uniform_integral_potential (ι := ι) hK hne) hbounded |
| (Filter.Eventually.of_forall hc0) (Filter.Eventually.of_forall hcmin) |
| intro a ha hamin |
| exact uniform_normalized_minimizer_unique hK hconv hne y hy m hm hsum |
| a (c y) ha (hc0 y) hamin (hcmin y) |
| |
| /-- A globally defined normalized minimizer selection exists, and is continuous |
| on the open set of pairwise distinct sites. -/ |
| theorem exists_continuous_uniform_intercept_selection {d : ℕ} {K : Set (E d)} |
| (hK : IsCompact K) (hconv : Convex ℝ K) (hne : (interior K).Nonempty) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) : |
| ∃ c : (ι → E d) → ι → ℝ, |
| (∀ y, ∑ i, m i * c y i = 0) ∧ |
| (∀ y (a : ι → ℝ), (∑ i, m i * a i) = 0 → |
| (∫ x, potential y (c y) x ∂uniformMeasure K) ≤ ∫ x, potential y a x ∂uniformMeasure K) ∧ |
| ContinuousOn c {y | Function.Injective y} := by |
| classical |
| let := UniformSource.probability hK hne |
| have hex : ∀ y : ι → E d, ∃ c : ι → ℝ, |
| (∑ i, m i * c i) = 0 ∧ |
| ∀ a : ι → ℝ, (∑ i, m i * a i) = 0 → |
| (∫ x, potential y c x ∂uniformMeasure K) ≤ ∫ x, potential y a x ∂uniformMeasure K := by |
| intro y |
| obtain ⟨c, hc0, hcmin⟩ := exists_minimum_objective (uniformMeasure K) y m hm hsum |
| ‖y‖₊ (fun i => norm_le_pi_norm y i) (UniformSource.integrable_norm hK) |
| refine ⟨c, hc0, ?_⟩ |
| intro a ha |
| simpa only [objective, hc0, ha, sub_zero] using hcmin a ha |
| choose c hc0 hcmin using hex |
| refine ⟨c, hc0, hcmin, ?_⟩ |
| intro y hy |
| exact (continuousAt_uniform_intercept_selection hK hconv hne m hm hsum c hc0 hcmin y hy).continuousWithinAt |
| |
| end Problem358.MaxPotential |
| |
| end |
| |
| noncomputable section |
| open Set MeasureTheory Filter |
| open scoped BigOperators RealInnerProductSpace Topology |
| |
| namespace Problem358.FixedMassIntercepts |
| |
| variable {d : ℕ} {ι : Type*} [Fintype ι] |
| |
| /-- Distinct finite site configurations form an open subset of the ambient coordinate space. -/ |
| theorem isOpen_injective_sites : IsOpen {y : ι → E d | Function.Injective y} := by |
| have heq : {y : ι → E d | Function.Injective y} = |
| ⋂ i : ι, ⋂ j : ι, {y | i ≠ j → y i ≠ y j} := by |
| ext y |
| simp only [mem_ofPred_eq, mem_iInter] |
| exact Function.injective_iff_pairwise_ne |
| rw [heq] |
| apply isOpen_iInter_of_finite |
| intro i |
| apply isOpen_iInter_of_finite |
| intro j |
| by_cases hij : i = j |
| · simp [hij] |
| · have hset : {y : ι → E d | i ≠ j → y i ≠ y j} = |
| {y : ι → E d | y i = y j}ᶜ := by |
| ext y |
| simp [hij] |
| rw [hset] |
| exact (isClosed_eq (show Continuous (fun y : ι → E d => y i) from continuous_apply i) |
| (show Continuous (fun y : ι → E d => y j) from continuous_apply j)).isOpen_compl |
| |
| variable [Nonempty ι] [MeasurableSpace ι] [MeasurableSingletonClass ι] |
| |
| /-- The actual prescribed intercepts minimize the integrated affine potential. -/ |
| theorem intercepts_minimize {K : Set (E d)} |
| (hK : IsCompact K) (hconv : Convex ℝ K) (hne : (interior K).Nonempty) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) |
| (y : ι → E d) (hy : Function.Injective y) (a : ι → ℝ) |
| (ha : ∑ i, m i * a i = 0) : |
| (∫ x, MaxPotential.potential y (intercepts hK hconv hne m hm hsum y) x |
| ∂uniformMeasure K) ≤ ∫ x, MaxPotential.potential y a x ∂uniformMeasure K := by |
| let := UniformSource.probability hK hne |
| let c := intercepts hK hconv hne m hm hsum y |
| obtain ⟨k, hk, _, hmax⟩ := exists_measurable_argmax |
| (fun i (x : E d) => ⟪x, y i⟫ + (c : ι → ℝ) i) |
| (fun i => ((continuous_id.inner continuous_const).add continuous_const).measurable) |
| apply MaxPotential.isMin_of_cell_masses (uniformMeasure K) y c m k c.property |
| (fun i => hk (measurableSet_singleton i)) (Eventually.of_forall hmax) |
| _ _ a ha |
| · intro i |
| exact (masses_eq_selector hK.measurableSet y hy c k hmax i).symm.trans |
| (congrFun (intercepts_spec hK hconv hne m hm hsum y hy) i) |
| · intro b |
| exact UniformSource.integrable_continuousOn hK |
| (MaxPotential.continuous_potential y b).continuousOn |
| |
| /-- Canonical closed-cell intercepts are continuous at every distinct-site configuration. |
| This compactness argument is independent of the moving-boundary C1 theorem. -/ |
| theorem continuousAt_intercepts_coe {K : Set (E d)} |
| (hK : IsCompact K) (hconv : Convex ℝ K) (hne : (interior K).Nonempty) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) |
| (y : ι → E d) (hy : Function.Injective y) : |
| ContinuousAt (fun z => (intercepts hK hconv hne m hm hsum z : ι → ℝ)) y := by |
| obtain ⟨b, hb0, hbmin, hbcont⟩ := |
| MaxPotential.exists_continuous_uniform_intercept_selection hK hconv hne m hm hsum |
| have heq : ∀ z : ι → E d, Function.Injective z → |
| (intercepts hK hconv hne m hm hsum z : ι → ℝ) = b z := by |
| intro z hz |
| apply MaxPotential.uniform_normalized_minimizer_unique hK hconv hne z hz m hm hsum |
| _ _ (intercepts hK hconv hne m hm hsum z).property (hb0 z) |
| · exact intercepts_minimize hK hconv hne m hm hsum z hz |
| · exact hbmin z |
| have h_nhds : {z : ι → E d | Function.Injective z} ∈ 𝓝 y := |
| isOpen_injective_sites.mem_nhds hy |
| apply (hbcont.continuousAt h_nhds).congr_of_eventuallyEq |
| filter_upwards [h_nhds] with z hz |
| exact heq z hz |
| |
| /-- On repeated-site configurations the chosen extension is exactly zero. -/ |
| theorem intercepts_eq_zero_of_not_injective {K : Set (E d)} |
| (hK : IsCompact K) (hconv : Convex ℝ K) (hne : (interior K).Nonempty) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) |
| (y : ι → E d) (hy : ¬Function.Injective y) : |
| intercepts hK hconv hne m hm hsum y = 0 := by |
| simp only [intercepts, dite_eq_right hy] |
| |
| /-- The actual normalized intercept choice is globally Borel measurable, including |
| at repeated-site configurations, where its value was explicitly set to zero. -/ |
| theorem measurable_intercepts_coe {K : Set (E d)} |
| (hK : IsCompact K) (hconv : Convex ℝ K) (hne : (interior K).Nonempty) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) : |
| Measurable (fun y => (intercepts hK hconv hne m hm hsum y : ι → ℝ)) := by |
| classical |
| let c : (ι → E d) → ι → ℝ := fun y => intercepts hK hconv hne m hm hsum y |
| have hc : ContinuousOn c {y | Function.Injective y} := fun y hy => |
| (continuousAt_intercepts_coe hK hconv hne m hm hsum y hy).continuousWithinAt |
| have hmeas := hc.measurable_piecewise (continuous_const.continuousOn : |
| ContinuousOn (fun _ : ι → E d => (0 : ι → ℝ)) {y | Function.Injective y}ᶜ) |
| isOpen_injective_sites.measurableSet |
| have heq : ({y : ι → E d | Function.Injective y}.piecewise c (fun _ => 0)) = c := by |
| funext y |
| by_cases hy : Function.Injective y |
| · simp only [piecewise, mem_ofPred_eq, ite_eq_left hy] |
| · simp only [piecewise, mem_ofPred_eq, ite_eq_right hy] |
| symm |
| exact congrArg Subtype.val (intercepts_eq_zero_of_not_injective hK hconv hne m hm hsum y hy) |
| rwa [heq] at hmeas |
| |
| end Problem358.FixedMassIntercepts |
| |
| end |
| |
| end OAI |
| |
