| import OAI.Analysis.Brenier.AffineOptimality |
| import OAI.Analysis.Brenier.MapLimits |
| import OAI.Analysis.Brenier.PotentialLimits |
| import OAI.Analysis.Brenier.PrescribedCells |
| import OAI.Analysis.Brenier.Quantization |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory |
| open scoped BigOperators ENNReal RealInnerProductSpace |
|
|
| namespace Problem358 |
|
|
| variable {d : ℕ} {ι : Type*} [Fintype ι] [Nonempty ι] |
| [MeasurableSpace ι] [MeasurableSingletonClass ι] |
|
|
| / |
| strictly positive finite atomic target, with the exact uniform source definition. |
| The proof constructs the cell masses variationally rather than assuming them. -/ |
| theorem exists_unique_optimal_map_uniform_atomic |
| (K : Set (E d)) [IsProbabilityMeasure (uniformMeasure K)] |
| (hK : IsCompact K) (y : ι → E d) (hinj : Function.Injective y) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) : |
| ∃ T : E d → E d, |
| IsUniqueQuadraticOptimalMap (uniformMeasure K) |
| (∑ i, ENNReal.ofReal (m i) • Measure.dirac (y i)) T := by |
| obtain ⟨c, k, hc, hk, hmax, hmass⟩ := |
| MaxPotential.exists_prescribed_cell_masses_uniform K hK y hinj m hm hsum |
| have hy : Measurable y := measurable_of_countable y |
| have hT : Measurable (fun x => y (k x)) := hy.comp hk |
| have hweights : ∀ i, uniformMeasure K (k ⁻¹' {i}) = ENNReal.ofReal (m i) := by |
| intro i |
| have hi : (uniformMeasure K).real (k ⁻¹' {i}) = m i := hmass i |
| rw [← hi] |
| exact (ENNReal.ofReal_toReal (measure_ne_top _ _)).symm |
| have hpush := map_finite_selector_of_masses (uniformMeasure K) k hk y |
| (fun i => ENNReal.ofReal (m i)) hweights |
| have hopt := isUniqueQuadraticOptimalMap_uniform_of_finite_affine K hK y c hinj hT |
| (Filter.Eventually.of_forall (fun x => ⟨k x, rfl, hmax x⟩)) |
| exact ⟨fun x => y (k x), hpush ▸ hopt⟩ |
|
|
| / |
| theorem exists_unique_optimal_map_uniform_finite_supported |
| (K : Set (E d)) (hK : IsCompact K) (hinterior : (interior K).Nonempty) |
| (μ : Measure (E d)) [IsProbabilityMeasure μ] |
| {F : Set (E d)} (hF : F.Finite) (hμ : IsSupported μ F) : |
| ∃ T : E d → E d, IsUniqueQuadraticOptimalMap (uniformMeasure K) μ T := by |
| let : IsProbabilityMeasure (uniformMeasure K) := UniformSource.probability hK hinterior |
| obtain ⟨n, hn, y, m, hy, _, hm, hsum, hrepr⟩ := |
| exists_positive_finite_atomic_presentation hF hμ |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| obtain ⟨T, hT⟩ := exists_unique_optimal_map_uniform_atomic K hK y hy m hm hsum |
| exact ⟨T, hrepr.symm ▸ hT⟩ |
|
|
| end Problem358 |
|
|
| namespace Problem358 |
|
|
| variable {d : ℕ} {ι : Type*} [Fintype ι] [Nonempty ι] |
| [MeasurableSpace ι] [MeasurableSingletonClass ι] |
|
|
| / |
| potential and its pointwise supporting-slope inequality. -/ |
| theorem exists_uniform_atomic_map_and_potential |
| (K : Set (E d)) [IsProbabilityMeasure (uniformMeasure K)] |
| (hK : IsCompact K) (y : ι → E d) (hinj : Function.Injective y) |
| (m : ι → ℝ) (hm : ∀ i, 0 < m i) (hsum : ∑ i, m i = 1) |
| (L : NNReal) (hL : ∀ i, ‖y i‖ ≤ L) (x₀ : E d) : |
| ∃ (u : E d → ℝ) (T : E d → E d), |
| IsUniqueQuadraticOptimalMap (uniformMeasure K) |
| (∑ i, ENNReal.ofReal (m i) • Measure.dirac (y i)) T ∧ |
| ConvexOn ℝ Set.univ u ∧ LipschitzWith L u ∧ u x₀ = 0 ∧ |
| (∀ x, T x ∈ Set.range y) ∧ |
| ∀ x z, u x + ⟪T x, z - x⟫ ≤ u z := by |
| obtain ⟨c, k, hc, hk, hmax, hmass⟩ := |
| MaxPotential.exists_prescribed_cell_masses_uniform K hK y hinj m hm hsum |
| let T : E d → E d := fun x => y (k x) |
| let u : E d → ℝ := fun x => MaxPotential.potential y c x - MaxPotential.potential y c x₀ |
| have hy : Measurable y := measurable_of_countable y |
| have hT : Measurable T := hy.comp hk |
| have hweights : ∀ i, uniformMeasure K (k ⁻¹' {i}) = ENNReal.ofReal (m i) := by |
| intro i |
| have hi : (uniformMeasure K).real (k ⁻¹' {i}) = m i := hmass i |
| rw [← hi] |
| exact (ENNReal.ofReal_toReal (measure_ne_top _ _)).symm |
| have hpush := map_finite_selector_of_masses (uniformMeasure K) k hk y |
| (fun i => ENNReal.ofReal (m i)) hweights |
| have hopt := isUniqueQuadraticOptimalMap_uniform_of_finite_affine K hK y c hinj hT |
| (Filter.Eventually.of_forall (fun x => ⟨k x, rfl, hmax x⟩)) |
| refine ⟨u, T, hpush ▸ hopt, ?_, ?_, ?_, ?_, ?_⟩ |
| · exact (MaxPotential.convexOn_potential y c).sub |
| (concaveOn_const _ convex_univ) |
| · apply LipschitzWith.of_dist_le_mul |
| intro x z |
| simpa only [u, Real.dist_eq, sub_sub_sub_cancel_right] using |
| (MaxPotential.lipschitzWith_potential y c L hL).dist_le_mul x z |
| · simp [u] |
| · intro x |
| exact ⟨k x, rfl⟩ |
| · intro x z |
| have h := MaxPotential.supporting_inequality y c x z (k x) (hmax x) |
| dsimp [u, T] |
| rw [real_inner_comm] at h |
| linarith |
|
|
| / |
| theorem exists_uniform_finite_map_and_potential |
| (K : Set (E d)) (hK : IsCompact K) (hinterior : (interior K).Nonempty) |
| (μ : Measure (E d)) [IsProbabilityMeasure μ] |
| {F Y : Set (E d)} (hF : F.Finite) (hFY : F ⊆ Y) (hμ : IsSupported μ F) |
| (L : NNReal) (hL : ∀ z ∈ Y, ‖z‖ ≤ L) (x₀ : E d) : |
| ∃ (u : E d → ℝ) (T : E d → E d), |
| IsUniqueQuadraticOptimalMap (uniformMeasure K) μ T ∧ |
| ConvexOn ℝ Set.univ u ∧ LipschitzWith L u ∧ u x₀ = 0 ∧ |
| (∀ x, T x ∈ Y) ∧ ∀ x z, u x + ⟪T x, z - x⟫ ≤ u z := by |
| let : IsProbabilityMeasure (uniformMeasure K) := UniformSource.probability hK hinterior |
| obtain ⟨n, hn, y, m, hy, hyF, hm, hsum, hrepr⟩ := |
| exists_positive_finite_atomic_presentation hF hμ |
| let : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| obtain ⟨u, T, hT, hu, hLip, hnorm, hrange, hsupport⟩ := |
| exists_uniform_atomic_map_and_potential K hK y hy m hm hsum L |
| (fun i => hL (y i) (hFY (hyF i))) x₀ |
| refine ⟨u, T, hrepr.symm ▸ hT, hu, hLip, hnorm, ?_, hsupport⟩ |
| intro x |
| obtain ⟨i, hi⟩ := hrange x |
| rw [← hi] |
| exact hFY (hyF i) |
|
|
| end Problem358 |
|
|
| end |
|
|
| noncomputable section |
|
|
| open MeasureTheory Filter |
| open scoped Topology NNReal |
|
|
| namespace Problem358 |
|
|
| / |
| unique optimal map with the prescribed target law. No stability estimate is required here. -/ |
| theorem exists_unique_optimal_map_of_approximating_potentials |
| {d : ℕ} {K Y : Set (E d)} {μ : Measure (E d)} |
| [IsFiniteMeasure (uniformMeasure K)] [IsFiniteMeasure μ] |
| (hK : IsCompact K) (hKconv : Convex ℝ K) (hY : IsCompact Y) |
| (hμ : IsSupported μ Y) (x₀ : E d) (hx₀ : x₀ ∈ K) |
| (uₙ : ℕ → E d → ℝ) (Tₙ Qₙ : ℕ → E d → E d) (L : ℝ≥0) |
| (hLip : ∀ n, LipschitzOnWith L (uₙ n) K) |
| (hconv : ∀ n, ConvexOn ℝ K (uₙ n)) |
| (hnormal : ∀ n, uₙ n x₀ = 0) |
| (hTmeas : ∀ n, Measurable (Tₙ n)) (hQmeas : ∀ n, Measurable (Qₙ n)) |
| (hmem : ∀ n, ∀ᵐ x ∂uniformMeasure K, Tₙ n x ∈ Y) |
| (hs : ∀ n, ∀ᵐ x ∂uniformMeasure K, ∀ z ∈ K, |
| uₙ n x + inner ℝ (Tₙ n x) (z - x) ≤ uₙ n z) |
| (hQlim : ∀ᵐ y ∂μ, Tendsto (fun n => Qₙ n y) atTop (𝓝 y)) |
| (hpush : ∀ n, Measure.map (Tₙ n) (uniformMeasure K) = Measure.map (Qₙ n) μ) : |
| ∃ T : E d → E d, IsUniqueQuadraticOptimalMap (uniformMeasure K) μ T := by |
| obtain ⟨u, T, φ, hφ, _, huLip, _, _, hT, hTlim, _, hsupport⟩ := |
| exists_potential_and_map_limit hK hKconv hY x₀ hx₀ uₙ Tₙ L |
| hLip hconv hnormal hTmeas hmem hs |
| have hQlim' : ∀ᵐ y ∂μ, Tendsto (fun n => Qₙ (φ n) y) atTop (𝓝 y) := by |
| filter_upwards [hQlim] with y hy |
| exact hy.comp hφ.tendsto_atTop |
| have hpushT : Measure.map T (uniformMeasure K) = μ := |
| map_eq_of_ae_limits (fun n => hTmeas (φ n)) hT (fun n => hQmeas (φ n)) |
| hTlim hQlim' (fun n => hpush (φ n)) |
| refine ⟨T, isUniqueQuadraticOptimalMap_uniform_of_supporting_potential |
| hK ⟨x₀, hx₀⟩ hKconv hY hμ huLip hT hpushT ?_⟩ |
| filter_upwards [hsupport] with x hx |
| intro z hz |
| simpa only [real_inner_comm] using hx z hz |
|
|
| / |
| construction for finite-range quantizations of the target. -/ |
| theorem exists_unique_optimal_map_of_simpleFunc_potentials |
| {d : ℕ} {K Y : Set (E d)} {μ : Measure (E d)} |
| [IsFiniteMeasure (uniformMeasure K)] [IsFiniteMeasure μ] |
| (hK : IsCompact K) (hKconv : Convex ℝ K) (hY : IsCompact Y) (hYne : Y.Nonempty) |
| (hμ : IsSupported μ Y) (x₀ : E d) (hx₀ : x₀ ∈ K) (L : ℝ≥0) |
| (hfinite : ∀ q : SimpleFunc (E d) (E d), (∀ x, q x ∈ Y) → |
| ∃ (u : E d → ℝ) (T : E d → E d), LipschitzOnWith L u K ∧ |
| ConvexOn ℝ K u ∧ u x₀ = 0 ∧ Measurable T ∧ |
| (∀ᵐ x ∂uniformMeasure K, T x ∈ Y) ∧ |
| (∀ᵐ x ∂uniformMeasure K, ∀ z ∈ K, u x + inner ℝ (T x) (z - x) ≤ u z) ∧ |
| Measure.map T (uniformMeasure K) = Measure.map q μ) : |
| ∃ T : E d → E d, IsUniqueQuadraticOptimalMap (uniformMeasure K) μ T := by |
| classical |
| have hq : ∀ n : ℕ, ∃ q : SimpleFunc (E d) (E d), (∀ x, q x ∈ Y) ∧ |
| ∀ x ∈ Y, dist (q x) x < (1 : ℝ) / ((n : ℝ) + 1) := by |
| intro n |
| exact exists_compact_quantizer hY hYne (by positivity) |
| choose q hqY hqdist using hq |
| choose u T hLip hconv hnormal hTmeas hmem hs hpush using |
| (fun n => hfinite (q n) (hqY n)) |
| apply exists_unique_optimal_map_of_approximating_potentials hK hKconv hY hμ |
| x₀ hx₀ u T (fun n => q n) L hLip hconv hnormal hTmeas |
| (fun n => (q n).measurable) hmem hs _ hpush |
| filter_upwards [ae_iff.mpr hμ] with y hy |
| apply tendsto_iff_dist_tendsto_zero.mpr |
| exact squeeze_zero (fun n => dist_nonneg) (fun n => (hqdist n y hy).le) |
| tendsto_one_div_add_atTop_nhds_zero_nat |
|
|
| / |
| uniform compact convex sources and arbitrary compactly supported probability targets. -/ |
| theorem exists_unique_optimal_map_uniform_compact |
| {d : ℕ} {K Y : Set (E d)} |
| (hK : IsCompact K) (hKconv : Convex ℝ K) (hInterior : (interior K).Nonempty) |
| (hY : IsCompact Y) (hYne : Y.Nonempty) |
| (μ : Measure (E d)) [IsProbabilityMeasure μ] (hμ : IsSupported μ Y) : |
| ∃ T : E d → E d, IsUniqueQuadraticOptimalMap (uniformMeasure K) μ T := by |
| let : IsProbabilityMeasure (uniformMeasure K) := UniformSource.probability hK hInterior |
| obtain ⟨x₀, hx₀⟩ := hInterior.mono interior_subset |
| obtain ⟨R, hR⟩ := hY.isBounded.exists_norm_le |
| let L : ℝ≥0 := ⟨max R 0, le_max_right _ _⟩ |
| have hL : ∀ y ∈ Y, ‖y‖ ≤ L := fun y hy => (hR y hy).trans (le_max_left _ _) |
| apply exists_unique_optimal_map_of_simpleFunc_potentials hK hKconv hY hYne hμ x₀ hx₀ L |
| intro q hqY |
| let : IsProbabilityMeasure (Measure.map q μ) := |
| (Measure.isProbabilityMeasure_map_iff q.measurable.aemeasurable).mpr inferInstance |
| have hFY : Set.range q ⊆ Y := by |
| rintro y ⟨x, rfl⟩ |
| exact hqY x |
| obtain ⟨u, T, hopt, hu, hLip, hnormal, hTY, hsupport⟩ := |
| exists_uniform_finite_map_and_potential K hK hInterior (Measure.map q μ) |
| q.finite_range hFY (simpleFunc_map_finite_support μ q).2 L hL x₀ |
| exact ⟨u, T, hLip.lipschitzOnWith, hu.subset (Set.subset_univ K) hKconv, |
| hnormal, hopt.1.1, ae_of_all _ hTY, |
| ae_of_all _ (fun x z _ => hsupport x z), hopt.1.2.1⟩ |
|
|
| end Problem358 |
|
|
| end |
|
|
| end OAI |
|
|
