| import Mathlib |
| import OAI.Probability.GaussianInformation.Kernel.Model |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Metric Set |
| open scoped ENNReal |
| namespace KernelPotential |
| open Filter |
| open scoped Topology NNReal |
| namespace Geometry |
| section Cover |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| lemma card_separated_le (s : Finset E) {R δ : ℝ} (hR : 0 ≤ R) (hδ : 0 < δ) |
| (hs : ∀ c ∈ s, ‖c‖ ≤ R) |
| (hsep : ∀ c ∈ s, ∀ d ∈ s, c ≠ d → δ ≤ ‖c - d‖) : |
| (s.card : ℝ) ≤ ((2 * R + δ) / δ) ^ Module.finrank ℝ E := by |
| borelize E |
| let μ : Measure E := Measure.addHaar |
| have hsmall : 0 < δ / 2 := by positivity |
| have hlarge : 0 < R + δ / 2 := by positivity |
| let A := ⋃ c ∈ s, ball c (δ / 2) |
| have hdisj : (s : Set E).PairwiseDisjoint (fun c => ball c (δ / 2)) := by |
| intro c hc d hd hcd |
| apply ball_disjoint_ball |
| rw [dist_eq_norm] |
| convert hsep c hc d hd hcd using 1 |
| ring |
| have hsub : A ⊆ ball (0 : E) (R + δ / 2) := by |
| refine iUnion₂_subset fun x hx => ball_subset_ball' ?_ |
| rw [dist_zero_right] |
| linarith [hs x hx] |
| have hb : (s.card : ℝ≥0∞) * ENNReal.ofReal ((δ / 2)^Module.finrank ℝ E) * |
| μ (ball 0 1) ≤ ENNReal.ofReal ((R + δ / 2)^Module.finrank ℝ E) * μ (ball 0 1) := by |
| calc |
| _ = μ A := by |
| rw [measure_biUnion_finset hdisj (fun _ _ => measurableSet_ball)] |
| simp only [μ.addHaar_ball_of_pos _ hsmall, Finset.sum_const, nsmul_eq_mul, mul_assoc] |
| _ ≤ μ (ball 0 (R + δ / 2)) := measure_mono hsub |
| _ = _ := μ.addHaar_ball_of_pos _ hlarge |
| have hb' := (ENNReal.mul_le_mul_iff_left (measure_ball_pos μ (0 : E) zero_lt_one).ne' |
| measure_ball_lt_top.ne).mp hb |
| have hbR : (s.card : ℝ) * (δ / 2)^Module.finrank ℝ E ≤ |
| (R + δ / 2)^Module.finrank ℝ E := by |
| simpa [ENNReal.toReal_ofReal (pow_nonneg hsmall.le _)] using |
| ENNReal.toReal_le_of_le_ofReal (pow_nonneg hlarge.le _) hb' |
| calc |
| (s.card : ℝ) ≤ (R + δ / 2)^Module.finrank ℝ E / (δ / 2)^Module.finrank ℝ E := |
| (le_div_iff₀ (pow_pos hsmall _)).mpr hbR |
| _ = ((2 * R + δ) / δ)^Module.finrank ℝ E := by |
| rw [← div_pow] |
| congr 1 |
| field_simp |
|
|
| lemma exists_ball_cover_card_le {R δ : ℝ} (hR : 0 ≤ R) (hδ : 0 < δ) : |
| ∃ s : Finset E, (∀ c ∈ s, ‖c‖ ≤ R) ∧ |
| (∀ x : E, ‖x‖ ≤ R → ∃ c ∈ s, dist x c ≤ δ) ∧ |
| (s.card : ℝ) ≤ ((2 * R + δ) / δ)^Module.finrank ℝ E := by |
| classical |
| let ε : ℝ≥0 := ⟨δ, hδ.le⟩ |
| let A := closedBall (0 : E) R |
| have hεpos : 0 < ε := hδ |
| have hε : ε ≠ 0 := ne_of_gt hεpos |
| obtain ⟨N, _, hNfin, hNcov⟩ := Metric.exists_finite_isCover_of_isCompact |
| (s := A) (ε := ε / 2) (by positivity) (isCompact_closedBall _ _) |
| have hp : Metric.packingNumber ε A < ⊤ := by |
| calc |
| _ = Metric.packingNumber (2 * (ε / 2)) A := by congr 1; ring |
| _ ≤ Metric.externalCoveringNumber (ε / 2) A := |
| Metric.packingNumber_two_mul_le_externalCoveringNumber _ _ |
| _ ≤ N.encard := hNcov.externalCoveringNumber_le_encard |
| _ < ⊤ := hNfin.encard_lt_top |
| let S := Metric.maximalSeparatedSet ε A |
| have hSfin : S.Finite := Set.encard_lt_top_iff.mp (by |
| change (Metric.maximalSeparatedSet ε A).encard < ⊤ |
| rw [Metric.encard_maximalSeparatedSet hp.ne] |
| exact hp) |
| have hSsub : ∀ c ∈ hSfin.toFinset, ‖c‖ ≤ R := by |
| intro c hc |
| have : c ∈ S := by simpa using hc |
| simpa [A] using (Metric.maximalSeparatedSet_subset this) |
| refine ⟨hSfin.toFinset, hSsub, ?_, card_separated_le _ hR hδ hSsub ?_⟩ |
| · intro x hx |
| have hcov := Metric.isCover_iff_subset_iUnion_closedBall.mp |
| (Metric.isCover_maximalSeparatedSet hp.ne) |
| have hm : x ∈ A := by simpa [A] using hx |
| obtain ⟨c, hc, hxc⟩ := mem_iUnion₂.mp (hcov hm) |
| exact ⟨c, by simpa using hc, hxc⟩ |
| · intro c hc d hd hcd |
| have hs := Metric.isSeparated_maximalSeparatedSet (ε := ε) (A := A) |
| (by simpa using hc) (by simpa using hd) hcd |
| have ht := ENNReal.toReal_mono (edist_ne_top c d) hs.le |
| change δ ≤ (edist c d).toReal at ht |
| simpa [ε, edist_dist, dist_eq_norm] using ht |
|
|
| end Cover |
|
|
| section PlaneCover |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] |
|
|
| lemma exists_affine_tube_cover (V : AffineSubspace ℝ E) (hV : (V : Set E).Nonempty) |
| {R δ : ℝ} (hR : 0 ≤ R) (hδ : 0 < δ) : |
| ∃ s : Finset E, |
| (∀ x : E, ‖x‖ ≤ R → infDist x (V : Set E) ≤ δ → ∃ c ∈ s, dist x c ≤ 2 * δ) ∧ |
| (s.card : ℝ) ≤ ((4 * R + 5 * δ) / δ)^Module.finrank ℝ V.direction := by |
| classical |
| by_cases hempty : {x : E | ‖x‖ ≤ R ∧ infDist x (V : Set E) ≤ δ}.Nonempty |
| · obtain ⟨x₀, hx₀n, hx₀d⟩ := hempty |
| obtain ⟨v₀, hv₀V, hv₀d⟩ := V.closed_of_finiteDimensional.exists_infDist_eq_dist hV x₀ |
| have hn₀ : ‖v₀‖ ≤ R + δ := by |
| have ht := dist_triangle v₀ x₀ 0 |
| rw [dist_zero_right, dist_zero_right, dist_comm v₀ x₀, ← hv₀d] at ht |
| linarith |
| obtain ⟨s, _, hcover, hcard⟩ := exists_ball_cover_card_le (E := V.direction) |
| (R := 2 * (R + δ)) (δ := δ) (by positivity) hδ |
| let f : V.direction ↪ E := ⟨fun v => (v : E) + v₀, by |
| intro v w h |
| exact Subtype.ext (add_right_cancel h)⟩ |
| refine ⟨s.map f, ?_, ?_⟩ |
| · intro x hxn hxd |
| obtain ⟨v, hvV, hvd⟩ := V.closed_of_finiteDimensional.exists_infDist_eq_dist hV x |
| have hnv : ‖v‖ ≤ R + δ := by |
| have ht := dist_triangle v x 0 |
| rw [dist_zero_right, dist_zero_right, dist_comm v x, ← hvd] at ht |
| linarith |
| let u : V.direction := ⟨v - v₀, V.vsub_mem_direction hvV hv₀V⟩ |
| have hnu : ‖u‖ ≤ 2 * (R + δ) := by |
| change ‖v - v₀‖ ≤ _ |
| exact (norm_sub_le v v₀).trans (by linarith) |
| obtain ⟨a, ha, hu⟩ := hcover u hnu |
| refine ⟨f a, Finset.mem_map.mpr ⟨a, ha, rfl⟩, ?_⟩ |
| have he : dist v (f a) = dist u a := by |
| rw [dist_eq_norm, dist_eq_norm] |
| change ‖v - ((a : E) + v₀)‖ = ‖(v - v₀) - (a : E)‖ |
| congr 1 |
| abel |
| calc |
| dist x (f a) ≤ dist x v + dist v (f a) := dist_triangle _ _ _ |
| _ ≤ δ + δ := add_le_add (hvd ▸ hxd) (he ▸ hu) |
| _ = 2 * δ := by ring |
| · simpa only [Finset.card_map, show 2 * (2 * (R + δ)) + δ = 4 * R + 5 * δ by ring] using hcard |
| · refine ⟨∅, ?_, ?_⟩ |
| · intro x hxn hxd |
| exact (hempty ⟨x, hxn, hxd⟩).elim |
| · simp only [Finset.card_empty, Nat.cast_zero] |
| positivity |
|
|
| end PlaneCover |
|
|
| end Geometry |
| end KernelPotential |
|
|
| end OAI |
|
|
