Datasets:

roseking
/

openAI-math-dataset

import OAI.Probability.GaussianRegression.Model

namespace OAI

/-!
Local mass bounds and quantitative finite Euclidean nets.
-/

open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology

noncomputable section

universe u

namespace NoiselessRegression

section FrostmanGeometry

variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

def HasLocalMassBound (ν : Measure E) (a : ℝ) : Prop :=
  ∀ t ∈ ν.support, ∀ r : ℝ, 0 < r →
    ν (Metric.closedBall t r) ≤ ENNReal.ofReal (r ^ a)

omit [BorelSpace E] in

theorem HasLocalMassBound.any_center {ν : Measure E} {a : ℝ}
    (hν : HasLocalMassBound ν a) (z : E) {r : ℝ} (hr : 0 < r) :
    ν (Metric.closedBall z r) ≤ ENNReal.ofReal ((2 * r) ^ a) := by
  by_cases hmass : ν (Metric.closedBall z r) = 0
  · simp [hmass]
  obtain ⟨t, htball, htsupp⟩ := ν.nonempty_inter_support_of_pos (pos_iff_ne_zero.mpr hmass)
  refine (measure_mono ?_).trans (hν t htsupp (2 * r) (by positivity))
  intro x hx
  apply Metric.mem_closedBall.mpr
  calc
    dist x t ≤ dist x z + dist z t := dist_triangle _ _ _
    _ ≤ r + r := add_le_add (Metric.mem_closedBall.mp hx)
      (by simpa [dist_comm] using Metric.mem_closedBall.mp htball)
    _ = 2 * r := by ring

theorem card_le_volume_ratio (s : Finset E) {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R)
    (hs : ∀ x ∈ s, ‖x‖ ≤ R)
    (hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) :
    (s.card : ℝ) ≤ ((R + r / 2) / (r / 2)) ^ Module.finrank ℝ E := by
  classical
  let ν : Measure E := Measure.addHaar
  let δ : ℝ := r / 2
  let ρ : ℝ := R + r / 2
  have hδ : 0 < δ := by dsimp [δ]; positivity
  have hρ : 0 < ρ := by dsimp [ρ]; positivity
  let A : Set E := ⋃ c ∈ s, Metric.ball c δ
  have hd : Set.Pairwise (s : Set E) (fun c d => Disjoint (Metric.ball c δ) (Metric.ball d δ)) := by
    intro c hc d hd hcd
    apply Metric.ball_disjoint_ball
    simpa [δ] using hsep c hc d hd hcd
  have hsub : A ⊆ Metric.ball (0 : E) ρ := by
    refine iUnion₂_subset fun x hx => ?_
    apply Metric.ball_subset_ball'
    dsimp [δ, ρ]
    rw [dist_zero_right]
    linarith [hs x hx]
  have I :
      (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) ≤
        ENNReal.ofReal (ρ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) := by
    calc
      (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) = ν A := by
        dsimp only [A]
        rw [measure_biUnion_finset hd (fun _ _ => Metric.isOpen_ball.measurableSet)]
        simp only [ν.addHaar_ball_of_pos _ hδ, Finset.sum_const, nsmul_eq_mul, mul_assoc]
      _ ≤ ν (Metric.ball (0 : E) ρ) := measure_mono hsub
      _ = _ := ν.addHaar_ball_of_pos 0 hρ
  have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) ≤
      ENNReal.ofReal (ρ ^ Module.finrank ℝ E) :=
    (ENNReal.mul_le_mul_iff_left (Metric.measure_ball_pos ν 0 zero_lt_one).ne'
      measure_ball_lt_top.ne).mp I
  have K : (s.card : ℝ) * δ ^ Module.finrank ℝ E ≤ ρ ^ Module.finrank ℝ E := by
    simpa only [ENNReal.toReal_mul, ENNReal.toReal_natCast,
      ENNReal.toReal_ofReal (pow_nonneg hδ.le _)] using
      ENNReal.toReal_le_of_le_ofReal (pow_nonneg hρ.le _) J
  rw [div_pow]
  exact (le_div_iff₀ (pow_pos hδ _)).mpr K

end FrostmanGeometry

section FiniteNets

variable {E : Type*} [PseudoMetricSpace E]

theorem exists_net_of_finite_packing_bound (A : Set E) {r : ℝ} (hr : 0 ≤ r) (N : ℕ)
    (hcard : ∀ s : Finset E, (∀ x ∈ s, x ∈ A) →
      (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → s.card ≤ N) :
    ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧
      (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧
      ∀ x ∈ A, ∃ y ∈ s, dist x y ≤ r := by
  classical
  let S : Set ℕ := {n | ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧
    (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ s.card = n}
  have hS : S.Finite := (Set.finite_Iic N).subset (by
    rintro n ⟨s, hs, hsep, rfl⟩
    exact hcard s hs hsep)
  have hS0 : S.Nonempty := ⟨0, ∅, by simp⟩
  obtain ⟨n, hn, hmax⟩ := Set.exists_max_image S id hS hS0
  obtain ⟨s, hs, hsep, rfl⟩ := hn
  refine ⟨s, hs, hsep, ?_⟩
  intro x hx
  by_contra! hnot
  have hxs : x ∉ s := by
    intro h
    have := hnot x h
    simpa using lt_of_le_of_lt hr this
  have hins : ∀ y ∈ insert x s, y ∈ A := by
    intro y hy
    rcases Finset.mem_insert.mp hy with rfl | hy
    · exact hx
    · exact hs y hy
  have hsep' : ∀ y ∈ insert x s, ∀ z ∈ insert x s, y ≠ z → r < dist y z := by
    intro y hy z hz hne
    rcases Finset.mem_insert.mp hy with hxy | hys
    · subst y
      rcases Finset.mem_insert.mp hz with hxz | hzs
      · exact (hne hxz.symm).elim
      · exact hnot z hzs
    · rcases Finset.mem_insert.mp hz with hxz | hzs
      · subst z
        simpa [dist_comm] using hnot y hys
      · exact hsep y hys z hzs hne
  have hinS : (insert x s).card ∈ S := ⟨insert x s, hins, hsep', rfl⟩
  have hh := hmax _ hinS
  simp only [id_eq, Finset.card_insert_of_notMem hxs] at hh
  omega

end FiniteNets

variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

theorem exists_unit_ball_net {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) :
    ∃ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) ∧
      (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E ∧
      ∀ x : E, ‖x‖ ≤ 1 → ∃ y ∈ s, dist x y ≤ r := by
  classical
  have hbound : ∀ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) →
      (∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) →
      (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E := by
    intro s hs hsep
    refine (card_le_volume_ratio s hr zero_le_one hs
      (fun x hx y hy hne => (hsep x hx y hy hne).le)).trans ?_
    apply pow_le_pow_left₀ (by positivity)
    apply (div_le_div_iff₀ (by positivity : 0 < r / 2) hr).mpr
    nlinarith
  obtain ⟨s, hs, hsep, hnet⟩ := exists_net_of_finite_packing_bound
    {x : E | ‖x‖ ≤ 1} hr.le ⌈(3 / r) ^ Module.finrank ℝ E⌉₊
    (by
      intro s hs hsep
      exact_mod_cast (hbound s hs hsep).trans (Nat.le_ceil _))
  exact ⟨s, hs, hbound s hs hsep, hnet⟩

end NoiselessRegression

end

end OAI

	import OAI.Probability.GaussianRegression.Model

	namespace OAI

	/-!
	Local mass bounds and quantitative finite Euclidean nets.
	-/

	open MeasureTheory ProbabilityTheory Set Filter
	open scoped ENNReal NNReal RealInnerProductSpace Topology

	noncomputable section

	universe u

	namespace NoiselessRegression

	section FrostmanGeometry

	variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
	[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

	def HasLocalMassBound (ν : Measure E) (a : ℝ) : Prop :=
	∀ t ∈ ν.support, ∀ r : ℝ, 0 < r →
	ν (Metric.closedBall t r) ≤ ENNReal.ofReal (r ^ a)

	omit [BorelSpace E] in

	theorem HasLocalMassBound.any_center {ν : Measure E} {a : ℝ}
	(hν : HasLocalMassBound ν a) (z : E) {r : ℝ} (hr : 0 < r) :
	ν (Metric.closedBall z r) ≤ ENNReal.ofReal ((2 * r) ^ a) := by
	by_cases hmass : ν (Metric.closedBall z r) = 0
	· simp [hmass]
	obtain ⟨t, htball, htsupp⟩ := ν.nonempty_inter_support_of_pos (pos_iff_ne_zero.mpr hmass)
	refine (measure_mono ?_).trans (hν t htsupp (2 * r) (by positivity))
	intro x hx
	apply Metric.mem_closedBall.mpr
	calc
	dist x t ≤ dist x z + dist z t := dist_triangle _ _ _
	_ ≤ r + r := add_le_add (Metric.mem_closedBall.mp hx)
	(by simpa [dist_comm] using Metric.mem_closedBall.mp htball)
	_ = 2 * r := by ring

	theorem card_le_volume_ratio (s : Finset E) {r R : ℝ} (hr : 0 < r) (hR : 0 ≤ R)
	(hs : ∀ x ∈ s, ‖x‖ ≤ R)
	(hsep : ∀ x ∈ s, ∀ y ∈ s, x ≠ y → r ≤ dist x y) :
	(s.card : ℝ) ≤ ((R + r / 2) / (r / 2)) ^ Module.finrank ℝ E := by
	classical
	let ν : Measure E := Measure.addHaar
	let δ : ℝ := r / 2
	let ρ : ℝ := R + r / 2
	have hδ : 0 < δ := by dsimp [δ]; positivity
	have hρ : 0 < ρ := by dsimp [ρ]; positivity
	let A : Set E := ⋃ c ∈ s, Metric.ball c δ
	have hd : Set.Pairwise (s : Set E) (fun c d => Disjoint (Metric.ball c δ) (Metric.ball d δ)) := by
	intro c hc d hd hcd
	apply Metric.ball_disjoint_ball
	simpa [δ] using hsep c hc d hd hcd
	have hsub : A ⊆ Metric.ball (0 : E) ρ := by
	refine iUnion₂_subset fun x hx => ?_
	apply Metric.ball_subset_ball'
	dsimp [δ, ρ]
	rw [dist_zero_right]
	linarith [hs x hx]
	have I :
	(s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) ≤
	ENNReal.ofReal (ρ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) := by
	calc
	(s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) * ν (Metric.ball 0 1) = ν A := by
	dsimp only [A]
	rw [measure_biUnion_finset hd (fun _ _ => Metric.isOpen_ball.measurableSet)]
	simp only [ν.addHaar_ball_of_pos _ hδ, Finset.sum_const, nsmul_eq_mul, mul_assoc]
	_ ≤ ν (Metric.ball (0 : E) ρ) := measure_mono hsub
	_ = _ := ν.addHaar_ball_of_pos 0 hρ
	have J : (s.card : ℝ≥0∞) * ENNReal.ofReal (δ ^ Module.finrank ℝ E) ≤
	ENNReal.ofReal (ρ ^ Module.finrank ℝ E) :=
	(ENNReal.mul_le_mul_iff_left (Metric.measure_ball_pos ν 0 zero_lt_one).ne'
	measure_ball_lt_top.ne).mp I
	have K : (s.card : ℝ) * δ ^ Module.finrank ℝ E ≤ ρ ^ Module.finrank ℝ E := by
	simpa only [ENNReal.toReal_mul, ENNReal.toReal_natCast,
	ENNReal.toReal_ofReal (pow_nonneg hδ.le _)] using
	ENNReal.toReal_le_of_le_ofReal (pow_nonneg hρ.le _) J
	rw [div_pow]
	exact (le_div_iff₀ (pow_pos hδ _)).mpr K

	end FrostmanGeometry

	section FiniteNets

	variable {E : Type*} [PseudoMetricSpace E]

	theorem exists_net_of_finite_packing_bound (A : Set E) {r : ℝ} (hr : 0 ≤ r) (N : ℕ)
	(hcard : ∀ s : Finset E, (∀ x ∈ s, x ∈ A) →
	(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) → s.card ≤ N) :
	∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧
	(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧
	∀ x ∈ A, ∃ y ∈ s, dist x y ≤ r := by
	classical
	let S : Set ℕ := {n \| ∃ s : Finset E, (∀ x ∈ s, x ∈ A) ∧
	(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) ∧ s.card = n}
	have hS : S.Finite := (Set.finite_Iic N).subset (by
	rintro n ⟨s, hs, hsep, rfl⟩
	exact hcard s hs hsep)
	have hS0 : S.Nonempty := ⟨0, ∅, by simp⟩
	obtain ⟨n, hn, hmax⟩ := Set.exists_max_image S id hS hS0
	obtain ⟨s, hs, hsep, rfl⟩ := hn
	refine ⟨s, hs, hsep, ?_⟩
	intro x hx
	by_contra! hnot
	have hxs : x ∉ s := by
	intro h
	have := hnot x h
	simpa using lt_of_le_of_lt hr this
	have hins : ∀ y ∈ insert x s, y ∈ A := by
	intro y hy
	rcases Finset.mem_insert.mp hy with rfl \| hy
	· exact hx
	· exact hs y hy
	have hsep' : ∀ y ∈ insert x s, ∀ z ∈ insert x s, y ≠ z → r < dist y z := by
	intro y hy z hz hne
	rcases Finset.mem_insert.mp hy with hxy \| hys
	· subst y
	rcases Finset.mem_insert.mp hz with hxz \| hzs
	· exact (hne hxz.symm).elim
	· exact hnot z hzs
	· rcases Finset.mem_insert.mp hz with hxz \| hzs
	· subst z
	simpa [dist_comm] using hnot y hys
	· exact hsep y hys z hzs hne
	have hinS : (insert x s).card ∈ S := ⟨insert x s, hins, hsep', rfl⟩
	have hh := hmax _ hinS
	simp only [id_eq, Finset.card_insert_of_notMem hxs] at hh
	omega

	end FiniteNets

	variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
	[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

	theorem exists_unit_ball_net {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) :
	∃ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) ∧
	(s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E ∧
	∀ x : E, ‖x‖ ≤ 1 → ∃ y ∈ s, dist x y ≤ r := by
	classical
	have hbound : ∀ s : Finset E, (∀ x ∈ s, ‖x‖ ≤ 1) →
	(∀ x ∈ s, ∀ y ∈ s, x ≠ y → r < dist x y) →
	(s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ E := by
	intro s hs hsep
	refine (card_le_volume_ratio s hr zero_le_one hs
	(fun x hx y hy hne => (hsep x hx y hy hne).le)).trans ?_
	apply pow_le_pow_left₀ (by positivity)
	apply (div_le_div_iff₀ (by positivity : 0 < r / 2) hr).mpr
	nlinarith
	obtain ⟨s, hs, hsep, hnet⟩ := exists_net_of_finite_packing_bound
	{x : E \| ‖x‖ ≤ 1} hr.le ⌈(3 / r) ^ Module.finrank ℝ E⌉₊
	(by
	intro s hs hsep
	exact_mod_cast (hbound s hs hsep).trans (Nat.le_ceil _))
	exact ⟨s, hs, hbound s hs hsep, hnet⟩

	end NoiselessRegression

	end

	end OAI