lean/OAI/Probability/ProjectionMoments/Model.lean

import Mathlib

namespace OAI

noncomputable section

section InlineMainProof

open MeasureTheory ProbabilityTheory Set Filter

open scoped ENNReal NNReal RealInnerProductSpace

universe u

namespace NoiselessRegression

abbrev Vector (d : ℕ) := EuclideanSpace ℝ (Fin d)

abbrev UnitSphere (d : ℕ) := Metric.sphere (0 : Vector d) 1

abbrev State (M : ℕ) := Fin (2 ^ M)

abbrev Samples (d T : ℕ) := Fin T → Vector d

def uniformSphere (d : ℕ) : Measure (UnitSphere d) :=
  let σ := (volume : Measure (Vector d)).toSphere
  (σ Set.univ)⁻¹ • σ

def sampleLaw (d T : ℕ) : Measure (Samples d T) :=
  Measure.pi fun _ => stdGaussian (Vector d)

def observationReference (d : ℕ) : Measure (Vector d × ℝ) :=
  (stdGaussian (Vector d)).prod volume

structure Learner (d M T : ℕ) (Ω : Type u) where
  initialChoice : Ω → Bool × State M
  transition : ℕ → Ω → State M → (Vector d × ℝ) → Bool × State M
  output : ℕ → Ω → State M → UnitSphere d

namespace Learner

abbrev RunStatus (M : ℕ) := Sum (State M) (ℕ × State M)

def start {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω) (ω : Ω) :
    RunStatus M :=
  let a := L.initialChoice ω
  if a.1 then .inr (0, a.2) else .inl a.2

def run {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω)
    (s : UnitSphere d) (ω : Ω) (x : Samples d T) : ℕ → RunStatus M
  | 0 => L.start ω
  | n + 1 =>
    match L.run s ω x n with
    | .inr done => .inr done
    | .inl state =>
      if hn : n < T then
        let row := x ⟨n, hn⟩
        let a := L.transition (n + 1) ω state (row, ⟪row, s.val⟫)
        if a.1 then .inr (n + 1, a.2) else .inl a.2
      else .inl state

def terminal {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω)
    (s : UnitSphere d) (ω : Ω) (x : Samples d T) : ℕ × State M :=
  match L.run s ω x T with
  | .inl state => (T, state)
  | .inr done => done

def estimate {d M T : ℕ} {Ω : Type u} (L : Learner d M T Ω)
    (s : UnitSphere d) (ω : Ω) (x : Samples d T) : UnitSphere d :=
  let a := L.terminal s ω x
  L.output a.1 ω a.2

structure Admissible {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
    (L : Learner d M T Ω) (ρ : Measure Ω) : Prop where
  initialChoice : AEMeasurable L.initialChoice ρ
  transition : ∀ t a, AEMeasurable (fun z : Ω × (Vector d × ℝ) =>
    L.transition t z.1 a z.2) (ρ.prod (observationReference d))
  output : ∀ t a, AEMeasurable (fun ω => L.output t ω a) ρ
  fixed_experiment : ∀ s, AEMeasurable (fun z : Ω × Samples d T =>
    L.estimate s z.1 z.2) (ρ.prod (sampleLaw d T))
  uniform_experiment : AEMeasurable
    (fun z : UnitSphere d × (Ω × Samples d T) => L.estimate z.1 z.2.1 z.2.2)
    ((uniformSphere d).prod (ρ.prod (sampleLaw d T)))

def angularError {d : ℕ} (s ŝ : UnitSphere d) : ℝ :=
  Real.arccos ⟪ŝ.val, s.val⟫

def uniformSuccess {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]
    (L : Learner d M T Ω) (ρ : Measure Ω) (ε : ℝ) : ℝ≥0∞ :=
  ((uniformSphere d).prod (ρ.prod (sampleLaw d T)))
    {z | angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε}

end Learner

open scoped Topology

open InnerProductSpace

section SphereBounds

open scoped Pointwise

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

def sphereLaw : Measure (Metric.sphere (0 : E) 1) :=
  let σ := (volume : Measure E).toSphere
  (σ Set.univ)⁻¹ • σ

end SphereBounds

open scoped Pointwise

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

def ambientSphereLaw : Measure E := (sphereLaw (E := E)).map Subtype.val

end NoiselessRegression

end InlineMainProof

open scoped ENNReal NNReal RealInnerProductSpace Topology MatrixOrder Matrix.Norms.L2Operator
open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace Matrix
open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace
open scoped ENNReal NNReal RealInnerProductSpace Topology
namespace ProjectionMoments

section GaussianGraph

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

def vectorSmoothing (ν : Measure E) (m : ℕ) (φ : (Fin m → ℝ) → ℝ≥0∞)
    (z : (Fin m → E) × (Fin m → ℝ)) : ℝ≥0∞ :=
  ∫⁻ t, φ (fun j => z.2 j - ⟪t, z.1 j⟫) ∂ν

end GaussianGraph

section

abbrev Label (m : ℕ) := EuclideanSpace ℝ (Fin m)

def ballKernel (m : ℕ) (η : ℝ) (y : Fin m → ℝ) : ℝ≥0∞ :=
  (Metric.closedBall (0 : Label m) η).indicator
    (fun _ => (volume (Metric.closedBall (0 : Label m) η))⁻¹) (WithLp.toLp 2 y)

section BallDensity

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

def rowMap {m : ℕ} (A : Fin m → E) : E →L[ℝ] Label m :=
  (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin m => ℝ)).symm.toContinuousLinearMap.comp
    (ContinuousLinearMap.pi (fun j => innerSL ℝ (A j)))

def ballAverage (ν : Measure E) (m : ℕ) (η : ℝ)
    (z : (Fin m → E) × Label m) : ℝ≥0∞ :=
  vectorSmoothing ν m (ballKernel m η) (z.1, WithLp.ofLp z.2)

def lowerDensity (ν : Measure E) (m : ℕ) (z : (Fin m → E) × Label m) : ℝ≥0∞ :=
  liminf (fun n : ℕ => ballAverage ν m ((n : ℝ) + 1)⁻¹ z) atTop

end BallDensity

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

def AllBallMass (ν : Measure E) (K a : ℝ) : Prop :=
  ∀ b : E, ∀ t : ℝ, 0 < t → ν (Metric.closedBall b t) ≤ ENNReal.ofReal (K * t ^ a)

end

section Rotations

variable (E : Type*) [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E]

abbrev Rotation := unitary (E →L[ℝ] E)

instance rotationMeasurableSpace : MeasurableSpace (Rotation E) := borel (Rotation E)

instance rotationBorelSpace : BorelSpace (Rotation E) := ⟨rfl⟩

instance rotationCompact : CompactSpace (Rotation E) := by
  apply isCompact_iff_compactSpace.mp
  apply Metric.isCompact_of_isClosed_isBounded isClosed_unitary
  apply (Metric.isBounded_closedBall (x := (0 : E →L[ℝ] E)) (r := 1)).subset (s := (unitary (E →L[ℝ] E) : Set (E →L[ℝ] E)))
  intro U hU
  rw [Metric.mem_closedBall, dist_zero_right]
  by_cases h : Subsingleton E
  · have := h
    simp
  · have : Nontrivial E := not_subsingleton_iff_nontrivial.mp h
    exact (CStarRing.norm_of_mem_unitary hU).le

def rotationLaw : Measure (Rotation E) :=
  ((Measure.haar : Measure (Rotation E)) Set.univ)⁻¹ • Measure.haar

variable {E}

variable [MeasurableSpace E] [BorelSpace E]

def rotateRows {ι : Type*} (O : Rotation E) (v : ι → E) : ι → E :=
  fun i => (O : E →L[ℝ] E) (v i)

end Rotations

section Triangular

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

def coordinateMap {m : ℕ} (A : Fin m → E) : E →ₗ[ℝ] (Fin m → ℝ) :=
  (WithLp.linearEquiv 2 ℝ (Fin m → ℝ)) ∘ₗ (rowMap A).toLinearMap

end Triangular

section CoordinateDensity

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

def coordinateDensity (ν : Measure E) (m : ℕ) (z : (Fin m → E) × (Fin m → ℝ)) : ℝ≥0∞ :=
  lowerDensity ν m (z.1, WithLp.toLp 2 z.2)

end CoordinateDensity

section

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

def frameLaw {m : ℕ} (u : Fin m → E) : Measure (Fin m → E) :=
  (rotationLaw E).map (fun O => rotateRows O u)

end

section

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

def rieszKernel (α : ℝ) (z s : E) : ℝ≥0∞ := (ENNReal.ofReal ‖s-z‖)^(-α)

def rieszPotential (α : ℝ) (h : E → ℝ≥0∞) : ℝ≥0∞ :=
  ⨆ z : E, ∫⁻ s, h s*rieszKernel α z s ∂NoiselessRegression.ambientSphereLaw

end

section

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

def mixRows {k : ℕ} (M : Matrix (Fin k) (Fin k) ℝ) (X : Fin k → E) : Fin k → E :=
  fun i => ∑ j, M i j • X j

def polarRadial {k : ℕ} (X : Fin k → E) : Matrix (Fin k) (Fin k) ℝ :=
  CFC.sqrt (Matrix.gram ℝ X)

def rawPolarRows {k : ℕ} (X : Fin k → E) : Fin k → E :=
  mixRows (polarRadial X)⁻¹ X

def polarRows {k : ℕ} (u X : Fin k → E) : Fin k → E := by
  classical
  exact if LinearIndependent ℝ X then rawPolarRows X else u

end

section

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

def unitBallVolume (k : ℕ) : ℝ≥0∞ := (volume : Measure (Label k)) (Metric.closedBall 0 1)

def unitBallLaw (k : ℕ) : Measure (Label k) :=
  (unitBallVolume k)⁻¹ • volume.restrict (Metric.closedBall 0 1)

def normalizedDensity (ν : Measure E) (k : ℕ) (u : Fin k → E)
    (z : (Fin k → E) × Label k) : ℝ≥0∞ :=
  unitBallVolume k * lowerDensity ν k (polarRows u z.1,z.2)

def sphereWeightedRestriction (h : E → ℝ≥0∞) (H : Set E) : Measure E :=
  (NoiselessRegression.ambientSphereLaw.withDensity h).restrict H

end

section

universe u

variable {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
  [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]

def capMeasure (c : E) (r : ℝ) : Measure E :=
  (NoiselessRegression.ambientSphereLaw (Metric.closedBall c r))⁻¹ •
    NoiselessRegression.ambientSphereLaw.restrict (Metric.closedBall c r)

structure CapDictionary (c : E) (r α : ℝ) where
  I : Type u
  countable : Countable I
  center : I → E
  radius : I → ℝ
  center_norm : ∀ i, ‖center i‖ = 1
  radius_pos : ∀ i, 0 < radius i
  radius_le : ∀ i, radius i ≤ r
  parent : I
  parent_center : center parent = c
  parent_radius : radius parent = r
  proper : ∀ R : ℝ, {i | (r/radius i)^α ≤ R}.Finite
  cover : ∀ (s : E), ‖s‖ = 1 → ∀ t : ℝ, 0 < t → t ≤ 1/4 →
    ∃ i, 4*r*t ≤ radius i ∧ radius i < 8*r*t ∧ dist s (center i) ≤ radius i/2

end

section

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

def radialLabel {k : ℕ} (X : Fin k → E) (y : Label k) : Label k :=
  WithLp.toLp 2 (polarRadial X *ᵥ y.ofLp)

def capKernelValue {k : ℕ} (K : (Fin k → E) × Label k → ℝ≥0∞) (s : E) : ℝ≥0∞ :=
  ∫⁻ X, K (X,rowMap X s) ∂Measure.pi (fun _ => stdGaussian E)

def capReferenceLabel {k : ℕ} (c : E) (r : ℝ) (X : Fin k → E) (z : Label k) : Label k :=
  rowMap X c + r • radialLabel X z

def capReference {k : ℕ} (K : (Fin k → E) × Label k → ℝ≥0∞) (c : E) (r : ℝ) : ℝ≥0∞ :=
  ∫⁻ X, ∫⁻ z, K (X,capReferenceLabel c r X z) ∂unitBallLaw k ∂Measure.pi (fun _ => stdGaussian E)

end

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

def fractionalBlock {J : Type*} [Fintype J] {k : ℕ}
    (Q : J → (Fin k → E) × Label k → ℝ≥0∞) (f : J → E → ℝ≥0∞) (s : E) : ℝ≥0∞ :=
  ∫⁻ X, ∑ j, Q j (X,rowMap X s)*f j s ∂Measure.pi (fun _ => stdGaussian E)

end ProjectionMoments

open MeasureTheory ProbabilityTheory Set Filter unitInterval
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace
namespace NoiselessRegression

namespace Learner

section

universe u

variable {d M T : ℕ} {Ω : Type u}

abbrev TaggedState (M T : ℕ) := Sum (State M) (Fin (T+1) × State M)

instance taggedMeasurableSpace : MeasurableSpace (TaggedState M T) := ⊤

def taggedStart (a : Bool × State M) : TaggedState M T :=
  if a.1 then .inr (⟨0,by omega⟩,a.2) else .inl a.2

end

universe u

variable {d M T : ℕ} {Ω : Type u} [MeasurableSpace Ω]

structure MeasurableRules (L : Learner d M T Ω) : Prop where
  initial : Measurable L.initialChoice
  transition : ∀ t j, Measurable (fun z : Ω × (Vector d × ℝ) => L.transition t z.1 j z.2)
  output : ∀ t j, Measurable (fun ω => L.output t ω j)

end Learner

structure FiniteKernelLearner (d M T : ℕ) where
  initial : Measure (Bool × State M)
  initial_probability : IsProbabilityMeasure initial
  transition : ℕ → State M → (Vector d × ℝ) → Measure (Bool × State M)
  transition_probability : ∀ t j obs, IsProbabilityMeasure (transition t j obs)
  output : ℕ → State M → Measure (UnitSphere d)
  output_probability : ∀ t j, IsProbabilityMeasure (output t j)

namespace FiniteKernelLearner

variable {d M T : ℕ}

structure CompletedRules (K : FiniteKernelLearner d M T) : Prop where
  transition : ∀ t j a, AEMeasurable (fun obs => K.transition t j obs {a}) (observationReference d)

end FiniteKernelLearner

section

variable {J O : Type*} [Fintype J] [MeasurableSpace J] [MeasurableSingletonClass J]
  [MeasurableSpace O]

def kernelRun (P : ℕ → J → O → Measure J) : (n : ℕ) → J → (Fin n → O) → Measure J
  | 0, j, _ => Measure.dirac j
  | n+1, j, obs => (kernelRun P n j (Fin.init obs)).bind (fun a => P (n+1) a (obs (Fin.last n)))

end

namespace FiniteKernelLearner

section

open Learner

variable {d M T : ℕ}

abbrev Tape (M T : ℕ) := (Bool × State M) × ((Fin T → I) × I)

def destination (t : ℕ) (h : t ≤ T) (a : Bool × State M) : TaggedState M T :=
  if a.1 then .inr (⟨t,by omega⟩,a.2) else .inl a.2

def taggedKernel (K : FiniteKernelLearner d M T) (t : ℕ) :
    TaggedState M T → (Vector d × ℝ) → Measure (TaggedState M T)
  | .inr a, _ => Measure.dirac (.inr a)
  | .inl j, obs => if ht : t ≤ T then (K.transition t j obs).map (destination t ht)
    else Measure.dirac (.inl j)

def terminalOutput (K : FiniteKernelLearner d M T) : TaggedState M T → Measure (UnitSphere d)
  | .inl j => K.output T j
  | .inr (t,j) => K.output t j

end

open Learner

variable {d M T : ℕ}

abbrev Result (d M : ℕ) := (ℕ × State M) × UnitSphere d

def terminalTag : TaggedState M T → ℕ × State M
  | .inl j => (T,j)
  | .inr (t,j) => (t,j)

def terminalJointOutput (K : FiniteKernelLearner d M T) (j : TaggedState M T) : Measure (Result d M) :=
  (K.terminalOutput j).map (fun y => (terminalTag j,y))

def jointExperimentLaw (K : FiniteKernelLearner d M T) (obs : Fin T → (Vector d × ℝ)) : Measure (Result d M) :=
  K.initial.bind (fun a => (kernelRun K.taggedKernel T (taggedStart a) obs).bind K.terminalJointOutput)

def uniformSuccess (K : FiniteKernelLearner d M T) (ε : ℝ) : ℝ≥0∞ :=
  ∫⁻ z : UnitSphere d × Samples d T,
    K.jointExperimentLaw (fun i => (z.2 i,⟪z.1.val,z.2 i⟫))
      {r | angularError z.1 r.2 ≤ ε} ∂((uniformSphere d).prod (sampleLaw d T))

def uniformFixedSuccess (K : FiniteKernelLearner d M T) (ε : ℝ) (t : ℕ) : ℝ≥0∞ :=
  ∫⁻ z : UnitSphere d × Samples d T,
    K.jointExperimentLaw (fun i => (z.2 i,⟪z.1.val,z.2 i⟫))
      {r | r.1.1 = t ∧ angularError z.1 r.2 ≤ ε} ∂((uniformSphere d).prod (sampleLaw d T))

end FiniteKernelLearner

section

universe u

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

namespace Learner

variable {d M T : ℕ} {Ω : Type u}

def uniformFixedSuccess [MeasurableSpace Ω] (L : Learner d M T Ω) (ρ : Measure Ω)
    (ε : ℝ) (t : ℕ) : ℝ≥0∞ :=
  ((uniformSphere d).prod (ρ.prod (sampleLaw d T)))
    {z | (L.terminal z.1 z.2.1 z.2.2).1 = t ∧ angularError z.1 (L.estimate z.1 z.2.1 z.2.2) ≤ ε}

end Learner

end

namespace FiniteKernelLearner

section

universe u

variable {d M T : ℕ}

def seededSuccess {Ξ : Type u} [MeasurableSpace Ξ]
    (K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) (ε : ℝ) : ℝ≥0∞ :=
  ∫⁻ ξ, (K ξ).uniformSuccess ε ∂ρ

def seededFixedSuccess {Ξ : Type u} [MeasurableSpace Ξ]
    (K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) (ε : ℝ) (t : ℕ) : ℝ≥0∞ :=
  ∫⁻ ξ, (K ξ).uniformFixedSuccess ε t ∂ρ

def FixedQuadraticMemory : Prop :=
  ∀ A₀ : ℝ, 0 < A₀ → ∃ c : ℝ, 0 < c ∧ ∃ d₀ : ℕ,
    ∀ d : ℕ, d₀ ≤ d → ∀ M T : ℕ, (M : ℝ) ≤ A₀ * (d : ℝ)^2 →
    ∀ ε : ℝ, 0 < ε → ε ≤ 1/10 →
    ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
    ∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
    (2/3 : ℝ≥0∞) ≤ seededSuccess K ρ ε →
    c*(d:ℝ)*Real.log (1/ε) ≤ (T:ℝ)

end

universe u

variable {d M T : ℕ} {Ξ : Type u} [MeasurableSpace Ξ]

def conditionalSuccess (K : FiniteKernelLearner d M T) (ε : ℝ)
    (s : UnitSphere d) (x : Samples d T) : ℝ≥0∞ :=
  K.jointExperimentLaw (fun i => (x i,⟪s.val,x i⟫)) {r | Learner.angularError s r.2 ≤ ε}

def seededPointwiseSuccess (K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ)
    (ε : ℝ) (s : UnitSphere d) : ℝ≥0∞ :=
  ∫⁻ z : Ξ × Samples d T, (K z.1).conditionalSuccess ε s z.2 ∂ρ.prod (sampleLaw d T)

def JointSuccessMeasurable (K : Ξ → FiniteKernelLearner d M T) (ρ : Measure Ξ) (ε : ℝ) : Prop :=
  AEMeasurable (fun z : UnitSphere d × (Ξ × Samples d T) =>
    (K z.2.1).conditionalSuccess ε z.1 z.2.2)
    ((uniformSphere d).prod (ρ.prod (sampleLaw d T)))

def SubquadraticMemoryWithPointwise : Prop :=
  ∃ c : ℝ, 0 < c ∧ ∀ M : ℕ → ℕ,
    Asymptotics.IsLittleO atTop (fun d => (M d : ℝ)) (fun d => (d:ℝ)^2) →
    ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
    0 < ε → ε ≤ 1/10 →
    ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
    ∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
    ((2/3 : ℝ≥0∞) ≤ seededSuccess K ρ ε ∨
      (JointSuccessMeasurable K ρ ε ∧ ∀ s, (2/3 : ℝ≥0∞) ≤ seededPointwiseSuccess K ρ ε s)) →
    c*(d:ℝ)*Real.log (1/ε) ≤ (T:ℝ)

end FiniteKernelLearner

end NoiselessRegression

open MeasureTheory ProbabilityTheory Set Filter InnerProductSpace

open NoiselessRegression NoiselessRegression.FiniteKernelLearner

open scoped ENNReal NNReal RealInnerProductSpace Topology

universe u

namespace ProjectionMoments

def IntegratedFrameMoment : Prop :=
    ∃ C : ℝ, 1 ≤ C ∧ ∀ (d k q : ℕ) (β K R : ℝ)
      (ν : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ν]
      (z : EuclideanSpace ℝ (Fin d)) (u : Fin k → EuclideanSpace ℝ (Fin d)),
      1 ≤ k → 2 ≤ q → k+q ≤ d → 0 < β → β ≤ d →
      1 ≤ β-k-((q:ℝ)-2) → 0 ≤ K → 0 < R →
      ν.support ⊆ Metric.closedBall z R → AllBallMass ν K β →
      Orthonormal ℝ u →
      (∀ᵐ A ∂frameLaw u, ν.map (coordinateMap A) ≪ (volume : Measure (Fin k → ℝ))) →
      Measurable (coordinateDensity ν k) ∧
      (∀ᵐ A ∂frameLaw u,
        (volume : Measure (Fin k → ℝ)).withDensity (fun y => coordinateDensity ν k (A,y)) =
          ν.map (coordinateMap A)) ∧
      (∀ (A : Fin k → EuclideanSpace ℝ (Fin d)), Orthonormal ℝ A →
        ∀ y : Fin k → ℝ,
          R < ‖WithLp.toLp 2 (y-coordinateMap A z)‖ → coordinateDensity ν k (A,y) = 0) ∧
      (∫⁻ A, ∫⁻ y : Fin k → ℝ, coordinateDensity ν k (A,y)^q ∂volume ∂frameLaw u) ^ (1/(q:ℝ)) ≤
        ENNReal.ofReal (C^d * (C*Real.sqrt d)^((k:ℝ)*(1-1/(q:ℝ))) *
          K * R^(β-(k:ℝ)*(1-1/(q:ℝ))))

def NormalizedUnitBallMoment : Prop :=
  ∀ D : ℝ, 0 < D →
    ∃ C_D : ℝ, 0 < C_D ∧ ∀ d : ℕ, 32 ≤ d →
      let k := d/16
      let lam : ℝ := ((d:ℝ)-1)/2
      ∀ (ν : Measure (EuclideanSpace ℝ (Fin d))) [IsFiniteMeasure ν]
        (u : Fin k → EuclideanSpace ℝ (Fin d)),
        ν Set.univ ≤ 1 → ν.support ⊆ Metric.closedBall 0 1 →
        AllBallMass ν (D^d) lam → Orthonormal ℝ u →
        (∀ᵐ X ∂Measure.pi (fun _ : Fin k => stdGaussian (EuclideanSpace ℝ (Fin d))),
          ν.map (rowMap (polarRows u X)) ≪ (volume : Measure (Label k))) →
        Measurable (normalizedDensity ν k u) ∧
        (∀ᵐ X ∂Measure.pi (fun _ : Fin k => stdGaussian (EuclideanSpace ℝ (Fin d))),
          ((unitBallVolume k)⁻¹ • (volume : Measure (Label k))).withDensity
            (fun y => normalizedDensity ν k u (X,y)) = ν.map (rowMap (polarRows u X))) ∧
        (∀ X y, 1 < ‖y‖ → normalizedDensity ν k u (X,y) = 0) ∧
        (∫⁻ X, ∫⁻ y, normalizedDensity ν k u (X,y)^k ∂unitBallLaw k
          ∂Measure.pi (fun _ => stdGaussian (EuclideanSpace ℝ (Fin d)))) ≤
          ENNReal.ofReal (Real.exp (C_D*d*k))

def AllRadiiSphereMoment : Prop :=
    ∃ C : ℝ, 1 ≤ C ∧ ∀ d : ℕ, 16 ≤ d →
      let k := d/4
      let β : ℝ := (d:ℝ)-1-k
      ∀ f : EuclideanSpace ℝ (Fin d) → ℝ≥0∞,
      Measurable f → (∀ s, f s ≤ 1) →
      ∀ K : ℝ, 0 ≤ K →
      AllBallMass (NoiselessRegression.ambientSphereLaw.withDensity f) K β →
      ∀ z : EuclideanSpace ℝ (Fin d), ∀ R : ℝ, 0 < R →
      ∀ u : Fin k → EuclideanSpace ℝ (Fin d), Orthonormal ℝ u →
      let μ := sphereWeightedRestriction f (Metric.closedBall z R)
      Measurable (coordinateDensity μ k) ∧
      (∀ A : Fin k → EuclideanSpace ℝ (Fin d), Orthonormal ℝ A →
        (volume : Measure (Fin k → ℝ)).withDensity (fun y => coordinateDensity μ k (A,y)) =
          μ.map (coordinateMap A)) ∧
      (∀ A : Fin k → EuclideanSpace ℝ (Fin d), Orthonormal ℝ A →
        ∀ y : Fin k → ℝ, R < ‖WithLp.toLp 2 (y-coordinateMap A z)‖ →
          coordinateDensity μ k (A,y) = 0) ∧
      (∫⁻ A, ∫⁻ y : Fin k → ℝ, coordinateDensity μ k (A,y)^k ∂volume ∂frameLaw u) ^ (1/(k:ℝ)) ≤
        ENNReal.ofReal (C^d*(C*Real.sqrt d)^((k:ℝ)*(1-1/(k:ℝ)))*
          K*R^(β-(k:ℝ)*(1-1/(k:ℝ))))

def PositiveCapDomination : Prop :=
    ∃ C : ℝ, 0 < C ∧ ∀ d : ℕ, 32 ≤ d →
      let k := d/16
      let α : ℝ := ((d:ℝ)-1)/2
      ∀ (c : EuclideanSpace ℝ (Fin d)), ‖c‖ = 1 →
      ∀ r : ℝ, 0 < r → r ≤ 2 → ∀ N : ℕ,
      ∀ K : Fin N → (Fin k → EuclideanSpace ℝ (Fin d)) × Label k → ℝ≥0∞,
      (∀ j, Measurable (K j)) → (∀ z, ∑ j, K j z ≤ 1) →
      (∑ j, capReference (K j) c r) ≤ 1 ∧
      ∀ j, ∃ D : CapDictionary c r α, ∃ a : D.I → ℝ,
        (∀ i, 0 ≤ a i) ∧ Summable (fun i => a i*(r/D.radius i)^α) ∧
        (∑' i, a i*(r/D.radius i)^α) ≤ Real.exp (C*d)*(capReference (K j) c r).toReal^(1-1/(k:ℝ)) ∧
        (capMeasure c r).withDensity (capKernelValue (K j)) ≤
          Measure.sum (fun i => ENNReal.ofReal (a i) • capMeasure (D.center i) (D.radius i)) ∧
        (Measure.sum (fun i => ENNReal.ofReal (a i) • capMeasure (D.center i) (D.radius i))) Set.univ < ∞

def DeterministicOffsetDensity : Prop :=
    ∃ C : ℝ, 1 ≤ C ∧ ∀ d : ℕ, 16 ≤ d →
      let k := d/8
      let α : ℝ := (d:ℝ)/2
      ∀ h : EuclideanSpace ℝ (Fin d) → ℝ≥0∞, Measurable h → (∀ s, h s ≤ 1) →
      ∀ z : EuclideanSpace ℝ (Fin d), ∀ ρ : ℝ, 0 < ρ →
      ∀ H : Set (EuclideanSpace ℝ (Fin d)), MeasurableSet H → H ⊆ Metric.closedBall z ρ →
      rieszPotential α h < ∞ ∧
      Measurable (lowerDensity (sphereWeightedRestriction h H) k) ∧
      (∀ A : Fin k → EuclideanSpace ℝ (Fin d), LinearIndependent ℝ A →
        volume.withDensity (fun y => lowerDensity (sphereWeightedRestriction h H) k (A,y)) =
          (sphereWeightedRestriction h H).map (rowMap A)) ∧
      (∀ b : Label k, (∫⁻ A, (lowerDensity (sphereWeightedRestriction h H) k (A,rowMap A z+b))^k
          ∂Measure.pi (fun _ => stdGaussian (EuclideanSpace ℝ (Fin d))))^(1/(k:ℝ)) ≤
        ENNReal.ofReal (C^d) * rieszPotential α h * ENNReal.ofReal (ρ^(α-k))) ∧
      (∀ A y, ‖rowMap A‖*ρ < ‖y-rowMap A z‖ →
        lowerDensity (sphereWeightedRestriction h H) k (A,y) = 0)

def AllRadiiBlockBound : Prop :=
    ∃ C : ℝ, 1 ≤ C ∧ ∀ d : ℕ, 16 ≤ d →
      let k := d/4
      let β : ℝ := (d:ℝ)-1-k
      ∀ N : ℕ, ∀ f : Fin N → EuclideanSpace ℝ (Fin d) → ℝ≥0∞,
      (∀ j, Measurable (f j)) → (∀ j s, f j s ≤ 1) →
      ∀ K : ℝ, 0 ≤ K →
      (∀ j, AllBallMass (NoiselessRegression.ambientSphereLaw.withDensity (f j)) K β) →
      ∀ Q : Fin N → (Fin k → EuclideanSpace ℝ (Fin d)) × Label k → ℝ≥0∞,
      (∀ j, Measurable (Q j)) → (∀ z, ∑ j, Q j z ≤ 1) →
      AllBallMass (NoiselessRegression.ambientSphereLaw.withDensity (fractionalBlock Q f))
        (C^d*(N:ℝ)^(1/(k:ℝ))*K) β

def HaarStreamingCost : Prop :=
    ∃ C : ℝ, 1 ≤ C ∧ ∀ d M T : ℕ, 16 ≤ d → ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      let k := d/4
      let Nstar : ℝ := ((k:ℝ)+2)*2^M
      seededSuccess K ρ ε ≤ ENNReal.ofReal
        (C^d*ε^k*(C^d*Nstar^(1/(k:ℝ)))^((T+k-1)/k))

def RieszStreamingCost : Prop :=
    ∃ C₀ : ℝ, 1 ≤ C₀ ∧ ∀ d M T : ℕ, 16 ≤ d → ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      let q := d/8
      let W : ℝ := ((T:ℝ)+2)*2^M
      seededSuccess K ρ ε ≤ ENNReal.ofReal
        (C₀^d*ε^((d:ℝ)/2-1)*(1+C₀^d*(1+Real.log (1/ε))*W^(1/(q:ℝ)))^((T+q-1)/q))

def CapFixedStoppingCost : Prop :=
    ∃ C : ℝ, 0 < C ∧ ∀ d M T : ℕ, 32 ≤ d → ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      ∀ t : ℕ, t ≤ T →
      let k := d/16
      let α : ℝ := ((d:ℝ)-1)/2
      seededFixedSuccess K ρ ε t ≤ ENNReal.ofReal
        (Real.exp (C*d)*(ε/2)^α*(Real.exp (C*d)*(2:ℝ)^((M:ℝ)/k))^((t+k-1)/k))

def CapTotalStoppingCost : Prop :=
    ∃ C : ℝ, 0 < C ∧ ∀ d M T : ℕ, 32 ≤ d → ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d M T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      let k := d/16
      let α : ℝ := ((d:ℝ)-1)/2
      seededSuccess K ρ ε ≤ ENNReal.ofReal
        (((T:ℝ)+1)*Real.exp (C*d)*(ε/2)^α*
          (Real.exp (C*d)*(2:ℝ)^((M:ℝ)/k))^((T+k-1)/k))

def CapFixedStoppingAsymptotic : Prop :=
    ∃ C : ℝ, 0 < C ∧ ∀ M : ℕ → ℕ,
      Asymptotics.IsLittleO atTop (fun d => (M d:ℝ)) (fun d => (d:ℝ)^2) →
      ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      ∀ t : ℕ, t ≤ T →
      let k := d/16
      let α : ℝ := ((d:ℝ)-1)/2
      seededFixedSuccess K ρ ε t ≤ ENNReal.ofReal
        (Real.exp (C*d*(1+((t+k-1)/k:ℕ)))*(ε/2)^α)

def CapTotalStoppingAsymptotic : Prop :=
    ∃ C₁ C₂ : ℝ, 0 < C₁ ∧ 0 < C₂ ∧ ∀ M : ℕ → ℕ,
      Asymptotics.IsLittleO atTop (fun d => (M d:ℝ)) (fun d => (d:ℝ)^2) →
      ∃ d₀ : ℕ, ∀ d : ℕ, d₀ ≤ d → ∀ T : ℕ, ∀ ε : ℝ,
      0 < ε → ε ≤ 1/10 →
      ∀ (Ξ : Type u) [MeasurableSpace Ξ] (ρ : Measure Ξ), IsProbabilityMeasure ρ →
      ∀ K : Ξ → FiniteKernelLearner d (M d) T, (∀ᵐ ξ ∂ρ, (K ξ).CompletedRules) →
      seededSuccess K ρ ε ≤ ENNReal.ofReal
        (((T:ℝ)+1)*Real.exp (C₁*d+C₂*T-(d:ℝ)*Real.log (1/ε)/3))

def KernelModelBridge : Prop :=
  ∀ d M T : ℕ, 3 ≤ d → ∀ K : FiniteKernelLearner d M T, K.CompletedRules →
    ∃ (ρ : Measure (Tape M T)) (L : Learner d M T (Tape M T)),
      IsProbabilityMeasure ρ ∧ L.MeasurableRules ∧ L.Admissible ρ ∧
      (∀ ε, L.uniformSuccess ρ ε = K.uniformSuccess ε) ∧
      (∀ ε t, L.uniformFixedSuccess ρ ε t = K.uniformFixedSuccess ε t)

def MainStatement : Prop :=
  IntegratedFrameMoment ∧ NormalizedUnitBallMoment ∧ AllRadiiSphereMoment ∧
  PositiveCapDomination ∧ DeterministicOffsetDensity ∧ AllRadiiBlockBound ∧
  HaarStreamingCost.{u} ∧ RieszStreamingCost.{u} ∧ CapFixedStoppingCost.{u} ∧
  CapTotalStoppingCost.{u} ∧ CapFixedStoppingAsymptotic.{u} ∧
  CapTotalStoppingAsymptotic.{u} ∧ KernelModelBridge ∧
  NoiselessRegression.FiniteKernelLearner.SubquadraticMemoryWithPointwise.{u} ∧
  NoiselessRegression.FiniteKernelLearner.FixedQuadraticMemory.{u}

end ProjectionMoments

end

end OAI