| import Mathlib |
| import OAI.Probability.SKGap.Localization.IntegrableResidualRecipeSq |
| import OAI.Probability.SKGap.Localization.RegeneratedAnswer |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory Metric Set |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Matrix |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise NNReal ENNReal |
| open scoped BigOperators |
| open Filter Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set Matrix |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| namespace SKGapCutoff.Regression |
|
|
| local instance iteratedMatrixMeasurable {n m : ℕ} : MeasurableSpace (Matrix (Fin n) (Fin m) ℝ) := |
| inferInstanceAs (MeasurableSpace (Fin n → Fin m → ℝ)) |
|
|
| theorem ExponentialEmpiricalConcentration.regenerated_normalized_query |
| {E : Type*} [PseudoMetricSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| (ρ : ∀ n, Measure (H n)) [∀ n, IsProbabilityMeasure (ρ n)] |
| (ν : Measure E) [IsProbabilityMeasure ν] |
| (X : ∀ n, H n → Fin n → E) (hXm : ∀ n, Measurable (X n)) |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| (r : ℕ) (U Y : ∀ n, H n → Matrix (Fin n) (Fin r) ℝ) (q : ∀ n, H n → Fin n → ℝ) |
| (hUm : ∀ n, Measurable (U n)) (hqm : ∀ n, Measurable (q n)) |
| (hU : ∀ n h a, ∑ i, U n h i a^2 ≤ 1) (hq : ∀ n h, ∑ i, q n h i^2 ≤ 1) |
| (u y : Fin r → E → ℝ) (g : E → ℝ) {K : ℝ≥0} |
| (hu : ∀ a, LipschitzWith K (u a)) (hy : ∀ a, LipschitzWith K (y a)) (hg : LipschitzWith K g) |
| (huT : ∀ a, ExponentialSquareTails ρ (fun n h i => u a (X n h i))) |
| (hyT : ∀ a, ExponentialSquareTails ρ (fun n h i => y a (X n h i))) |
| (hgT : ExponentialSquareTails ρ (fun n h i => g (X n h i))) |
| (hui : ∀ a, Integrable (fun x => u a x^2) ν) |
| (hyi : ∀ a, Integrable (fun x => y a x^2) ν) (hgi : Integrable (fun x => g x^2) ν) |
| (heu : ∀ (n : ℕ) h i a, Real.sqrt n*U n h i a=u a (X n h i)) |
| (hey : ∀ (n : ℕ) h i a, Real.sqrt n*Y n h i a=y a (X n h i)) |
| (heq : ∀ (n : ℕ) h i, Real.sqrt n*q n h i=g (X n h i)) : |
| let W := fun n (z : H n × ((Fin n ⊕ Unit) → ℝ)) i => |
| Real.sqrt n*regeneratedAnswer (U n z.1) (Y n z.1) (q n z.1) z.2 i |
| let law := (ν.prod (gaussianReal 0 1)).map |
| (fun z => (z.1,z.2+∑ a, (∫ x, y a x*g x ∂ν)*u a z.1)) |
| ExponentialEmpiricalConcentration (fun n => (ρ n).prod (standardArrayLaw (Fin n ⊕ Unit))) |
| (fun n z i => (X n z.1 i,W n z i)) law ∧ |
| ExponentialSquareTails (fun n => (ρ n).prod (standardArrayLaw (Fin n ⊕ Unit))) W := by |
| let C : ∀ n, H n → Option (Fin r) → ℝ := fun n h a => a.elim 1 |
| (fun a => (∑ i, y a (X n h i)*g (X n h i))/(n:ℝ)) |
| let c : Option (Fin r) → ℝ := fun a => a.elim 1 (fun a => ∫ x, y a x*g x ∂ν) |
| have hC : ExponentialConvergence ρ C c := by |
| apply ExponentialConvergence.pi_finite |
| intro a |
| cases a with |
| | none => exact ExponentialConvergence.constant ρ 1 |
| | some a => exact hX.product_average (y a) g (hy a) hg (hyT a) hgT (hyi a) hgi |
| have hp := hX.adaptive_affine_query ρ ν X hXm r U q hUm hqm hU hq |
| u hu huT hui C c hC |
| simp only [C,c,Option.elim_none,Option.elim_some,one_mul] at hp |
| let φ : (E × ℝ) × ℝ → E × ℝ := fun z => (z.1.1,z.2) |
| have hφ : LipschitzWith 1 φ := by |
| simpa using (LipschitzWith.prod_fst.comp |
| (LipschitzWith.prod_fst (α := E × ℝ) (β := ℝ))).prodMk LipschitzWith.prod_snd |
| have hpE := hp.1.map φ hφ |
| have hm : Measurable (fun z : E × ℝ => |
| (z,z.2+∑ a, (∫ x, y a x*g x ∂ν)*u a z.1)) := by |
| apply measurable_id.prodMk |
| exact measurable_snd.add (Finset.measurable_sum _ (fun a _ => |
| measurable_const.mul ((hu a).continuous.measurable.comp measurable_fst))) |
| rw [Measure.map_map hφ.continuous.measurable hm] at hpE |
| have he (n : ℕ) (z : H n × ((Fin n ⊕ Unit) → ℝ)) (i : Fin n) : |
| Real.sqrt n*regeneratedAnswer (U n z.1) (Y n z.1) (q n z.1) z.2 i = |
| Real.sqrt n*queryInnovation (residualProjection (U n z.1)) (q n z.1) z.2 i+ |
| ∑ a, ((∑ j, y a (X n z.1 j)*g (X n z.1 j))/(n:ℝ))*u a (X n z.1 i) := by |
| have hn : 0<n := Nat.pos_of_ne_zero (fun hn => by subst n |
| simp only [regeneratedAnswer,Pi.add_apply,mul_add,scaled_query_predictor hn,heu,hey,heq] |
| simp only [mul_comm,add_comm] |
| dsimp only |
| simp only [he] |
| exact ⟨hpE,hp.2⟩ |
|
|
| theorem ExponentialEmpiricalConcentration.regenerated_nondegenerate_step |
| {E : Type*} [PseudoMetricSpace E] [MeasurableSpace E] [BorelSpace E] |
| [SecondCountableTopology E] |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| (ρ : ∀ n, Measure (H n)) [∀ n, IsProbabilityMeasure (ρ n)] |
| (ν : Measure E) [IsProbabilityMeasure ν] |
| (X : ∀ n, H n → Fin n → E) (hXm : ∀ n, Measurable (X n)) |
| (hX : ExponentialEmpiricalConcentration ρ X ν) |
| (r : ℕ) (U Y : ∀ n, H n → Matrix (Fin n) (Fin r) ℝ) |
| (hUm : ∀ n, Measurable (U n)) |
| (hU : ∀ n h a, ∑ i, U n h i a^2 ≤ 1) |
| (u y : Fin r → E → ℝ) (g : E → ℝ) {K : ℝ≥0} (hK : 1 ≤ K) |
| (hu : ∀ a, LipschitzWith K (u a)) (hy : ∀ a, LipschitzWith K (y a)) |
| (hg : LipschitzWith K g) |
| (huT : ∀ a, ExponentialSquareTails ρ (fun n h i => u a (X n h i))) |
| (hyT : ∀ a, ExponentialSquareTails ρ (fun n h i => y a (X n h i))) |
| (hgT : ExponentialSquareTails ρ (fun n h i => g (X n h i))) |
| (hui : ∀ a, Integrable (fun x => u a x^2) ν) |
| (hyi : ∀ a, Integrable (fun x => y a x^2) ν) (hgi : Integrable (fun x => g x^2) ν) |
| (heu : ∀ (n : ℕ) h i a, Real.sqrt n*U n h i a=u a (X n h i)) |
| (hey : ∀ (n : ℕ) h i a, Real.sqrt n*Y n h i a=y a (X n h i)) |
| (hs : 0 < ∫ x, residualRecipe u g (fun a => ∫ z, u a z*g z ∂ν) x^2 ∂ν) : |
| let R := fun n h => residualProjection (U n h) *ᵥ (fun i => g (X n h i)) |
| let q := fun n h => unitVector (R n h) |
| let s := ∫ x, residualRecipe u g (fun a => ∫ z, u a z*g z ∂ν) x^2 ∂ν |
| let φ := fun x => (x,residualRecipe u g (fun a => ∫ z, u a z*g z ∂ν) x/Real.sqrt s) |
| let ν' := ν.map φ |
| let W := fun n (z : H n × ((Fin n ⊕ Unit) → ℝ)) i => |
| Real.sqrt n*(regeneratedAnswer (U n z.1) (Y n z.1) (q n z.1) z.2) i |
| ExponentialConvergence ρ (fun n h => (∑ i, R n h i^2)/(n:ℝ)) s ∧ |
| ExponentialEmpiricalConcentration |
| (fun n => (ρ n).prod (standardArrayLaw (Fin n ⊕ Unit))) |
| (fun n z i => ((X n z.1 i,Real.sqrt n*q n z.1 i),W n z i)) |
| ((ν'.prod (gaussianReal 0 1)).map |
| (fun z => (z.1,z.2+∑ a, (∫ x, y a x.1*x.2 ∂ν')*u a z.1.1))) ∧ |
| ExponentialSquareTails ρ (fun n h i => Real.sqrt n*q n h i) ∧ |
| ExponentialSquareTails (fun n => (ρ n).prod (standardArrayLaw (Fin n ⊕ Unit))) W := by |
| dsimp only |
| let R := fun n h => residualProjection (U n h) *ᵥ (fun i => g (X n h i)) |
| let q := fun n h => unitVector (R n h) |
| let S := fun n h => (∑ i, R n h i^2)/(n:ℝ) |
| let c := fun a => ∫ z, u a z*g z ∂ν |
| let s := ∫ x, residualRecipe u g c x^2 ∂ν |
| let φ := fun x => (x,residualRecipe u g c x/Real.sqrt s) |
| let ν' := ν.map φ |
| have hφ : Measurable φ := by |
| apply measurable_id.prodMk |
| apply Measurable.div_const |
| exact hg.continuous.measurable.sub (Finset.measurable_sum _ (fun a _ => |
| (hu a).continuous.measurable.const_mul (c a))) |
| let : IsProbabilityMeasure ν' := inferInstance |
| have hh := hX.normalized_projected_query U u g hu hg huT hgT hui hgi heu hs |
| change ExponentialConvergence ρ S s ∧ |
| ExponentialEmpiricalConcentration ρ |
| (fun n h i => (X n h i,Real.sqrt n*q n h i)) ν' ∧ |
| ExponentialSquareTails ρ (fun n h i => Real.sqrt n*q n h i) at hh |
| let X' := fun n h i => (X n h i,Real.sqrt n*q n h i) |
| have hRm n : Measurable (R n) := by |
| have hP := measurable_residualProjection_comp (U n) (hUm n) |
| have hgX : Measurable (fun h i => g (X n h i)) := by |
| exact Measurable.of_eval (fun i => hg.continuous.measurable.comp |
| ((measurable_pi_apply i).comp (hXm n))) |
| apply Measurable.of_eval |
| intro i |
| simp only [R,Matrix.mulVec,dotProduct] |
| apply Finset.measurable_sum |
| intro j _ |
| exact (((measurable_pi_apply j).comp (measurable_pi_apply i)).comp hP).mul |
| ((measurable_pi_apply j).comp hgX) |
| have hqm n : Measurable (q n) := measurable_unitVector.comp (hRm n) |
| have hX'm n : Measurable (X' n) := by |
| apply Measurable.of_eval |
| intro i |
| exact ((measurable_pi_apply i).comp (hXm n)).prodMk |
| (((measurable_pi_apply i).comp (hqm n)).const_mul (Real.sqrt n)) |
| have hu' a : LipschitzWith K (fun z : E × ℝ => u a z.1) := by |
| simpa only [Function.comp_def,mul_one] using (hu a).comp (LipschitzWith.prod_fst (α := E) (β := ℝ)) |
| have hy' a : LipschitzWith K (fun z : E × ℝ => y a z.1) := by |
| simpa only [Function.comp_def,mul_one] using (hy a).comp (LipschitzWith.prod_fst (α := E) (β := ℝ)) |
| have hg' : LipschitzWith K (Prod.snd : E × ℝ → ℝ) := |
| LipschitzWith.prod_snd.weaken hK |
| have hui' a : Integrable (fun z : E × ℝ => u a z.1^2) ν' := by |
| apply (integrable_map_measure (((hu a).continuous.measurable.comp measurable_fst).pow_const 2).aestronglyMeasurable hφ.aemeasurable).2 |
| exact hui a |
| have hyi' a : Integrable (fun z : E × ℝ => y a z.1^2) ν' := by |
| apply (integrable_map_measure (((hy a).continuous.measurable.comp measurable_fst).pow_const 2).aestronglyMeasurable hφ.aemeasurable).2 |
| exact hyi a |
| have hgi' : Integrable (fun z : E × ℝ => z.2^2) ν' := by |
| rw [integrable_map_measure (by fun_prop) hφ.aemeasurable] |
| simp only [φ,Function.comp_def,div_pow] |
| exact (integrable_residualRecipe_sq u g c |
| (fun a => (hu a).continuous.measurable.aestronglyMeasurable) |
| hg.continuous.measurable.aestronglyMeasurable hui hgi).div_const _ |
| have hqle n h : ∑ i, q n h i^2 ≤ 1 := unitVector_sq_le _ |
| have hstep := hh.2.1.regenerated_normalized_query ρ ν' X' hX'm r U Y q |
| hUm hqm hU hqle |
| (fun a z => u a z.1) (fun a z => y a z.1) Prod.snd hu' hy' hg' |
| huT hyT hh.2.2 hui' hyi' hgi' heu hey (fun _ _ _ => rfl) |
| exact ⟨hh.1,hstep.1,hh.2.2,hstep.2⟩ |
| |
| @[fun_prop] lemma measurable_extendFrame_comp {n r : ℕ} {H : Type*} [MeasurableSpace H] |
| (U : H → Matrix (Fin n) (Fin r) ℝ) (q : H → Fin n → ℝ) |
| (hU : Measurable U) (hq : Measurable q) : Measurable (fun h => extendFrame (U h) (q h)) := by |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro a |
| obtain rfl | ⟨a,rfl⟩ := Fin.eq_zero_or_eq_succ a |
| · exact (measurable_pi_apply i).comp hq |
| · exact ((measurable_pi_apply a).comp (measurable_pi_apply i)).comp hU |
| |
| lemma extended_normalized_columns_bound {n r : ℕ} (U : Matrix (Fin n) (Fin r) ℝ) |
| (v : Fin n → ℝ) (hU : ∀ a, ∑ i, U i a^2 ≤ 1) (a : Fin (r+1)) : |
| ∑ i, extendFrame U (unitVector v) i a^2 ≤ 1 := by |
| obtain rfl | ⟨a,rfl⟩ := Fin.eq_zero_or_eq_succ a |
| · exact unitVector_sq_le v |
| · exact hU a |
| |
| theorem regenerated_good_history |
| {H : ℕ → Type*} [∀ n, MeasurableSpace (H n)] |
| (ρ : ∀ n, Measure (H n)) [∀ n, IsProbabilityMeasure (ρ n)] |
| (r : ℕ) (U Y : ∀ n, H n → Matrix (Fin n) (Fin r) ℝ) |
| (v : ∀ n, H n → Fin n → ℝ) |
| (G : ∀ n, Set (H n)) (hrare : ExponentiallyRare ρ (fun n => (G n)ᶜ)) |
| (hgood : ∀ n h, h ∈ G n → (U n h)ᵀ*U n h=1 ∧ (U n h)ᵀ*Y n h=(Y n h)ᵀ*U n h) |
| (s : ℝ) (hs : 0 < s) |
| (hS : ExponentialConvergence ρ |
| (fun n h => (∑ i, (residualProjection (U n h) *ᵥ v n h) i^2)/(n:ℝ)) s) : |
| let q := fun n h => unitVector (residualProjection (U n h) *ᵥ v n h) |
| let G' := fun n => G n ∩ {h | 0 < (∑ i, (residualProjection (U n h) *ᵥ v n h) i^2)/(n:ℝ)} |
| ExponentiallyRare (fun n => (ρ n).prod (standardArrayLaw (Fin n ⊕ Unit))) |
| (fun n => {z | z.1 ∉ G' n}) ∧ |
| (∀ n (z : H n × ((Fin n ⊕ Unit) → ℝ)), z.1 ∈ G' n → |
| let U' := extendFrame (U n z.1) (q n z.1) |
| let Y' := extendFrame (Y n z.1) (regeneratedAnswer (U n z.1) (Y n z.1) (q n z.1) z.2) |
| U'ᵀ*U'=1 ∧ U'ᵀ*Y'=Y'ᵀ*U') := by |
| dsimp only |
| let S := fun n h => (∑ i, (residualProjection (U n h) *ᵥ v n h) i^2)/(n:ℝ) |
| have hr : ExponentiallyRare ρ (fun n => (G n ∩ {h | 0 < S n h})ᶜ) := by |
| apply (hrare.union (hS.positive_rare hs)).mono |
| refine Filter.Eventually.of_forall ?_ |
| intro n h he |
| by_cases hh : h ∈ G n |
| · apply Or.inr |
| change S n h ≤ 0 |
| exact le_of_not_gt (fun hp => he ⟨hh,hp⟩) |
| · exact Or.inl hh |
| refine ⟨hr.prod_fst (fun n => standardArrayLaw (Fin n ⊕ Unit)), ?_⟩ |
| intro n z hz |
| obtain ⟨hU,hY⟩ := hgood n z.1 hz.1 |
| have ho := normalized_residual_perpendicular (U n z.1) hU (v n z.1) |
| have hq : ∑ i, (unitVector (residualProjection (U n z.1) *ᵥ v n z.1)) i^2=1 := by |
| apply unitVector_sq_eq |
| by_contra! hn |
| have hnp := div_nonpos_of_nonpos_of_nonneg hn (Nat.cast_nonneg n : (0:ℝ) ≤ n) |
| exact not_le_of_gt hz.2 hnp |
| exact ⟨extendFrame_orthonormal _ _ hU hq ho, |
| regenerated_frame_compatible _ _ _ hU hY ho z.2⟩ |
| |
| end SKGapCutoff.Regression |
| |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| |
| end |
| |
| end OAI |
| |
