| import Mathlib |
| import OAI.Probability.SKGap.Localization.IntegrableLinearCombinationSq |
| import OAI.Probability.SKGap.Localization.ProdEventSectionLaw |
|
|
| 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 |
| namespace SKGapCutoff.Regression |
|
|
| local instance {n m : ℕ} : MeasurableSpace (Matrix (Fin n) (Fin m) ℝ) := |
| inferInstanceAs (MeasurableSpace (Fin n → Fin m → ℝ)) |
|
|
| noncomputable def completedMatrix {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (g : (Fin n × Fin n) → ℝ) : Matrix (Fin n) (Fin n) ℝ := |
| revealedCompletion U Y + residualProjection U*goe g*residualProjection U |
|
|
| lemma completedMatrix_symmetric {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (hY : Uᵀ*Y=Yᵀ*U) (g : (Fin n × Fin n) → ℝ) : |
| (completedMatrix U Y g)ᵀ=completedMatrix U Y g := by |
| simp only [completedMatrix,Matrix.transpose_add,Matrix.transpose_mul, |
| revealedCompletion_symmetric U Y hY,residualProjection_symmetric,goe_symmetric, |
| Matrix.mul_assoc] |
| |
| lemma completedMatrix_answer {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (hU : Uᵀ*U=1) (hY : Uᵀ*Y=Yᵀ*U) (g : (Fin n × Fin n) → ℝ) : |
| completedMatrix U Y g*U=Y := by |
| rw [completedMatrix,Matrix.add_mul,revealedCompletion_answer U Y hU hY, |
| Matrix.mul_assoc,residualProjection_mul U hU,Matrix.mul_zero,add_zero] |
| |
| lemma mul_extendFrame {n r : ℕ} (A : Matrix (Fin n) (Fin n) ℝ) |
| (U : Matrix (Fin n) (Fin r) ℝ) (q : Fin n → ℝ) : |
| A*extendFrame U q=extendFrame (A*U) (A *ᵥ q) := by |
| funext i a |
| obtain rfl | ⟨a,rfl⟩ := Fin.eq_zero_or_eq_succ a |
| · rfl |
| · rfl |
| |
| lemma extendedProjection_mul_old {n r : ℕ} (U : Matrix (Fin n) (Fin r) ℝ) |
| (q : Fin n → ℝ) (hU : Uᵀ*U=1) (ho : Uᵀ *ᵥ q=0) : |
| residualProjection (extendFrame U q)*residualProjection U= |
| residualProjection (extendFrame U q) := by |
| have hqU : (queryColumn q)ᵀ*U=0 := by |
| ext a b |
| fin_cases a |
| have hh := congrFun ho b |
| simpa only [Matrix.mul_apply,Matrix.transpose_apply,queryColumn,Matrix.mulVec, |
| dotProduct,Matrix.zero_apply,Pi.zero_apply,mul_comm] using hh |
| have hqP : (queryColumn q)ᵀ*residualProjection U=(queryColumn q)ᵀ := by |
| rw [residualProjection,Matrix.mul_sub,Matrix.mul_one, |
| ← Matrix.mul_assoc,hqU,Matrix.zero_mul,sub_zero] |
| rw [extendFrame_projection,queryComplement,Matrix.sub_mul, |
| residualProjection_idem U hU,Matrix.mul_assoc,hqP] |
| |
| lemma old_mul_extendedProjection {n r : ℕ} (U : Matrix (Fin n) (Fin r) ℝ) |
| (q : Fin n → ℝ) (hU : Uᵀ*U=1) (ho : Uᵀ *ᵥ q=0) : |
| residualProjection U*residualProjection (extendFrame U q)= |
| residualProjection (extendFrame U q) := by |
| have hh := congrArg Matrix.transpose (extendedProjection_mul_old U q hU ho) |
| simpa only [Matrix.transpose_mul,residualProjection_symmetric] using hh |
| |
| lemma completedMatrix_new_residual {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (q : Fin n → ℝ) (hU : Uᵀ*U=1) (ho : Uᵀ *ᵥ q=0) |
| (g : (Fin n × Fin n) → ℝ) : |
| residualProjection (extendFrame U q)*completedMatrix U Y g* |
| residualProjection (extendFrame U q)= |
| residualProjection (extendFrame U q)*goe g*residualProjection (extendFrame U q) := by |
| let P := residualProjection U |
| let Q := residualProjection (extendFrame U q) |
| have hQP : Q*P=Q := extendedProjection_mul_old U q hU ho |
| have hPQ : P*Q=Q := old_mul_extendedProjection U q hU ho |
| have hR : Q*revealedCompletion U Y*Q=0 := by |
| calc |
| _ = Q*(P*revealedCompletion U Y*P)*Q := by |
| simp only [← Matrix.mul_assoc,hQP] |
| rw [Matrix.mul_assoc (Q*revealedCompletion U Y) P Q,hPQ] |
| _ = 0 := by rw [revealedCompletion_residual U Y hU,Matrix.mul_zero,Matrix.zero_mul] |
| change Q*(revealedCompletion U Y+P*goe g*P)*Q=Q*goe g*Q |
| rw [Matrix.mul_add,Matrix.add_mul,hR,zero_add] |
| simp only [← Matrix.mul_assoc,hQP] |
| rw [Matrix.mul_assoc (Q*goe g) P Q,hPQ] |
| |
| theorem completedMatrix_update {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (q : Fin n → ℝ) (hU : Uᵀ*U=1) (hY : Uᵀ*Y=Yᵀ*U) (ho : Uᵀ *ᵥ q=0) |
| (g : (Fin n × Fin n) → ℝ) : |
| completedMatrix U Y g = |
| revealedCompletion (extendFrame U q) (extendFrame Y (completedMatrix U Y g *ᵥ q))+ |
| residualProjection (extendFrame U q)*goe g*residualProjection (extendFrame U q) := by |
| have hh := regression_decomposition (completedMatrix U Y g) (extendFrame U q) |
| (completedMatrix_symmetric U Y hY g) |
| simpa only [mul_extendFrame,completedMatrix_answer U Y hU hY, |
| revealedCompletion,completedMatrix_new_residual U Y q hU ho] using hh |
| |
| lemma completedMatrix_new_compatibility {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (q : Fin n → ℝ) (hU : Uᵀ*U=1) (hY : Uᵀ*Y=Yᵀ*U) (g : (Fin n × Fin n) → ℝ) : |
| (extendFrame U q)ᵀ*extendFrame Y (completedMatrix U Y g *ᵥ q)= |
| (extendFrame Y (completedMatrix U Y g *ᵥ q))ᵀ*extendFrame U q := by |
| have he : completedMatrix U Y g*extendFrame U q= |
| extendFrame Y (completedMatrix U Y g *ᵥ q) := by |
| rw [mul_extendFrame,completedMatrix_answer U Y hU hY g] |
| rw [← he,Matrix.transpose_mul,completedMatrix_symmetric U Y hY g,Matrix.mul_assoc] |
| |
| lemma completedMatrix_input_decomposition {n r : ℕ} (U Y : Matrix (Fin n) (Fin r) ℝ) |
| (hU : Uᵀ*U=1) (hY : Uᵀ*Y=Yᵀ*U) (v : Fin n → ℝ) |
| (g : (Fin n × Fin n) → ℝ) : |
| completedMatrix U Y g *ᵥ v = Y *ᵥ (Uᵀ *ᵥ v)+ |
| completedMatrix U Y g *ᵥ (residualProjection U *ᵥ v) := by |
| simp only [residualProjection,Matrix.sub_mulVec,Matrix.one_mulVec, |
| Matrix.mulVec_sub,Matrix.mulVec_mulVec] |
| rw [← Matrix.mul_assoc,completedMatrix_answer U Y hU hY g] |
| abel |
| |
| @[fun_prop] lemma measurable_revealedCompletion {n r : ℕ} : |
| Measurable (fun z : Matrix (Fin n) (Fin r) ℝ × Matrix (Fin n) (Fin r) ℝ => |
| revealedCompletion z.1 z.2) := by |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro j |
| simp only [revealedCompletion,Matrix.add_apply,Matrix.sub_apply, |
| Matrix.mul_apply,Matrix.transpose_apply] |
| fun_prop |
| |
| @[fun_prop] lemma measurable_extendFrame {n r : ℕ} : |
| Measurable (fun z : Matrix (Fin n) (Fin r) ℝ × (Fin n → ℝ) => extendFrame z.1 z.2) := 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 measurable_snd |
| · exact (measurable_pi_apply a).comp ((measurable_pi_apply i).comp measurable_fst) |
|
|
| theorem completedMatrix_regeneration {n r : ℕ} (hn : 0 < n) {H : Type*} |
| [MeasurableSpace H] (ρ : Measure H) [SFinite ρ] |
| (U Y : H → Matrix (Fin n) (Fin r) ℝ) (q : H → Fin n → ℝ) |
| (hUm : Measurable U) (hYm : Measurable Y) (hqm : Measurable q) |
| (hU : ∀ h, (U h)ᵀ*U h=1) (hY : ∀ h, (U h)ᵀ*Y h=(Y h)ᵀ*U h) |
| (hq : ∀ h, ∑ i, q h i^2=1) (ho : ∀ h, (U h)ᵀ *ᵥ q h=0) : |
| let p := fun h => U h *ᵥ ((Y h)ᵀ *ᵥ q h) |
| (ρ.prod (standardArrayLaw (Fin n × Fin n))).map |
| (fun z => ((z.1,completedMatrix (U z.1) (Y z.1) z.2 *ᵥ q z.1), |
| completedMatrix (U z.1) (Y z.1) z.2)) = |
| ((answerHistoryLaw ρ (fun h => residualProjection (U h)) q).prod |
| (standardArrayLaw (Fin n × Fin n))).map |
| (fun z => ((z.1.1,p z.1.1+z.1.2), |
| completedMatrix (extendFrame (U z.1.1) (q z.1.1)) |
| (extendFrame (Y z.1.1) (p z.1.1+z.1.2)) z.2)) := by |
| dsimp only |
| let P := fun h => residualProjection (U h) |
| let p := fun h => U h *ᵥ ((Y h)ᵀ *ᵥ q h) |
| have hU' : Measurable (fun h => fun i j => U h i j) := hUm |
| have hY' : Measurable (fun h => fun i j => Y h i j) := hYm |
| have hPm : Measurable P := by |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro j |
| simp only [P,residualProjection,Matrix.sub_apply,Matrix.mul_apply,Matrix.transpose_apply] |
| fun_prop |
| have hpm : Measurable p := by |
| apply Measurable.of_eval |
| intro i |
| simp only [p,Matrix.mulVec,dotProduct,Matrix.transpose_apply] |
| fun_prop |
| have hPq h : P h *ᵥ q h=q h := by |
| dsimp [P,residualProjection] |
| rw [Matrix.sub_mulVec,Matrix.one_mulVec,← Matrix.mulVec_mulVec,ho, |
| Matrix.mulVec_zero,sub_zero] |
| have hPqc h : P h*queryColumn (q h)=queryColumn (q h) := by |
| ext i a |
| fin_cases a |
| exact congrFun (hPq h) i |
| have hj := adaptive_query_regeneration hn ρ P q hPm hqm |
| (fun h => residualProjection_symmetric (U h)) |
| (fun h => residualProjection_idem (U h) (hU h)) hq hPqc |
| let out : (H × (Fin n → ℝ)) × ((Fin n × Fin n) → ℝ) → |
| (H × (Fin n → ℝ)) × Matrix (Fin n) (Fin n) ℝ := fun z => |
| ((z.1.1,p z.1.1+z.1.2), |
| revealedCompletion (extendFrame (U z.1.1) (q z.1.1)) |
| (extendFrame (Y z.1.1) (p z.1.1+z.1.2))+Matrix.of (fun i j => z.2 (i,j))) |
| have hout : Measurable out := by |
| have ha : Measurable (fun z : (H × (Fin n → ℝ)) × ((Fin n × Fin n) → ℝ) => |
| p z.1.1+z.1.2) := (hpm.comp measurable_fst.fst).add measurable_fst.snd |
| have hu' : Measurable (fun z : (H × (Fin n → ℝ)) × ((Fin n × Fin n) → ℝ) => |
| extendFrame (U z.1.1) (q z.1.1)) := measurable_extendFrame.comp |
| ((hUm.comp measurable_fst.fst).prodMk (hqm.comp measurable_fst.fst)) |
| have hy' : Measurable (fun z : (H × (Fin n → ℝ)) × ((Fin n × Fin n) → ℝ) => |
| extendFrame (Y z.1.1) (p z.1.1+z.1.2)) := measurable_extendFrame.comp ((hYm.comp measurable_fst.fst).prodMk ha) |
| have hc := measurable_revealedCompletion.comp (hu'.prodMk hy') |
| apply (measurable_fst.fst.prodMk ha).prodMk |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro j |
| exact (((measurable_pi_apply j).comp (measurable_pi_apply i)).comp hc).add |
| ((measurable_pi_apply (i,j)).comp measurable_snd) |
| have hmL : Measurable (fun z : H × ((Fin n × Fin n) → ℝ) => |
| ((z.1, queryAnswer (P z.1) (q z.1) z.2),queryResidual (P z.1) (q z.1) z.2)) := |
| (measurable_fst.prodMk (measurable_queryAnswer_comp _ _ _ |
| (hPm.comp measurable_fst) (hqm.comp measurable_fst) measurable_snd)).prodMk |
| (measurable_queryResidual_comp _ _ _ (hPm.comp measurable_fst) |
| (hqm.comp measurable_fst) measurable_snd) |
| have hmR : Measurable (fun z : (H × (Fin n → ℝ)) × ((Fin n × Fin n) → ℝ) => |
| (z.1,queryResidual (P z.1.1) (q z.1.1) z.2)) := |
| measurable_fst.prodMk (measurable_queryResidual_comp _ _ _ |
| (hPm.comp measurable_fst.fst) (hqm.comp measurable_fst.fst) measurable_snd) |
| have he := congrArg (Measure.map out) hj |
| rw [Measure.map_map hout hmL,Measure.map_map hout hmR] at he |
| have hof (A : Matrix (Fin n) (Fin n) ℝ) : Matrix.of (fun i j => A i j)=A := rfl |
| have hpoint (z : H × ((Fin n × Fin n) → ℝ)) : out ((z.1,queryAnswer (P z.1) (q z.1) z.2), |
| queryResidual (P z.1) (q z.1) z.2) = |
| ((z.1,completedMatrix (U z.1) (Y z.1) z.2 *ᵥ q z.1), |
| completedMatrix (U z.1) (Y z.1) z.2) := by |
| have ha := completed_query_answer (U z.1) (Y z.1) (q z.1) z.2 (ho z.1) (hPq z.1) |
| change completedMatrix (U z.1) (Y z.1) z.2 *ᵥ q z.1=p z.1+queryAnswer (P z.1) (q z.1) z.2 at ha |
| dsimp only [out] |
| rw [← ha] |
| congr 1 |
| simpa only [extendFrame_projection,P,queryResidual,hof] using |
| (completedMatrix_update (U z.1) (Y z.1) (q z.1) (hU z.1) (hY z.1) (ho z.1) z.2).symm |
| simp_rw [Function.comp_def,hpoint] at he |
| simpa only [out,completedMatrix,queryResidual,← extendFrame_projection,P,p,hof] using he |
|
|
| end SKGapCutoff.Regression |
|
|
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| |
| end |
|
|
| end OAI |
|
|
