| import Mathlib |
| import OAI.Probability.SKGap.Matrix.CompletedMatrix |
|
|
| 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 |
| namespace SKGapCutoff.Regression |
|
|
| local instance completedMatrixMeasurableSpace {n m : ℕ} : |
| MeasurableSpace (Matrix (Fin n) (Fin m) ℝ) := |
| inferInstanceAs (MeasurableSpace (Fin n → Fin m → ℝ)) |
|
|
| @[fun_prop] lemma measurable_completedMatrix {n r : ℕ} : |
| Measurable (fun z : (Matrix (Fin n) (Fin r) ℝ × Matrix (Fin n) (Fin r) ℝ) × |
| ((Fin n × Fin n) → ℝ) => completedMatrix z.1.1 z.1.2 z.2) := by |
| apply Measurable.of_eval |
| intro i |
| apply Measurable.of_eval |
| intro j |
| simp only [completedMatrix,revealedCompletion,residualProjection,goe, |
| Matrix.add_apply,Matrix.sub_apply,Matrix.mul_apply,Matrix.transpose_apply] |
| fun_prop |
|
|
| @[fun_prop] lemma measurable_revealedCompletion_comp {n r : ℕ} {H : Type*} |
| [MeasurableSpace H] (U Y : H → Matrix (Fin n) (Fin r) ℝ) |
| (hU : Measurable U) (hY : Measurable Y) : |
| Measurable (fun h => revealedCompletion (U h) (Y h)) := by |
| have hU' : Measurable (fun h i j => U h i j) := hU |
| have hY' : Measurable (fun h i j => Y h i j) := hY |
| 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 |
|
|
| theorem completedMatrix_section_law {n r : ℕ} (hn : 0 < n) |
| (U Y : Matrix (Fin n) (Fin r) ℝ) (q : Fin n → ℝ) |
| (hU : Uᵀ*U=1) (hY : Uᵀ*Y=Yᵀ*U) |
| (hq : ∑ i, q i^2=1) (ho : Uᵀ *ᵥ q=0) : |
| let p := U *ᵥ (Yᵀ *ᵥ q) |
| (standardArrayLaw (Fin n × Fin n)).map |
| (fun g => (completedMatrix U Y g *ᵥ q,completedMatrix U Y g)) = |
| ((standardArrayLaw (Fin n × Fin n)).prod (standardArrayLaw (Fin n ⊕ Unit))).map |
| (fun z => (p+queryInnovation (residualProjection U) q z.2, |
| completedMatrix (extendFrame U q) |
| (extendFrame Y (p+queryInnovation (residualProjection U) q z.2)) z.1)) := by |
| dsimp only |
| let P := residualProjection U |
| let p := U *ᵥ (Yᵀ *ᵥ q) |
| have hPq : P *ᵥ q=q := by |
| simp only [P,residualProjection,Matrix.sub_mulVec,Matrix.one_mulVec, |
| ← Matrix.mulVec_mulVec,ho,Matrix.mulVec_zero,sub_zero] |
| have hqc : P*queryColumn q=queryColumn q := by |
| ext i a |
| fin_cases a |
| exact congrFun hPq i |
| have hj := query_joint_law hn P (residualProjection_symmetric U) |
| (residualProjection_idem U hU) q hq hqc |
| let out : ((Fin n × Fin n) → ℝ) × (Fin n → ℝ) → |
| (Fin n → ℝ) × Matrix (Fin n) (Fin n) ℝ := fun z => |
| (p+z.2,revealedCompletion (extendFrame U q) (extendFrame Y (p+z.2))+ |
| Matrix.of (fun i j => z.1 (i,j))) |
| have ha : Measurable (fun z : ((Fin n × Fin n) → ℝ) × (Fin n → ℝ) => p+z.2) := |
| measurable_const.add measurable_snd |
| have hy : Measurable (fun z : ((Fin n × Fin n) → ℝ) × (Fin n → ℝ) => |
| extendFrame Y (p+z.2)) := measurable_extendFrame.comp (measurable_const.prodMk ha) |
| have hc : Measurable (fun z : ((Fin n × Fin n) → ℝ) × (Fin n → ℝ) => |
| revealedCompletion (extendFrame U q) (extendFrame Y (p+z.2))) := |
| measurable_revealedCompletion_comp _ _ measurable_const hy |
| have hout : Measurable out := by |
| apply 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_fst) |
| have hm1 := (queryResidual_measurable P q).prodMk (queryAnswer_measurable P q) |
| have hm2 : Measurable (fun z : ((Fin n × Fin n) → ℝ) × ((Fin n ⊕ Unit) → ℝ) => |
| (queryResidual P q z.1,queryInnovation P q z.2)) := |
| ((queryResidual_measurable P q).comp measurable_fst).prodMk |
| ((queryInnovation_measurable P q).comp measurable_snd) |
| have he := congrArg (Measure.map out) hj |
| rw [Measure.map_map hout hm1,Measure.map_map hout hm2] at he |
| have hof (A : Matrix (Fin n) (Fin n) ℝ) : Matrix.of (fun i j => A i j)=A := rfl |
| have hpoint g : out (queryResidual P q g,queryAnswer P q g)= |
| (completedMatrix U Y g *ᵥ q,completedMatrix U Y g) := by |
| have ha := completed_query_answer U Y q g ho hPq |
| change completedMatrix U Y g *ᵥ q=p+queryAnswer P q g at ha |
| dsimp only [out] |
| rw [← ha] |
| congr 1 |
| simpa only [extendFrame_projection,P,queryResidual,hof] using |
| (completedMatrix_update U Y q hU hY ho g).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 |
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|
