| import Mathlib |
| import OAI.Probability.SKGap.Localization.SampledFlatCLM |
| import OAI.Probability.SKGap.Matrix.DoublePlantLogNorm |
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| namespace OAI |
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| 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 |
| namespace SKGapCutoff.Regression |
|
|
| noncomputable def doublePlantedGOEArray {n : ℕ} (β : ℝ) (x y : Spin n) |
| (g : GaussianCoordinates n) : GaussianCoordinates n := |
| plantedGOEArray β x g + fun p => if p.1=p.2 then 0 else β^2/(n:ℝ)*spin y p.1*spin y p.2 |
|
|
| lemma doublePlantedGOEArray_gaussian {n : ℕ} (β : ℝ) (x y : Spin n) : |
| HasGaussianLaw (doublePlantedGOEArray β x y) (standardArrayLaw (Fin n × Fin n)) := |
| gaussian_add_constant (plantedGOEArray_gaussian β x) _ |
|
|
| lemma sampled_double_planted_gaussian (β : ℝ) {n : ℕ} (hn : 0 < n) (x y : Spin n) : |
| HasGaussianLaw (fun g : GaussianCoordinates n => fun p : Fin n × Fin n => |
| sampledInteraction g p.1 p.2) (doubleSpinPlantedLaw β x y) := by |
| rw [doubleSpinPlantedLaw_eq_gaussianPi β hn] |
| have hh := (piGaussian_coordinates (fun p => β^2/(n:ℝ)*(upperCoefficient x p + upperCoefficient y p)) |
| (Real.toNNReal (β^2/(n:ℝ)))).map_fun (sampledFlatCLM n) |
| exact hh.congr (Eventually.of_forall fun g => funext fun p => sampledFlatCLM_apply g p) |
|
|
| lemma doublePlantedGOEArray_mean {n : ℕ} (β : ℝ) (x y : Spin n) (p : Fin n × Fin n) : |
| (∫ g, doublePlantedGOEArray β x y g p ∂standardArrayLaw (Fin n × Fin n)) = |
| if p.1=p.2 then 0 else β^2/(n:ℝ)*(spin x p.1*spin x p.2+spin y p.1*spin y p.2) := by |
| change (∫ g, plantedGOEArray β x g p + _ ∂standardArrayLaw (Fin n × Fin n)) = _ |
| rw [integral_add ((plantedGOEArray_gaussian β x).eval p).integrable |
| (integrable_const _), plantedGOEArray_mean, integral_const] |
| by_cases h : p.1=p.2 |
| · simp [h] |
| · simp [h] |
| ring |
|
|
| lemma sampled_double_planted_mean (β : ℝ) {n : ℕ} (hn : 0 < n) (x y : Spin n) |
| (p : Fin n × Fin n) : |
| (∫ g, sampledInteraction g p.1 p.2 ∂doubleSpinPlantedLaw β x y) = |
| if p.1=p.2 then 0 else β^2/(n:ℝ)*(spin x p.1*spin x p.2+spin y p.1*spin y p.2) := by |
| rw [doubleSpinPlantedLaw_eq_gaussianPi β hn] |
| by_cases h : p.1=p.2 |
| · simp [sampledInteraction, h] |
| · simp only [sampledInteraction, h, ite_false] |
| rw [piGaussian_coordinate_mean, upperCoefficient_min_max x _ _ h, |
| upperCoefficient_min_max y _ _ h] |
|
|
| lemma doublePlantedGOEArray_covariance {n : ℕ} (hn : 0 < n) (β : ℝ) (x y : Spin n) |
| (p q : Fin n × Fin n) : |
| cov[fun g => doublePlantedGOEArray β x y g p, fun g => doublePlantedGOEArray β x y g q; |
| standardArrayLaw (Fin n × Fin n)] = |
| if p.1=p.2 ∨ q.1=q.2 then 0 else |
| (β^2/(n:ℝ)) * ((if p.1=q.1 ∧ p.2=q.2 then 1 else 0) + |
| (if p.1=q.2 ∧ p.2=q.1 then 1 else 0)) := by |
| change cov[fun g => plantedGOEArray β x g p + _, fun g => plantedGOEArray β x g q + _ |
| rw [covariance_add_const_left ((plantedGOEArray_gaussian β x).eval p).integrable, |
| covariance_add_const_right ((plantedGOEArray_gaussian β x).eval q).integrable, |
| plantedGOEArray_covariance hn] |
|
|
| lemma sampled_double_planted_covariance (β : ℝ) {n : ℕ} (hn : 0 < n) (x y : Spin n) |
| (p q : Fin n × Fin n) : |
| cov[fun g => sampledInteraction g p.1 p.2, fun g => sampledInteraction g q.1 q.2; |
| doubleSpinPlantedLaw β x y] = |
| if p.1=p.2 ∨ q.1=q.2 then 0 else |
| (β^2/(n:ℝ)) * ((if p.1=q.1 ∧ p.2=q.2 then 1 else 0) + |
| (if p.1=q.2 ∧ p.2=q.1 then 1 else 0)) := by |
| rw [doubleSpinPlantedLaw_eq_gaussianPi β hn] |
| by_cases hp : p.1=p.2 |
| · simp [sampledInteraction, hp] |
| by_cases hq : q.1=q.2 |
| · simp [sampledInteraction, hq] |
| simp only [sampledInteraction, hp, hq, false_or, ite_false] |
| rw [piGaussian_coordinate_covariance, Real.coe_toNNReal _ (by positivity)] |
| simp only [minmax_pair_eq_iff] |
| by_cases h1 : p.1=q.1 ∧ p.2=q.2 |
| · have h2 : ¬ (p.1=q.2 ∧ p.2=q.1) := by omega |
| rw [ite_eq_left (Or.inl h1), ite_eq_left h1, ite_eq_right h2, add_zero, mul_one] |
| by_cases h2 : p.1=q.2 ∧ p.2=q.1 |
| · rw [ite_eq_left (Or.inr h2), ite_eq_right h1, ite_eq_left h2, zero_add, mul_one] |
| · rw [ite_eq_right (not_or.mpr ⟨h1,h2⟩), ite_eq_right h1, ite_eq_right h2, add_zero, mul_zero] |
| |
| lemma double_planted_matrix_representation (β : ℝ) {n : ℕ} (hn : 0 < n) (x y : Spin n) : |
| (doubleSpinPlantedLaw β x y).map (fun g : GaussianCoordinates n => |
| fun p : Fin n × Fin n => sampledInteraction g p.1 p.2) = |
| (standardArrayLaw (Fin n × Fin n)).map (doublePlantedGOEArray β x y) := by |
| apply gaussian_vector_law_ext (sampled_double_planted_gaussian β hn x y) |
| (doublePlantedGOEArray_gaussian β x y) |
| · intro p |
| rw [sampled_double_planted_mean β hn, doublePlantedGOEArray_mean] |
| · intro p q |
| rw [sampled_double_planted_covariance β hn, doublePlantedGOEArray_covariance hn] |
|
|
| end SKGapCutoff.Regression |
|
|
| end |
|
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| end OAI |
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