Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Probability.SKGap.Localization.DiagonalSum
import OAI.Probability.SKGap.Matrix.MatrixQuad

namespace OAI

section
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
open Filter
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius NNReal
noncomputable section
open MeasureTheory ProbabilityTheory Real Set Filter
open MeasureTheory.Measure
open scoped ENNReal NNReal MeasureTheory Topology
open MeasureTheory
noncomputable section
noncomputable section
open MeasureTheory Set NormedSpace
open scoped Topology
noncomputable section
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
noncomputable section
open Set Real
open scoped Topology
noncomputable section
open Matrix Set Filter
open scoped Topology Matrix.Norms.Frobenius
noncomputable section
open Matrix NormedSpace ContinuousLinearMap
open scoped Matrix.Norms.Frobenius
noncomputable section
open Matrix
noncomputable section
open MeasureTheory ProbabilityTheory Real Set
open scoped ENNReal NNReal
noncomputable section
open MeasureTheory ProbabilityTheory InformationTheory Real Set
open scoped NNReal ENNReal
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory
open Real
noncomputable section
open scoped BigOperators Topology
open Filter Real
namespace SKGap
section GOEPath
variable {ι : Type*} [Fintype ι] [DecidableEq ι]

def stabilityMatrix (a : ι → ℝ) (z q : ℝ) (M : Matrix ι ι ℝ) : Matrix ι ι ℝ :=
  fun i k => (if i = k then 1 + z^2*q*a i else 0) - z*sqrt (a i)*M i k*sqrt (a k)

lemma stabilityMatrix_quad (a : ι → ℝ) (ha : ∀ i, 0 ≤ a i) (z q : ℝ)
    (M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) :
    matrixQuad (stabilityMatrix a z q M) p =
      ‖p‖^2 + z^2*q*‖diagonalSqrt a p‖^2 - z*matrixQuad M (diagonalSqrt a p) := by
  rw [diagonalSqrt_norm_sq a ha, ← sum_coord_sq p]
  simp only [matrixQuad, stabilityMatrix, mul_sub, sub_mul, Finset.sum_sub_distrib]
  congr 1
  · simp only [mul_ite, ite_mul, mul_zero, zero_mul, Finset.mul_sum, ← Finset.sum_add_distrib]
    apply Finset.sum_congr rfl
    intro i _
    rw [Finset.sum_ite_eq]
    simp only [Finset.mem_univ, ite_true]
    ring
  · simp only [Finset.mul_sum, diagonalSqrt]
    apply Finset.sum_congr rfl
    intro i _
    apply Finset.sum_congr rfl
    intro k _
    ring

omit [Fintype ι] in
lemma stabilityMatrix_symm {M : Matrix ι ι ℝ} (hM : M.IsHermitian)
    (a : ι → ℝ) (z q : ℝ) : (stabilityMatrix a z q M).IsHermitian := by
  ext i k
  have hmk : M k i = M i k := by simpa using congrFun (congrFun hM i) k
  simp only [Matrix.conjTranspose_apply, star_trivial, stabilityMatrix, hmk]
  by_cases h : i = k
  · subst k; rfl
  · simp only [h, Ne.symm h, ↓reduceIte]
    ring

lemma diagonalProcess_coercive {r z q c : ℝ} {a : ι → ℝ}
    (ha : ∀ i, 0 ≤ a i) {g : MatrixCoordinates ι → ℝ}
    (hg : diagonalProcess r a z q g ≤ 1-c) (p : EuclideanSpace ℝ ι) :
    c*‖p‖^2 ≤ matrixQuad (stabilityMatrix a z q (goeMatrix r g)) p := by
  by_cases hp : p = 0
  · simp [hp, matrixQuad]
  have hp0 : 0 < ‖p‖ := norm_pos_iff.mpr hp
  let v : EuclideanSpace ℝ ι := ‖p‖⁻¹ • p
  have hv : ‖v‖ = 1 := by
    dsimp [v]
    rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hp0, inv_mul_cancel₀ hp0.ne']
  let v' : UnitBall ι := ⟨v, by simp [Metric.mem_closedBall, hv]⟩
  have h := (gaussianAffine_le_gaussianSup (continuous_diagonalOffset a z q)
    (continuous_diagonalCoeff (continuous_quadraticCoeff r) a z) g v').trans hg
  rw [diagonal_affine_matrix] at h
  have hs := stabilityMatrix_quad a ha z q (goeMatrix r g) v
  change z*matrixQuad (goeMatrix r g) (diagonalSqrt a v) -
    z^2*q*‖diagonalSqrt a v‖^2 ≤ 1-c at h
  rw [hv, one_pow] at hs
  have hc : c ≤ matrixQuad (stabilityMatrix a z q (goeMatrix r g)) v := by linarith
  rw [show v = ‖p‖⁻¹ • p from rfl, matrixQuad_smul] at hc
  have hc' : c ≤ matrixQuad (stabilityMatrix a z q (goeMatrix r g)) p / ‖p‖^2 := by
    simpa only [inv_pow, div_eq_mul_inv, mul_comm] using hc
  exact (le_div_iff₀ (sq_pos_of_pos hp0)).mp hc'
variable [Nonempty ι]
theorem goe_diagonal_matrix_path_tail {j A : ℝ} {a : ι → ℝ}
    (hj : 0 < j) (hA0 : 0 < A) (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A)
    (hsub : sqrt j*A < 1)
    (herr : 2*sqrt A*sqrt (sqrt (2*j^2*A^2/(Fintype.card ι:ℝ))) ≤
      (1-sqrt j*A)^2/4) :
    (gaussianCoordinates (MatrixCoordinates ι)).real
      {g | ∃ z ∈ Icc (0:ℝ) 1, ∃ p : EuclideanSpace ℝ ι,
        matrixQuad (stabilityMatrix a z ((j/(Fintype.card ι:ℝ))*∑ i, a i)
          (goeMatrix (j/(Fintype.card ι:ℝ)) g)) p < (1-sqrt j*A)^2/2 * ‖p‖^2} ≤
      Real.exp (-((1-sqrt j*A)^2)^2 * (Fintype.card ι:ℝ) / (16*π^2*j*A^2)) := by
  apply (measureReal_mono ?_).trans (scaled_diagonal_path_tail hj hA0 ha hA hsub herr)
  rintro g ⟨z, hz, p, hp⟩
  refine ⟨z, hz, ?_⟩
  by_contra hh
  have hc := diagonalProcess_coercive ha (le_of_not_gt hh) p
  exact (not_lt_of_ge hc) hp
end GOEPath
end SKGap
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI

	import Mathlib
	import OAI.Probability.SKGap.Localization.DiagonalSum
	import OAI.Probability.SKGap.Matrix.MatrixQuad

	namespace OAI

	section
	noncomputable section
	open MeasureTheory ProbabilityTheory InformationTheory Real Set
	open scoped NNReal ENNReal
	open Filter
	open scoped Topology
	noncomputable section
	open Matrix Real
	open scoped BigOperators Matrix.Norms.Frobenius ENNReal NNReal
	noncomputable section
	open Matrix Real
	open scoped BigOperators Matrix.Norms.Frobenius NNReal
	noncomputable section
	open MeasureTheory ProbabilityTheory Real Set Filter
	open MeasureTheory.Measure
	open scoped ENNReal NNReal MeasureTheory Topology
	open MeasureTheory
	noncomputable section
	noncomputable section
	open MeasureTheory Set NormedSpace
	open scoped Topology
	noncomputable section
	open Matrix Real
	open scoped BigOperators Matrix.Norms.Frobenius
	noncomputable section
	open Set Real
	open scoped Topology
	noncomputable section
	open Matrix Set Filter
	open scoped Topology Matrix.Norms.Frobenius
	noncomputable section
	open Matrix NormedSpace ContinuousLinearMap
	open scoped Matrix.Norms.Frobenius
	noncomputable section
	open Matrix
	noncomputable section
	open MeasureTheory ProbabilityTheory Real Set
	open scoped ENNReal NNReal
	noncomputable section
	open MeasureTheory ProbabilityTheory InformationTheory Real Set
	open scoped NNReal ENNReal
	noncomputable section
	open scoped BigOperators
	open MeasureTheory ProbabilityTheory
	open Real
	noncomputable section
	open scoped BigOperators Topology
	open Filter Real
	namespace SKGap
	section GOEPath
	variable {ι : Type*} [Fintype ι] [DecidableEq ι]

	def stabilityMatrix (a : ι → ℝ) (z q : ℝ) (M : Matrix ι ι ℝ) : Matrix ι ι ℝ :=
	fun i k => (if i = k then 1 + z^2qa i else 0) - zsqrt (a i)M i k*sqrt (a k)

	lemma stabilityMatrix_quad (a : ι → ℝ) (ha : ∀ i, 0 ≤ a i) (z q : ℝ)
	(M : Matrix ι ι ℝ) (p : EuclideanSpace ℝ ι) :
	matrixQuad (stabilityMatrix a z q M) p =
	‖p‖^2 + z^2q‖diagonalSqrt a p‖^2 - z*matrixQuad M (diagonalSqrt a p) := by
	rw [diagonalSqrt_norm_sq a ha, ← sum_coord_sq p]
	simp only [matrixQuad, stabilityMatrix, mul_sub, sub_mul, Finset.sum_sub_distrib]
	congr 1
	· simp only [mul_ite, ite_mul, mul_zero, zero_mul, Finset.mul_sum, ← Finset.sum_add_distrib]
	apply Finset.sum_congr rfl
	intro i _
	rw [Finset.sum_ite_eq]
	simp only [Finset.mem_univ, ite_true]
	ring
	· simp only [Finset.mul_sum, diagonalSqrt]
	apply Finset.sum_congr rfl
	intro i _
	apply Finset.sum_congr rfl
	intro k _
	ring

	omit [Fintype ι] in
	lemma stabilityMatrix_symm {M : Matrix ι ι ℝ} (hM : M.IsHermitian)
	(a : ι → ℝ) (z q : ℝ) : (stabilityMatrix a z q M).IsHermitian := by
	ext i k
	have hmk : M k i = M i k := by simpa using congrFun (congrFun hM i) k
	simp only [Matrix.conjTranspose_apply, star_trivial, stabilityMatrix, hmk]
	by_cases h : i = k
	· subst k; rfl
	· simp only [h, Ne.symm h, ↓reduceIte]
	ring

	lemma diagonalProcess_coercive {r z q c : ℝ} {a : ι → ℝ}
	(ha : ∀ i, 0 ≤ a i) {g : MatrixCoordinates ι → ℝ}
	(hg : diagonalProcess r a z q g ≤ 1-c) (p : EuclideanSpace ℝ ι) :
	c*‖p‖^2 ≤ matrixQuad (stabilityMatrix a z q (goeMatrix r g)) p := by
	by_cases hp : p = 0
	· simp [hp, matrixQuad]
	have hp0 : 0 < ‖p‖ := norm_pos_iff.mpr hp
	let v : EuclideanSpace ℝ ι := ‖p‖⁻¹ • p
	have hv : ‖v‖ = 1 := by
	dsimp [v]
	rw [norm_smul, Real.norm_eq_abs, abs_inv, abs_of_pos hp0, inv_mul_cancel₀ hp0.ne']
	let v' : UnitBall ι := ⟨v, by simp [Metric.mem_closedBall, hv]⟩
	have h := (gaussianAffine_le_gaussianSup (continuous_diagonalOffset a z q)
	(continuous_diagonalCoeff (continuous_quadraticCoeff r) a z) g v').trans hg
	rw [diagonal_affine_matrix] at h
	have hs := stabilityMatrix_quad a ha z q (goeMatrix r g) v
	change z*matrixQuad (goeMatrix r g) (diagonalSqrt a v) -
	z^2q‖diagonalSqrt a v‖^2 ≤ 1-c at h
	rw [hv, one_pow] at hs
	have hc : c ≤ matrixQuad (stabilityMatrix a z q (goeMatrix r g)) v := by linarith
	rw [show v = ‖p‖⁻¹ • p from rfl, matrixQuad_smul] at hc
	have hc' : c ≤ matrixQuad (stabilityMatrix a z q (goeMatrix r g)) p / ‖p‖^2 := by
	simpa only [inv_pow, div_eq_mul_inv, mul_comm] using hc
	exact (le_div_iff₀ (sq_pos_of_pos hp0)).mp hc'

	variable [Nonempty ι]

	theorem goe_diagonal_matrix_path_tail {j A : ℝ} {a : ι → ℝ}
	(hj : 0 < j) (hA0 : 0 < A) (ha : ∀ i, 0 ≤ a i) (hA : ∀ i, a i ≤ A)
	(hsub : sqrt j*A < 1)
	(herr : 2sqrt Asqrt (sqrt (2j^2A^2/(Fintype.card ι:ℝ))) ≤
	(1-sqrt j*A)^2/4) :
	(gaussianCoordinates (MatrixCoordinates ι)).real
	{g \| ∃ z ∈ Icc (0:ℝ) 1, ∃ p : EuclideanSpace ℝ ι,
	matrixQuad (stabilityMatrix a z ((j/(Fintype.card ι:ℝ))*∑ i, a i)
	(goeMatrix (j/(Fintype.card ι:ℝ)) g)) p < (1-sqrt jA)^2/2 ‖p‖^2} ≤
	Real.exp (-((1-sqrt jA)^2)^2 (Fintype.card ι:ℝ) / (16π^2j*A^2)) := by
	apply (measureReal_mono ?_).trans (scaled_diagonal_path_tail hj hA0 ha hA hsub herr)
	rintro g ⟨z, hz, p, hp⟩
	refine ⟨z, hz, ?_⟩
	by_contra hh
	have hc := diagonalProcess_coercive ha (le_of_not_gt hh) p
	exact (not_lt_of_ge hc) hp

	end GOEPath
	end SKGap

	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end

	end OAI