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a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Probability.InvariantIsing.Gaussian.BilinearGaussianMinmaxMean
import OAI.Probability.InvariantIsing.Core.UnitInnerMaximum
namespace OAI
/-! Passing the row tests to the full unit sphere in Gordon's matrix comparison. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped BigOperators RealInnerProductSpace Topology
namespace InvariantIsing
lemma continuous_gaussianPattern_apply {N m : ℕ} (v : EuclideanSpace ℝ (Fin m)) :
Continuous (fun z : EuclideanSpace ℝ (Fin N × Fin m) => gaussianPatternEuclideanOperator z v) := by
change Continuous (fun z : EuclideanSpace ℝ (Fin N × Fin m) =>
WithLp.toLp 2 (fun i => ∑ j, z (i,j)*v j))
exact (PiLp.continuous_toLp 2 (fun _ : Fin N => ℝ)).comp
(continuous_pi (fun i => by fun_prop))
theorem gaussian_finite_column_minimum {V : Type*} [Fintype V] [Nonempty V]
{N m : ℕ} (hN : 0 < N) (v : V → EuclideanSpace ℝ (Fin m)) (hv : ∀ b, ‖v b‖ = 1) :
(∫ z : EuclideanSpace ℝ (Fin N), ‖z‖ ∂stdGaussian _)-Real.sqrt m ≤
∫ z : EuclideanSpace ℝ (Fin N × Fin m),
Finset.univ.inf' Finset.univ_nonempty (fun b => ‖gaussianPatternEuclideanOperator z (v b)‖)
∂stdGaussian _ := by
let : NeZero N := ⟨Nat.ne_of_gt hN⟩
let : Nonempty (UnitVector (EuclideanSpace ℝ (Fin N))) := unitVector_nonempty _
let S := UnitVector (EuclideanSpace ℝ (Fin N))
let s : ℕ → S := TopologicalSpace.denseSeq S
have hs : DenseRange s := TopologicalSpace.denseRange_denseSeq S
let μ := Measure.pi (fun _ : Fin N => gaussianReal 0 1)
let ν := Measure.pi (fun _ : Fin N × Fin m => gaussianReal 0 1)
let Z (g : (Fin N × Fin m) → ℝ) : EuclideanSpace ℝ (Fin N × Fin m) := WithLp.toLp 2 g
let W (g : Fin N → ℝ) : EuclideanSpace ℝ (Fin N) := WithLp.toLp 2 g
let A (n : ℕ) (g : Fin N → ℝ) : ℝ :=
indexedGaussianMaximum (fun i : Fin (n+1) => fun j => (s i).1 j) g
let C (n : ℕ) (g : (Fin N × Fin m) → ℝ) : ℝ :=
finiteMinmax (fun x : Fin (n+1) × V => ∑ ij, ((s x.1).1 ij.1*v x.2 ij.2)*g ij)
let M (z : EuclideanSpace ℝ (Fin N × Fin m)) :=
Finset.univ.inf' Finset.univ_nonempty (fun b => ‖gaussianPatternEuclideanOperator z (v b)‖)
have hAlim (g : Fin N → ℝ) : Tendsto (fun n => A n g) atTop (𝓝 ‖W g‖) := by
have he (n : ℕ) : A n g = prefixMaximum (fun i => ⟪(s i).1,W g⟫) n := by
unfold A indexedGaussianMaximum prefixMaximum
congr 1
funext i
simp only [EuclideanSpace.inner_eq_star_dotProduct,dotProduct,star_trivial,W,
mul_comm]
simp_rw [he]
exact unitInner_prefix_tendsto s hs (W g)
have hClim (g : (Fin N × Fin m) → ℝ) : Tendsto (fun n => C n g) atTop (𝓝 (M (Z g))) := by
have he (n : ℕ) : C n g = Finset.univ.inf' Finset.univ_nonempty
(fun b => prefixMaximum (fun i => ⟪(s i).1,gaussianPatternEuclideanOperator (Z g) (v b)⟫) n) := by
unfold C finiteMinmax finiteRowMax prefixMaximum
congr 1
funext b
congr 1
funext i
exact (gaussianPattern_unitBilinear (Z g) (s i).1 (v b)).symm
simp_rw [he]
exact Tendsto.finset_inf'_nhds_apply Finset.univ_nonempty
(fun b _ => unitInner_prefix_tendsto s hs (gaussianPatternEuclideanOperator (Z g) (v b)))
have hAbound (n : ℕ) (g : Fin N → ℝ) : |A n g| ≤ ‖W g‖ := by
obtain ⟨i,_,he⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty
(fun i : Fin (n+1) => ∑ j, (s i).1 j*g j)
change |Finset.univ.sup' Finset.univ_nonempty _| ≤ _
rw [he]
have h := abs_real_inner_le_norm (s i).1 (W g)
rw [(s i).2,one_mul] at h
simpa only [EuclideanSpace.inner_eq_star_dotProduct,dotProduct,star_trivial,W,
PiLp.toLp_apply,mul_comm] using h
have hCbound (n : ℕ) (g : (Fin N × Fin m) → ℝ) : |C n g| ≤ ‖Z g‖ := by
apply finiteMinmax_abs_le
intro x
rw [← gaussianPattern_unitBilinear (Z g) (s x.1).1 (v x.2)]
calc
_ ≤ ‖(s x.1).1‖*‖gaussianPatternEuclideanOperator (Z g) (v x.2)‖ := abs_real_inner_le_norm _ _
_ = ‖gaussianPatternEuclideanOperator (Z g) (v x.2)‖ := by rw [(s x.1).2,one_mul]
_ ≤ ‖gaussianPatternEuclideanOperator (Z g)‖*‖v x.2‖ :=
(gaussianPatternEuclideanOperator (Z g)).le_opNorm _
_ ≤ ‖Z g‖ := by rw [hv,mul_one]; exact gaussianPatternEuclideanOperator_norm_le _
have hAint := tendsto_integral_of_dominated_convergence (μ := μ) (fun g => ‖W g‖)
(fun n => (measurable_indexedGaussianMaximum _).aestronglyMeasurable)
(((gaussianCoordinateVector_memLp (id : Fin N → Fin N)).integrable (by norm_num)).norm)
(fun n => ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hAbound n g))
(ae_of_all _ hAlim)
have hCint := tendsto_integral_of_dominated_convergence (μ := ν) (fun g => ‖Z g‖)
(fun n => (indexedGaussianMinmax_integrable _).aestronglyMeasurable)
(((gaussianCoordinateVector_memLp (id : (Fin N × Fin m) → (Fin N × Fin m))).integrable
(by norm_num)).norm)
(fun n => ae_of_all _ (fun g => by simpa only [Real.norm_eq_abs] using hCbound n g))
(ae_of_all _ hClim)
have hbound (n : ℕ) : (∫ g, A n g ∂μ)-Real.sqrt m ≤ ∫ g, C n g ∂ν :=
bilinear_gaussian_minmax_mean (fun i : Fin (n+1) => (s i).1) v (fun i => (s i).2) hv
have hh := le_of_tendsto_of_tendsto (hAint.sub_const (Real.sqrt m)) hCint
(Filter.Eventually.of_forall hbound)
have hW : HasLaw W (stdGaussian _) μ :=
⟨(PiLp.continuous_toLp 2 (fun _ : Fin N => ℝ)).measurable.aemeasurable,map_pi_eq_stdGaussian⟩
have hZ : HasLaw Z (stdGaussian _) ν :=
⟨(PiLp.continuous_toLp 2 (fun _ : Fin N × Fin m => ℝ)).measurable.aemeasurable,
map_pi_eq_stdGaussian⟩
have heW : (∫ g, ‖W g‖ ∂μ) = ∫ z : EuclideanSpace ℝ (Fin N), ‖z‖ ∂stdGaussian _ := by
simpa only [Function.comp_def] using hW.integral_comp (f := norm) (by fun_prop)
have heZ : (∫ g, M (Z g) ∂ν) = ∫ z, M z ∂stdGaussian _ := by
have hM : Continuous M := Continuous.finset_inf'_apply Finset.univ_nonempty
(fun b _ => (continuous_gaussianPattern_apply (v b)).norm)
simpa only [Function.comp_def] using hZ.integral_comp hM.aestronglyMeasurable
rw [heW,heZ] at hh
exact hh
end InvariantIsing
end
end OAI
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