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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 | import Mathlib
namespace OAI
open MeasureTheory ProbabilityTheory Metric Set
open scoped ENNReal
open MeasureTheory ProbabilityTheory Set
open scoped ENNReal
namespace GaussianTools
open MeasureTheory ProbabilityTheory
open scoped ENNReal
noncomputable def scalarRow (d : ℕ) := Measure.pi (fun _ : Fin d => gaussianReal 0 1)
lemma row_memLp {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) :
MemLp (fun a : Fin d → ℝ => ∑ j,a j*v j) 2 (scalarRow d) := by
apply memLp_finsetSum
intro j _
exact ((memLp_id_gaussianReal (2 : NNReal)).comp_measurePreserving
(measurePreserving_eval (fun _ : Fin d => gaussianReal 0 1) j)).mul_const (v j)
lemma row_integral {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) :
(∫ a : Fin d → ℝ,∑ j,a j*v j ∂scalarRow d)=0 := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
have hi (j : Fin d) : Integrable (fun a : Fin d → ℝ => a j*v j) (scalarRow d) :=
(((memLp_id_gaussianReal (2 : NNReal)).comp_measurePreserving
(measurePreserving_eval (fun _ : Fin d => gaussianReal 0 1) j)).mul_const (v j)).integrable (by norm_num)
rw [integral_finsetSum _ (fun j _ => hi j)]
simp only [integral_mul_const,scalarRow,integral_eval,integral_id_gaussianReal,zero_mul,Finset.sum_const_zero]
lemma row_energy {d : ℕ} (v : EuclideanSpace ℝ (Fin d)) :
(∫ a : Fin d → ℝ,(∑ j,a j*v j)^2 ∂scalarRow d)=‖v‖^2 := by
have h := variance_sum_pi (μ := fun _ : Fin d => gaussianReal 0 1)
(X := fun j x => x*v j) (fun j => (memLp_id_gaussianReal (2 : NNReal)).mul_const (v j))
have hm : Measurable (fun a : Fin d → ℝ => ∑ j,a j*v j) := by fun_prop
simp only [Finset.sum_fn] at h
rw [variance_eq_integral hm.aemeasurable] at h
change (∫ a,(∑ j,a j*v j - ∫ a,∑ j,a j*v j ∂scalarRow d)^2 ∂scalarRow d)=_ at h
rw [row_integral] at h
simp only [sub_zero,variance_mul_const] at h
rw [h,EuclideanSpace.norm_sq_eq]
apply Finset.sum_congr rfl
intro j _
simp
noncomputable def labelDensity {p : ℕ} (r : ℝ) (y : Fin p → ℝ) : ℝ :=
∏ i,gaussianPDFReal 0 (NNReal.mk (r^2) (sq_nonneg r)) (y i)
lemma labelDensity_pos {p : ℕ} {r : ℝ} (hr : 0 < r) (y : Fin p → ℝ) :
0 < labelDensity r y := by
apply Finset.prod_pos
intro i _
exact gaussianPDFReal_pos _ _ _ (by intro hh; have he := congrArg (fun t : NNReal => (t : ℝ)) hh; exact (pow_ne_zero 2 hr.ne') he)
lemma measurable_labelDensity {p : ℕ} (r : ℝ) : Measurable (@labelDensity p r) := by
unfold labelDensity
fun_prop
lemma log_labelDensity {p : ℕ} {r : ℝ} (hr : 0 < r) (y : Fin p → ℝ) :
Real.log (labelDensity r y) =
-(p:ℝ)*Real.log r-(p:ℝ)/2*Real.log (2*Real.pi) -
(∑ i,(y i)^2)/(2*r^2) := by
unfold labelDensity
rw [Real.log_prod (fun i _ => (gaussianPDFReal_pos 0 (NNReal.mk (r^2) (sq_nonneg r)) (y i) (by intro hh; have he := congrArg (fun t : NNReal => (t : ℝ)) hh; exact (pow_ne_zero 2 hr.ne') he)).ne')]
simp only [gaussianPDFReal,NNReal.coe_mk,sub_zero]
have hs : Real.sqrt (2*Real.pi*r^2)=Real.sqrt (2*Real.pi)*r := by
rw [Real.sqrt_mul (by positivity),Real.sqrt_sq_eq_abs,abs_of_pos hr]
simp_rw [Real.log_mul (by positivity : (Real.sqrt (2*Real.pi*r^2))⁻¹ ≠ 0) (Real.exp_pos _).ne',
Real.log_inv,hs,Real.log_mul (by positivity : Real.sqrt (2*Real.pi) ≠ 0) hr.ne',
Real.log_sqrt (by positivity : 0 ≤ 2*Real.pi),Real.log_exp]
simp_rw [neg_div]
simp only [Finset.sum_add_distrib,Finset.sum_neg_distrib,← Finset.sum_div,
Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]
ring
lemma rows_energy {p d : ℕ} (v : EuclideanSpace ℝ (Fin d)) :
(∫ X : Fin p → Fin d → ℝ, ∑ i,(∑ j,X i j*v j)^2
∂Measure.pi (fun _ : Fin p => scalarRow d)) = (p:ℝ)*‖v‖^2 := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
have hi (i : Fin p) : Integrable (fun X : Fin p → Fin d → ℝ => (∑ j,X i j*v j)^2)
(Measure.pi (fun _ : Fin p => scalarRow d)) :=
((row_memLp v).comp_measurePreserving
(measurePreserving_eval (fun _ : Fin p => scalarRow d) i)).integrable_sq
rw [integral_finsetSum _ (fun i _ => hi i)]
have he (i : Fin p) : (∫ X : Fin p → Fin d → ℝ,(∑ j,X i j*v j)^2
∂Measure.pi (fun _ : Fin p => scalarRow d)) = ‖v‖^2 := by
rw [← row_energy v]
exact integral_comp_eval (μ := fun _ : Fin p => scalarRow d) (i := i)
(f := fun a : Fin d → ℝ => (∑ j,a j*v j)^2) ((by fun_prop : Measurable
(fun a : Fin d → ℝ => (∑ j,a j*v j)^2)).aestronglyMeasurable)
simp_rw [he]
simp
lemma rows_shift_energy {p d : ℕ} (v : EuclideanSpace ℝ (Fin d)) (e : Fin p → ℝ) :
(∫ X : Fin p → Fin d → ℝ, ∑ i,((∑ j,X i j*v j)+e i)^2
∂Measure.pi (fun _ : Fin p => scalarRow d)) ≤
2*(p:ℝ)*‖v‖^2+2*∑ i,(e i)^2 := by
have : IsProbabilityMeasure (scalarRow d) := by unfold scalarRow; infer_instance
have hi (i : Fin p) : MemLp (fun X : Fin p → Fin d → ℝ => ∑ j,X i j*v j) 2
(Measure.pi (fun _ : Fin p => scalarRow d)) :=
(row_memLp v).comp_measurePreserving (measurePreserving_eval (fun _ : Fin p => scalarRow d) i)
have hil : Integrable (fun X : Fin p → Fin d → ℝ => ∑ i,((∑ j,X i j*v j)+e i)^2)
(Measure.pi (fun _ : Fin p => scalarRow d)) :=
integrable_finsetSum _ (fun i _ => ((hi i).add (memLp_const (e i))).integrable_sq)
have hir : Integrable (fun X : Fin p → Fin d → ℝ => 2*(∑ i,(∑ j,X i j*v j)^2) + 2*∑ i,(e i)^2)
(Measure.pi (fun _ : Fin p => scalarRow d)) :=
((integrable_finsetSum _ (fun i _ => (hi i).integrable_sq)).const_mul 2).add (integrable_const _)
calc
_ ≤ ∫ X : Fin p → Fin d → ℝ, 2*(∑ i,(∑ j,X i j*v j)^2) + 2*∑ i,(e i)^2
∂Measure.pi (fun _ : Fin p => scalarRow d) := by
apply integral_mono hil hir
intro X
dsimp only
simp only [Finset.mul_sum,← Finset.sum_add_distrib]
apply Finset.sum_le_sum
intro i _
nlinarith [sq_nonneg ((∑ j,X i j*v j)-e i)]
_ = _ := by
rw [integral_add ((integrable_finsetSum _ (fun i _ => (hi i).integrable_sq)).const_mul 2)
(integrable_const _),integral_const_mul, rows_energy]
simp
ring
end GaussianTools
end OAI
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