File size: 6,967 Bytes
a2ee19a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import Mathlib
import OAI.Probability.Perceptron.Sphere.GaussianRotation
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped ENNReal NNReal Topology BigOperators BoundedContinuousFunction
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal BigOperators Topology RealInnerProductSpace
namespace SphericalPerceptronFreeEnergy
variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
[FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
variable {S : Type*} [MeasurableSpace S] (ρ : Measure S) [IsProbabilityMeasure ρ]
{W : S → ℝ} {V : S → StrongDual ℝ E}
def vectorPartition (W : S → ℝ) (V : S → StrongDual ℝ E) (x : E) : ℝ :=
∫ s, Real.exp (W s + V s x) ∂ρ
def vectorLogPartition (W : S → ℝ) (V : S → StrongDual ℝ E) (x : E) : ℝ :=
Real.log (vectorPartition ρ W V x)
def vectorLogDerivative (W : S → ℝ) (V : S → StrongDual ℝ E) (x : E) : StrongDual ℝ E :=
(vectorPartition ρ W V x)⁻¹ • (∫ s, Real.exp (W s + V s x) • V s ∂ρ)
omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [MeasurableSpace S] in
lemma vectorEnergy_bound {K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C)
(x : E) (s : S) : |W s + V s x| ≤ K+C*‖x‖ := by
apply (abs_add_le _ _).trans
apply add_le_add (hW s)
exact (ContinuousLinearMap.le_opNorm _ _).trans
(mul_le_mul_of_nonneg_right (hV s) (norm_nonneg x))
omit [MeasurableSpace E] [BorelSpace E] in
lemma vectorPartition_integrable (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) :
Integrable (fun s => Real.exp (W s + V s x)) ρ := by
refine Integrable.of_bound (by fun_prop) (Real.exp (K+C*‖x‖)) (ae_of_all _ fun s => ?_)
rw [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
exact Real.exp_le_exp.mpr ((le_abs_self _).trans (vectorEnergy_bound hW hV x s))
omit [MeasurableSpace E] [BorelSpace E] in
lemma vectorPartition_pos (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) :
0 < vectorPartition ρ W V x := by
apply integral_pos_iff_support_of_nonneg (fun s => (Real.exp_pos _).le)
(vectorPartition_integrable ρ hWm hVm hW hV x) |>.mpr
simp [Function.support,Real.exp_ne_zero]
omit [MeasurableSpace E] [BorelSpace E] in
lemma vectorPartition_hasFDerivAt (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) :
HasFDerivAt (vectorPartition ρ W V) (∫ s, Real.exp (W s+V s x) • V s ∂ρ) x := by
apply hasFDerivAt_integral_of_dominated_of_fderiv_le
(s := Metric.ball x 1) (bound := fun _ => Real.exp (K+C*(‖x‖+1))*C)
(F' := fun y s => Real.exp (W s+V s y) • V s) (Metric.ball_mem_nhds x zero_lt_one)
(Filter.Eventually.of_forall fun y => by fun_prop)
(vectorPartition_integrable ρ hWm hVm hW hV x) (by fun_prop)
(ae_of_all _ fun s y hy => ?_) (integrable_const _) (ae_of_all _ fun s y _ => ?_)
· have hy' : ‖y‖ ≤ ‖x‖+1 := by
have hh : ‖y-x‖ < 1 := by simpa only [Metric.mem_ball,dist_eq_norm] using hy
have hn := norm_add_le (y-x) x
rw [sub_add_cancel] at hn
linarith
rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
apply mul_le_mul _ (hV _) (norm_nonneg _) (Real.exp_pos _).le
apply Real.exp_le_exp.mpr
exact ((le_abs_self _).trans (vectorEnergy_bound hW hV y s)).trans
(add_le_add le_rfl (mul_le_mul_of_nonneg_left hy' hC))
· exact (((V s).hasFDerivAt).const_add (W s)).exp
omit [MeasurableSpace E] [BorelSpace E] in
lemma vectorLogPartition_hasFDerivAt (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) :
HasFDerivAt (vectorLogPartition ρ W V) (vectorLogDerivative ρ W V x) x :=
(vectorPartition_hasFDerivAt ρ hWm hVm hC hW hV x).log
(vectorPartition_pos ρ hWm hVm hW hV x).ne'
omit [MeasurableSpace E] [BorelSpace E] in
lemma vectorLogDerivative_bound (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) (x : E) :
‖vectorLogDerivative ρ W V x‖ ≤ C := by
have hp := vectorPartition_pos ρ hWm hVm hW hV x
have hn : ‖∫ s, Real.exp (W s+V s x) • V s ∂ρ‖ ≤ C*vectorPartition ρ W V x := by
have hh := norm_integral_le_of_norm_le (f := fun s => Real.exp (W s+V s x) • V s)
((vectorPartition_integrable ρ hWm hVm hW hV x).const_mul C)
(ae_of_all _ fun s => by
rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
exact (mul_le_mul_of_nonneg_left (hV s) (Real.exp_pos _).le).trans_eq (mul_comm ..))
simpa only [integral_const_mul,vectorPartition] using hh
rw [vectorLogDerivative,norm_smul,Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hp)]
calc
_ ≤ (vectorPartition ρ W V x)⁻¹*(C*vectorPartition ρ W V x) :=
mul_le_mul_of_nonneg_left hn (inv_nonneg.mpr hp.le)
_ = C := by field_simp
omit [MeasurableSpace E] [BorelSpace E] in
lemma vectorLogDerivative_continuous (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) :
Continuous (vectorLogDerivative ρ W V) := by
apply continuous_iff_continuousAt.mpr
intro x
have hp := (vectorPartition_hasFDerivAt ρ hWm hVm hC hW hV x).continuousAt
apply (hp.inv₀ (vectorPartition_pos ρ hWm hVm hW hV x).ne').smul
apply continuousAt_of_dominated (bound := fun _ => Real.exp (K+C*(‖x‖+1))*C)
· exact Filter.Eventually.of_forall fun y => by fun_prop
· filter_upwards [Metric.ball_mem_nhds x zero_lt_one] with y hy
exact ae_of_all _ fun s => by
have hy' : ‖y‖ ≤ ‖x‖+1 := by
have hh : ‖y-x‖ < 1 := by simpa only [Metric.mem_ball,dist_eq_norm] using hy
have hn := norm_add_le (y-x) x
rw [sub_add_cancel] at hn
linarith
rw [norm_smul,Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)]
apply mul_le_mul _ (hV _) (norm_nonneg _) (Real.exp_pos _).le
apply Real.exp_le_exp.mpr
exact ((le_abs_self _).trans (vectorEnergy_bound hW hV y s)).trans
(add_le_add le_rfl (mul_le_mul_of_nonneg_left hy' hC))
· exact integrable_const _
· exact ae_of_all _ fun s => by fun_prop
lemma vectorLogPartition_gaussian_variance (hWm : Measurable W) (hVm : Measurable V)
{K C : ℝ} (hC : 0 ≤ C) (hW : ∀ s, |W s| ≤ K) (hV : ∀ s, ‖V s‖ ≤ C) :
variance (vectorLogPartition ρ W V) (stdGaussian E) ≤ (Real.pi^2/8)*C^2 :=
stdGaussian_variance_le_of_fderiv_bound
(vectorLogPartition_hasFDerivAt ρ hWm hVm hC hW hV)
(vectorLogDerivative_continuous ρ hWm hVm hC hW hV) hC
(vectorLogDerivative_bound ρ hWm hVm hW hV)
end SphericalPerceptronFreeEnergy
end
end OAI
|