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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 | import Mathlib
import OAI.Probability.Perceptron.Variational.IntegratedConvex
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology BigOperators
namespace SphericalPerceptronFreeEnergy
lemma euclideanGaussian_replica_direction_bound {I : Type} [Fintype I] {S : Type*} [MeasurableSpace S]
(μ : Measure S) [IsProbabilityMeasure μ] (r : ℕ)
{H : S→ℝ} {V : S→EuclideanSpace ℝ I} {W : (Fin r→S)→EuclideanSpace ℝ I}
(hH : Measurable H) (hV : Measurable V) (hW : Measurable W)
{A C D K : ℝ} (hC : 0≤C) (hD : 0≤D) (hK : 0≤K)
(hA : ∀ x, |H x|≤A) (hVC : ∀ x, ‖V x‖≤C) (hWD : ∀ x, ‖W x‖≤D)
(hWK : ∀ x y, |inner ℝ (W x) (V y)|≤K) :
|∫ g, gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r
(fun x => inner ℝ (W x) g) ∂stdGaussian (EuclideanSpace ℝ I)|≤2*r*K := by
let ν := Measure.pi (fun _ : Fin r => μ)
let HH := replicaPotential H r
let VV := fun x : Fin r→S => ∑ l, V (x l)
have hHH : Measurable HH := replicaPotential_measurable hH r
have hVV : Measurable VV := Finset.measurable_sum _ fun l _ => hV.comp (measurable_pi_apply l)
have hHHA (x : Fin r→S) : |HH x|≤r*A := by
apply (Finset.abs_sum_le_sum_abs _ _).trans
calc
_ ≤ ∑ _ : Fin r, A := Finset.sum_le_sum fun l _ => hA (x l)
_ = _ := by simp
have hVVC (x : Fin r→S) : ‖VV x‖^2≤(r*C)^2 := by
have hb : ‖VV x‖≤r*C := by
apply (norm_sum_le _ _).trans
calc
_ ≤ ∑ _ : Fin r, C := Finset.sum_le_sum fun l _ => hVC (x l)
_ = _ := by simp
exact (sq_le_sq₀ (norm_nonneg _) (mul_nonneg (Nat.cast_nonneg r) hC)).2 hb
have hWWD (x : Fin r→S) : ‖W x‖^2≤D^2 :=
(sq_le_sq₀ (norm_nonneg _) hD).2 (hWD x)
have hc (x y : Fin r→S) : |inner ℝ (W x) (VV y)|≤r*K := by
dsimp only [VV]
rw [inner_sum]
apply (Finset.abs_sum_le_sum_abs _ _).trans
calc
_ ≤ ∑ _ : Fin r, K := Finset.sum_le_sum fun l _ => hWK x (y l)
_ = _ := by simp
have he := euclideanGaussian_tiltMean_ibp ν hHH hVV hW hHHA hVVC hWWD
have hham (g : EuclideanSpace ℝ I) : (fun x => HH x+inner ℝ (VV x) g)=
replicaPotential (fun x => H x+inner ℝ (V x) g) r := by
funext x
simp only [HH,VV,replicaPotential,sum_inner,Finset.sum_add_distrib]
have hl (g : EuclideanSpace ℝ I) :
gibbsReplicaMean μ (fun x => H x+inner ℝ (V x) g) r (fun x => inner ℝ (W x) g)=
tiltMean ν (fun x => HH x+inner ℝ (VV x) g) (fun x => inner ℝ (W x) g) 1 := by
rw [hham]
rfl
simp_rw [hl]
rw [he]
apply abs_integral_le_integral_abs.trans
calc
_ ≤ ∫ _ : EuclideanSpace ℝ I, (2*r*K:ℝ) ∂stdGaussian (EuclideanSpace ℝ I) := by
apply integral_mono_of_nonneg (ae_of_all _ fun _ => abs_nonneg _) (integrable_const _)
apply ae_of_all
intro g
have hm : Measurable (fun x => HH x+inner ℝ (VV x) g) := hHH.add (hVV.inner measurable_const)
have hm1 : Measurable (fun x => inner ℝ (W x) (VV x)) := hW.inner hVV
have hm2 : Measurable (fun x : Fin 2→(Fin r→S) => inner ℝ (W (x 1)) (VV (x 0))) :=
(hW.comp (measurable_pi_apply 1)).inner (hVV.comp (measurable_pi_apply 0))
have hb1 := tiltMean_bound_general ν hm hm1 (t:=1) (by positivity : 0≤(r:ℝ)*K) (fun x => hc x x)
have hb2 : |gibbsReplicaMean ν (fun x => HH x+inner ℝ (VV x) g) 2
(fun x => inner ℝ (W (x 1)) (VV (x 0)))|≤(r:ℝ)*K :=
tiltMean_bound_general (Measure.pi (fun _ : Fin 2 => ν))
(replicaPotential_measurable hm 2) hm2 (by positivity)
(fun x => hc (x 1) (x 0))
exact (abs_sub _ _).trans (by linarith)
_ = _ := by simp
end SphericalPerceptronFreeEnergy
end
end OAI
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