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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 | import OAI.Probability.InvariantIsing.Cavity.CavityMagneticBlockModulus
namespace OAI
/-! L1 replacement transports the physical constrained overlap tests
to the magnetic block paths used in the variational problem. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped Topology BoundedContinuousFunction
namespace InvariantIsing
lemma integrable_overlap_weighted_test (q a : OverlapPath) (Φ : ℝ →ᵇ ℝ) :
Integrable (fun s => Φ (q s)*(a s-q s)) pathMeasure := by
apply integrable_of_measurable_abs_le
((Φ.continuous.measurable.comp q.measurable).mul (a.measurable.sub q.measurable))
intro s
change |Φ (q s)*(a s-q s)| ≤ ‖Φ‖
rw [abs_mul]
have hd : |a s-q s| ≤ 1 := abs_le.mpr
⟨by linarith [a.nonneg s,q.le_one s], by linarith [a.le_one s,q.nonneg s]⟩
exact (mul_le_mul (Φ.norm_coe_le_norm _) hd (abs_nonneg _) (norm_nonneg Φ)).trans_eq (mul_one _)
lemma overlap_weighted_test_sub_le (q a b : OverlapPath) (Φ : ℝ →ᵇ ℝ) :
|(∫ s, Φ (q s)*(a s-q s) ∂pathMeasure) -
∫ s, Φ (q s)*(b s-q s) ∂pathMeasure| ≤
‖Φ‖ * ∫ s, |a s-b s| ∂pathMeasure := by
rw [← integral_sub (integrable_overlap_weighted_test q a Φ)
(integrable_overlap_weighted_test q b Φ)]
have he : (fun s => Φ (q s)*(a s-q s)-Φ (q s)*(b s-q s)) =
(fun s => Φ (q s)*(a s-b s)) := by funext s; ring
rw [he]
calc
_ ≤ ∫ s, |Φ (q s)*(a s-b s)| ∂pathMeasure := abs_integral_le_integral_abs
_ ≤ ∫ s, ‖Φ‖ * |a s-b s| ∂pathMeasure := by
apply integral_mono
· have hi := (integrable_overlap_weighted_test q a Φ).sub
(integrable_overlap_weighted_test q b Φ)
change Integrable (fun s => Φ (q s)*(a s-q s)-Φ (q s)*(b s-q s)) pathMeasure at hi
rw [he] at hi
exact hi.abs
· exact (a.integrable.sub b.integrable).abs.const_mul _
· intro s
change |Φ (q s)*(a s-b s)| ≤ ‖Φ‖ * |a s-b s|
rw [abs_mul]
exact mul_le_mul_of_nonneg_right (Φ.norm_coe_le_norm _) (abs_nonneg _)
_ = _ := integral_const_mul _ _
theorem overlap_weighted_test_l1_transport (q a b : ℕ → OverlapPath)
(hab : Tendsto (fun n => ∫ s, |a n s-b n s| ∂pathMeasure) atTop (𝓝 0))
(Φ : ℝ →ᵇ ℝ) :
Tendsto (fun n => (∫ s, Φ (q n s)*(a n s-q n s) ∂pathMeasure) -
∫ s, Φ (q n s)*(b n s-q n s) ∂pathMeasure) atTop (𝓝 0) := by
apply squeeze_zero_norm (fun n => ?_) (by simpa using hab.const_mul ‖Φ‖)
simpa only [Real.norm_eq_abs] using overlap_weighted_test_sub_le (q n) (a n) (b n) Φ
end InvariantIsing
end
end OAI
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