| import Mathlib |
| import OAI.Probability.Perceptron.Interpolation.PerturbationContact |
| import OAI.Probability.Perceptron.Cascade.PoissonDensityDerivative |
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| namespace OAI |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
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| def sourceCountExpectedLog (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) (M : ℕ) : ℝ := |
| ∫ a, enrichedPoissonLog n k f p d u h (M,a) ∂enrichedRowsDisorderLaw n k p z |
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| lemma sourceCountExpectedLog_increment (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) (M : ℕ) : |
| |sourceCountExpectedLog n k f p d h z u (M+1)-sourceCountExpectedLog n k f p d h z u M|≤‖f‖ := |
| enrichedPoissonLog_mean_succ n k f p d u h z hz hz0 hz1 M |
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| lemma sourceExpectedPressure_poissonMean (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) (t : ℝ≥0) : |
| sourceExpectedPressure n k f p d h z t u = |
| (1/(n+1:ℕ))*poissonRealMean (sourceCountExpectedLog n k f p d h z u) ((n+1:ℕ)*(t:ℝ)) := by |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| have hr := sourceDisorderReassoc_preserving n k z t |
| have hs := sourceDisorderRealization_preserving n k p z t |
| have hi := (enrichedPoissonLog_variance n k f p d u h z hz hz0 hz1 ((n+1:ℕ)*t)).1 |
| have hi' : AEStronglyMeasurable (enrichedRandomPressure n k f p d u h) |
| (enrichedPoissonDisorderLaw n k p z t) := (hi.const_mul (1/(n+1:ℕ))).aestronglyMeasurable |
| have hm := sourceIndexedPressure_memLp n k f p d u h z hz hz0 hz1 t |
| rw [sourceExpectedPressure] |
| rw [integral_congr_ae ((sourceKernelPressure_ae n k f p d h z hz hz0 hz1 t).mono fun a ha => ha u)] |
| rw [← integral_map hr.measurable.aemeasurable (by rw [hr.map_eq]; exact hm.aestronglyMeasurable),hr.map_eq] |
| unfold sourceIndexedPressure |
| change (∫ a, enrichedRandomPressure n k f p d u h (sourceDisorderRealization n k p a) |
| ∂sourceIndexedDisorderLaw n k z t)=_ |
| rw [← integral_map hs.measurable.aemeasurable (by rw [hs.map_eq]; exact hi'),hs.map_eq] |
| simp only [enrichedRandomPressure,integral_const_mul] |
| apply congrArg ((1/(n+1:ℕ):ℝ)* ·) |
| change (∫ a, enrichedPoissonLog n k f p d u h a ∂enrichedPoissonDisorderLaw n k p z t) = |
| poissonRealMean (sourceCountExpectedLog n k f p d h z u) (((n+1:ℕ)*t:ℝ≥0):ℝ) |
| rw [poissonRealMean_eq_integral (norm_nonneg f) |
| (sourceCountExpectedLog_increment n k f p d h z u hz hz0 hz1) ((n+1:ℕ)*t)] |
| exact integral_prod _ (hi.integrable (by norm_num)) |
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| def sourceDensityIncrement (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) (t : ℝ≥0) : ℝ := |
| ∫ M, sourceCountExpectedLog n k f p d h z u (M+1)-sourceCountExpectedLog n k f p d h z u M |
| ∂poissonMeasure ((n+1:ℕ)*t) |
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| lemma sourceDensityIncrement_bound (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) (t : ℝ≥0) : |
| |sourceDensityIncrement n k f p d h z u t|≤‖f‖ := by |
| simpa only [sourceDensityIncrement,Real.norm_eq_abs,probReal_univ,mul_one] using |
| norm_integral_le_of_norm_le_const (μ := poissonMeasure ((n+1:ℕ)*t)) |
| (f := fun M => sourceCountExpectedLog n k f p d h z u (M+1)-sourceCountExpectedLog n k f p d h z u M) |
| (ae_of_all _ fun M => by simpa only [Real.norm_eq_abs] using |
| sourceCountExpectedLog_increment n k f p d h z u hz hz0 hz1 M) |
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| lemma sourceExpectedPressure_density_derivative (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) (t : ℝ≥0) (ht : 0<t) : |
| HasDerivAt (fun s : ℝ => sourceExpectedPressure n k f p d h z s.toNNReal u) |
| (sourceDensityIncrement n k f p d h z u t) t := by |
| let F := sourceCountExpectedLog n k f p d h z u |
| have hF := sourceCountExpectedLog_increment n k f p d h z u hz hz0 hz1 |
| have hd := ((hasDerivAt_poissonRealMean (norm_nonneg f) hF ((n+1:ℕ)*(t:ℝ))).comp (t:ℝ) |
| ((hasDerivAt_id (t:ℝ)).const_mul ((n+1:ℕ):ℝ))).const_mul (1/(n+1:ℕ):ℝ) |
| have hD : poissonRealMean (fun M => F (M+1)-F M) ((n+1:ℕ)*(t:ℝ))= |
| sourceDensityIncrement n k f p d h z u t := by |
| have hinc : ∀ M, |(F (M+1+1)-F (M+1))-(F (M+1)-F M)|≤2*‖f‖ := fun M => |
| (abs_sub _ _).trans (by linarith [hF (M+1),hF M]) |
| simpa only [sourceDensityIncrement,NNReal.coe_mul,NNReal.coe_natCast] using |
| poissonRealMean_eq_integral (by positivity : 0≤2*‖f‖) hinc ((n+1:ℕ)*t) |
| have hd' : HasDerivAt (fun s : ℝ => (1/(n+1:ℕ))*poissonRealMean F ((n+1:ℕ)*s)) |
| (sourceDensityIncrement n k f p d h z u t) t := by |
| apply hd.congr_deriv |
| change (1/(n+1:ℕ))*(poissonRealMean _ _ *((n+1:ℕ)*1))=_ |
| rw [hD] |
| field_simp |
| apply hd'.congr_of_eventuallyEq |
| filter_upwards [eventually_gt_nhds (show (0:ℝ)<(t:ℝ) from ht)] with s hs |
| rw [sourceExpectedPressure_poissonMean n k f p d h z u hz hz0 hz1, |
| Real.coe_toNNReal _ hs.le] |
|
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| lemma sourceExpectedPressure_density_continuous (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1)→ℕ) |
| (h : Fin (k+1)→ℝ) (z : Fin k→ℝ) (u : Fin (n+1)→ℝ) |
| (hz : StrictMono z) (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) : |
| Continuous (fun t : ℝ≥0 => sourceExpectedPressure n k f p d h z t u) := by |
| have hc : Continuous (poissonRealMean (sourceCountExpectedLog n k f p d h z u)) := |
| continuous_iff_continuousAt.mpr fun r => |
| (hasDerivAt_poissonRealMean (norm_nonneg f) |
| (sourceCountExpectedLog_increment n k f p d h z u hz hz0 hz1) r).continuousAt |
| simp_rw [sourceExpectedPressure_poissonMean n k f p d h z u hz hz0 hz1] |
| exact continuous_const.mul (hc.comp (continuous_const.mul NNReal.continuous_coe)) |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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