| import Mathlib |
| import OAI.Probability.Perceptron.Variational.DensityDerivative |
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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 |
|
|
| lemma poisson_abs_count_difference (r : ℝ≥0) (M : ℕ) : |
| (∫ j : ℕ, |(j:ℝ)-(M:ℝ)| ∂poissonMeasure r) ≤ Real.sqrt r+|(r:ℝ)-(M:ℝ)| := by |
| have hi := (poisson_nat_memLp r).integrable (by norm_num) |
| have hc := integral_centered_abs_le_sqrt_variance (poissonMeasure r) (poisson_nat_memLp r) |
| rw [poisson_nat_mean,poisson_nat_variance] at hc |
| calc |
| _ ≤ ∫ j : ℕ, |(j:ℝ)-(r:ℝ)|+|(r:ℝ)-(M:ℝ)| ∂poissonMeasure r := by |
| apply integral_mono (hi.sub (integrable_const _)).abs |
| ((hi.sub (integrable_const _)).abs.add (integrable_const _)) |
| intro j |
| exact abs_sub_le _ _ _ |
| _ = (∫ j : ℕ, |(j:ℝ)-(r:ℝ)| ∂poissonMeasure r)+|(r:ℝ)-(M:ℝ)| := by |
| rw [integral_add (f := fun j : ℕ => |(j:ℝ)-(r:ℝ)|) |
| (g := fun _ : ℕ => |(r:ℝ)-(M:ℝ)|) (hi.sub (integrable_const _)).abs (integrable_const _)] |
| simp |
| _ ≤ _ := add_le_add hc le_rfl |
|
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| lemma poisson_bounded_increment_mean_compare (r : ℝ≥0) {F : ℕ→ℝ} {C : ℝ} |
| (hC : 0≤C) (hF : ∀ m, |F (m+1)-F m|≤C) (M : ℕ) : |
| |(∫ j, F j ∂poissonMeasure r)-F M|≤C*(Real.sqrt r+|(r:ℝ)-(M:ℝ)|) := by |
| have hi := (poisson_bounded_increment_memLp r hC hF).integrable (by norm_num) |
| have hj := (poisson_nat_memLp r).integrable (by norm_num) |
| calc |
| _ = |∫ j, F j-F M ∂poissonMeasure r| := by rw [integral_sub hi (integrable_const _)]; simp |
| _ ≤ ∫ j, |F j-F M| ∂poissonMeasure r := abs_integral_le_integral_abs |
| _ ≤ ∫ j : ℕ, C*|(j:ℝ)-(M:ℝ)| ∂poissonMeasure r := |
| integral_mono (hi.sub (integrable_const _)).abs |
| ((hj.sub (integrable_const _)).abs.const_mul C) |
| (fun j => nat_bounded_increment_difference hC hF j M) |
| _ ≤ _ := by rw [integral_const_mul]; exact mul_le_mul_of_nonneg_left (poisson_abs_count_difference r M) hC |
|
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| lemma sourceExpectedPressure_fixed_count_compare (n k M : 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))*sourceCountExpectedLog n k f p d h z u M| ≤ |
| ‖f‖/(n+1:ℕ)*(Real.sqrt ((n+1:ℕ)*(t:ℝ))+|((n+1:ℕ)*(t:ℝ))-(M:ℝ)|) := by |
| rw [sourceExpectedPressure_poissonMean n k f p d h z u hz hz0 hz1] |
| have hF := sourceCountExpectedLog_increment n k f p d h z u hz hz0 hz1 |
| have H := poisson_bounded_increment_mean_compare ((n+1:ℕ)*t) (norm_nonneg f) hF M |
| rw [← poissonRealMean_eq_integral (norm_nonneg f) hF] at H |
| simp only [NNReal.coe_mul,NNReal.coe_natCast] at H |
| rw [← mul_sub,abs_mul,abs_of_nonneg (by positivity : (0:ℝ)≤1/(n+1:ℕ))] |
| calc |
| _ ≤ (1/(n+1:ℕ))*(‖f‖*(Real.sqrt ((n+1:ℕ)*(t:ℝ))+|((n+1:ℕ)*(t:ℝ))-(M:ℝ)|)) := |
| mul_le_mul_of_nonneg_left H (by positivity) |
| _ = _ := by ring |
|
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| lemma sourceExpectedPressure_floor_count_compare (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))*sourceCountExpectedLog n k f p d h z u ⌊(n+1:ℕ)*(t:ℝ)⌋₊| ≤ |
| ‖f‖/(n+1:ℕ)*(Real.sqrt ((n+1:ℕ)*(t:ℝ))+1) := by |
| apply (sourceExpectedPressure_fixed_count_compare n k _ f p d h z u hz hz0 hz1 t).trans |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| apply add_le_add le_rfl |
| rw [abs_of_nonneg (sub_nonneg.mpr (Nat.floor_le (by positivity)))] |
| linarith [Nat.lt_floor_add_one ((n+1:ℕ)*(t:ℝ))] |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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