| import OAI.Probability.InvariantIsing.Fields.CascadeMarking |
| import OAI.Probability.IsingPerceptron.RetainedDisplacement |
|
|
| namespace OAI |
|
|
| /-! Gaussian `log cosh` marking preserves the unmarked cascade law. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory |
| open scoped NNReal |
|
|
| namespace InvariantIsing |
|
|
| def gaussianCascadeMarks (v : ℕ → ℝ≥0) (i : ℕ) : ProbabilityMeasure ℝ := |
| ⟨gaussianReal 0 (v i), inferInstance⟩ |
|
|
| def gaussianCascadeUpdate (n : ℕ) : ℕ → ℝ × ℝ → ℝ := |
| IsingPerceptron.stoppedUpdate n (fun _ p => p.1 + p.2) |
|
|
| def gaussianCascadeBackward (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (i : ℕ) : ℝ → ℝ := |
| IsingPerceptron.backwardValue n b (gaussianCascadeMarks v) |
| (fun _ p => p.1 + p.2) (fun x => Real.log (Real.cosh x)) i |
|
|
| def gaussianCascadeMultiplier (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) |
| (i : ℕ) (p : ℝ × ℝ) : ℝ := |
| Real.exp (gaussianCascadeBackward n b v (i + 1) (gaussianCascadeUpdate n i p) - |
| gaussianCascadeBackward n b v i p.1) |
|
|
| lemma measurable_gaussianCascadeUpdate (n i : ℕ) : Measurable (gaussianCascadeUpdate n i) := |
| IsingPerceptron.measurable_stoppedUpdate n (fun _ => measurable_fst.add measurable_snd) i |
|
|
| lemma measurable_gaussianCascadeBackward (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (i : ℕ) : |
| Measurable (gaussianCascadeBackward n b v i) := |
| IsingPerceptron.measurable_backwardValue n b (gaussianCascadeMarks v) |
| (fun _ => measurable_fst.add measurable_snd) IsingPerceptron.measurable_logCosh i |
|
|
| lemma measurable_gaussianCascadeMultiplier (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (i : ℕ) : |
| Measurable (gaussianCascadeMultiplier n b v i) := |
| (((measurable_gaussianCascadeBackward n b v (i + 1)).comp |
| (measurable_gaussianCascadeUpdate n i)).sub |
| ((measurable_gaussianCascadeBackward n b v i).comp measurable_fst)).exp |
|
|
| lemma gaussianCascadeMultiplier_pos (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) |
| (i : ℕ) (x g : ℝ) : 0 < gaussianCascadeMultiplier n b v i (x, g) := |
| Real.exp_pos _ |
|
|
| / |
| lemma gaussianCascadeMultiplier_moment (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) |
| (hb : IsingPerceptron.CascadeExponents n b) (i : ℕ) (x : ℝ) : |
| (∫⁻ g, ENNReal.ofReal (gaussianCascadeMultiplier n b v i (x, g) ^ b i) |
| ∂gaussianReal 0 (v i)) = 1 := by |
| have hadm := IsingPerceptron.backwardValue_linearGrowth_moments n b (gaussianCascadeMarks v) |
| (fun j _ => IsingPerceptron.gaussianReal_exponentialNormMoments 0 (v j)) |
| IsingPerceptron.measurable_logCosh IsingPerceptron.logCosh_linearGrowth |
| (fun j hj => (hb.1 j hj).1) |
| exact IsingPerceptron.backwardValue_multiplier_moment_admissible n b |
| (gaussianCascadeMarks v) (fun _ p => p.1 + p.2) (fun z => Real.log (Real.cosh z)) |
| (fun j hj => (hb.1 j hj).1) hadm i x |
|
|
| def gaussianCascadeDisplacement (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (x : ℝ) : |
| IsingPerceptron.NoiseTree ℝ n → IsingPerceptron.RawTree n := |
| IsingPerceptron.noiseTreeDisplace n b (gaussianCascadeMarks v) |
| (gaussianCascadeMultiplier n b v) (gaussianCascadeUpdate n) x |
|
|
| / |
| theorem gaussianCascadeDisplacement_law (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) |
| (hb : IsingPerceptron.CascadeExponents n b) (x : ℝ) : |
| (gaussianCascadeLaw n b v).map (gaussianCascadeDisplacement n b v x) = |
| IsingPerceptron.rawCascadeLaw n b := by |
| exact IsingPerceptron.noiseTreeDisplace_law n b (gaussianCascadeMarks v) |
| (measurable_gaussianCascadeMultiplier n b v) (measurable_gaussianCascadeUpdate n) |
| (gaussianCascadeMultiplier_pos n b v) (gaussianCascadeMultiplier_moment n b v hb) x |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
