File size: 3,840 Bytes
c4cdbbd | 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 | 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 _
/-- Every edge multiplier is normalized under the exponent of its cascade level. -/
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
/-- The exact unmarked-law part of `arr:rpc` for Gaussian `log cosh` marks. -/
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
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