| import Mathlib |
| import OAI.Probability.IsingPerceptron.ArraySynchronization |
| import OAI.Probability.IsingPerceptron.EnrichedGGFinite |
| import OAI.Probability.IsingPerceptron.SamplingIndependent |
|
|
| namespace OAI |
|
|
| /-! Joint Shapes. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
|
|
| lemma retained_joint_shape_ae (n r : ℕ) (b a : ℕ → ℝ) : |
| ∀ᵐ p ∂retainedLabelJoint n b a, pathPrefixPattern n r (sampledMarkPath n p) = |
| labelPrefixPattern n r (fun i => p.2 i) := by |
| have hg : ∀ᵐ p ∂retainedLabelBase n b, Function.Injective p.2 := by |
| have hp := measurePreserving_snd (μ := (labeledCascadeLaw n b : Measure (LabeledTree n))) |
| (ν := Measure.infinitePi (fun _ : ForestVertex n => gaussianReal 0 1)) |
| exact hp.quasiMeasurePreserving.tendsto_ae.eventually (gaussian_marks_injective n (fun _ => 0)) |
| have hf : MeasurePreserving Prod.fst (retainedLabelJoint n b a) (retainedLabelBase n b) := |
| ⟨measurable_fst,Measure.fst_compProd _ _⟩ |
| have hh := hf.quasiMeasurePreserving.tendsto_ae.eventually hg |
| filter_upwards [hh] with p hp |
| exact pathPrefixPattern_coordinates n r p.2 p.1.2 hp |
|
|
| lemma measurable_sampledLabelPattern (n r : ℕ) : |
| Measurable (fun p : (LabeledTree n × (ForestVertex n → ℝ)) × (ℕ → LabeledLeaf n) => |
| labelPrefixPattern n r (fun i => p.2 i)) := |
| (measurable_infiniteLabelPattern n r).comp measurable_snd |
|
|
| theorem retained_joint_shape (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (a : ℕ → ℝ) (D : PrefixPattern n r → ℝ) : |
| (∫ p, D (labelPrefixPattern n r (fun i => p.2 i)) ∂retainedLabelJoint n b a) = |
| ∫ σ, D (labelPrefixPattern n r (fun i => σ i)) ∂cascadeReplicaLaw n b := by |
| have hF : Measurable (fun z => D (pathPrefixPattern n r z)) := |
| (measurable_of_countable D).comp (measurable_pathPrefixPattern n r) |
| calc |
| _ = ∫ p, D (pathPrefixPattern n r (sampledMarkPath n p)) ∂retainedLabelJoint n b a := by |
| apply integral_congr_ae |
| filter_upwards [retained_joint_shape_ae n r b a] with p hp |
| rw [hp] |
| _ = ∫ z, D (pathPrefixPattern n r z) ∂gaussianMarkedPathLaw n b (fun i => b i*a i) := by |
| rw [← integral_map (measurable_sampledMarkPath n).aemeasurable hF.aestronglyMeasurable, |
| retained_joint_path_law n b hb a] |
| rfl |
| _ = _ := gaussian_marked_shape n r b hb _ D |
|
|
| theorem retained_joint_shape_score (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (a : ℕ → ℝ) (d : Fin n) (D : PrefixPattern n r → ℝ) : |
| (∫ p, D (labelPrefixPattern n r (fun i => p.2 i))*p.1.2 (edgeAt n (p.2 0) d) |
| ∂retainedLabelJoint n b a) = |
| (∫ σ, D (labelPrefixPattern n r (fun i => σ i)) ∂cascadeReplicaLaw n b)*(b d*a d) := by |
| have hF : Measurable (fun z : ℕ → Fin n → ℝ => D (pathPrefixPattern n r z)*z 0 d) := |
| ((measurable_of_countable D).comp (measurable_pathPrefixPattern n r)).mul |
| ((measurable_pi_apply d).comp (measurable_pi_apply 0)) |
| calc |
| _ = ∫ p, D (pathPrefixPattern n r (sampledMarkPath n p))*sampledMarkPath n p 0 d |
| ∂retainedLabelJoint n b a := by |
| apply integral_congr_ae |
| filter_upwards [retained_joint_shape_ae n r b a] with p hp |
| rw [hp]; rfl |
| _ = ∫ z, D (pathPrefixPattern n r z)*z 0 d ∂gaussianMarkedPathLaw n b (fun i => b i*a i) := by |
| rw [← integral_map (measurable_sampledMarkPath n).aemeasurable hF.aestronglyMeasurable, |
| retained_joint_path_law n b hb a] |
| rfl |
| _ = _ := gaussian_marked_shape_score n r b hb _ d D |
|
|
| end IsingPerceptron |
|
|
| |
|
|
| |
|
|
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
|
|
| lemma prefixPattern_bound (n r : ℕ) (D : PrefixPattern n r → ℝ) : |
| ∃ c : ℝ, 0 ≤ c ∧ ∀ x, |D x| ≤ c := by |
| refine ⟨∑ x, |D x|,Finset.sum_nonneg (fun _ _ => abs_nonneg _),?_⟩ |
| intro x |
| exact Finset.single_le_sum (f := fun x => |D x|) (fun _ _ => abs_nonneg _) (Finset.mem_univ x) |
|
|
| lemma retained_joint_shape_score_integrable (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (a : ℕ → ℝ) (d : Fin n) (D : PrefixPattern n r → ℝ) : |
| Integrable (fun p => D (labelPrefixPattern n r (fun i => p.2 i))*p.1.2 (edgeAt n (p.2 0) d)) |
| (retainedLabelJoint n b a) := by |
| obtain ⟨c,hc,hD⟩ := prefixPattern_bound n r D |
| have hi := gaussian_marked_shape_score_integrable n r b hb (fun i => b i*a i) d D hD |
| have hF : Measurable (fun z : ℕ → Fin n → ℝ => D (pathPrefixPattern n r z)*z 0 d) := |
| ((measurable_of_countable D).comp (measurable_pathPrefixPattern n r)).mul |
| ((measurable_pi_apply d).comp (measurable_pi_apply 0)) |
| change Integrable _ ((markedReplicaLaw n b (fun i => ⟨gaussianReal (b i*a i) 1,inferInstance⟩)).map (replicaMarkPath n)) at hi |
| rw [← retained_joint_path_law n b hb a] at hi |
| have hj := (integrable_map_measure hF.aestronglyMeasurable (measurable_sampledMarkPath n).aemeasurable).mp hi |
| apply hj.congr |
| filter_upwards [retained_joint_shape_ae n r b a] with p hp |
| change D (pathPrefixPattern n r (sampledMarkPath n p))*sampledMarkPath n p 0 d = _ |
| rw [hp]; rfl |
|
|
| lemma raw_joint_shape_score_integrable (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (t : ℝ) (D : PrefixPattern n r → ℝ) : |
| Integrable (fun p => D (labelPrefixPattern n r (fun i => p.2 i))* |
| cylinderField (leafLevelCoefficients n d (p.2 0)) p.1.2) (rawLabelJoint n b d t) := by |
| have hi := retained_joint_shape_score_integrable n r b hb (fun i => if i = d then t else 0) d D |
| have hp := raw_retained_joint_preserving n b d t |
| have hi' : Integrable (fun p => D (labelPrefixPattern n r (fun i => p.2 i))*p.1.2 (edgeAt n (p.2 0) d)) |
| ((rawLabelJoint n b d t).map (rawLabelToRetained n)) := by rwa [hp.map_eq] |
| have hj := (integrable_map_measure hi'.aestronglyMeasurable hp.measurable.aemeasurable).mp hi' |
| change Integrable (fun p : (LabeledTree n × (ℕ → ℝ)) × (ℕ → LabeledLeaf n) => D (labelPrefixPattern n r (fun i => p.2 i))*edgeGaussianCoordinates n p.1.2 (edgeAt n (p.2 0) d)) _ at hj |
| simpa only [leafLevelCoefficients_field] using hj |
| |
| lemma raw_joint_shape (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (t : ℝ) (D : PrefixPattern n r → ℝ) : |
| (∫ p, D (labelPrefixPattern n r (fun i => p.2 i)) ∂rawLabelJoint n b d t) = |
| ∫ σ, D (labelPrefixPattern n r (fun i => σ i)) ∂cascadeReplicaLaw n b := by |
| have hF : Measurable (fun p : (LabeledTree n × (ForestVertex n → ℝ)) × (ℕ → LabeledLeaf n) => |
| D (labelPrefixPattern n r (fun i => p.2 i))) := |
| (measurable_of_countable D).comp (measurable_sampledLabelPattern n r) |
| have h := (HasLaw.mk (raw_retained_joint_preserving n b d t).measurable.aemeasurable (raw_retained_joint_preserving n b d t).map_eq).integral_comp hF.aestronglyMeasurable |
| exact h.trans (retained_joint_shape n r b hb _ D) |
| |
| lemma raw_joint_shape_score (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (t : ℝ) (D : PrefixPattern n r → ℝ) : |
| (∫ p, D (labelPrefixPattern n r (fun i => p.2 i))* |
| cylinderField (leafLevelCoefficients n d (p.2 0)) p.1.2 ∂rawLabelJoint n b d t) = |
| (∫ σ, D (labelPrefixPattern n r (fun i => σ i)) ∂cascadeReplicaLaw n b)*(b d*t) := by |
| have hi := retained_joint_shape_score_integrable n r b hb (fun i => if i = d then t else 0) d D |
| have h := (HasLaw.mk (raw_retained_joint_preserving n b d t).measurable.aemeasurable (raw_retained_joint_preserving n b d t).map_eq).integral_comp hi.aestronglyMeasurable |
| have hh := h.trans (retained_joint_shape_score n r b hb (fun i => if i = d then t else 0) d D) |
| simpa only [Function.comp_apply,rawLabelToRetained,leafLevelCoefficients_field,ite_eq_left rfl,ite_true] using hh |
| |
| lemma randomReplicaAverage_shape (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (t : ℝ) (D : PrefixPattern n r → ℝ) : |
| randomReplicaAverage (labeledCascadeLaw n b) (labeledLeafLaw n) (leafLevelCoefficients n d) t |
| (fun σ => D (labelPrefixPattern n r σ)) = |
| ∫ σ, D (labelPrefixPattern n r (fun i => σ i)) ∂cascadeReplicaLaw n b := by |
| obtain ⟨c,hc,hD⟩ := prefixPattern_bound n r D |
| have hi : Integrable (fun p => D (labelPrefixPattern n r (fun i => p.2 i))) (rawLabelJoint n b d t) := |
| integrable_of_measurable_abs_le |
| ((measurable_of_countable (fun σ : Fin r → LabeledLeaf n => D (labelPrefixPattern n r σ))).comp (by fun_prop)) |
| (fun _ => hD _) |
| exact (rawJoint_integral_reference n b d t r (fun _ σ => D (labelPrefixPattern n r σ)) hi).symm.trans |
| (raw_joint_shape n r b hb d t D) |
| |
| lemma randomReplicaEnergy_shape (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (t : ℝ) (D : PrefixPattern n (r+1) → ℝ) : |
| randomReplicaEnergy (labeledCascadeLaw n b) (labeledLeafLaw n) (leafLevelCoefficients n d) t |
| (fun σ => D (labelPrefixPattern n (r+1) σ)) = |
| (∫ σ, D (labelPrefixPattern n (r+1) (fun i => σ i)) ∂cascadeReplicaLaw n b)*(b d*t) := by |
| have hi := raw_joint_shape_score_integrable n (r+1) b hb d t D |
| have h := rawJoint_integral_reference n b d t (r+1) |
| (fun p σ => D (labelPrefixPattern n (r+1) σ)*cylinderField (leafLevelCoefficients n d (σ 0)) p.2) hi |
| simpa only [randomReplicaEnergy,rawLabelBase,mul_comm] using h.symm.trans (raw_joint_shape_score n (r+1) b hb d t D) |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| def prefixPatternMean (n r : ℕ) (b : ℕ → ℝ) (D : PrefixPattern n r → ℝ) : ℝ := |
| ∫ σ, D (labelPrefixPattern n r (fun i => σ i)) ∂cascadeReplicaLaw n b |
| |
| def patternReindex {n r s : ℕ} (Q : PrefixPattern n r) (e : Fin s → Fin r) : PrefixPattern n s := |
| fun i j d => Q (e i) (e j) d |
| |
| def boolReal (b : Bool) : ℝ := if b then 1 else 0 |
| |
| lemma patternReindex_label {n r s : ℕ} (σ : Fin r → LabeledLeaf n) (e : Fin s → Fin r) : |
| patternReindex (labelPrefixPattern n r σ) e = labelPrefixPattern n s (σ ∘ e) := rfl |
| |
| lemma leafLevelCoefficients_cross_bool (n r : ℕ) (d : Fin n) (σ : Fin r → LabeledLeaf n) (i j : Fin r) : |
| cylinderCross (leafLevelCoefficients n d (σ i)) (leafLevelCoefficients n d (σ j)) = |
| boolReal (labelPrefixPattern n r σ i j d) := by |
| rw [leafLevelCoefficients_cross] |
| simp only [labelPrefixPattern,boolReal] |
| split <;> simp |
| |
| lemma prefixPatternMean_const (n r : ℕ) (b : ℕ → ℝ) (c : ℝ) : |
| prefixPatternMean n r b (fun _ => c) = c := by simp [prefixPatternMean] |
| |
| |
| theorem cascade_prefix_gg (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (D : PrefixPattern n (r+1) → ℝ) : |
| (r+1 : ℕ)*prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.succ)*boolReal (Q 1 0 d)) = |
| (1-b d)*prefixPatternMean n (r+1) b D + |
| prefixPatternMean n (r+1) b (fun Q => D Q*(∑ j : Fin r, boolReal (Q 0 j.succ d))) := by |
| obtain ⟨c,hc,hD⟩ := prefixPattern_bound n (r+1) D |
| have h := randomReplicaEnergy_identity (P := (labeledCascadeLaw n b : Measure (LabeledTree n))) |
| (measurable_labeledLeafLaw n) (leafLevelCoefficients n d) (leafLevelCoefficients_bound n d) |
| (leafLevelCoefficients_diag n d) 1 (fun σ => D (labelPrefixPattern n (r+1) σ)) hc (fun σ => hD _) |
| have hE : (fun σ : Fin (r+1) → LabeledLeaf n => D (labelPrefixPattern n (r+1) σ)* |
| (∑ l : Fin r, cylinderCross (leafLevelCoefficients n d (σ 0)) (leafLevelCoefficients n d (σ l.succ)))) = |
| (fun σ => (fun Q : PrefixPattern n (r+1) => D Q*(∑ l : Fin r, boolReal (Q 0 l.succ d))) |
| (labelPrefixPattern n (r+1) σ)) := by |
| funext σ |
| simp only [leafLevelCoefficients_cross_bool] |
| have hF : (fun τ : Fin (r+1+1) → LabeledLeaf n => D (labelPrefixPattern n (r+1) (Fin.tail τ))* |
| cylinderCross (leafLevelCoefficients n d ((Fin.tail τ) 0)) (leafLevelCoefficients n d (τ 0))) = |
| (fun τ => (fun Q : PrefixPattern n (r+1+1) => D (patternReindex Q Fin.succ)*boolReal (Q 1 0 d)) |
| (labelPrefixPattern n (r+1+1) τ)) := by |
| funext τ |
| change D (labelPrefixPattern n (r+1) (Fin.tail τ))*cylinderCross (leafLevelCoefficients n d (τ 1)) (leafLevelCoefficients n d (τ 0)) = _ |
| rw [leafLevelCoefficients_cross_bool n (r+1+1) d τ 1 0] |
| rfl |
| rw [hE,hF,randomReplicaEnergy_shape n r b hb d 1 D, |
| randomReplicaAverage_shape n (r+1) b hb d 1 D, |
| randomReplicaAverage_shape n (r+1) b hb d 1 (fun Q => D Q*(∑ l : Fin r, boolReal (Q 0 l.succ d))), |
| randomReplicaAverage_shape n (r+1+1) b hb d 1 (fun Q => D (patternReindex Q Fin.succ)*boolReal (Q 1 0 d))] at h |
| change prefixPatternMean n (r+1) b D*(b d*1) = _ at h |
| change _ = (1-b d)*prefixPatternMean n (r+1) b D+_ |
| dsimp only [prefixPatternMean] at * |
| linarith |
| |
| lemma prefix_pair_mean (n : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) (d : Fin n) : |
| prefixPatternMean n 2 b (fun Q => boolReal (Q 1 0 d)) = 1-b d := by |
| have h := cascade_prefix_gg n 0 b hb d (fun _ => 1) |
| simpa only [Nat.zero_add,Nat.cast_one,one_mul,Fin.sum_univ_zero,mul_zero, |
| prefixPatternMean_const,mul_one,add_zero] using h |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| lemma replica_injective_map_reindex {X : Type*} [MeasurableSpace X] (ν : Measure X) [IsProbabilityMeasure ν] |
| {r : ℕ} {e : Fin r → ℕ} (he : Function.Injective e) : |
| (Measure.infinitePi (fun _ : ℕ => ν)).map (fun σ => fun i : Fin r => σ (e i)) = |
| Measure.pi (fun _ : Fin r => ν) := by |
| have hi : iIndepFun (fun i : ℕ => fun σ : ℕ → X => σ i) |
| (Measure.infinitePi (fun _ : ℕ => ν)) := iIndepFun_infinitePi (fun _ => measurable_id) |
| have h := (hi.precomp (fun {_ _} h => he h)).map_fun_eq_pi_map |
| (fun _ => (measurable_pi_apply _).aemeasurable) |
| simpa only [Measure.infinitePi_map_eval] using h |
| |
| lemma replicaLaw_integral_injective_eq {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| [Countable X] [MeasurableSingletonClass X] (P : Measure Ω) [IsProbabilityMeasure P] |
| (ν : Ω → Measure X) (hν : Measurable ν) [∀ ω, IsProbabilityMeasure (ν ω)] |
| {r : ℕ} {e : Fin r → ℕ} (he : Function.Injective e) |
| (F : (Fin r → X) → ℝ) {c : ℝ} (hF : ∀ σ, |F σ| ≤ c) : |
| (∫ σ, F (fun i => σ (e i)) ∂replicaLaw P ν hν) = |
| ∫ σ, F (fun i => σ i) ∂replicaLaw P ν hν := by |
| have h1m : Measurable (fun σ : ℕ → X => F (fun i => σ (e i))) := |
| (measurable_of_countable F).comp (by fun_prop) |
| have h2m : Measurable (fun σ : ℕ → X => F (fun i : Fin r => σ i)) := |
| (measurable_of_countable F).comp (by fun_prop) |
| rw [replicaLaw_integral_bounded P ν hν h1m (fun σ => hF _), |
| replicaLaw_integral_bounded P ν hν h2m (fun σ => hF _)] |
| apply integral_congr_ae |
| filter_upwards [] with ω |
| have h1 : HasLaw (fun σ : ℕ → X => fun i : Fin r => σ (e i)) |
| (Measure.pi (fun _ : Fin r => ν ω)) (Measure.infinitePi (fun _ : ℕ => ν ω)) := |
| ⟨(by fun_prop : Measurable _).aemeasurable,replica_injective_map_reindex (ν ω) he⟩ |
| have h2 : HasLaw (fun σ : ℕ → X => fun i : Fin r => σ i) |
| (Measure.pi (fun _ : Fin r => ν ω)) (Measure.infinitePi (fun _ : ℕ => ν ω)) := |
| ⟨(by fun_prop : Measurable _).aemeasurable,replica_prefix_map (ν ω) r⟩ |
| exact (h1.integral_comp (measurable_of_countable F).aestronglyMeasurable).trans |
| (h2.integral_comp (measurable_of_countable F).aestronglyMeasurable).symm |
| |
| lemma cascadeReplicaLaw_eq_replicaLaw (n : ℕ) (b : ℕ → ℝ) : |
| cascadeReplicaLaw n b = replicaLaw (labeledCascadeLaw n b) (labeledLeafLaw n) (measurable_labeledLeafLaw n) := by |
| unfold cascadeReplicaLaw replicaLaw |
| congr 1 |
| |
| lemma cascade_integral_injective (n : ℕ) (b : ℕ → ℝ) {r : ℕ} {e : Fin r → ℕ} |
| (he : Function.Injective e) (F : (Fin r → LabeledLeaf n) → ℝ) {c : ℝ} (hF : ∀ σ, |F σ| ≤ c) : |
| (∫ σ, F (fun i => σ (e i)) ∂cascadeReplicaLaw n b) = |
| ∫ σ, F (fun i => σ i) ∂cascadeReplicaLaw n b := by |
| simp only [cascadeReplicaLaw_eq_replicaLaw] |
| exact replicaLaw_integral_injective_eq (Ω := LabeledTree n) (X := LabeledLeaf n) |
| (labeledCascadeLaw n b) (labeledLeafLaw n) (measurable_labeledLeafLaw n) (r := r) (e := e) he F hF |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| lemma prefixPatternMean_injective (n r : ℕ) (b : ℕ → ℝ) {e : Fin r → ℕ} |
| (he : Function.Injective e) (D : PrefixPattern n r → ℝ) : |
| (∫ σ, D (labelPrefixPattern n r (fun i => σ (e i))) ∂cascadeReplicaLaw n b) = |
| prefixPatternMean n r b D := by |
| obtain ⟨c,hc,hD⟩ := prefixPattern_bound n r D |
| exact cascade_integral_injective n b he (fun τ => D (labelPrefixPattern n r τ)) (fun σ => hD _) |
| |
| lemma prefixPatternMean_congr (n r : ℕ) (b : ℕ → ℝ) {D E : PrefixPattern n r → ℝ} |
| (h : ∀ σ : Fin r → LabeledLeaf n, D (labelPrefixPattern n r σ) = E (labelPrefixPattern n r σ)) : |
| prefixPatternMean n r b D = prefixPatternMean n r b E := |
| integral_congr_ae (ae_of_all _ (fun σ => h (fun index => σ index))) |
| |
| lemma prefixPatternMean_integrable (n r : ℕ) (b : ℕ → ℝ) (D : PrefixPattern n r → ℝ) : |
| Integrable (fun σ => D (labelPrefixPattern n r (fun i => σ i))) (cascadeReplicaLaw n b) := by |
| obtain ⟨c,hc,hD⟩ := prefixPattern_bound n r D |
| exact integrable_of_measurable_abs_le |
| ((measurable_of_countable D).comp (measurable_infiniteLabelPattern n r)) (fun σ => hD _) |
| |
| lemma prefixPatternMean_add (n r : ℕ) (b : ℕ → ℝ) (D E : PrefixPattern n r → ℝ) : |
| prefixPatternMean n r b (fun Q => D Q+E Q) = prefixPatternMean n r b D+prefixPatternMean n r b E := |
| integral_add (prefixPatternMean_integrable n r b D) (prefixPatternMean_integrable n r b E) |
| |
| lemma prefixPatternMean_sub (n r : ℕ) (b : ℕ → ℝ) (D E : PrefixPattern n r → ℝ) : |
| prefixPatternMean n r b (fun Q => D Q-E Q) = prefixPatternMean n r b D-prefixPatternMean n r b E := |
| integral_sub (prefixPatternMean_integrable n r b D) (prefixPatternMean_integrable n r b E) |
| |
| lemma prefixPatternMean_mul (n r : ℕ) (b : ℕ → ℝ) (c : ℝ) (D : PrefixPattern n r → ℝ) : |
| prefixPatternMean n r b (fun Q => c*D Q) = c*prefixPatternMean n r b D := integral_const_mul _ _ |
| |
| lemma prefixPatternMean_sum {I : Type*} (s : Finset I) (n r : ℕ) (b : ℕ → ℝ) (D : I → PrefixPattern n r → ℝ) : |
| prefixPatternMean n r b (fun Q => ∑ i ∈ s, D i Q) = ∑ i ∈ s, prefixPatternMean n r b (D i) := |
| integral_finsetSum s (fun i _ => prefixPatternMean_integrable n r b (D i)) |
| |
| def lastToFront (r : ℕ) : Fin (r+1+1) → ℕ := Fin.cases (r+1) Fin.val |
| |
| lemma lastToFront_injective (r : ℕ) : Function.Injective (lastToFront r) := by |
| intro i j |
| refine Fin.cases ?_ (fun i => ?_) i <;> refine Fin.cases ?_ (fun j => ?_) j |
| · intro _; rfl |
| · simp only [lastToFront,Fin.cases_zero,Fin.cases_succ]; intro h; have := j.isLt; omega |
| · simp only [lastToFront,Fin.cases_zero,Fin.cases_succ]; intro h; have := i.isLt; omega |
| · simp only [lastToFront,Fin.cases_succ]; intro h; exact congrArg Fin.succ (Fin.ext h) |
| |
| lemma prefix_fresh_reindex (n r : ℕ) (b : ℕ → ℝ) (d : Fin n) |
| (D : PrefixPattern n (r+1) → ℝ) : |
| prefixPatternMean n (r+1+1) b (fun Q => D (patternReindex Q Fin.succ)*boolReal (Q 1 0 d)) = |
| prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.castSucc)*boolReal (Q 0 (Fin.last (r+1)) d)) := by |
| have h := prefixPatternMean_injective n (r+1+1) b (lastToFront_injective r) |
| (fun Q => D (patternReindex Q Fin.succ)*boolReal (Q 1 0 d)) |
| rw [← h] |
| apply integral_congr_ae |
| filter_upwards [] with σ |
| rfl |
| |
| lemma cascade_prefix_gg_first (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (d : Fin n) (D : PrefixPattern n (r+1) → ℝ) : |
| (r+1 : ℕ)*prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.castSucc)*boolReal (Q 0 (Fin.last (r+1)) d)) = |
| (1-b d)*prefixPatternMean n (r+1) b D + |
| prefixPatternMean n (r+1) b (fun Q => D Q*(∑ j : Fin r, boolReal (Q 0 j.succ d))) := by |
| rw [← prefix_fresh_reindex] |
| exact cascade_prefix_gg n r b hb d D |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| lemma prefixPatternMean_perm (n r : ℕ) (b : ℕ → ℝ) (e : Equiv.Perm (Fin r)) |
| (D : PrefixPattern n r → ℝ) : |
| prefixPatternMean n r b (fun Q => D (patternReindex Q e)) = prefixPatternMean n r b D := by |
| have h := prefixPatternMean_injective n r b (e := fun i => (e i).val) |
| (fun _ _ h => e.injective (Fin.ext h)) D |
| exact h |
| |
| lemma patternReindex_perm_cancel {n r : ℕ} (Q : PrefixPattern n r) (e : Equiv.Perm (Fin r)) : |
| patternReindex (patternReindex Q e) e.symm = Q := by |
| funext i j d |
| simp [patternReindex] |
| |
| lemma prefix_fresh_permuted (n r : ℕ) (b : ℕ → ℝ) (d : Fin n) |
| (e : Equiv.Perm (Fin (r+1))) (D : PrefixPattern n (r+1) → ℝ) : |
| prefixPatternMean n (r+1+1) b (fun Q => D (patternReindex (patternReindex Q Fin.succ) e.symm)*boolReal (Q 1 0 d)) = |
| prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.castSucc)*boolReal (Q (e 0).castSucc (Fin.last (r+1)) d)) := by |
| have h := prefixPatternMean_injective n (r+1+1) b (newReplicaEmbedding_injective (r+1) e) |
| (fun Q => D (patternReindex (patternReindex Q Fin.succ) e.symm)*boolReal (Q 1 0 d)) |
| rw [← h] |
| apply integral_congr_ae |
| filter_upwards [] with σ |
| congr 1 |
| · apply congrArg D |
| funext i j d |
| simp [patternReindex,labelPrefixPattern,newReplicaEmbedding] |
| |
| lemma prefix_old_permuted (n r : ℕ) (b : ℕ → ℝ) (d : Fin n) |
| (e : Equiv.Perm (Fin (r+1))) (D : PrefixPattern n (r+1) → ℝ) : |
| prefixPatternMean n (r+1) b |
| (fun Q => D (patternReindex Q e.symm)*(∑ j : Fin r, boolReal (Q 0 j.succ d))) = |
| prefixPatternMean n (r+1) b |
| (fun Q => D Q*(∑ j ∈ Finset.univ.erase (e 0), boolReal (Q (e 0) j d))) := by |
| rw [← prefixPatternMean_perm n (r+1) b e] |
| congr 1 |
| funext Q |
| rw [patternReindex_perm_cancel] |
| dsimp only [patternReindex] |
| rw [sum_nonzero_permutation e (fun j => boolReal (Q (e 0) j d))] |
| |
| lemma cascade_prefix_gg_all (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (i : Fin (r+1)) (d : Fin n) (D : PrefixPattern n (r+1) → ℝ) : |
| (r+1 : ℕ)*prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.castSucc)*boolReal (Q i.castSucc (Fin.last (r+1)) d)) = |
| (1-b d)*prefixPatternMean n (r+1) b D + |
| prefixPatternMean n (r+1) b |
| (fun Q => D Q*(∑ j ∈ Finset.univ.erase i, boolReal (Q i j d))) := by |
| have h := cascade_prefix_gg n r b hb d (fun Q => D (patternReindex Q (Equiv.swap 0 i).symm)) |
| rw [prefix_fresh_permuted,prefixPatternMean_perm,prefix_old_permuted,Equiv.swap_apply_left] at h |
| exact h |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| def prefixVectorValue (n : ℕ) (a : ℕ → ℝ) (v : Fin n → Bool) : ℝ := |
| a 0+∑ d : Fin n, (a (d+1)-a d)*boolReal (v d) |
| |
| lemma sum_prefix_telescope (n k : ℕ) (hk : k ≤ n) (a : ℕ → ℝ) : |
| a 0+(∑ d : Fin n, (a (d+1)-a d)*(if d.val+1 ≤ k then 1 else 0)) = a k := by |
| induction n generalizing k with |
| | zero => |
| have : k = 0 := by omega |
| subst k |
| simp |
| | succ n ih => |
| rw [Fin.sum_univ_castSucc] |
| by_cases h : k ≤ n |
| · have he : (∑ d : Fin n, (a (d.castSucc+1)-a d.castSucc)*(if d.castSucc.val+1 ≤ k then 1 else 0)) = |
| ∑ d : Fin n, (a (d+1)-a d)*(if d.val+1 ≤ k then 1 else 0) := rfl |
| rw [he] |
| simp only [Fin.val_last,show ¬ n+1 ≤ k by omega,ite_false,mul_zero,add_zero] |
| exact ih k h |
| · have hkn : k = n+1 := by omega |
| subst k |
| have hs : (∑ d : Fin n, (a (d.castSucc+1)-a d.castSucc)*(if d.castSucc.val+1 ≤ n+1 then 1 else 0)) = |
| ∑ d : Fin n, (a (d+1)-a d)*(if d.val+1 ≤ n then 1 else 0) := by |
| apply Finset.sum_congr rfl |
| intro d hd |
| have hd := d.isLt |
| simp only [Fin.val_castSucc,show d.val+1 ≤ n+1 by omega,show d.val+1 ≤ n by omega,ite_true] |
| rw [hs] |
| simp only [Fin.val_last,le_refl,ite_true,mul_one] |
| have hi := ih n (le_refl n) |
| linarith |
| |
| lemma prefixVectorValue_label (n : ℕ) (a : ℕ → ℝ) (v w : LabeledLeaf n) : |
| prefixVectorValue n a (fun d => if (labeledAddress n v).take (d+1) = (labeledAddress n w).take (d+1) then true else false) = |
| a (labeledCommonDepth n v w) := by |
| unfold prefixVectorValue |
| have hf : (fun d : Fin n => boolReal (if (labeledAddress n v).take (d+1) = (labeledAddress n w).take (d+1) then true else false)) = |
| (fun d => if d.val+1 ≤ labeledCommonDepth n v w then 1 else 0) := by |
| funext d |
| simp only [labeled_prefix_eq_iff n v w (d := d.val+1) (by omega), boolReal] |
| split <;> simp |
| simp_rw [congrFun hf] |
| exact sum_prefix_telescope n _ (labeledCommonDepth_le n v w) a |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal |
| namespace IsingPerceptron |
| |
| lemma prefixPatternMean_restrict (n r : ℕ) (b : ℕ → ℝ) (D : PrefixPattern n r → ℝ) : |
| prefixPatternMean n (r+1) b (fun Q => D (patternReindex Q Fin.castSucc)) = prefixPatternMean n r b D := rfl |
| |
| def prefixGGResidual (n r : ℕ) (b : ℕ → ℝ) (i : Fin (r+1)) (D : PrefixPattern n (r+1) → ℝ) : |
| ((Fin n → Bool) → ℝ) →ₗ[ℝ] ℝ where |
| toFun V := (r+1 : ℕ)*prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.castSucc)*V (Q i.castSucc (Fin.last (r+1)))) - |
| prefixPatternMean n (r+1) b D*prefixPatternMean n 2 b (fun Q => V (Q 0 1)) - |
| ∑ j ∈ Finset.univ.erase i, prefixPatternMean n (r+1) b (fun Q => D Q*V (Q i j)) |
| map_add' V W := by |
| simp only [Pi.add_apply,mul_add,prefixPatternMean_add,Finset.sum_add_distrib] |
| ring |
| map_smul' c V := by |
| simp only [Pi.smul_apply,smul_eq_mul,RingHom.id_apply,show ∀ x y : ℝ, x*(c*y) = c*(x*y) from fun x y => by ring, |
| prefixPatternMean_mul,← Finset.mul_sum] |
| ring |
| |
| lemma prefixGGResidual_const (n r : ℕ) (b : ℕ → ℝ) (i : Fin (r+1)) (D : PrefixPattern n (r+1) → ℝ) : |
| prefixGGResidual n r b i D (fun _ => 1) = 0 := by |
| simp only [prefixGGResidual,LinearMap.coe_mk,AddHom.coe_mk,mul_one, |
| prefixPatternMean_restrict,prefixPatternMean_const,Finset.sum_const, |
| Finset.card_erase_of_mem (Finset.mem_univ i),Finset.card_univ,Fintype.card_fin, |
| Nat.add_sub_cancel,nsmul_eq_mul,Nat.cast_add,Nat.cast_one] |
| ring |
| |
| lemma prefix_pair_mean_01 (n : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) (d : Fin n) : |
| prefixPatternMean n 2 b (fun Q => boolReal (Q 0 1 d)) = 1-b d := by |
| rw [← prefix_pair_mean n b hb d] |
| apply prefixPatternMean_congr |
| intro σ |
| simp [labelPrefixPattern,eq_comm] |
| |
| lemma prefixGGResidual_level (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (i : Fin (r+1)) (d : Fin n) (D : PrefixPattern n (r+1) → ℝ) : |
| prefixGGResidual n r b i D (fun v => boolReal (v d)) = 0 := by |
| have h := cascade_prefix_gg_all n r b hb i d D |
| have hs : prefixPatternMean n (r+1) b |
| (fun Q => D Q*(∑ j ∈ Finset.univ.erase i, boolReal (Q i j d))) = |
| ∑ j ∈ Finset.univ.erase i, prefixPatternMean n (r+1) b (fun Q => D Q*boolReal (Q i j d)) := by |
| simp only [Finset.mul_sum,prefixPatternMean_sum] |
| rw [hs] at h |
| change (r+1 : ℕ)*prefixPatternMean n (r+1+1) b _ - |
| prefixPatternMean n (r+1) b D*prefixPatternMean n 2 b (fun Q => boolReal (Q 0 1 d)) - _ = 0 |
| rw [prefix_pair_mean_01 n b hb d,h] |
| ring |
| |
| lemma prefixGGResidual_value (n r : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) |
| (i : Fin (r+1)) (D : PrefixPattern n (r+1) → ℝ) (a : ℕ → ℝ) : |
| prefixGGResidual n r b i D (prefixVectorValue n a) = 0 := by |
| have he : prefixVectorValue n a = a 0 • (fun _ : Fin n → Bool => (1:ℝ))+ |
| ∑ d : Fin n, (a (d+1)-a d) • (fun v : Fin n → Bool => boolReal (v d)) := by |
| funext v |
| simp [prefixVectorValue,Finset.sum_apply] |
| rw [he,map_add,map_smul,map_sum,prefixGGResidual_const] |
| simp only [map_smul,prefixGGResidual_level n r b hb i,smul_zero,Finset.sum_const_zero,add_zero] |
| |
| end IsingPerceptron |
| |
| |
| |
| |
| |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Topology ENNReal NNReal Classical |
| namespace IsingPerceptron |
| |
| def cascadeScalarArray (n : ℕ) (a : ℕ → ℝ) (σ : ℕ → LabeledLeaf n) : RealArray := |
| fun i j => a (labeledCommonDepth n (σ i) (σ j)) |
| |
| lemma measurable_cascadeScalarArray (n : ℕ) (a : ℕ → ℝ) : Measurable (cascadeScalarArray n a) := by |
| unfold cascadeScalarArray |
| apply Measurable.of_eval; intro i |
| apply Measurable.of_eval; intro j |
| have h : Measurable (fun p : LabeledLeaf n × LabeledLeaf n => a (labeledCommonDepth n p.1 p.2)) := measurable_of_countable _ |
| exact h.comp (show Measurable (fun σ : ℕ → LabeledLeaf n => (σ i,σ j)) by fun_prop) |
| |
| lemma prefixVectorValue_pattern (n r : ℕ) (a : ℕ → ℝ) (σ : Fin r → LabeledLeaf n) (i j : Fin r) : |
| prefixVectorValue n a (labelPrefixPattern n r σ i j) = a (labeledCommonDepth n (σ i) (σ j)) := |
| prefixVectorValue_label n a (σ i) (σ j) |
| |
| lemma integral_ite_one_eq {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (A : Set Ω) |
| (hA : MeasurableSet A) : (∫ ω, (if ω ∈ A then (1:ℝ) else 0) ∂μ) = μ.real A := by |
| classical |
| simpa only [Set.indicator,Pi.one_apply] using integral_indicator_one (μ := μ) hA |
| |
| lemma integral_ite_one_mul_eq {Ω : Type*} [MeasurableSpace Ω] (μ : Measure Ω) (A B : Set Ω) |
| (hA : MeasurableSet A) (hB : MeasurableSet B) : |
| (∫ ω, (if ω ∈ A then (1:ℝ) else 0)*(if ω ∈ B then 1 else 0) ∂μ) = μ.real (A ∩ B) := by |
| classical |
| rw [← integral_ite_one_eq μ (A ∩ B) (hA.inter hB)] |
| apply integral_congr_ae |
| filter_upwards [] with ω |
| by_cases ha : ω ∈ A <;> by_cases hb : ω ∈ B <;> simp [ha,hb] |
| |
| |
| |
| theorem cascade_scalar_gg (n : ℕ) (b : ℕ → ℝ) (hb : CascadeExponents n b) (a : ℕ → ℝ) : |
| HasGhirlandaGuerra (cascadeScalarArray n a) (cascadeReplicaLaw n b) := by |
| classical |
| intro m hm i A hA t ht |
| obtain ⟨r,rfl⟩ := Nat.exists_eq_succ_of_ne_zero (by omega : m ≠ 0) |
| let D : PrefixPattern n (r+1) → ℝ := fun Q => |
| if (fun i j => prefixVectorValue n a (Q i j)) ∈ A then 1 else 0 |
| let c : ℕ → ℝ := fun k => if a k ∈ t then 1 else 0 |
| have h := prefixGGResidual_value n r b hb i D c |
| change (r+1 : ℕ)*prefixPatternMean n (r+1+1) b |
| (fun Q => D (patternReindex Q Fin.castSucc)*prefixVectorValue n c (Q i.castSucc (Fin.last (r+1)))) - |
| prefixPatternMean n (r+1) b D*prefixPatternMean n 2 b (fun Q => prefixVectorValue n c (Q 0 1)) - |
| (∑ j ∈ Finset.univ.erase i, prefixPatternMean n (r+1) b (fun Q => D Q*prefixVectorValue n c (Q i j))) = 0 at h |
| simp only [prefixPatternMean,D] at h |
| simp only [patternReindex_label,prefixVectorValue_pattern,Function.comp_apply,Fin.val_castSucc,Fin.val_last,c] at h |
| have hAm : MeasurableSet (arrayBlock (cascadeScalarArray n a) (r+1) ⁻¹' A) := |
| hA.preimage (measurable_arrayBlock (measurable_cascadeScalarArray n a) _) |
| have htm : ∀ i j, MeasurableSet {σ | cascadeScalarArray n a σ i j ∈ t} := |
| fun i j => ht.preimage ((measurable_pi_apply j).comp ((measurable_pi_apply i).comp (measurable_cascadeScalarArray n a))) |
| change (r+1 : ℕ)*(∫ σ, (if arrayBlock (cascadeScalarArray n a) (r+1) σ ∈ A then (1:ℝ) else 0)* |
| (if cascadeScalarArray n a σ i (r+1) ∈ t then 1 else 0) ∂cascadeReplicaLaw n b) - |
| (∫ σ, (if arrayBlock (cascadeScalarArray n a) (r+1) σ ∈ A then (1:ℝ) else 0) ∂cascadeReplicaLaw n b)* |
| (∫ σ, (if cascadeScalarArray n a σ 0 1 ∈ t then (1:ℝ) else 0) ∂cascadeReplicaLaw n b) - |
| (∑ j ∈ Finset.univ.erase i, ∫ σ, (if arrayBlock (cascadeScalarArray n a) (r+1) σ ∈ A then (1:ℝ) else 0)* |
| (if cascadeScalarArray n a σ i j ∈ t then 1 else 0) ∂cascadeReplicaLaw n b) = 0 at h |
| have hint (u v : ℕ) : (∫ σ, (if arrayBlock (cascadeScalarArray n a) (r+1) σ ∈ A then (1:ℝ) else 0)* |
| (if cascadeScalarArray n a σ u v ∈ t then 1 else 0) ∂cascadeReplicaLaw n b) = |
| (cascadeReplicaLaw n b).real ((arrayBlock (cascadeScalarArray n a) (r+1) ⁻¹' A) ∩ {σ | cascadeScalarArray n a σ u v ∈ t}) := |
| integral_ite_one_mul_eq _ _ _ hAm (htm u v) |
| have htest : (∫ σ, (if arrayBlock (cascadeScalarArray n a) (r+1) σ ∈ A then (1:ℝ) else 0) ∂cascadeReplicaLaw n b) = |
| (cascadeReplicaLaw n b).real (arrayBlock (cascadeScalarArray n a) (r+1) ⁻¹' A) := integral_ite_one_eq _ _ hAm |
| have hpair : (∫ σ, (if cascadeScalarArray n a σ 0 1 ∈ t then (1:ℝ) else 0) ∂cascadeReplicaLaw n b) = |
| (cascadeReplicaLaw n b).real {σ | cascadeScalarArray n a σ 0 1 ∈ t} := integral_ite_one_eq _ _ (htm 0 1) |
| simp only [hint,htest,hpair] at h |
| have hr : ((r+1 : ℕ) : ℝ) ≠ 0 := by positivity |
| rw [← add_div] |
| apply (eq_div_iff hr).2 |
| nlinarith [h] |
|
|
| end IsingPerceptron |
|
|
| end |
|
|
| end OAI |
|
|
