| import Mathlib |
| import OAI.Probability.Perceptron.Variational.IndexedDiagonalField |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators |
|
|
| lemma diagonalMark_norm_le {I : Type} [Fintype I] (c : I → ℝ) {C : ℝ} |
| (hC : 0 ≤ C) (hc : ∀ i, |c i| ≤ C) : ‖diagonalMark c‖ ≤ C := by |
| apply ContinuousLinearMap.opNorm_le_bound _ hC |
| intro x |
| have hs : ‖diagonalMark c x‖^2 ≤ (C*‖x‖)^2 := by |
| rw [mul_pow,EuclideanSpace.norm_sq_eq,EuclideanSpace.norm_sq_eq,Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro i _ |
| simp only [Real.norm_eq_abs,sq_abs,diagonalMark_apply,mul_pow] |
| exact mul_le_mul_of_nonneg_right (by |
| simpa only [sq_abs] using pow_le_pow_left₀ (abs_nonneg (c i)) (hc i) 2) (sq_nonneg (x i)) |
| nlinarith [norm_nonneg (diagonalMark c x),mul_nonneg hC (norm_nonneg x)] |
|
|
| lemma enrichedRootMap_norm_le {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) |
| {H : ℝ} (hH : 0 ≤ H) (hh : h 0 ≤ H) : |
| ‖enrichedRootMap p d h‖ ≤ Real.sqrt (2*H+1) := by |
| apply diagonalMark_norm_le _ (Real.sqrt_nonneg _) |
| intro i |
| rw [abs_of_nonneg (Real.sqrt_nonneg _)] |
| apply Real.sqrt_le_sqrt |
| cases i with |
| | inl i => dsimp [enrichedCoordinateLevel]; linarith |
| | inr i => |
| simp only [enrichedCoordinateLevel,Fin.val_zero,Nat.cast_zero,zero_div] |
| by_cases hd : d i.1=0 |
| · rw [hd,pow_zero]; linarith |
| · rw [zero_pow hd]; positivity |
|
|
| lemma perturbationScale_le_one (N : ℕ) (hN : 1 ≤ N) : perturbationScale N ≤ 1 := |
| Real.rpow_le_one_of_one_le_of_nonpos (by exact_mod_cast hN) (by norm_num) |
|
|
| lemma perturbationWeight_eq (j : ℕ) : (2:ℝ)^(-((j+1:ℕ):ℤ))=(1/2:ℝ)^(j+1) := by |
| simp only [zpow_neg,zpow_natCast,one_div,inv_pow] |
|
|
| lemma half_power_sum_le_one (N : ℕ) : (∑ j : Fin N, (1/2:ℝ)^(j.val+1)) ≤ 1 := by |
| rw [Fin.sum_univ_eq_sum_range (fun j => (1/2:ℝ)^(j+1)) N] |
| simp_rw [pow_succ] |
| rw [← Finset.sum_mul] |
| nlinarith [sum_geometric_two_le N] |
|
|
| lemma perturbationAmplitude_square_le (N : ℕ) (hN : 1 ≤ N) (u : Fin N → ℝ) |
| (hu : ∀ j, u j ∈ Icc 1 2) (j : Fin N) : |
| perturbationAmplitude N u j^2 ≤ 4*(N:ℝ)*(1/2:ℝ)^(j.val+1) := by |
| let w : ℝ := (1/2:ℝ)^(j.val+1) |
| have hw0 : 0 ≤ w := pow_nonneg (by norm_num) _ |
| have hw1 : w ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| have he0 : 0 ≤ perturbationScale N := Real.rpow_nonneg (Nat.cast_nonneg N) _ |
| have he1 := perturbationScale_le_one N hN |
| have he2 : perturbationScale N^2 ≤ 1 := by nlinarith |
| have hu2 : (u j)^2 ≤ 4 := by have h1 := (hu j).1; have h2 := (hu j).2; nlinarith |
| have hw2 : w^2 ≤ w := by nlinarith |
| unfold perturbationAmplitude |
| rw [perturbationWeight_eq] |
| change (Real.sqrt (N:ℝ)*perturbationScale N*w*u j)^2 ≤ _ |
| simp only [mul_pow,Real.sq_sqrt (Nat.cast_nonneg N)] |
| calc |
| _ ≤ (N:ℝ)*1*w^2*4 := by |
| apply mul_le_mul _ hu2 (sq_nonneg _) (by positivity) |
| exact mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left he2 (Nat.cast_nonneg N)) (sq_nonneg w) |
| _ ≤ 4*(N:ℝ)*w := by nlinarith [mul_le_mul_of_nonneg_left hw2 (show 0 ≤ (N:ℝ) by positivity)] |
|
|
| lemma enrichedFeatureBound_sq_le (N : ℕ) (hN : 1 ≤ N) (u : Fin N → ℝ) |
| (hu : ∀ j, u j ∈ Icc 1 2) : (enrichedFeatureBound N u:ℝ)^2 ≤ 5*(N:ℝ) := by |
| change (Real.sqrt ((N:ℝ)+∑ j, perturbationAmplitude N u j^2))^2 ≤ _ |
| rw [Real.sq_sqrt (by positivity)] |
| have hs : (∑ j : Fin N, perturbationAmplitude N u j^2) ≤ 4*(N:ℝ) := by |
| calc |
| _ ≤ ∑ j : Fin N, 4*(N:ℝ)*(1/2:ℝ)^(j.val+1) := |
| Finset.sum_le_sum fun j _ => perturbationAmplitude_square_le N hN u hu j |
| _ = 4*(N:ℝ)*(∑ j : Fin N, (1/2:ℝ)^(j.val+1)) := (Finset.mul_sum _ _ _).symm |
| _ ≤ 4*(N:ℝ)*1 := mul_le_mul_of_nonneg_left (half_power_sum_le_one N) (by positivity) |
| _ = _ := mul_one _ |
| linarith |
|
|
| lemma finite_profile_telescope {k : ℕ} (q : Fin (k+1) → ℝ) : |
| q 0 + ∑ i : Fin k, (q i.succ-q i.castSucc) = q (Fin.last k) := by |
| rw [Finset.sum_sub_distrib] |
| have h1 := Fin.sum_univ_succ q |
| have h2 := Fin.sum_univ_castSucc q |
| linarith |
|
|
| def profileGaussianRoot {I : Type} {k : ℕ} (q : Fin (k+1) → I → ℝ) (i : I) : ℝ := |
| Real.sqrt (q 0 i) |
|
|
| def profileGaussianStep {I : Type} {k : ℕ} (q : Fin (k+1) → I → ℝ) |
| (j : ℕ) (i : I) : ℝ := |
| if hj : j<k then Real.sqrt (q ⟨k-j,by omega⟩ i-q ⟨k-j-1,by omega⟩ i) else 0 |
|
|
| lemma profileGaussian_coefficient {I : Type} {k : ℕ} (q : Fin (k+1) → I → ℝ) : |
| hierarchyRowCoefficients k (profileGaussianRoot q) (profileGaussianStep q) = |
| Fin.cases (fun i => Real.sqrt (q 0 i)) |
| (fun l i => Real.sqrt (q l.succ i-q l.castSucc i)) := by |
| funext l i |
| induction l using Fin.cases with |
| | zero => rfl |
| | succ l => |
| simp only [hierarchyRowCoefficients,Fin.cases_succ,profileGaussianStep, |
| dite_eq_left (show k-1-l.val<k by omega)] |
| congr 2 <;> congr 1 <;> apply Fin.ext <;> simp only [Fin.val_succ,Fin.val_castSucc] <;> omega |
|
|
| lemma profileGaussian_coefficient_squares {I : Type} {k : ℕ} (q : Fin (k+1) → I → ℝ) |
| (h0 : ∀ i, 0 ≤ q 0 i) (hq : ∀ i, Monotone (fun l => q l i)) (i : I) : |
| (∑ l : Fin (k+1), hierarchyRowCoefficients k (profileGaussianRoot q) (profileGaussianStep q) l i^2) = |
| q (Fin.last k) i := by |
| rw [profileGaussian_coefficient,Fin.sum_univ_succ] |
| simp only [Fin.cases_zero,Fin.cases_succ,Real.sq_sqrt (h0 i)] |
| have he (j : Fin k) : (Real.sqrt (q j.succ i-q j.castSucc i))^2 = q j.succ i-q j.castSucc i := |
| Real.sq_sqrt (sub_nonneg.mpr (hq i (by simp [Fin.le_iff_val_le_val]))) |
| simp_rw [he] |
| exact finite_profile_telescope (fun l => q l i) |
|
|
| lemma enrichedCoordinateLevel_nonneg {N k : ℕ} (p d : Fin N → ℕ) |
| {h : Fin (k+1) → ℝ} (hh : ∀ l, 0 ≤ h l) (l : Fin (k+1)) (i : EnrichedIndex N N p) : |
| 0 ≤ enrichedCoordinateLevel p d h l i := by |
| cases i with |
| | inl i => exact mul_nonneg (by norm_num) (hh l) |
| | inr i => exact pow_nonneg (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) _ |
|
|
| lemma enrichedCoordinateLevel_monotone {N k : ℕ} (p d : Fin N → ℕ) |
| {h : Fin (k+1) → ℝ} (hh : Monotone h) (i : EnrichedIndex N N p) : |
| Monotone (fun l => enrichedCoordinateLevel p d h l i) := by |
| intro a b hab |
| cases i with |
| | inl i => exact mul_le_mul_of_nonneg_left (hh hab) (by norm_num) |
| | inr i => |
| apply pow_le_pow_left₀ (div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)) |
| exact div_le_div_of_nonneg_right (by exact_mod_cast hab) (Nat.cast_nonneg _) |
|
|
| lemma enrichedCoordinateLevel_le {N k : ℕ} (p d : Fin N → ℕ) |
| {h : Fin (k+1) → ℝ} {H : ℝ} (hH : 0 ≤ H) (hh : ∀ l, h l ≤ H) |
| (l : Fin (k+1)) (i : EnrichedIndex N N p) : |
| enrichedCoordinateLevel p d h l i ≤ 2*H+1 := by |
| cases i with |
| | inl i => dsimp [enrichedCoordinateLevel]; linarith [hh l] |
| | inr i => |
| have ht : ((l.val:ℝ)/(k+1:ℕ)) ≤ 1 := by |
| apply (div_le_one (by positivity)).mpr |
| exact_mod_cast l.isLt.le |
| have ht0 : 0 ≤ ((l.val:ℝ)/(k+1:ℕ)) := by positivity |
| exact (pow_le_one₀ ht0 ht).trans (by linarith) |
|
|
| lemma enrichedIncrementMap_profile {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) (j : ℕ) : |
| enrichedIncrementMap p d h j = diagonalMark (profileGaussianStep (enrichedCoordinateLevel p d h) j) := by |
| by_cases hj : j<k |
| · rw [enrichedIncrementMap,dite_eq_left hj] |
| congr 1 |
| funext i |
| simp only [profileGaussianStep,dite_eq_left hj] |
| · apply ContinuousLinearMap.ext |
| intro x |
| ext i |
| simp [enrichedIncrementMap,profileGaussianStep,hj,diagonalMark_apply] |
|
|
| def sourceGaussianRow {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) (u : Fin N → ℝ) : |
| ℕ → NormalizedSpin N × IndexedLeaf k → ℝ := |
| indexedGaussianRow k (hierarchyRowCoefficients k |
| (profileGaussianRoot (enrichedCoordinateLevel p d h)) |
| (profileGaussianStep (enrichedCoordinateLevel p d h))) (sourceEnrichedFeature N p u) |
|
|
| lemma sourceGaussianRow_measurable {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) |
| (u : Fin N → ℝ) (i : ℕ) : Measurable (sourceGaussianRow p d h u i) := |
| indexedGaussianRow_measurable _ _ (enrichedFeature_continuous _ _ _ _ _).measurable i |
|
|
| lemma sourceGaussianRow_square_bound {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) |
| (u : Fin N → ℝ) (hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) {H : ℝ} |
| (hH : 0 ≤ H) (hhH : ∀ l, h l ≤ H) (x : NormalizedSpin N × IndexedLeaf k) : |
| (∑ i : Fin (indexedGaussianRowLength (I := EnrichedIndex N N p) k x), |
| sourceGaussianRow p d h u i.val x^2) ≤ (2*H+1)*(enrichedFeatureBound N u:ℝ)^2 := by |
| apply indexedGaussianRow_norm_le |
| · intro i |
| rw [profileGaussian_coefficient_squares _ (enrichedCoordinateLevel_nonneg p d hh0 0) |
| (enrichedCoordinateLevel_monotone p d hh)] |
| exact enrichedCoordinateLevel_le p d hH hhH _ i |
| · positivity |
| · intro y |
| have he := congrArg (fun t : ℝ => t^2) (sourceEnrichedFeature_norm N p u y) |
| simpa only [EuclideanSpace.norm_sq_eq,Real.norm_eq_abs,sq_abs] using he.le |
|
|
| lemma sourceGaussianRow_identity {N k : ℕ} (p d : Fin N → ℕ) (h : Fin (k+1) → ℝ) |
| (u : Fin N → ℝ) (g : ℕ → ℝ) (x : NormalizedSpin N × IndexedLeaf k) : |
| countableGaussianField (sourceGaussianRow p d h u) |
| (indexedGaussianRowLength (I := EnrichedIndex N N p) k) g x = |
| inner ℝ (sourceEnrichedFeature N p u x.1) |
| (indexedLeafState (gaussianLinearMarkStep (enrichedIncrementMap p d h)) k |
| ((fun _ => enrichedRootMap p d h (indexedGaussianDisorder k (EnrichedIndex N N p) g).1), |
| (indexedGaussianDisorder k (EnrichedIndex N N p) g).2) x.2 0) := by |
| unfold sourceGaussianRow |
| rw [indexedGaussianRow_diagonal_identity] |
| have he : enrichedIncrementMap p d h = fun j => diagonalMark |
| (profileGaussianStep (enrichedCoordinateLevel p d h) j) := |
| funext (enrichedIncrementMap_profile p d h) |
| rw [he] |
| rfl |
|
|
| end SphericalPerceptronFreeEnergy |
|
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| end |
|
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| end OAI |
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