| import Mathlib |
| import OAI.Probability.SKGap.Localization.DiagonalHessianJoined |
| import OAI.Probability.SKGap.Localization.ConditionalEmpiricalLaw |
| import OAI.Probability.SKGap.Entropy.ConditionalBadSet |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| lemma scalarQMoment_empirical_overlap {n : ℕ} [NeZero n] (y : Fin n → ℝ) : |
| scalarQMoment (empiricalLaw y)=overlap y := by |
| simp only [scalarQMoment,integral_empiricalLaw,Fintype.card_fin,overlap,magnetization, |
| div_eq_mul_inv,mul_comm] |
|
|
| lemma empirical_columns_zero_outer {n : ℕ} [NeZero n] (y : Fin n → ℝ) (d s j : ℝ) : |
| j • vecMulVec (empiricalGramColumns y d s 0).ofLp (empiricalGramColumns y d s 0).ofLp= |
| (j/(n:ℝ)) • vecMulVec (fun _=>1) (fun _=>1) := by |
| have hnR : (n:ℝ) ≠ 0 := by exact_mod_cast NeZero.ne n |
| have hn : sqrt (n:ℝ) ≠ 0 := (sqrt_pos.mpr (by exact_mod_cast NeZero.pos n)).ne' |
| ext i k |
| simp only [empiricalGramColumns,gramCoordinates,Matrix.cons_val,WithLp.ofLp_toLp, |
| Matrix.smul_apply,smul_eq_mul,vecMulVec_apply,mul_one] |
| field_simp |
| rw [Real.sq_sqrt (Nat.cast_nonneg n)] |
|
|
| lemma empirical_unit_outer {n : ℕ} [NeZero n] (y : Fin n → ℝ) {q : ℝ} (hq : 0 < q) (j : ℝ) : |
| (2*j*q) • vecMulVec (empiricalUnit y q).ofLp (empiricalUnit y q).ofLp= |
| (2*j/(n:ℝ)) • vecMulVec (magnetization y) (magnetization y) := by |
| have hn : (n:ℝ) ≠ 0 := by exact_mod_cast NeZero.ne n |
| have hc : sqrt ((n:ℝ)*q) ≠ 0 := (sqrt_pos.mpr (mul_pos (by exact_mod_cast NeZero.pos n) hq)).ne' |
| ext i k |
| simp only [empiricalUnit,WithLp.ofLp_toLp,Matrix.smul_apply,smul_eq_mul,vecMulVec_apply,magnetization] |
| field_simp |
| rw [Real.sq_sqrt (mul_nonneg hq.le (Nat.cast_nonneg n))] |
| ring |
|
|
| lemma empiricalConditionalMiddle_literal {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) |
| (hq : 0 < scalarQMoment (empiricalLaw y)) (g : MatrixCoordinates (Fin n) → ℝ) : |
| empiricalConditionalMiddle j t σ y g= |
| plantedInteraction j (empiricalConditionalGoe j t σ y g)+ |
| (2*j/(n:ℝ)) • vecMulVec (magnetization y) (magnetization y)- |
| onsager j y • (1 : Matrix (Fin n) (Fin n) ℝ) := by |
| dsimp only [empiricalConditionalMiddle,empiricalConditionalGoe,plantedInteraction] |
| rw [empirical_columns_zero_outer,empirical_unit_outer y hq, |
| scalarBMoment,scalarQMoment_empirical_overlap] |
| rfl |
|
|
| lemma fieldHessian_diagonalHessian {n : ℕ} (j : ℝ) (W : Matrix (Fin n) (Fin n) ℝ) |
| (y a : Fin n → ℝ) : |
| fieldHessian j (plantedInteraction j W) y a = diagonalHessian a |
| (plantedInteraction j W+(2*j/(n:ℝ)) • vecMulVec (magnetization y) (magnetization y)- |
| onsager j y • (1 : Matrix (Fin n) (Fin n) ℝ)) := by |
| ext i k |
| simp only [fieldHessian,hessianCore,diagonalHessian,diagonalRoot,Matrix.sub_apply, |
| Matrix.one_apply,Matrix.diagonal_mul,Matrix.mul_diagonal,Matrix.add_apply, |
| Matrix.smul_apply,smul_eq_mul,vecMulVec_apply] |
| ring |
|
|
| lemma empiricalConditionalGoe_hessian {n : ℕ} [NeZero n] (j t σ : ℝ) (y a : Fin n → ℝ) |
| (hq : 0 < scalarQMoment (empiricalLaw y)) (g : MatrixCoordinates (Fin n) → ℝ) : |
| fieldHessian j (plantedInteraction j (empiricalConditionalGoe j t σ y g)) y a= |
| diagonalHessian a (empiricalConditionalMiddle j t σ y g) := by |
| rw [empiricalConditionalMiddle_literal j t σ y hq g,fieldHessian_diagonalHessian] |
|
|
| lemma quadraticForm_dotProduct {n : ℕ} (M : Matrix (Fin n) (Fin n) ℝ) (x : Fin n → ℝ) : |
| quadraticForm M x=x⬝ᵥ(M*ᵥx) := by |
| simp only [quadraticForm,dotProduct,Matrix.mulVec,Finset.mul_sum] |
| apply Finset.sum_congr rfl; intro i _ |
| apply Finset.sum_congr rfl; intro k _; ring |
| |
| lemma quadraticForm_margin {n : ℕ} {M : Matrix (Fin n) (Fin n) ℝ} {m : ℝ} |
| (hM : (M-m • (1 : Matrix (Fin n) (Fin n) ℝ)).PosSemidef) |
| (x : EuclideanSpace ℝ (Fin n)) (hx : ‖x‖=1) : m ≤ quadraticForm M x.ofLp := by |
| have hh := hM.dotProduct_mulVec_nonneg x.ofLp |
| have hxx : x.ofLp⬝ᵥx.ofLp=1 := by |
| have he := EuclideanSpace.real_norm_sq_eq x |
| rw [hx,one_pow] at he |
| simpa only [dotProduct,pow_two] using he.symm |
| simp only [star_trivial,Matrix.sub_mulVec,Matrix.smul_mulVec,Matrix.one_mulVec, |
| dotProduct_sub,dotProduct_smul,smul_eq_mul,hxx,mul_one] at hh |
| rw [quadraticForm_dotProduct] |
| linarith |
|
|
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius ENNReal |
|
|
| lemma measurable_conditionalGoe_map {n : ℕ} [NeZero n] (j t σ : ℝ) (y : Fin n → ℝ) : |
| Measurable (fun g (p : Fin n × Fin n) => empiricalConditionalGoe j t σ y g p.1 p.2) := by |
| dsimp [empiricalConditionalGoe,conditionalBlockMatrix,goeBlockResidual,goeBilinear, |
| Matrix.mulVec,dotProduct,goeMatrix] |
| fun_prop |
|
|
| lemma measurable_badField_fiber {n : ℕ} (j A ε c : ℝ) (y : Fin n → ℝ) : |
| MeasurableSet {W : (Fin n × Fin n) → ℝ | ((fun i k=>W (i,k)),y) ∈ badFieldSet n j A ε c} := |
| ((isClosed_badFieldSet n j A ε c).preimage (by fun_prop)).measurableSet |
|
|
| lemma conditional_bad_le_freshbad {n : ℕ} [NeZero n] {j t σ A ε c m : ℝ} |
| (hj : 0 ≤ j) (ht : 0 < t) (hcm : c < m) (y : Fin n → ℝ) |
| (hq : 0 < scalarQMoment (empiricalLaw y)) : |
| (conditionalGoeLaw (j/(n:ℝ)) (t+σ^2) (magnetization y) (empiricalObservation j t σ y)).real |
| {W | ((fun i k=>W (i,k)),y) ∈ badFieldSet n j A ε c} ≤ |
| (gaussianCoordinates (MatrixCoordinates (Fin n))).real |
| {g | ∃ a, (∀ i,a i ∈ Icc 0 A) ∧ |
| (∑ i,(a i-(1-tanh (y i)^2))^2) ≤ ε^2*(n:ℝ) ∧ |
| ¬(diagonalHessian a (empiricalConditionalMiddle j t σ y g)- |
| m • (1 : Matrix (Fin n) (Fin n) ℝ)).PosSemidef} := by |
| rw [show magnetization y=(fun i=>tanh (y i)) from rfl,empirical_conditional_law hj ht y hq] |
| unfold Measure.real |
| rw [Measure.map_apply (measurable_conditionalGoe_map j t σ y) (measurable_badField_fiber j A ε c y)] |
| apply ENNReal.toReal_mono (by finiteness) |
| apply measure_mono |
| intro g hg |
| obtain ⟨a,ha,hclose,x,hx,hquad⟩ := hg |
| refine ⟨a,ha,hclose,?_⟩ |
| intro hpsd |
| rw [empiricalConditionalGoe_hessian j t σ y a hq g] at hquad |
| exact (not_le_of_gt hcm) ((quadraticForm_margin hpsd x hx).trans hquad) |
|
|
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
