| import Mathlib |
| import OAI.Probability.SKGap.Matrix.GOESparse |
| import OAI.Probability.SKGap.Matrix.TanhMatrixBounds |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real |
| open scoped BigOperators RealInnerProductSpace Matrix.Norms.L2Operator |
| variable {n : ℕ} |
|
|
| lemma coupling_symmetric (g : Disorder n) (i k : Fin n) : coupling g i k=coupling g k i := by |
| by_cases h : i=k |
| · subst k;rfl |
| · rcases lt_or_gt_of_ne h with h | h <;> simp [coupling,h,not_lt_of_gt h] |
|
|
| lemma coupling_norm_from_goe (r d M : ℝ) (g : MatrixCoordinates (Fin n)→ℝ) |
| (hd : 0 ≤ d) (hg : ‖matrixOperator (goeMatrix r g)‖ ≤ M) |
| (hdiag : ∀ i,|goeMatrix r g i i| ≤ d) : |
| ‖Matrix.of (coupling (goeDisorder r g))‖ ≤ M+d := by |
| rw [coupling_goeDisorder] |
| apply (norm_sub_le _ _).trans |
| apply add_le_add _ (diagonal_norm_le hd _ hdiag) |
| rwa [matrixOperator_eq_toEuclideanCLM,Matrix.l2_opNorm_toEuclideanCLM] at hg |
|
|
| lemma coupling_principal_norm_from_goe (r d ε : ℝ) (g : MatrixCoordinates (Fin n)→ℝ) |
| (S : Finset (Fin n)) (hd : 0 ≤ d) |
| (hg : principalOperatorNorm (goeMatrix r g) S ≤ ε) |
| (hdiag : ∀ i,|goeMatrix r g i i| ≤ d) : |
| ‖(Matrix.of (coupling (goeDisorder r g))).submatrix (fun i : S=>i.1) (fun i : S=>i.1)‖ ≤ ε+d := by |
| rw [coupling_goeDisorder] |
| have he : (goeMatrix r g-Matrix.diagonal (fun i=>goeMatrix r g i i)).submatrix |
| (fun i : S=>i.1) (fun i : S=>i.1)= |
| (goeMatrix r g).submatrix (fun i : S=>i.1) (fun i : S=>i.1)- |
| Matrix.diagonal (fun i : S=>goeMatrix r g i.1 i.1) := by |
| ext i k |
| simp only [submatrix_apply,Matrix.sub_apply] |
| congr 1 |
| simp only [Matrix.diagonal_apply,Subtype.val_inj] |
| rw [he] |
| apply (norm_sub_le _ _).trans |
| have hb : ‖Matrix.diagonal (fun i : S=>goeMatrix r g i.1 i.1)‖ ≤ d := |
| diagonal_norm_le (ι:=S) hd _ (fun i=>hdiag i.1) |
| apply add_le_add _ hb |
| simpa only [principalOperatorNorm,matrixOperator_eq_toEuclideanCLM,Matrix.l2_opNorm_toEuclideanCLM] using hg |
|
|
| lemma row_square_restrict (A : Matrix (Fin n) (Fin n) ℝ) (S : Finset (Fin n)) |
| {R : ℝ} (hr : ∀ i,∑ k,A i k^2 ≤ R) (i : S) : |
| (∑ k : S,A i.1 k.1^2) ≤ R := by |
| rw [Finset.sum_coe_sort S (fun k=>A i.1 k^2)] |
| exact (Finset.sum_le_sum_of_subset_of_nonneg (Finset.subset_univ S) |
| (fun k _ _=>sq_nonneg (A i.1 k))).trans (hr i.1) |
|
|
| lemma tanh_principal_norm (g : Disorder n) (S : Finset (Fin n)) {d R ε : ℝ} |
| (hd : 0 ≤ d) (hR : 0 ≤ R) |
| (hma : ∀ i k,|coupling g i k| ≤ d) (hrow : ∀ i,∑ k,coupling g i k^2 ≤ R) |
| (hJ : ‖(Matrix.of (coupling g)).submatrix (fun i : S=>i.1) (fun i : S=>i.1)‖ ≤ ε) : |
| ‖Matrix.of (fun i k : S=>tanh (coupling g i.1 k.1))‖ ≤ ε+d*R/3 := by |
| let A : Matrix S S ℝ := (Matrix.of (coupling g)).submatrix (fun i : S=>i.1) (fun i : S=>i.1) |
| have hh := tanh_matrix_remainder_norm A (fun i k=>coupling_symmetric g i.1 k.1) |
| hR hd (fun i k=>hma i.1 k.1) (row_square_restrict _ S hrow) |
| have he : Matrix.of (fun i k : S=>tanh (coupling g i.1 k.1))= |
| A+(Matrix.of (fun i k=>tanh (A i k))-A) := by ext i k;simp [A] |
| rw [he] |
| exact (norm_add_le _ _).trans (add_le_add hJ hh) |
|
|
| lemma tanh_square_principal_norm (g : Disorder n) (S : Finset (Fin n)) {ε : ℝ} |
| (hε : 0 ≤ ε) (hrows : ∀ i,∑ k∈S,coupling g i k^2 ≤ ε) : |
| ‖Matrix.of (fun i k : S=>tanh (coupling g i.1 k.1)^2)‖ ≤ ε := by |
| apply tanh_matrix_Q_norm (Matrix.of (fun i k : S=>coupling g i.1 k.1)) |
| (fun i k=>coupling_symmetric g i.1 k.1) hε |
| intro i |
| change (∑ k : S,coupling g i.1 k.1^2) ≤ ε |
| rw [Finset.sum_coe_sort S (fun k=>coupling g i.1 k^2)] |
| exact hrows i.1 |
| end SKGap |
|
|
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Real |
| open scoped BigOperators |
| variable {n : ℕ} |
|
|
| lemma conditionalExpectation_flip (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f : Spin n→ℝ) (x : Spin n) : |
| conditionalExpectation g h i f (flip i x)=conditionalExpectation g h i f x := by |
| simp only [conditionalExpectation,flip_flip] |
| rw [add_comm (weight g h (flip i x)*f (flip i x)),add_comm (weight g h (flip i x))] |
|
|
| lemma conditionalExpectation_idem (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f : Spin n→ℝ) (x : Spin n) : |
| conditionalExpectation g h i (conditionalExpectation g h i f) x=conditionalExpectation g h i f x := by |
| unfold conditionalExpectation at * |
| rw [flip_flip] |
| have hd : weight g h x+weight g h (flip i x) ≠ 0 := (add_pos (weight_pos _ _ _) (weight_pos _ _ _)).ne' |
| have hd' : weight g h (flip i x)+weight g h x ≠ 0 := by simpa only [add_comm] using hd |
| field_simp [hd,hd'] |
| ring |
| |
| lemma expectation_flip_average (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (F : Spin n→ℝ) : |
| expectation g h F=(1/2)*∑ x,(mass g h x*F x+mass g h (flip i x)*F (flip i x)) := by |
| rw [Finset.sum_add_distrib,sum_flip i (fun x=>mass g h x*F x)] |
| unfold expectation |
| ring |
| |
| lemma conditionalExpectation_adjoint (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f k : Spin n→ℝ) : |
| expectation g h (fun x=>conditionalExpectation g h i f x*k x)= |
| expectation g h (fun x=>f x*conditionalExpectation g h i k x) := by |
| rw [expectation_flip_average g h i,expectation_flip_average g h i] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro x _ |
| rw [conditionalExpectation_flip,conditionalExpectation_flip] |
| unfold mass conditionalExpectation |
| have hd : weight g h x+weight g h (flip i x) ≠ 0 := (add_pos (weight_pos _ _ _) (weight_pos _ _ _)).ne' |
| have hZ := (partition_pos g h).ne' |
| field_simp [hd,hZ] |
| |
| lemma conditionalExpectation_mul_invariant (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f k : Spin n→ℝ) (hk : ∀ x,k (flip i x)=k x) (x : Spin n) : |
| conditionalExpectation g h i (fun x=>f x*k x) x=conditionalExpectation g h i f x*k x := by |
| unfold conditionalExpectation |
| dsimp only |
| rw [hk] |
| ring |
| |
| lemma conditionalExpectation_const (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (c : ℝ) (x : Spin n) : conditionalExpectation g h i (fun _=>c) x=c := by |
| unfold conditionalExpectation |
| have hd : weight g h x+weight g h (flip i x) ≠ 0 := (add_pos (weight_pos _ _ _) (weight_pos _ _ _)).ne' |
| field_simp [hd] |
|
|
| lemma expectation_conditionalExpectation (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f : Spin n→ℝ) : expectation g h (conditionalExpectation g h i f)=expectation g h f := by |
| have he := conditionalExpectation_adjoint g h i f (fun _=>1) |
| simpa only [conditionalExpectation_const,mul_one] using he |
|
|
| lemma expectation_projection_square (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f : Spin n→ℝ) : |
| expectation g h (fun x=>(f x-conditionalExpectation g h i f x)^2)= |
| expectation g h (fun x=>f x*(f x-conditionalExpectation g h i f x)) := by |
| have he := conditionalExpectation_adjoint g h i f (conditionalExpectation g h i f) |
| simp only [conditionalExpectation_idem] at he |
| have hp (x : Spin n) : (f x-conditionalExpectation g h i f x)^2= |
| f x*(f x-conditionalExpectation g h i f x)+ |
| (conditionalExpectation g h i f x*conditionalExpectation g h i f x- |
| f x*conditionalExpectation g h i f x) := by ring |
| simp_rw [hp] |
| rw [expectation_add,expectation_sub,he] |
| ring |
|
|
| def heatBathGenerator (g : Disorder n) (h : Fin n→ℝ) (f : Spin n→ℝ) (x : Spin n) : ℝ := |
| ∑ i,(f x-conditionalExpectation g h i f x) |
|
|
| lemma expectation_sum (g : Disorder n) (h : Fin n→ℝ) {ι : Type*} [Fintype ι] |
| (F : ι→Spin n→ℝ) : expectation g h (fun x=>∑ i,F i x)=∑ i,expectation g h (F i) := by |
| unfold expectation |
| simp_rw [Finset.mul_sum] |
| rw [Finset.sum_comm] |
|
|
| lemma heatBathGenerator_form (g : Disorder n) (h : Fin n→ℝ) (f : Spin n→ℝ) : |
| expectation g h (fun x=>f x*heatBathGenerator g h f x)=dirichlet g h f := by |
| unfold heatBathGenerator dirichlet |
| simp_rw [Finset.mul_sum] |
| rw [expectation_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact (expectation_projection_square g h i f).symm |
|
|
| lemma heatBathGenerator_adjoint (g : Disorder n) (h : Fin n→ℝ) (f k : Spin n→ℝ) : |
| expectation g h (fun x=>heatBathGenerator g h f x*k x)= |
| expectation g h (fun x=>f x*heatBathGenerator g h k x) := by |
| unfold heatBathGenerator |
| simp_rw [Finset.sum_mul,Finset.mul_sum] |
| rw [expectation_sum,expectation_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp_rw [sub_mul,mul_sub] |
| rw [expectation_sub,expectation_sub,conditionalExpectation_adjoint] |
|
|
| lemma conditionalDifference_zero_iff (g : Disorder n) (h : Fin n→ℝ) (i : Fin n) |
| (f : Spin n→ℝ) (x : Spin n) : f x-conditionalExpectation g h i f x=0 ↔ f x=f (flip i x) := by |
| have hd : weight g h x+weight g h (flip i x) ≠ 0 := (add_pos (weight_pos _ _ _) (weight_pos _ _ _)).ne' |
| have hw := (weight_pos g h (flip i x)).ne' |
| unfold conditionalExpectation |
| rw [sub_eq_zero,eq_div_iff hd] |
| constructor |
| · intro hh |
| have he : weight g h (flip i x)*(f x-f (flip i x))=0 := by nlinarith only [hh] |
| exact sub_eq_zero.mp ((mul_eq_zero.mp he).resolve_left hw) |
| · intro hh |
| rw [hh] |
| ring |
|
|
| lemma flip_invariant_constant (f : Spin n→ℝ) (hf : ∀ i x,f (flip i x)=f x) : |
| ∀ x y,f x=f y := by |
| classical |
| intro x y |
| have hs : ∀ S : Finset (Fin n),f (fun i=>if i∈S then y i else x i)=f x := by |
| intro S |
| induction S using Finset.induction_on with |
| | empty => simp |
| | @insert i S hi ih => |
| let z : Spin n := fun k=>if k∈S then y k else x k |
| have he : (fun k=>if k∈insert i S then y k else x k)=Function.update z i (y i) := by |
| funext k |
| by_cases hk : k=i |
| · subst k;simp |
| · simp [z,hk] |
| rw [he] |
| have hz : f (Function.update z i (y i))=f z := by |
| by_cases hiy : y i=z i |
| · rw [hiy,Function.update_eq_self] |
| · have hnot : y i= !(z i) := by |
| cases hy : y i <;> cases hz : z i <;> simp_all |
| rw [hnot] |
| exact hf i z |
| exact hz.trans ih |
| simpa using (hs Finset.univ).symm |
|
|
| lemma dirichlet_eq_zero_iff (g : Disorder n) (h : Fin n→ℝ) (f : Spin n→ℝ) : |
| dirichlet g h f=0 ↔ ∀ x y,f x=f y := by |
| constructor |
| · intro hh |
| have hi : ∀ i,expectation g h (fun x=>(f x-conditionalExpectation g h i f x)^2)=0 := by |
| intro i |
| exact (Finset.sum_eq_zero_iff_of_nonneg (fun j _=>expectation_nonneg g h (fun _=>sq_nonneg _))).mp hh i (Finset.mem_univ _) |
| have hf : ∀ i x,f (flip i x)=f x := by |
| intro i x |
| have hx : mass g h x*(f x-conditionalExpectation g h i f x)^2=0 := |
| (Finset.sum_eq_zero_iff_of_nonneg (fun y _=>mul_nonneg (mass_nonneg g h y) (sq_nonneg _))).mp |
| (hi i) x (Finset.mem_univ _) |
| have hd := (mul_eq_zero.mp hx).resolve_left (mass_pos g h x).ne' |
| exact ((conditionalDifference_zero_iff g h i f x).mp (sq_eq_zero_iff.mp hd)).symm |
| exact flip_invariant_constant f hf |
| · intro hf |
| unfold dirichlet |
| apply Finset.sum_eq_zero |
| intro i _ |
| have he : ∀ x,f x-conditionalExpectation g h i f x=0 := fun x=>(conditionalDifference_zero_iff g h i f x).mpr (hf x (flip i x)) |
| simp only [he,zero_pow (by omega : (2:ℕ) ≠ 0)] |
| simp [expectation] |
| end SKGap |
| |
| end |
| end |
| |
| end OAI |
| |
