| import Mathlib |
| import OAI.Probability.SKRatio.Calculus.PlusSpin |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology Matrix |
| open ContinuousLinearMap |
| namespace SKRatio.Calculus |
| open Real Set |
|
|
| lemma pairSum_sub {n : ℕ} (F G : Fin n → Fin n → ℝ) : |
| pairSum (fun i j => F i j - G i j) = pairSum F - pairSum G := by |
| simp only [pairSum, Finset.sum_sub_distrib] |
|
|
| lemma pairSum_mul {n : ℕ} (a : ℝ) (F : Fin n → Fin n → ℝ) : |
| pairSum (fun i j => a * F i j) = a * pairSum F := by |
| simp only [pairSum, Finset.mul_sum] |
| |
| lemma pairSum_le {n : ℕ} {F G : Fin n → Fin n → ℝ} |
| (h : ∀ i j, i < j → F i j ≤ G i j) : pairSum F ≤ pairSum G := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j hj |
| exact h i j (Finset.mem_filter.mp hj).2 |
|
|
| def rowControl {n : ℕ} (J : Interaction n) (k : ℕ) : ℝ := |
| (Finset.univ.sup (fun i => ∑ j, ‖J i j‖₊ ^ k) : NNReal) |
|
|
| lemma rowControl_nonneg {n : ℕ} (J : Interaction n) (k : ℕ) : 0 ≤ rowControl J k := |
| NNReal.coe_nonneg _ |
|
|
| lemma row_le_rowControl {n : ℕ} (J : Interaction n) (k : ℕ) (i : Fin n) : |
| (∑ j, |J i j|^k) ≤ rowControl J k := by |
| have h := Finset.le_sup (f := fun i => ∑ j, ‖J i j‖₊^k) (Finset.mem_univ i) |
| simpa only [rowControl, NNReal.coe_sum, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs] using |
| (NNReal.coe_le_coe.mpr h) |
|
|
| abbrev R2 {n : ℕ} (J : Interaction n) : ℝ := rowControl J 2 |
| abbrev R3 {n : ℕ} (J : Interaction n) : ℝ := rowControl J 3 |
|
|
| lemma pair_row_bound {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j = J j i) |
| (p : Fin n → ℝ) (k : ℕ) : |
| pairSum (fun i j => |J i j|^k * (p i^2+p j^2)) ≤ rowControl J k * ∑ i, p i^2 := by |
| have hid : pairSum (fun i j => |J i j|^k * (p i^2+p j^2)) = |
| ∑ i, ∑ j ∈ Finset.univ.erase i, |J i j|^k * p i^2 := by |
| rw [sum_offdiag_pairs] |
| unfold pairSum |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| dsimp only |
| rw [← hsymm i j] |
| ring |
| rw [hid, Finset.mul_sum] |
| apply Finset.sum_le_sum |
| intro i _ |
| rw [← Finset.sum_mul] |
| apply mul_le_mul_of_nonneg_right _ (sq_nonneg _) |
| exact (Finset.sum_le_sum_of_subset_of_nonneg (Finset.erase_subset _ _) |
| (fun _ _ _ => pow_nonneg (abs_nonneg _) k)).trans (row_le_rowControl J k i) |
|
|
| lemma offdiag_abs_row_bound {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j = J j i) |
| (p : Fin n → ℝ) (k : ℕ) : |
| (∑ i, ∑ j ∈ Finset.univ.erase i, |J i j|^k * |p i| * |p j|) ≤ |
| rowControl J k * ∑ i, p i^2 := by |
| rw [sum_offdiag_pairs] |
| refine le_trans (pairSum_le (fun i j _ => ?_)) (pair_row_bound J hsymm p k) |
| rw [← hsymm i j] |
| have h : 2*|p i| * |p j| ≤ p i^2+p j^2 := by |
| nlinarith only [sq_nonneg (|p i|-|p j|), sq_abs (p i), sq_abs (p j)] |
| have hm := mul_le_mul_of_nonneg_left h (pow_nonneg (abs_nonneg (J i j)) k) |
| nlinarith only [hm] |
|
|
| lemma eliminate_second_differences {n : ℕ} (p w : Fin n → ℝ) |
| (a r : Fin n → Fin n → ℝ) (hw : ∀ i, 0 < w i) (hr : ∀ i j, r i j = r j i) : |
| (∑ i, ∑ j ∈ Finset.univ.erase i, p i*a i j*(p j-2*r i j)) - |
| (∑ i, ∑ j ∈ Finset.univ.erase i, w j*(r i j)^2) ≤ |
| (∑ i, ∑ j ∈ Finset.univ.erase i, p i*a i j*p j) + |
| pairSum (fun i j => (a i j*p i+a j i*p j)^2/(w i+w j)) := by |
| rw [sum_offdiag_pairs, sum_offdiag_pairs, sum_offdiag_pairs, ← pairSum_sub, ← pairSum_add] |
| apply pairSum_le |
| intro i j _ |
| rw [← hr i j] |
| have h := complete_pair_square (a i j*p i+a j i*p j) (r i j) (w i+w j) |
| (add_pos (hw i) (hw j)) |
| linarith only [h] |
|
|
| def gradientForm {n : ℕ} (J : Interaction n) (p : Fin n → ℝ) : ℝ := |
| (∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j * p i * conditionalVariance J (plusSpin n) j * p j) + |
| 2 * (∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j^2 * p i * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j * p j) + |
| pairSum (fun i j => |
| J i j^2 * (conditionalVariance J (plusSpin n) j * p i + |
| conditionalVariance J (plusSpin n) i * p j)^2 / |
| (gradientWeight J (plusSpin n) i + gradientWeight J (plusSpin n) j)) |
| |
| lemma field_difference_error {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j = J j i) |
| (hsmall : ∀ i j, |J i j| ≤ 1/10) (i j : Fin n) : |
| |halfDiff i (fun y => mean J y j) (plusSpin n) - |
| J i j * conditionalVariance J (plusSpin n) j - |
| 2*(J i j)^2 * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j| ≤ |
| 7*|J i j|^3 := by |
| let v := conditionalVariance J (plusSpin n) j |
| let m := mean J (plusSpin n) j |
| have hv : 0 ≤ v := (conditionalVariance_pos J (plusSpin n) j).le |
| have hv1 : v ≤ 1 := conditionalVariance_le_one J (plusSpin n) j |
| rw [halfDiff_mean_plus J hsymm] |
| change |v*q (J i j) m - J i j*v - 2*(J i j)^2*m*v| ≤ _ |
| rw [show v*q (J i j) m - J i j*v - 2*(J i j)^2*m*v = |
| v*(q (J i j) m - J i j - 2*m*(J i j)^2) by ring, abs_mul, abs_of_nonneg hv] |
| calc |
| _ ≤ v*(7*|J i j|^3) := mul_le_mul_of_nonneg_left |
| (q_cubic_bound (hsmall i j) (mean_abs_le_one J (plusSpin n) j)) hv |
| _ ≤ 7*|J i j|^3 := by nlinarith only [mul_le_mul_of_nonneg_right hv1 (by positivity : 0 ≤ 7*|J i j|^3)] |
|
|
| lemma linear_field_replacement {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j = J j i) |
| (hsmall : ∀ i j, |J i j| ≤ 1/10) (p : Fin n → ℝ) : |
| (∑ i, ∑ j ∈ Finset.univ.erase i, |
| p i * halfDiff i (fun y => mean J y j) (plusSpin n) * p j) ≤ |
| (∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j * p i * conditionalVariance J (plusSpin n) j * p j) + |
| 2 * (∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j^2 * p i * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j * p j) + |
| 7*R3 J*(∑ i, p i^2) := by |
| have hpoint (i j : Fin n) : p i * halfDiff i (fun y => mean J y j) (plusSpin n) * p j ≤ |
| J i j * p i * conditionalVariance J (plusSpin n) j * p j + |
| 2 * (J i j^2 * p i * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j * p j) + |
| 7*|J i j|^3 * |p i| * |p j| := by |
| let e := halfDiff i (fun y => mean J y j) (plusSpin n) - |
| J i j * conditionalVariance J (plusSpin n) j - |
| 2*(J i j)^2 * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j |
| have he : |e| ≤ 7*|J i j|^3 := field_difference_error J hsymm hsmall i j |
| have hprod : p i*e*p j ≤ 7*|J i j|^3 * |p i| * |p j| := by |
| calc |
| _ ≤ |p i*e*p j| := le_abs_self _ |
| _ = |e| * |p i| * |p j| := by simp only [abs_mul] |
| _ ≤ _ := mul_le_mul_of_nonneg_right |
| (mul_le_mul_of_nonneg_right he (abs_nonneg _)) (abs_nonneg _) |
| dsimp only [e] at hprod |
| linarith only [hprod] |
| have hsum := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin n))) => |
| Finset.sum_le_sum (fun j (_ : j ∈ Finset.univ.erase i) => hpoint i j)) |
| simp only [Finset.sum_add_distrib, ← Finset.mul_sum, mul_assoc] at hsum |
| have hrow := mul_le_mul_of_nonneg_left (offdiag_abs_row_bound J hsymm p 3) |
| (by norm_num : (0:ℝ) ≤ 7) |
| simp only [← Finset.mul_sum, mul_assoc] at hrow ⊢ |
| linarith only [hsum, hrow] |
|
|
| lemma completed_squares_replacement {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j = J j i) |
| (hsmall : ∀ i j, |J i j| ≤ 1/10) (p : Fin n → ℝ) : |
| pairSum (fun i j => |
| (halfDiff i (fun y => mean J y j) (plusSpin n)*p i + |
| halfDiff j (fun y => mean J y i) (plusSpin n)*p j)^2 / |
| (gradientWeight J (plusSpin n) i+gradientWeight J (plusSpin n) j)) ≤ |
| pairSum (fun i j => |
| J i j^2 * (conditionalVariance J (plusSpin n) j * p i + |
| conditionalVariance J (plusSpin n) i * p j)^2 / |
| (gradientWeight J (plusSpin n) i + gradientWeight J (plusSpin n) j)) + |
| (224/27)*R3 J*(∑ i, p i^2) := by |
| calc |
| _ ≤ pairSum (fun i j => |
| J i j^2 * (conditionalVariance J (plusSpin n) j * p i + |
| conditionalVariance J (plusSpin n) i * p j)^2 / |
| (gradientWeight J (plusSpin n) i + gradientWeight J (plusSpin n) j) + |
| (224/27)*|J i j|^3*(p i^2+p j^2)) := by |
| apply pairSum_le |
| intro i j _ |
| rw [halfDiff_mean_plus J hsymm, halfDiff_mean_plus J hsymm, ← hsymm i j] |
| simp only [mean_weight_plus, conditionalVariance] |
| exact pair_square_replacement (hsmall i j) |
| (Real.neg_one_lt_tanh _).le (Real.tanh_lt_one _) |
| (Real.neg_one_lt_tanh _).le (Real.tanh_lt_one _) |
| _ = _ + (224/27) * pairSum (fun i j => |J i j|^3*(p i^2+p j^2)) := by |
| rw [pairSum_add] |
| congr 1 |
| simp only [mul_assoc, pairSum_mul] |
| _ ≤ _ := by |
| have h := mul_le_mul_of_nonneg_left (pair_row_bound J hsymm p 3) |
| (by norm_num : (0:ℝ) ≤ 224/27) |
| simp only [mul_assoc] |
| exact add_le_add le_rfl h |
|
|
| theorem pointwise_gradient_bound_plus {n : ℕ} (J : Interaction n) |
| (hsymm : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) |
| (hsmall : ∀ i j, |J i j| ≤ 1/10) (f : Observables n) (t : ℝ) : |
| (1/2:ℝ) * (deriv (fun s => unweightedGradient (semigroup J s f) (plusSpin n)) t - |
| generator J (unweightedGradient (semigroup J t f)) (plusSpin n)) ≤ |
| -unweightedGradient (semigroup J t f) (plusSpin n) + |
| gradientForm J (fun i => halfDiff i (semigroup J t f) (plusSpin n)) + |
| 16*R3 J*unweightedGradient (semigroup J t f) (plusSpin n) := by |
| rw [pointwise_gradient_identity J hdiag] |
| let p := fun i => halfDiff i (semigroup J t f) (plusSpin n) |
| let a := fun i j => halfDiff i (fun y => mean J y j) (plusSpin n) |
| let r := fun i j => halfDiff i (halfDiff j (semigroup J t f)) (plusSpin n) |
| have hc := eliminate_second_differences p (gradientWeight J (plusSpin n)) a r |
| (gradientWeight_pos J (plusSpin n)) (fun i j => halfDiff_comm i j _ _) |
| have hl := linear_field_replacement J hsymm hsmall p |
| have hs := completed_squares_replacement J hsymm hsmall p |
| have hU : 0 ≤ unweightedGradient (semigroup J t f) (plusSpin n) := |
| Finset.sum_nonneg (fun i _ => sq_nonneg _) |
| have hR := rowControl_nonneg J 3 |
| have hbudget := mul_nonneg hR hU |
| dsimp only [p, a, r] at hc hl hs |
| unfold gradientForm unweightedGradient at * |
| linarith only [hc, hl, hs, hbudget] |
|
|
| def signProduct {n : ℕ} (x y : Spin n) : Spin n := |
| fun i => if x i then y i else !(y i) |
|
|
| @[simp] lemma signProduct_plus {n : ℕ} (x : Spin n) : signProduct x (plusSpin n) = x := by |
| funext i |
| cases hi : x i < |
|
|
| @[simp] lemma plus_signProduct {n : ℕ} (x : Spin n) : signProduct (plusSpin n) x = x := by |
| funext i |
| simp [signProduct, plusSpin] |
|
|
| lemma signProduct_involutive {n : ℕ} (x : Spin n) : Function.Involutive (signProduct x) := by |
| intro y |
| funext i |
| cases h : x i < |
|
|
| lemma spin_signProduct {n : ℕ} (x y : Spin n) (i : Fin n) : |
| spin (signProduct x y) i = spin x i * spin y i := by |
| cases hx : x i < |
|
|
| @[simp] lemma abs_spin {n : ℕ} (x : Spin n) (i : Fin n) : |spin x i| = 1 := by |
| cases hx : x i < |
|
|
| lemma signProduct_replace {n : ℕ} (x y : Spin n) (i : Fin n) (b : Bool) : |
| signProduct x (replace y i b) = replace (signProduct x y) i (if x i then b else !b) := by |
| funext j |
| by_cases hj : j = i |
| · subst j; simp [replace, signProduct] |
| · simp [replace, signProduct, hj] |
|
|
| def signConjugate {n : ℕ} (x : Spin n) (J : Interaction n) : Interaction n := |
| fun i j => spin x i * J i j * spin x j |
| |
| def signPullback {n : ℕ} (x : Spin n) (f : Observables n) : Observables n := |
| fun y => f (signProduct x y) |
|
|
| lemma halfDiff_signPullback {n : ℕ} (x y : Spin n) (f : Observables n) (i : Fin n) : |
| halfDiff i (signPullback x f) y = spin x i * halfDiff i f (signProduct x y) := by |
| simp only [halfDiff, signPullback, signProduct_replace] |
| cases hi : x i < |
| Bool.false_eq_true, neg_mul, one_mul] |
| · ring |
|
|
| lemma field_signConjugate {n : ℕ} (J : Interaction n) (x y : Spin n) (i : Fin n) : |
| field (signConjugate x J) y i = spin x i * field J (signProduct x y) i := by |
| simp only [field, signConjugate, spin_signProduct, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
|
|
| lemma mean_signConjugate {n : ℕ} (J : Interaction n) (x y : Spin n) (i : Fin n) : |
| mean (signConjugate x J) y i = spin x i * mean J (signProduct x y) i := by |
| simp only [mean, field_signConjugate] |
| cases hi : x i < |
| neg_mul, one_mul, Real.tanh_neg] |
|
|
| lemma generator_signConjugate {n : ℕ} (J : Interaction n) (x : Spin n) (f : Observables n) : |
| generator (signConjugate x J) (signPullback x f) = signPullback x (generator J f) := by |
| funext y |
| simp only [generator, halfDiff_signPullback, mean_signConjugate, signPullback] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [spin_signProduct] |
| linear_combination mean J (signProduct x y) i * halfDiff i f (signProduct x y) * spin_sq x i |
|
|
| lemma unweightedGradient_signPullback {n : ℕ} (x : Spin n) (f : Observables n) : |
| unweightedGradient (signPullback x f) = signPullback x (unweightedGradient f) := by |
| funext y |
| simp only [unweightedGradient, halfDiff_signPullback, mul_pow, spin_sq, one_mul, signPullback] |
|
|
| lemma gradient_drift_signPullback {n : ℕ} (J : Interaction n) (x y : Spin n) (f : Observables n) : |
| (∑ i, 2 * halfDiff i (signPullback x f) y * |
| halfDiff i (generator (signConjugate x J) (signPullback x f)) y) = |
| ∑ i, 2 * halfDiff i f (signProduct x y) * halfDiff i (generator J f) (signProduct x y) := by |
| rw [generator_signConjugate] |
| simp only [halfDiff_signPullback] |
| apply Finset.sum_congr rfl |
| intro i _ |
| linear_combination 2 * halfDiff i f (signProduct x y) * halfDiff i (generator J f) (signProduct x y) * spin_sq x i |
|
|
| lemma signConjugate_symm {n : ℕ} (J : Interaction n) (hsymm : ∀ i j, J i j = J j i) |
| (x : Spin n) : ∀ i j, signConjugate x J i j = signConjugate x J j i := by |
| intro i j |
| simp only [signConjugate, ← hsymm i j] |
| ring |
|
|
| lemma signConjugate_diag {n : ℕ} (J : Interaction n) (hdiag : ∀ i, J i i = 0) |
| (x : Spin n) : ∀ i, signConjugate x J i i = 0 := by |
| intro i |
|
|
| @[simp] lemma abs_signConjugate {n : ℕ} (J : Interaction n) (x : Spin n) (i j : Fin n) : |
| |signConjugate x J i j| = |J i j| := by |
| simp [signConjugate, abs_mul] |
|
|
| lemma rowControl_signConjugate {n : ℕ} (J : Interaction n) (x : Spin n) (k : ℕ) : |
| rowControl (signConjugate x J) k = rowControl J k := by |
| have hn (i j : Fin n) : ‖signConjugate x J i j‖₊ = ‖J i j‖₊ := by |
| apply NNReal.coe_injective |
| simp only [coe_nnnorm, Real.norm_eq_abs, abs_signConjugate] |
| simp only [rowControl, hn] |
|
|
| lemma instantaneous_gradient_bound_plus {n : ℕ} (J : Interaction n) |
| (hsymm : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) |
| (hsmall : ∀ i j, |J i j| ≤ 1/10) (f : Observables n) : |
| (1/2:ℝ) * ((∑ i, 2 * halfDiff i f (plusSpin n) * |
| halfDiff i (generator J f) (plusSpin n)) - generator J (unweightedGradient f) (plusSpin n)) ≤ |
| -unweightedGradient f (plusSpin n) + gradientForm J (fun i => halfDiff i f (plusSpin n)) + |
| 16*R3 J*unweightedGradient f (plusSpin n) := by |
| have h := pointwise_gradient_bound_plus J hsymm hdiag hsmall f 0 |
| rw [(unweightedGradient_hasDerivAt J f (plusSpin n) 0).deriv] at h |
| simpa only [semigroup_zero, one_apply_eq_self] using h |
|
|
| theorem pointwise_gradient_local_bound {n : ℕ} (J : Interaction n) |
| (hsymm : ∀ i j, J i j = J j i) (hdiag : ∀ i, J i i = 0) |
| (hsmall : ∀ i j, |J i j| ≤ 1/10) (f : Observables n) (t : ℝ) (x : Spin n) : |
| (1/2:ℝ) * (deriv (fun s => unweightedGradient (semigroup J s f) x) t - |
| generator J (unweightedGradient (semigroup J t f)) x) ≤ |
| -unweightedGradient (semigroup J t f) x + |
| gradientForm (signConjugate x J) (fun i => spin x i * halfDiff i (semigroup J t f) x) + |
| 16*R3 J*unweightedGradient (semigroup J t f) x := by |
| have h := instantaneous_gradient_bound_plus (signConjugate x J) (signConjugate_symm J hsymm x) |
| (signConjugate_diag J hdiag x) (by simpa only [abs_signConjugate] using hsmall) |
| (signPullback x (semigroup J t f)) |
| rw [gradient_drift_signPullback, unweightedGradient_signPullback, |
| generator_signConjugate] at h |
| simp only [halfDiff_signPullback, signPullback, signProduct_plus, R3, |
| rowControl_signConjugate] at h |
| rw [(unweightedGradient_hasDerivAt J f x t).deriv] |
| exact h |
|
|
| def euclideanOpNorm {n : ℕ} (J : Interaction n) : ℝ := ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) J‖ |
| |
| lemma euclideanOpNorm_nonneg {n : ℕ} (J : Interaction n) : 0 ≤ euclideanOpNorm J := |
| norm_nonneg _ |
|
|
| lemma euclideanNorm_variance_le {n : ℕ} (J : Interaction n) (x : Spin n) |
| (p : Fin n → ℝ) : |
| ‖WithLp.toLp 2 (fun i => conditionalVariance J x i * p i)‖ ≤ |
| ‖(WithLp.toLp 2 p : EuclideanSpace ℝ (Fin n))‖ := by |
| apply (sq_le_sq₀ (norm_nonneg _) (norm_nonneg _)).mp |
| simp only [EuclideanSpace.real_norm_sq_eq] |
| apply Finset.sum_le_sum |
| intro i _ |
| have hv := (conditionalVariance_pos J x i).le |
| have hv1 := conditionalVariance_le_one J x i |
| have hv2 : (conditionalVariance J x i)^2 ≤ 1 := by nlinarith only [hv, hv1] |
| nlinarith only [mul_le_mul_of_nonneg_right hv2 (sq_nonneg (p i))] |
|
|
| lemma gradientForm_linear_bound {n : ℕ} (J : Interaction n) (hdiag : ∀ i, J i i = 0) |
| (p : Fin n → ℝ) : |
| (∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j * p i * conditionalVariance J (plusSpin n) j * p j) ≤ |
| euclideanOpNorm J * ∑ i, p i^2 := by |
| let P : EuclideanSpace ℝ (Fin n) := WithLp.toLp 2 p |
| let Q : EuclideanSpace ℝ (Fin n) := |
| WithLp.toLp 2 (fun j => conditionalVariance J (plusSpin n) j * p j) |
| have hid : (∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j * p i * conditionalVariance J (plusSpin n) j * p j) = |
| inner ℝ P (Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) J Q) := by |
| rw [Matrix.inner_toEuclideanCLM (J : Matrix (Fin n) (Fin n) ℝ) P Q] |
| simp only [dotProduct, Matrix.mulVec, P, Q, Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [← Finset.sum_erase_add _ _ (Finset.mem_univ i)] |
| simp only [hdiag, zero_mul, mul_zero, add_zero] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| rw [hid] |
| calc |
| _ ≤ ‖P‖ * ‖Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) J Q‖ := real_inner_le_norm _ _ |
| _ ≤ ‖P‖ * (euclideanOpNorm J * ‖Q‖) := |
| mul_le_mul_of_nonneg_left ((Matrix.toEuclideanCLM (𝕜 := ℝ) (n := Fin n) J).le_opNorm _) (norm_nonneg _) |
| _ ≤ ‖P‖ * (euclideanOpNorm J * ‖P‖) := |
| mul_le_mul_of_nonneg_left |
| (mul_le_mul_of_nonneg_left (euclideanNorm_variance_le J (plusSpin n) p) |
| (euclideanOpNorm_nonneg J)) (norm_nonneg _) |
| _ = euclideanOpNorm J * ∑ i, p i^2 := by |
| rw [show ‖P‖ * (euclideanOpNorm J * ‖P‖) = euclideanOpNorm J * ‖P‖^2 by ring, |
| EuclideanSpace.real_norm_sq_eq] |
|
|
| lemma abs_mean_variance_bound {m : ℝ} (hm : |m| ≤ 1) : |
| |m*(1-m^2)| ≤ 2/(3*Real.sqrt 3) := by |
| have hs : 0 < Real.sqrt 3 := Real.sqrt_pos.2 (by norm_num) |
| have hs2 : (Real.sqrt 3)^2 = 3 := Real.sq_sqrt (by norm_num) |
| have hmv : 0 ≤ 1-m^2 := by nlinarith only [sq_abs m, abs_nonneg m, hm] |
| rw [abs_mul, abs_of_nonneg hmv, le_div_iff₀ (by positivity : 0 < 3*Real.sqrt 3)] |
| have h : 0 ≤ (Real.sqrt 3 * |m| - 1)^2 * (Real.sqrt 3 * |m| + 2) := |
| mul_nonneg (sq_nonneg _) (by positivity) |
| have hc : (Real.sqrt 3 * |m|)^3 = 3*m^2*(Real.sqrt 3*|m|) := by |
| rw [pow_succ, mul_pow, hs2, sq_abs] |
| nlinarith only [h, hc] |
|
|
| lemma mean_variance_bound {n : ℕ} (J : Interaction n) (x : Spin n) (j : Fin n) : |
| |mean J x j * conditionalVariance J x j| ≤ 2/(3*Real.sqrt 3) := |
| abs_mean_variance_bound (mean_abs_le_one J x j) |
| |
| lemma gradientForm_quadratic_bound {n : ℕ} (J : Interaction n) |
| (hsymm : ∀ i j, J i j = J j i) (p : Fin n → ℝ) : |
| 2*(∑ i, ∑ j ∈ Finset.univ.erase i, |
| J i j^2 * p i * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j * p j) ≤ |
| (4/(3*Real.sqrt 3))* R2 J * ∑ i, p i^2 := by |
| have hpoint (i j : Fin n) : |
| J i j^2 * p i * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j * p j ≤ |
| (2/(3*Real.sqrt 3)) * (|J i j|^2*|p i| *|p j|) := by |
| calc |
| _ ≤ |J i j^2 * p i * mean J (plusSpin n) j * conditionalVariance J (plusSpin n) j * p j| := |
| le_abs_self _ |
| _ = |mean J (plusSpin n) j*conditionalVariance J (plusSpin n) j| * |
| (|J i j|^2*|p i| *|p j|) := by simp only [abs_mul, abs_pow] |
| _ ≤ _ := mul_le_mul_of_nonneg_right (mean_variance_bound J (plusSpin n) j) (by positivity) |
| have hsum := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin n))) => |
| Finset.sum_le_sum (fun j (_ : j ∈ Finset.univ.erase i) => hpoint i j)) |
| simp only [← Finset.mul_sum] at hsum |
| have hrow := mul_le_mul_of_nonneg_left (offdiag_abs_row_bound J hsymm p 2) |
| (by positivity : 0 ≤ 2/(3*Real.sqrt 3)) |
| calc |
| _ ≤ 2*((2/(3*Real.sqrt 3))* (R2 J * ∑ i, p i^2)) := |
| mul_le_mul_of_nonneg_left (hsum.trans hrow) (by norm_num) |
| _ = _ := by ring |
|
|
| end SKRatio.Calculus |
| end |
| end |
|
|
| end OAI |
|
|
