| import Mathlib |
| import OAI.Probability.SKGap.Localization.BilinearCoefficient |
| import OAI.Probability.SKGap.Gaussian.SquaredGaussianFourth |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory Metric Set |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SKGapCutoff |
|
|
| lemma tanh_cubic_remainder (d : ℝ) : |Real.tanh d - d| ≤ |d|^3 := by |
| let f (z : ℝ) := Real.tanh z - z |
| have hd (z : ℝ) : HasDerivAt f (-(Real.tanh z)^2) z := by |
| convert! (tanh_hasDerivAt z).sub (hasDerivAt_id z) using 1 |
| unfold scalarVariance |
| ring |
| have hb (z : ℝ) (hz : z ∈ segment ℝ 0 d) : ‖-(Real.tanh z)^2‖ ≤ |d|^2 := by |
| have h1 : |Real.tanh z| ≤ |z| := by simpa using tanh_lipschitz 0 z |
| have h2 : |z| ≤ |d| := by |
| simpa only [sub_zero, Real.norm_eq_abs] using norm_sub_le_of_mem_segment hz |
| rw [norm_neg, norm_pow, Real.norm_eq_abs] |
| exact pow_le_pow_left₀ (abs_nonneg _) (h1.trans h2) 2 |
| have h := Convex.norm_image_sub_le_of_norm_hasDerivWithin_le |
| (fun z (_ : z ∈ segment ℝ 0 d) => (hd z).hasDerivWithinAt) hb |
| (convex_segment 0 d) (left_mem_segment ℝ 0 d) (right_mem_segment ℝ 0 d) |
| simpa only [f, Real.tanh_zero, sub_zero, Real.norm_eq_abs, pow_succ] using h |
|
|
| lemma normalized_tanh_difference (h d : ℝ) : |
| (Real.tanh h - Real.tanh (h-d)) / scalarVariance h = |
| Real.tanh d / (1 - Real.tanh h * Real.tanh d) := by |
| have hc : Real.cosh (h-d) ≠ 0 := (Real.cosh_pos _).ne' |
| have hh : Real.cosh h ≠ 0 := (Real.cosh_pos _).ne' |
| have hd : Real.cosh d ≠ 0 := (Real.cosh_pos _).ne' |
| have hm : Real.cosh h * Real.cosh d - Real.sinh h * Real.sinh d ≠ 0 := by |
| simpa only [Real.cosh_sub] using hc |
| rw [tanh_difference_eq, show h-(h-d)=d by ring, scalarVariance_eq] |
| simp only [Real.tanh_eq_sinh_div_cosh] |
| rw [Real.cosh_sub] |
| field_simp |
| |
| lemma rational_cubic_remainder (m u d : ℝ) (hm : |m| ≤ 1) |
| (hd : |d| ≤ 1/2) (hu : |u| ≤ |d|) (hud : |u-d| ≤ |d|^3) : |
| |u/(1-m*u) - d - m*d^2| ≤ 4*|d|^3 := by |
| have hmud : |m*u| ≤ |d| := by |
| rw [abs_mul] |
| exact (mul_le_mul hm hu (abs_nonneg _) (by norm_num)).trans_eq (one_mul _) |
| have hden : 1/2 ≤ 1-m*u := by linarith [le_abs_self (m*u)] |
| have hdenpos : 0 < 1-m*u := by linarith |
| have hd0 := abs_nonneg d |
| have he : u/(1-m*u) - d - m*d^2 = |
| (u-d) + m*(u^2-d^2) + m^2*u^3/(1-m*u) := by |
| apply mul_right_cancel₀ hdenpos.ne' |
| simp only [sub_mul, add_mul, div_mul_cancel₀ _ hdenpos.ne'] |
| ring |
| have hs : |u^2-d^2| ≤ |d|^3 := by |
| have hh : |u^2-d^2| = |u-d| * |u+d| := by rw [← abs_mul]; congr 1; ring |
| rw [hh] |
| have ha : |u+d| ≤ 1 := (abs_add_le _ _).trans (by linarith) |
| nlinarith [mul_le_mul hud ha (abs_nonneg _) (pow_nonneg hd0 3)] |
| have ht : |m*(u^2-d^2)| ≤ |d|^3 := by |
| rw [abs_mul] |
| simpa only [one_mul] using mul_le_mul hm hs (abs_nonneg _) (by norm_num : (0:ℝ) ≤ 1) |
| have hq : |m^2*u^3/(1-m*u)| ≤ 2*|d|^3 := by |
| rw [abs_div, abs_mul, abs_pow, abs_pow, abs_of_pos hdenpos, div_le_iff₀ hdenpos] |
| have hm2 : |m|^2 ≤ 1 := by nlinarith [abs_nonneg m] |
| have hu3 : |u|^3 ≤ |d|^3 := pow_le_pow_left₀ (abs_nonneg _) hu 3 |
| have hp := mul_le_mul hm2 hu3 (pow_nonneg (abs_nonneg _) _) (by norm_num : (0:ℝ) ≤ 1) |
| nlinarith [mul_le_mul_of_nonneg_left hden (pow_nonneg hd0 3)] |
| rw [he] |
| exact (abs_add_le _ _).trans ((add_le_add ((abs_add_le _ _).trans (add_le_add hud ht)) hq).trans (by ring_nf; rfl)) |
| |
| lemma normalized_tanh_cubic (h d : ℝ) (hd : |d| ≤ 1/2) : |
| |(Real.tanh h - Real.tanh (h-d))/scalarVariance h - d - Real.tanh h*d^2| ≤ |
| 4*|d|^3 := by |
| rw [normalized_tanh_difference] |
| exact rational_cubic_remainder _ _ _ (Real.abs_tanh_lt_one h).le hd |
| (by simpa using tanh_lipschitz 0 d) (tanh_cubic_remainder d) |
| |
| lemma normalized_influence_cubic {n : ℕ} (J : Interaction n) (x : Spin n) (i j : Fin n) |
| (hJ : |J j i| ≤ 1/4) : |
| |influence J x i j / siteVariance J x j - J j i - 2*spin x i*mean J x j*(J j i)^2| ≤ |
| 16*|J j i|^3 := by |
| have ha : |2*J j i*spin x i| = 2*|J j i| := by |
| rw [abs_mul, abs_mul, abs_spin_eq_one]; norm_num |
| have hd : |2*J j i*spin x i| ≤ 1/2 := by rw [ha]; linarith |
| have hh := normalized_tanh_cubic (field J x j) (2*J j i*spin x i) hd |
| have hv : siteVariance J x j ≠ 0 := (siteVariance_pos J x j).ne' |
| have he : influence J x i j / siteVariance J x j - J j i - 2*spin x i*mean J x j*(J j i)^2 = |
| (spin x i / 2) * ((Real.tanh (field J x j) - |
| Real.tanh (field J x j - 2*J j i*spin x i))/scalarVariance (field J x j) - |
| 2*J j i*spin x i - Real.tanh (field J x j)*(2*J j i*spin x i)^2) := by |
| simp only [influence, halfDiff_as_flip, mean, field_flip_general, scalarVariance, siteVariance] |
| have hs := spin_sq x i |
| have ht : 1 - Real.tanh (field J x j)^2 ≠ 0 := (scalarVariance_pos _).ne' |
| field_simp [ht] |
| linear_combination (2 * J j i * (1 - Real.tanh (field J x j)^2) + |
| 4 * spin x i * Real.tanh (field J x j) * (J j i)^2 * (1 - Real.tanh (field J x j)^2)) * hs |
| rw [he, abs_mul, abs_div, abs_spin_eq_one] |
| rw [ha] at hh |
| norm_num only [abs_of_pos (show (0:ℝ) < 2 by norm_num)] |
| nlinarith |
| |
| lemma matrix_opNorm_le_schur {n : ℕ} (A : Interaction n) (R : ℝ) (hR : 0 ≤ R) |
| (hr : ∀ i, ∑ j, |A i j| ≤ R) (hc : ∀ j, ∑ i, |A i j| ≤ R) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A‖ ≤ R := by |
| apply ContinuousLinearMap.opNorm_le_of_re_inner_le hR |
| intro x y hx hy |
| change inner (𝕜 := ℝ) (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) A x) y ≤ R |
| rw [real_inner_comm, Matrix.inner_toEuclideanCLM] |
| change (∑ i, y i * ∑ j, A i j*x j) ≤ R |
| simp only [Finset.mul_sum] |
| calc |
| _ ≤ ∑ i, ∑ j, |A i j| * ((y i)^2 + (x j)^2)/2 := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| have hab : 2*|y i| *|x j| ≤ (y i)^2+(x j)^2 := by |
| nlinarith [sq_nonneg (|y i|-|x j|), sq_abs (y i), sq_abs (x j)] |
| have hp := mul_le_mul_of_nonneg_left hab (abs_nonneg (A i j)) |
| have hh : y i*(A i j*x j) ≤ |y i| *|A i j| *|x j| := by |
| simpa only [abs_mul, mul_assoc] using le_abs_self (y i*(A i j*x j)) |
| nlinarith |
| _ = ((∑ i, (y i)^2 * ∑ j, |A i j|) + ∑ j, (x j)^2 * ∑ i, |A i j|)/2 := by |
| simp only [mul_add, add_div, Finset.sum_add_distrib, ← Finset.sum_div] |
| rw [Finset.sum_comm (f := fun i j => |A i j| * (x j)^2)] |
| simp only [← Finset.sum_mul] |
| congr 2 <;> apply Finset.sum_congr rfl <;> intro i _ <;> ring |
| _ ≤ ((∑ i, (y i)^2 * R) + ∑ j, (x j)^2*R)/2 := by |
| apply div_le_div_of_nonneg_right _ (by norm_num) |
| exact add_le_add |
| (Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hr i) (sq_nonneg _))) |
| (Finset.sum_le_sum (fun i _ => mul_le_mul_of_nonneg_left (hc i) (sq_nonneg _))) |
| _ = R := by |
| rw [← Finset.sum_mul, ← Finset.sum_mul, ← EuclideanSpace.real_norm_sq_eq, |
| ← EuclideanSpace.real_norm_sq_eq, hx, hy] |
| ring |
| |
| lemma matrix_diagonal_opNorm_le {n : ℕ} (d : Fin n → ℝ) (R : ℝ) (hR : 0 ≤ R) |
| (hd : ∀ i, |d i| ≤ R) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal d)‖ ≤ R := by |
| apply matrix_opNorm_le_schur _ R hR |
| · intro i |
| simpa [Matrix.diagonal, apply_ite, abs_zero] using hd i |
| · intro i |
| simpa [Matrix.diagonal, apply_ite, abs_zero] using hd i |
| |
| noncomputable def normalizedInfluence {n : ℕ} (J : Interaction n) (x : Spin n) : Interaction n := |
| fun i j => influence J x i j / siteVariance J x j |
| |
| noncomputable def influenceCubicError {n : ℕ} (J : Interaction n) (x : Spin n) : Interaction n := |
| fun i j => normalizedInfluence J x i j - J j i - 2*spin x i*mean J x j*(J j i)^2 |
| |
| lemma influenceCubicError_norm_le {n : ℕ} (J : Interaction n) |
| (hsymm : ∀ i j, J i j=J j i) (x : Spin n) (c B : ℝ) (hc : 0 ≤ c) (hc1 : c ≤ 1/4) |
| (hB : 0 ≤ B) (hmax : ∀ i j, |J i j| ≤ c) |
| (hcol : ∀ i, ∑ j, J j i^2 ≤ B) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (influenceCubicError J x)‖ ≤ 16*c*B := by |
| have hp (i j : Fin n) : |influenceCubicError J x i j| ≤ 16*c*(J j i)^2 := by |
| have hh := normalized_influence_cubic J x i j ((hmax j i).trans hc1) |
| change |influenceCubicError J x i j| ≤ _ at hh |
| apply hh.trans |
| have hmul := mul_le_mul_of_nonneg_right (hmax j i) (sq_nonneg |J j i|) |
| nlinarith [sq_abs (J j i)] |
| apply matrix_opNorm_le_schur _ (16*c*B) (by positivity) |
| · intro i |
| calc |
| _ ≤ ∑ j, 16*c*(J j i)^2 := Finset.sum_le_sum (fun j _ => hp i j) |
| _ = 16*c*∑ j, J j i^2 := by rw [Finset.mul_sum] |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hcol i) (by positivity) |
| · intro j |
| calc |
| _ ≤ ∑ i, 16*c*(J j i)^2 := Finset.sum_le_sum (fun i _ => hp i j) |
| _ = 16*c*∑ i, J i j^2 := by simp only [hsymm j, Finset.mul_sum] |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hcol j) (by positivity) |
| |
| noncomputable def normalizedInfluenceApprox {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : Interaction n := |
| fun i j => J i j + 2*β^2/(n:ℝ)*spin x i*mean J x j |
| |
| lemma normalizedInfluence_sub_approx {n : ℕ} (β : ℝ) (g : GaussianCoordinates n) (x : Spin n) : |
| normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x = |
| influenceCubicError (sampledInteraction g) x + (2:ℝ) • |
| (Matrix.diagonal (spin x) * RandomMatrix.sampledSquaredDeviation β g * |
| Matrix.diagonal (mean (sampledInteraction g) x)) := by |
| ext i j |
| simp only [Matrix.sub_apply, Matrix.add_apply, Matrix.smul_apply, smul_eq_mul, |
| Matrix.mul_diagonal, Matrix.diagonal_mul, normalizedInfluenceApprox, |
| influenceCubicError, RandomMatrix.sampledSquaredDeviation, sampledInteraction_symm g j i] |
| ring |
| |
| lemma normalizedInfluence_approx_norm_le {n : ℕ} (β : ℝ) (g : GaussianCoordinates n) (x : Spin n) |
| (c B : ℝ) (hc : 0 ≤ c) (hc1 : c ≤ 1/4) (hB : 0 ≤ B) |
| (hmax : ∀ i j, |sampledInteraction g i j| ≤ c) |
| (hcol : ∀ i, ∑ j, sampledInteraction g j i^2 ≤ B) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖ ≤ |
| 16*c*B+2*‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖ := by |
| have hs : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (spin x))‖ ≤ 1 := |
| matrix_diagonal_opNorm_le _ 1 (by norm_num) (fun i => (abs_spin_eq_one x i).le) |
| have hm : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (Matrix.diagonal (mean (sampledInteraction g) x))‖ ≤ 1 := |
| matrix_diagonal_opNorm_le _ 1 (by norm_num) (fun i => (Real.abs_tanh_lt_one _).le) |
| have hp : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (Matrix.diagonal (spin x) * RandomMatrix.sampledSquaredDeviation β g * |
| Matrix.diagonal (mean (sampledInteraction g) x))‖ ≤ |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖ := by |
| rw [map_mul, map_mul] |
| calc |
| _ ≤ _ * _ := norm_mul_le _ _ |
| _ ≤ (‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (spin x))‖ * |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖) * |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (mean (sampledInteraction g) x))‖ := by |
| exact mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _) |
| _ ≤ (1 * ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖) * 1 := by |
| gcongr |
| _ = _ := by ring |
| rw [normalizedInfluence_sub_approx, map_add] |
| calc |
| _ ≤ _ + _ := norm_add_le _ _ |
| _ ≤ _ := add_le_add |
| (influenceCubicError_norm_le _ (sampledInteraction_symm g) x c B hc hc1 hB hmax hcol) |
| (by rw [map_smul, norm_smul]; norm_num only [Real.norm_eq_abs, abs_of_pos (show (0:ℝ)<2 by norm_num)]; exact mul_le_mul_of_nonneg_left hp (by norm_num)) |
| |
| lemma normalizedInfluence_approx_tendsto (β ε : ℝ) (hε : 0 < ε) : |
| Tendsto (fun n => disorderLaw β n {g | ∃ x : Spin n, ε < |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖}) |
| atTop (nhds 0) := by |
| let B : ℝ := (2*(β^2+10))^2 |
| let c : ℝ := min (1/4) (ε/(64*(B+1))) |
| have hB : 0 ≤ B := sq_nonneg _ |
| have hc : 0 < c := lt_min (by norm_num) (by positivity) |
| have hc1 : c ≤ 1/4 := min_le_left _ _ |
| have hcd : 16*c*B ≤ ε/4 := by |
| calc |
| _ ≤ 16*(ε/(64*(B+1)))*(B+1) := mul_le_mul |
| (mul_le_mul_of_nonneg_left (min_le_right _ _) (by norm_num)) (by linarith) |
| hB (by positivity) |
| _ = ε/4 := by field_simp; ring |
| have hl := (RandomMatrix.sampled_opNorm_tendsto β).add |
| ((RandomMatrix.sampled_maxEntry_tendsto β c hc).add |
| (RandomMatrix.sampledSquaredDeviation_tendsto β (ε/8) (by positivity))) |
| simp only [add_zero] at hl |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hl |
| (Eventually.of_forall (fun _ => bot_le)) |
| apply Eventually.of_forall |
| intro n |
| let N : Set (GaussianCoordinates n) := {g | 2*(β^2+10) < |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖} |
| let E : Set (GaussianCoordinates n) := {g | ∃ i j, c ≤ |sampledInteraction g i j|} |
| let V : Set (GaussianCoordinates n) := {g | ε/8 < |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖} |
| have hs : {g | ∃ x : Spin n, ε < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖} ⊆ |
| N ∪ (E ∪ V) := by |
| intro g hg |
| by_cases hN : g ∈ N |
| · exact Or.inl hN |
| by_cases hE : g ∈ E |
| · exact Or.inr (Or.inl hE) |
| by_cases hV : g ∈ V |
| · exact Or.inr (Or.inr hV) |
| have hn : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖ ≤ 2*(β^2+10) := |
| le_of_not_gt hN |
| have he (i j : Fin n) : |sampledInteraction g i j| ≤ c := |
| (not_le.mp (fun h => hE ⟨i,j,h⟩)).le |
| have hv : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (RandomMatrix.sampledSquaredDeviation β g)‖ ≤ ε/8 := |
| le_of_not_gt hV |
| obtain ⟨x,hx⟩ := hg |
| have hp := normalizedInfluence_approx_norm_le β g x c B hc.le hc1 hB he (fun i => |
| (RandomMatrix.matrix_column_sq_le _ i).trans (pow_le_pow_left₀ (norm_nonneg _) hn 2)) |
| exfalso |
| linarith |
| exact (measure_mono hs).trans ((measure_union_le N (E ∪ V)).trans |
| (add_le_add le_rfl (measure_union_le E V))) |
|
|
| noncomputable def influenceApprox {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : Interaction n := |
| normalizedInfluenceApprox β J x * Matrix.diagonal (siteVariance J x) |
|
|
| lemma influence_eq_normalized_mul {n : ℕ} (J : Interaction n) (x : Spin n) : |
| influence J x = normalizedInfluence J x * Matrix.diagonal (siteVariance J x) := by |
| ext i j |
| simp only [Matrix.mul_diagonal, normalizedInfluence] |
| rw [div_mul_cancel₀ _ (siteVariance_pos J x j).ne'] |
| |
| lemma influenceApprox_apply {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) (i j : Fin n) : |
| influenceApprox β J x i j = J i j*siteVariance J x j + |
| 2*β^2/(n:ℝ)*spin x i*(mean J x j*siteVariance J x j) := by |
| simp only [influenceApprox, Matrix.mul_diagonal, normalizedInfluenceApprox] |
| ring |
|
|
| lemma influence_approx_error_le {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (influence J x - influenceApprox β J x)‖ ≤ |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence J x - normalizedInfluenceApprox β J x)‖ := by |
| have hv : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (Matrix.diagonal (siteVariance J x))‖ ≤ 1 := by |
| apply matrix_diagonal_opNorm_le _ 1 (by norm_num) |
| intro i |
| rw [abs_of_pos (siteVariance_pos J x i)] |
| exact siteVariance_le_one J x i |
| rw [influence_eq_normalized_mul, influenceApprox, ← Matrix.sub_mul, map_mul] |
| exact (norm_mul_le _ _).trans (by simpa only [mul_one] using mul_le_mul_of_nonneg_left hv (norm_nonneg _)) |
|
|
| lemma influence_approx_tendsto (β ε : ℝ) (hε : 0 < ε) : |
| Tendsto (fun n => disorderLaw β n {g | ∃ x : Spin n, ε < |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (influence (sampledInteraction g) x - influenceApprox β (sampledInteraction g) x)‖}) |
| atTop (nhds 0) := by |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds |
| (normalizedInfluence_approx_tendsto β ε hε) (Eventually.of_forall (fun _ => bot_le)) |
| apply Eventually.of_forall |
| intro n |
| apply measure_mono |
| rintro g ⟨x,hx⟩ |
| exact ⟨x,hx.trans_le (influence_approx_error_le β (sampledInteraction g) x)⟩ |
| |
| lemma normalizedInfluenceApprox_norm_le {n : ℕ} (β : ℝ) (J : Interaction n) (x : Spin n) : |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluenceApprox β J x)‖ ≤ |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) J‖ + 2*β^2 := by |
| let R : Interaction n := fun i j => 2*β^2/(n:ℝ)*spin x i*mean J x j |
| have he : normalizedInfluenceApprox β J x = J+R := rfl |
| have hr : ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) R‖ ≤ 2*β^2 := by |
| by_cases hn : n=0 |
| · subst n |
| have hz : R=0 := Subsingleton.elim _ _ |
| rw [hz, map_zero, norm_zero]; positivity |
| have hn0 : 0 < (n:ℝ) := Nat.cast_pos.mpr (Nat.pos_of_ne_zero hn) |
| have hp (i j : Fin n) : |R i j| ≤ 2*β^2/(n:ℝ) := by |
| change |2*β^2/(n:ℝ)*spin x i*mean J x j| ≤ _ |
| rw [abs_mul, abs_mul, abs_spin_eq_one, mul_one, abs_of_nonneg (by positivity : 0 ≤ 2*β^2/(n:ℝ))] |
| exact (mul_le_mul_of_nonneg_left (Real.abs_tanh_lt_one _).le (by positivity)).trans_eq (mul_one _) |
| apply matrix_opNorm_le_schur _ (2*β^2) (by positivity) |
| · intro i |
| calc |
| _ ≤ ∑ j : Fin n, 2*β^2/(n:ℝ) := Finset.sum_le_sum (fun j _ => hp i j) |
| _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| · intro j |
| calc |
| _ ≤ ∑ i : Fin n, 2*β^2/(n:ℝ) := Finset.sum_le_sum (fun i _ => hp i j) |
| _ = _ := by simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] |
| rw [he, map_add] |
| exact (norm_add_le _ _).trans (add_le_add le_rfl hr) |
|
|
| lemma normalizedInfluence_opNorm_tight (β : ℝ) : |
| Tendsto (fun n => disorderLaw β n {g | ∃ x : Spin n, |
| 2*(β^2+10)+2*β^2+1 < |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x)‖}) |
| atTop (nhds 0) := by |
| have hl := (RandomMatrix.sampled_opNorm_tendsto β).add |
| (normalizedInfluence_approx_tendsto β 1 (by norm_num)) |
| simp only [add_zero] at hl |
| apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hl |
| (Eventually.of_forall (fun _ => bot_le)) |
| apply Eventually.of_forall |
| intro n |
| refine (measure_mono (show {g | ∃ x : Spin n, 2*(β^2+10)+2*β^2+1 < |
| ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x)‖} ⊆ |
| {g | 2*(β^2+10) < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖} ∪ |
| {g | ∃ x : Spin n, 1 < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) |
| (normalizedInfluence (sampledInteraction g) x - normalizedInfluenceApprox β (sampledInteraction g) x)‖} |
| from ?_)).trans (measure_union_le _ _) |
| rintro g ⟨x,hx⟩ |
| by_cases hJ : 2*(β^2+10) < ‖Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (sampledInteraction g)‖ |
| · exact Or.inl hJ |
| apply Or.inr |
| refine ⟨x,?_⟩ |
| have hh := normalizedInfluenceApprox_norm_le β (sampledInteraction g) x |
| have ht := norm_le_norm_sub_add |
| (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluence (sampledInteraction g) x)) |
| (Matrix.toEuclideanCLM (n := Fin n) (𝕜 := ℝ) (normalizedInfluenceApprox β (sampledInteraction g) x)) |
| rw [← map_sub] at ht |
| linarith [le_of_not_gt hJ] |
|
|
| end SKGapCutoff |
|
|
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set |
|
|
| end |
|
|
| end OAI |
|
|
