| import Mathlib |
| import OAI.MathematicalPhysics.PEPSFilters.LocalOperators |
|
|
| namespace OAI |
|
|
| /-! Dimension-free row-column and scalar modular distortion bounds. -/ |
|
|
| noncomputable section |
| open scoped BigOperators ComplexOrder |
| open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator |
| open scoped BigOperators |
| open scoped Topology |
| open Filter |
| open scoped MatrixOrder |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open PolynomialPEPS.PinnedEntropy |
|
|
| open scoped BigOperators Matrix.Norms.L2Operator |
| namespace PolynomialPEPS.Subvolume.MatrixRows |
| open Matrix |
| variable {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ] |
|
|
| omit [DecidableEq ι] in |
| lemma column_l2 (A : Matrix ι κ ℂ) (j : κ) : |
| Real.sqrt (∑ i, ‖A i j‖^2) ≤ ‖A‖ := by |
| have h := Matrix.l2_opNorm_mulVec A (PiLp.single 2 j (1 : ℂ)) |
| have hx : ‖(PiLp.single 2 j (1 : ℂ) : EuclideanSpace ℂ κ)‖ = 1 := by simp |
| rw [hx,mul_one] at h |
| have he : (EuclideanSpace.equiv ι ℂ).symm (A *ᵥ (PiLp.single 2 j (1 : ℂ))) = |
| WithLp.toLp 2 (fun i => A i j) := by |
| ext i |
| change (A *ᵥ Pi.single j (1 : ℂ)) i = A i j |
| simp only [Matrix.mulVec_single,MulOpposite.op_one,one_smul] |
| rfl |
| rw [he,EuclideanSpace.norm_eq] at h |
| exact h |
|
|
| lemma row_l2 (A : Matrix ι κ ℂ) (i : ι) : |
| Real.sqrt (∑ j, ‖A i j‖^2) ≤ ‖A‖ := by |
| simpa [Matrix.conjTranspose_apply, Matrix.l2_opNorm_conjTranspose] using column_l2 A.conjTranspose i |
|
|
| lemma row_product (A B : Matrix ι κ ℂ) (i : ι) : |
| ∑ j, ‖A i j‖ * ‖B i j‖ ≤ ‖A‖ * ‖B‖ := by |
| have h := Real.sum_mul_le_sqrt_mul_sqrt Finset.univ (fun j => ‖A i j‖) (fun j => ‖B i j‖) |
| exact h.trans (mul_le_mul (row_l2 A i) (row_l2 B i) (Real.sqrt_nonneg _) (norm_nonneg _)) |
|
|
| lemma column_product (A B : Matrix ι κ ℂ) (j : κ) : |
| ∑ i, ‖A i j‖ * ‖B i j‖ ≤ ‖A‖ * ‖B‖ := by |
| simpa [Matrix.conjTranspose_apply, Matrix.l2_opNorm_conjTranspose] using |
| row_product A.conjTranspose B.conjTranspose j |
|
|
| theorem weighted_sum (A B : Matrix ι ι ℂ) (p : ι → ℝ) (hp : ∀ i, 0 ≤ p i) : |
| ∑ i, ∑ j, max (p i) (p j) * (‖A i j‖ * ‖B i j‖) ≤ |
| 2 * (∑ i, p i) * (‖A‖ * ‖B‖) := by |
| calc |
| _ ≤ ∑ i, ∑ j, (p i+p j) * (‖A i j‖ * ‖B i j‖) := by |
| gcongr with i j |
| exact max_le (le_add_of_nonneg_right (hp j)) (le_add_of_nonneg_left (hp i)) |
| _ = (∑ i, p i * (∑ j, ‖A i j‖ * ‖B i j‖)) + |
| (∑ j, p j * (∑ i, ‖A i j‖ * ‖B i j‖)) := by |
| simp_rw [add_mul,Finset.sum_add_distrib,← Finset.mul_sum] |
| rw [Finset.sum_comm] |
| simp_rw [Finset.mul_sum] |
| _ ≤ (∑ i, p i * (‖A‖*‖B‖)) + (∑ j, p j * (‖A‖*‖B‖)) := by |
| exact add_le_add |
| (Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left (row_product A B i) (hp i)) |
| (Finset.sum_le_sum fun j _ => mul_le_mul_of_nonneg_left (column_product A B j) (hp j)) |
| _ = _ := by rw [← Finset.sum_mul]; ring |
|
|
| theorem coefficient_sum (A B : Matrix ι ι ℂ) (p : ι → ℝ) (hp : ∀ i, 0 ≤ p i) |
| (d : ι → ι → ℂ) (C : ℝ) (hC : 0 ≤ C) |
| (hd : ∀ i j, ‖d i j‖ ≤ C * max (p i) (p j)) : |
| ‖∑ i, ∑ j, d i j * A i j * B i j‖ ≤ |
| 2*C*(∑ i, p i)*(‖A‖*‖B‖) := by |
| calc |
| _ ≤ ∑ i, ∑ j, ‖d i j * A i j * B i j‖ := |
| (norm_sum_le _ _).trans (Finset.sum_le_sum fun i _ => norm_sum_le _ _) |
| _ ≤ ∑ i, ∑ j, C*max (p i) (p j) * (‖A i j‖*‖B i j‖) := by |
| simp only [norm_mul] |
| exact Finset.sum_le_sum fun i _ => Finset.sum_le_sum fun j _ => by |
| simpa only [mul_assoc] using |
| mul_le_mul_of_nonneg_right (hd i j) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) |
| _ = C * (∑ i, ∑ j, max (p i) (p j) * (‖A i j‖*‖B i j‖)) := by |
| simp_rw [Finset.mul_sum] |
| congr 1 |
| ext i |
| congr 1 |
| ext j |
| ring |
| _ ≤ C * (2*(∑ i, p i)*(‖A‖*‖B‖)) := mul_le_mul_of_nonneg_left (weighted_sum A B p hp) hC |
| _ = _ := by ring |
|
|
| end PolynomialPEPS.Subvolume.MatrixRows |
|
|
| namespace PolynomialPEPS.Subvolume.ModularScalar |
|
|
| lemma even_exp_remainder (x : ℝ) (hx : 0 ≤ x) : |
| Real.exp x + Real.exp (-x) - 2 ≤ x^2 * Real.exp x := by |
| let y := Real.exp (x/2) |
| let z := Real.exp (-x/2) |
| have hy : 0 < y := Real.exp_pos _ |
| have hz : 0 < z := Real.exp_pos _ |
| have hyz : y*z=1 := by |
| dsimp [y,z] |
| rw [← Real.exp_add, ← add_div, add_neg_cancel, zero_div, Real.exp_zero] |
| have hysq : y^2 = Real.exp x := by |
| dsimp [y] |
| rw [sq,← Real.exp_add] |
| congr 1 |
| ring |
| have hzsq : z^2 = Real.exp (-x) := by |
| dsimp [z] |
| rw [sq,← Real.exp_add] |
| congr 1 |
| ring |
| have hd0 : 0 ≤ y-z := sub_nonneg.mpr (Real.exp_le_exp.mpr (by linarith)) |
| have he := Real.add_one_le_exp (-x) |
| have hmul : y*Real.exp (-x) = z := by |
| dsimp [y,z] |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
| have hd : y-z ≤ x*y := by |
| have ht := mul_le_mul_of_nonneg_left he hy.le |
| rw [hmul] at ht |
| nlinarith |
| have hs := sq_le_sq₀ hd0 (mul_nonneg hx hy.le) |>.mpr hd |
| rw [mul_pow,hysq] at hs |
| nlinarith [sq_nonneg (y-z)] |
|
|
| lemma polynomial_exp_decay (x : ℝ) (hx : 0 ≤ x) : |
| x^2 * Real.exp (-x/4) ≤ 32 := by |
| have hp := Real.pow_div_factorial_le_exp (x/4) (show 0 ≤ x/4 by positivity) 2 |
| norm_num at hp |
| have he : Real.exp (x/4)*Real.exp (-x/4) = 1 := by |
| rw [← Real.exp_add, ← add_div, add_neg_cancel, zero_div, Real.exp_zero] |
| have hh := mul_le_mul_of_nonneg_right hp (Real.exp_pos (-x/4)).le |
| nlinarith |
|
|
| theorem quadratic_log_ratio (x a : ℝ) (hx : 0 ≤ x) (ha : |a| ≤ 1/4) : |
| Real.exp (-x/2) * (Real.exp (a*x) + Real.exp (-a*x) - 2) ≤ 32*a^2 := by |
| have heven : Real.exp (a*x) + Real.exp (-a*x) = |
| Real.exp (|a| * x) + Real.exp (-(|a| * x)) := by |
| rcases le_total 0 a with h|h |
| · simp [abs_of_nonneg h] |
| · simp [abs_of_nonpos h,add_comm] |
| rw [heven] |
| have ht := mul_le_mul_of_nonneg_left (even_exp_remainder (|a| * x) (by positivity)) |
| (Real.exp_pos (-x/2)).le |
| have hcomb : Real.exp (-x/2) * ((|a| * x)^2 * Real.exp (|a| * x)) = |
| a^2 * (x^2 * Real.exp ((|a|-1/2)*x)) := by |
| rw [mul_pow,sq_abs] |
| calc |
| _ = a^2*x^2*(Real.exp (-x/2)*Real.exp (|a| * x)) := by ring |
| _ = _ := by |
| rw [← Real.exp_add, |
| show -x/2 + |a| * x = (|a|-1/2)*x by ring] |
| ring |
| rw [hcomb] at ht |
| have he : Real.exp ((|a|-1/2)*x) ≤ Real.exp (-x/4) := |
| Real.exp_le_exp.mpr (by nlinarith) |
| calc |
| _ ≤ a^2 * (x^2 * Real.exp ((|a|-1/2)*x)) := ht |
| _ ≤ a^2 * (x^2 * Real.exp (-x/4)) := by gcongr |
| _ ≤ a^2 * 32 := mul_le_mul_of_nonneg_left (polynomial_exp_decay x hx) (sq_nonneg a) |
| _ = 32*a^2 := by ring |
|
|
| end PolynomialPEPS.Subvolume.ModularScalar |
|
|
| namespace PolynomialPEPS.Subvolume.ModularScalar |
|
|
| lemma positive_weight_bound (s t a : ℝ) (hs : 0 < s) (ht : 0 < t) |
| (ha : |a| ≤ 1/4) : |
| Real.sqrt (s*t) * ((s/t)^a+(t/s)^a-2) ≤ 32*a^2*max s t := by |
| suffices hordered : ∀ s t : ℝ, 0 < s → 0 < t → t ≤ s → |
| Real.sqrt (s*t) * ((s/t)^a+(t/s)^a-2) ≤ 32*a^2*s by |
| rcases le_total t s with h|h |
| · simpa only [max_eq_left h] using hordered s t hs ht h |
| · simpa only [max_eq_right h,mul_comm s t,add_comm ((s/t)^a) ((t/s)^a)] using |
| hordered t s ht hs h |
| intro s t hs ht hts |
| let x := Real.log s - Real.log t |
| have hx : 0 ≤ x := sub_nonneg.mpr (Real.log_le_log ht hts) |
| have hst : (s/t)^a = Real.exp (a*x) := by |
| rw [Real.rpow_def_of_pos (div_pos hs ht),Real.log_div hs.ne' ht.ne'] |
| congr 1 |
| dsimp [x] |
| ring |
| have hts' : (t/s)^a = Real.exp (-a*x) := by |
| rw [Real.rpow_def_of_pos (div_pos ht hs),Real.log_div ht.ne' hs.ne'] |
| congr 1 |
| dsimp [x] |
| ring |
| have hroot : Real.sqrt (s*t) = s*Real.exp (-x/2) := by |
| rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (mul_pos hs ht),Real.log_mul hs.ne' ht.ne'] |
| nth_rw 2 [← Real.exp_log hs] |
| rw [← Real.exp_add] |
| congr 1 |
| dsimp [x] |
| ring |
| rw [hst,hts',hroot,mul_assoc] |
| calc |
| _ ≤ s*(32*a^2) := mul_le_mul_of_nonneg_left (quadratic_log_ratio x a hx ha) hs.le |
| _ = _ := by ring |
|
|
| def modularCoefficient (s t a : ℝ) : ℝ := |
| if 0 < s ∧ 0 < t then Real.sqrt (s*t) * ((s/t)^a+(t/s)^a-2)/2 else 0 |
|
|
| lemma modularCoefficient_nonneg (s t a : ℝ) : 0 ≤ modularCoefficient s t a := by |
| unfold modularCoefficient |
| split_ifs with h |
| · have h₁ := Real.add_one_le_exp (Real.log (s/t)*a) |
| have h₂ := Real.add_one_le_exp (Real.log (t/s)*a) |
| have hlog : Real.log (t/s) = -Real.log (s/t) := by |
| rw [Real.log_div h.2.ne' h.1.ne',Real.log_div h.1.ne' h.2.ne'] |
| ring |
| have hexp : 0 ≤ (s/t)^a+(t/s)^a-2 := by |
| rw [Real.rpow_def_of_pos (div_pos h.1 h.2),Real.rpow_def_of_pos (div_pos h.2 h.1)] |
| rw [hlog] at h₂ ⊢ |
| nlinarith |
| positivity |
| · rfl |
|
|
| theorem modularCoefficient_bound (s t a : ℝ) (hs : 0 ≤ s) (ht : 0 ≤ t) (ha : |a| ≤ 1/4) : |
| |modularCoefficient s t a| ≤ 16*a^2*max s t := by |
| rw [abs_of_nonneg (modularCoefficient_nonneg s t a)] |
| unfold modularCoefficient |
| split_ifs with h |
| · have H := positive_weight_bound s t a h.1 h.2 ha |
| linarith |
| · exact mul_nonneg (mul_nonneg (by norm_num) (sq_nonneg a)) |
| (by simpa using max_le_max hs ht) |
|
|
| end PolynomialPEPS.Subvolume.ModularScalar |
|
|
| namespace PolynomialPEPS.Subvolume.ModularScalar |
|
|
| lemma modularCoefficient_formula (s t a : ℝ) (hs : 0 ≤ s) (ht : 0 ≤ t) : |
| Real.sqrt s * Real.sqrt t * (((s^a)*(t^a)⁻¹ + (s^a)⁻¹*(t^a))/2 - 1) = |
| modularCoefficient s t a := by |
| by_cases hs0 : s=0 |
| · simp [hs0,modularCoefficient] |
| by_cases ht0 : t=0 |
| · simp [ht0,modularCoefficient] |
| have hs' : 0<s := lt_of_le_of_ne hs (Ne.symm hs0) |
| have ht' : 0<t := lt_of_le_of_ne ht (Ne.symm ht0) |
| rw [modularCoefficient,ite_eq_left ⟨hs',ht'⟩,← Real.sqrt_mul hs, |
| Real.div_rpow hs ht,Real.div_rpow ht hs] |
| ring |
|
|
| end PolynomialPEPS.Subvolume.ModularScalar |
|
|
| end |
|
|
| end OAI |
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|
