| import Mathlib |
| import OAI.Probability.SKGap.Stability.OrdinaryClosure |
| import OAI.Probability.SKGap.Stability.ImplicitUniformControl |
| import OAI.Probability.SKGap.Localization.LiteralSystemBridge |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Matrix.Norms.Frobenius |
| namespace SKGapCutoff.Recipe |
| open SKGap.Stein Primary SKGap.ImplicitSystem Matrix |
| variable {n : ℕ} |
|
|
| structure LiteralStableAt (J : Interaction n) (j : ℝ) |
| (z m : VectorFields n) (a : Spin n→ℝ) (r : Fin n→ℝ) |
| (R ρ δ : ℝ) (x : Spin n) : Prop where |
| a_bound : |a x|≤R |
| a_small : |a x|≤ρ |
| z_near : vectorNorm (z x-r)≤ρ*Real.sqrt n |
| m_near : vectorNorm (m x-(fun i=>Real.tanh (r i)))≤ρ*Real.sqrt n |
| m_bound : ∀i,|m x i|≤1 |
| stable : ∀v,δ*‖v‖^2≤ inner ℝ v |
| (stableMatrix j ((∑i,scalarVariance (z x i))/(n:ℝ)) (n:ℝ)⁻¹ |
| (phiSqrt (z x) r (a x)) (toEuclideanCLM (𝕜:=ℝ) J) |
| (WithLp.toLp 2 (fun i=>Real.tanh (z x i))) v) |
|
|
| lemma LiteralStableAt.perturbation {J : Interaction n} {j : ℝ} |
| {z m : VectorFields n} {a : Spin n→ℝ} {r : Fin n→ℝ} |
| {R ρ δ : ℝ} {x : Spin n} (H : LiteralStableAt J j z m a r R ρ δ x) |
| (hn : 0<n) (hδ : 0<δ) (hρ : 0≤ρ) |
| (hsmall : ρ≤δ/(2*(|j| *Real.exp (R/2)*(3*Real.exp (R/2)+16)+1))) : |
| |j| *|siteMean (fun v i=>phi (z v i) (r i) (a v)) x- |
| (∑i,scalarVariance (z x i))/(n:ℝ)| *‖phiSqrt (z x) r (a x)‖^2+ |
| |j*(n:ℝ)⁻¹| *‖phiSqrt (z x) r (a x)‖^2* |
| (‖(WithLp.toLp 2 (m x) : Euclid n)+WithLp.toLp 2 (fun i=>Real.tanh (r i))- |
| (2:ℝ) • WithLp.toLp 2 (fun i=>Real.tanh (z x i))‖* |
| ‖(WithLp.toLp 2 (fun i=>logSlope (z x i) (r i) (a x)) : Euclid n)‖+ |
| 2*‖(WithLp.toLp 2 (fun i=>Real.tanh (z x i)) : Euclid n)‖* |
| ‖(WithLp.toLp 2 (fun i=>logSlope (z x i) (r i) (a x)) : Euclid n)+ |
| WithLp.toLp 2 (fun i=>Real.tanh (z x i))‖)≤δ/2 := |
| (implicit_normalized_perturbation (z x) r (m x) (a x) j (phiSqrt (z x) r (a x)) |
| hn hρ H.a_bound H.a_small H.z_near H.m_near (phiSqrt_bound _ _ _ H.a_bound)).trans |
| (implicit_rho_threshold j R δ ρ hδ hsmall) |
|
|
| lemma LiteralStableAt.m_size {J : Interaction n} {j : ℝ} |
| {z m : VectorFields n} {a : Spin n→ℝ} {r : Fin n→ℝ} |
| {R ρ δ : ℝ} {x : Spin n} (H : LiteralStableAt J j z m a r R ρ δ x) : |
| vectorNorm (m x)≤Real.sqrt n := by |
| have h:=vectorNorm_mono (u:=m x) (v:=fun _=>1) (fun i=>by simpa using H.m_bound i) |
| simpa only [vectorNorm_const,abs_one,one_mul] using h |
|
|
| lemma frobenius_le_opNorm_sqrt (D : Interaction n) : ‖D‖≤SKGap.opNorm D*Real.sqrt n := |
| frobenius_by_rows D (norm_nonneg _) (fun i=>SKGap.SourceError.matrix_row_square_le D i) |
|
|
| structure LiteralBudget (f : KernelExpr) (R T L B : ℝ) : Prop where |
| nonneg : 0≤B |
| two : 2≤B |
| root : Real.sqrt (Real.exp (R/2))≤B |
| square : Real.exp (R/2)≤B^2 |
| matrix : L≤B |
| primary : T≤B |
| source : ratioSmallBudget (.kernel f) T≤B |
| scalar : ratioSmallBudget ((RatioExpr.kernel f).d .z) T*kernelSmallBudget phiExpr R T≤B |
| phi_row : (phiExpr.dBudget R+phiExpr.ddBudget R)*T≤B |
| slope_row : ((RatioExpr.kernel phiExpr.dz).dBudget+(RatioExpr.kernel phiExpr.dz).ddBudget)*T≤B |
|
|
| lemma literal_initial_control (J : Interaction n) (j : ℝ) (f : KernelExpr) |
| (z m w y : VectorFields n) (a c : Spin n→ℝ) (r e : Fin n→ℝ) (x : Spin n) |
| (hn : 0<n) {R ρ δ T L B : ℝ} (hδ : 0<δ) (hρ : 0≤ρ) (hT : 0≤T) |
| (hsmall : ρ≤δ/(2*(|j| *Real.exp (R/2)*(3*Real.exp (R/2)+16)+1))) |
| (hB : LiteralBudget f R T L B) (hJ : SKGap.opNorm J≤L) (he : vectorNorm e≤1) |
| (H : LiteralStableAt J j z m a r R ρ δ x) |
| (Hf : ∀k,LiteralStableAt J j z m a r R ρ δ (flip x k)) |
| (hD : SKGap.opNorm (derivativeMatrix z x)+‖derivativeVector a x‖≤T) |
| (hDm : SKGap.opNorm (derivativeMatrix m x)≤T) |
| (h : LiteralEquations J j f z m w y a c r e x) |
| (hf : ∀k,LiteralEquations J j f z m w y a c r e (flip x k)) : |
| SmallBound w x (uniformControlBudget j δ B) ∧ |
| SmallBound y x (uniformControlBudget j δ B) ∧ |
| |Real.sqrt n*c x|≤uniformControlBudget j δ B ∧ |
| ‖derivativeVector (fun v=>Real.sqrt n*c v) x‖≤uniformControlBudget j δ B := by |
| let A : VectorFields n:=fun v i=>phi (z v i) (r i) (a v) |
| let ell : VectorFields n:=fun v i=>logSlope (z v i) (r i) (a v) |
| have hnR : (0:ℝ)<n:=Nat.cast_pos.mpr hn |
| have hd : 0<Real.sqrt (n:ℝ):=Real.sqrt_pos.mpr hnR |
| have hscale : (n:ℝ)⁻¹*(Real.sqrt (n:ℝ))^2=1 := by rw [Real.sq_sqrt hnR.le,inv_mul_cancel₀ hnR.ne'] |
| have ha:=initial_kernel_bound phiExpr z a r x hT H.a_bound (fun k=>(Hf k).a_bound) hD |
| have ha' : CoefficientBound A x (Real.exp (R/2)) ((phiExpr.dBudget R+phiExpr.ddBudget R)*T) := by |
| convert! ha using 1 |
| · ext v i; exact (moment_base _ _ _).symm |
| · simp [phiExpr,KernelExpr.mass] |
| have hel:=initial_ratio_bound (.kernel phiExpr.dz) z a r x hT hD |
| have hel' : CoefficientBound ell x (RatioExpr.kernel phiExpr.dz).mass |
| (((RatioExpr.kernel phiExpr.dz).dBudget+(RatioExpr.kernel phiExpr.dz).ddBudget)*T) := by |
| convert! hel using 1 |
| ext v i; exact logSlope_ratio _ _ _ |
| have hu:=(initialU_small f z a r e x hT he hD).mono hB.source |
| have hp:=(initialP_small f z a r e x hT he H.a_bound (fun k=>(Hf k).a_bound) hD).mono hB.scalar |
| have hp':=scaledMean_norm hn _ x hp |
| have hX {v : Spin n} (HV : LiteralStableAt J j z m a r R ρ δ v) : ‖phiSqrt (z v) r (a v)‖≤B := by |
| apply (Real.le_sqrt (norm_nonneg _) (Real.exp_pos _).le).mpr (phiSqrt_bound _ _ _ HV.a_bound) |>.trans hB.root |
| have hχ {v : Spin n} (HV : LiteralStableAt J j z m a r R ρ δ v) : |siteMean A v|≤B^2 := by |
| apply (coefficient_mean_value A v (Real.exp_pos _).le (fun i=>?_) ).trans hB.square |
| rw [abs_of_pos (phi_pos _ _ _)] |
| exact (phi_le_exp ..).trans (Real.exp_le_exp.mpr (by linarith [HV.a_bound])) |
| have hell {v : Spin n} : ‖(WithLp.toLp 2 (ell v) : Euclid n)‖≤B*Real.sqrt n := |
| (vectorNorm_logSlope (z v) r (a v)).trans (mul_le_mul_of_nonneg_right hB.two hd.le) |
| have hDA (V : Euclid n) : columnNorm (fun k=>((spin x k/2) • |
| (phiSqrt (z x) r (a x)*phiSqrt (z x) r (a x)- |
| phiSqrt (z (flip x k)) r (a (flip x k))*phiSqrt (z (flip x k)) r (a (flip x k)))) V)≤B*‖V‖ := by |
| simp only [phiSqrt_square] |
| exact (diagonal_halfDiff_columns A x ha'.row_nonneg ha'.difference V).trans |
| (mul_le_mul_of_nonneg_right hB.phi_row (norm_nonneg _)) |
| have hDu : columnNorm (fun k=>(spin x k/2) • |
| ((WithLp.toLp 2 (initialU f z a r e x) : Euclid n)-WithLp.toLp 2 (initialU f z a r e (flip x k))))≤B := by |
| rw [columnNorm_halfDiff]; exact hu.derivative |
| have hDχ : columnNorm (fun k=>(spin x k/2)*(siteMean A x-siteMean A (flip x k)))≤B := by |
| rw [columnNorm_scalar_halfDiff] |
| exact (coefficient_mean_difference A x ha'.row_nonneg ha'.difference).trans hB.phi_row |
| have hDm' : columnNorm (fun k=>(spin x k/2) • |
| ((WithLp.toLp 2 (m x) : Euclid n)-WithLp.toLp 2 (m (flip x k))))≤B*Real.sqrt n := by |
| rw [columnNorm_halfDiff] |
| exact (frobenius_le_opNorm_sqrt _).trans (mul_le_mul_of_nonneg_right (hDm.trans hB.primary) hd.le) |
| have hDell : columnNorm (fun k=>(spin x k/2) • |
| ((WithLp.toLp 2 (ell x) : Euclid n)-WithLp.toLp 2 (ell (flip x k))))≤B*Real.sqrt n := by |
| rw [columnNorm_halfDiff] |
| exact (frobenius_by_rows _ hel'.row_nonneg hel'.difference).trans |
| (mul_le_mul_of_nonneg_right hB.slope_row hd.le) |
| have hDp : Real.sqrt n*columnNorm (fun k=>(spin x k/2)* |
| (siteMean (initialP f z a r e) x-siteMean (initialP f z a r e) (flip x k)))≤B := by |
| rw [columnNorm_scalar_halfDiff]; exact hp'.2 |
| have hEq : Equations j (siteMean A x) (n:ℝ)⁻¹ (siteMean (initialP f z a r e) x) |
| (phiSqrt (z x) r (a x)*phiSqrt (z x) r (a x)) (toEuclideanCLM (𝕜:=ℝ) J) |
| (WithLp.toLp 2 (initialU f z a r e x)) (WithLp.toLp 2 (m x)) |
| (WithLp.toLp 2 (fun i=>Real.tanh (r i))) (WithLp.toLp 2 (ell x)) |
| (WithLp.toLp 2 (w x)) (WithLp.toLp 2 (y x)) (c x) := by |
| rw [phiSqrt_square]; exact h |
| have hEqf k : Equations j (siteMean A (flip x k)) (n:ℝ)⁻¹ (siteMean (initialP f z a r e) (flip x k)) |
| (phiSqrt (z (flip x k)) r (a (flip x k))*phiSqrt (z (flip x k)) r (a (flip x k))) (toEuclideanCLM (𝕜:=ℝ) J) |
| (WithLp.toLp 2 (initialU f z a r e (flip x k))) (WithLp.toLp 2 (m (flip x k))) |
| (WithLp.toLp 2 (fun i=>Real.tanh (r i))) (WithLp.toLp 2 (ell (flip x k))) |
| (WithLp.toLp 2 (w (flip x k))) (WithLp.toLp 2 (y (flip x k))) (c (flip x k)) := by |
| rw [phiSqrt_square]; exact hf k |
| have ht:=implicit_uniform_control j (siteMean A x) ((∑i,scalarVariance (z x i))/(n:ℝ)) (n:ℝ)⁻¹ |
| (siteMean (initialP f z a r e) x) (phiSqrt (z x) r (a x)) (toEuclideanCLM (𝕜:=ℝ) J) |
| (WithLp.toLp 2 (initialU f z a r e x)) (WithLp.toLp 2 (m x)) |
| (WithLp.toLp 2 (fun i=>Real.tanh (r i))) (WithLp.toLp 2 (ell x)) |
| (WithLp.toLp 2 (fun i=>Real.tanh (z x i))) (WithLp.toLp 2 (w x)) (WithLp.toLp 2 (y x)) (c x) |
| (fun k=>spin x k/2) (fun k=>siteMean A (flip x k)) (fun k=>siteMean (initialP f z a r e) (flip x k)) |
| (fun k=>c (flip x k)) (fun k=>(∑i,scalarVariance (z (flip x k) i))/(n:ℝ)) |
| (fun k=>phiSqrt (z (flip x k)) r (a (flip x k))) |
| (fun k=>WithLp.toLp 2 (initialU f z a r e (flip x k))) (fun k=>WithLp.toLp 2 (m (flip x k))) |
| (fun k=>WithLp.toLp 2 (ell (flip x k))) (fun k=>WithLp.toLp 2 (w (flip x k))) |
| (fun k=>WithLp.toLp 2 (y (flip x k))) (fun k=>WithLp.toLp 2 (fun i=>Real.tanh (z (flip x k) i))) |
| hδ hd (inv_nonneg.mpr hnR.le) hscale hB.nonneg (hX H) (fun k=>hX (Hf k)) (hJ.trans hB.matrix) |
| (hχ H) (fun k=>hχ (Hf k)) hu.size H.m_size (fun k=>(Hf k).m_size) (vectorNorm_tanh r) |
| hell (fun _=>hell) hp'.1 H.stable (fun k=>(Hf k).stable) |
| (H.perturbation hn hδ hρ hsmall) (fun k=>(Hf k).perturbation hn hδ hρ hsmall) |
| hEq hEqf hDA hDu hDχ hDm' hDell hDp |
| rw [columnNorm_halfDiff,columnNorm_halfDiff,columnNorm_scalar_halfDiff] at ht |
| exact implicit_tuple_small w y c x hd.le (uniformControlBudget_nonneg hδ hB.nonneg) ht |
| end SKGapCutoff.Recipe |
| |
| end |
| end |
| |
| end OAI |
| |
