| import OAI.Analysis.FreeEntropy.MatrixRotationBounds |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Finset Set |
| open |
| namespace |
| variable ℕ |
| theorem |
| Continuous → ℂ → |
| unfold |
| apply |
| apply |
| rintro ⟨⟩ |
| dsimp |
| split_ifs ;> fun_prop |
| theorem rotateCoordinates_continuous_joint : |
| Continuous (fun z : unitary (Matrix (Fin d) (Fin d) ℂ) × MatrixCoordinates n d => rotateCoordinates z.1 z.2) := by |
| unfold rotateCoordinates |
| apply matrixCoordinatesOfHermitian_continuous.comp |
| apply continuous_pi |
| intro i |
| exact ((continuous_subtype_val.comp continuous_fst).star.mul ((hermitianMatrix_continuous i).comp continuous_snd)).mul |
| (continuous_subtype_val.comp continuous_fst) |
| def blockEvent (p d : ℕ) (r M : ℝ) : Set (MatrixCoordinates |
| {a | ‖‖≤ ∧ ∃ → ∧ |
| ∃U ℂ≤‖‖ |
| theorem ℝ |
| have ℂ |
| have → | |
| ∃U ℂ≤‖‖ |
| have ℂ × | |
| r≤‖coordinateMask ‖ |
| isClosed_le |
| convert |
| ext |
| simp |
| constructor |
| · rintro ⟨⟩; exact ⟨U,a,hU,rfl⟩ |
| · rintro ⟨U,b,hU,rfl⟩; exact ⟨U,hU⟩ |
| have he : blockEvent p d r M = {a | ‖a‖≤M} ∩ |
| ⋃ f : Fin d → Fin p, ⋃ (_ : balancedBlocks f), |
| {a | ∃U : unitary (Matrix (Fin d) (Fin d) ℂ),r≤‖coordinateMask (blockMask f) (rotateCoordinates U a)‖} := by |
| ext a; simp [blockEvent] |
| rw [he] |
| exact (isClosed_le continuous_norm continuous_const).inter (isClosed_iUnion_of_finite (fun f => isClosed_iUnion_of_finite (fun _ => hf f))) |
| theorem blockEvent_subset_net (hd : 0<d) {r M ε : ℝ} (hε : 0≤ε) |
| (s : Finset (unitary (Matrix (Fin d) (Fin d) ℂ))) |
| (hs : ∀U : unitary (Matrix (Fin d) (Fin d) ℂ),∃V∈s,matrixOpNorm ((U:Matrix (Fin d) (Fin d) ℂ)-(V:Matrix (Fin d) (Fin d) ℂ))≤ε) : |
| blockEvent p d r M ⊆ ⋃ f : Fin d → Fin p, ⋃ (_ : balancedBlocks f), ⋃ V∈s, |
| {a | r-2*ε*M≤‖coordinateMask (blockMask f) (rotateCoordinates V a)‖} := by |
| intro a ha |
| obtain ⟨haM,f,hf,U,hU⟩ := ha |
| obtain ⟨V,hV,hUV⟩ := hs U |
| refine mem_iUnion.mpr ⟨f,mem_iUnion.mpr ⟨hf,mem_iUnion.mpr ⟨V,mem_iUnion.mpr ⟨hV,?_⟩⟩⟩⟩ |
| have hh := coordinateMask_norm_sub (blockMask f) (rotateCoordinates U a) (rotateCoordinates V a) |
| have hr := rotateCoordinates_sub_norm hd U V a |
| have hb : 2*matrixOpNorm ((U:Matrix (Fin d) (Fin d) ℂ)-(V:Matrix (Fin d) (Fin d) ℂ))*‖a‖≤2*ε*M := by |
| exact mul_le_mul (mul_le_mul_of_nonneg_left hUV (by norm_num)) haM (norm_nonneg _) (by positivity) |
| have hh' := (abs_le.mp (hh.trans (hr.trans hb))).2 |
| dsimp at * |
| linarith |
| theorem normalLaw_rotated_mask_tail {s r : ℝ} (hs : 0<s) (hr : 0≤r) |
| (f : Fin d → Fin p) (U : unitary (Matrix (Fin d) (Fin d) ℂ)) : |
| (normalLaw s) {a | r≤‖coordinateMask (blockMask f) (rotateCoordinates U a)‖} ≤ |
| ENNReal.ofReal (Real.exp (-r^2/(4*s))*Real.sqrt 2^(blockMask f).card) := by |
| have hm := normalLaw_measurePreserving s (rotateLinearIsometryEquiv (n := p+1) U) |
| have hE : MeasurableSet {a : MatrixCoordinates (p+1) d | r≤‖coordinateMask (blockMask f) a‖} := |
| (isClosed_le continuous_const (coordinateMask_continuous _).norm).measurableSet |
| have he := hm.measure_preimage hE.nullMeasurableSet |
| change (normalLaw s) {a | r≤‖coordinateMask (blockMask f) (rotateCoordinates U a)‖} = _ at he |
| rw [he] |
| exact normalLaw_coordinateMask_tail _ hs hr |
| theorem blockEvent_gaussian_bound (hd : 0<d) {s r M ε : ℝ} (hs : 0<s) (hε : 0<ε) |
| (hr : 0≤r-2*ε*M) : |
| (normalLaw s) (blockEvent p d r M) ≤ ENNReal.ofReal |
| ((p:ℝ)^d*(1+2/ε)^(2*d^2)*Real.exp (-(r-2*ε*M)^2/(4*s))*Real.sqrt 2^(2*d^2)) := by |
| classical |
| obtain ⟨S,hS,hcard⟩ := unitary_matrix_net hd hε |
| have hsub := blockEvent_subset_net (p := p) (r := r) (M := M) hd hε.le S hS |
| let K : ℝ := Real.exp (-(r-2*ε*M)^2/(4*s))*Real.sqrt 2^(2*d^2) |
| have hK : 0≤K := by dsimp [K]; positivity |
| have hbound (f : Fin d → Fin p) (hf : balancedBlocks f) (V : unitary (Matrix (Fin d) (Fin d) ℂ)) : |
| (normalLaw s) {a | r-2*ε*M≤‖coordinateMask (blockMask f) (rotateCoordinates V a)‖}≤ENNReal.ofReal K := by |
| apply (normalLaw_rotated_mask_tail hs hr f V).trans |
| apply ENNReal.ofReal_le_ofReal |
| apply mul_le_mul_of_nonneg_left _ (Real.exp_pos _).le |
| exact pow_le_pow_right₀ (by nlinarith [Real.sq_sqrt (show (0:ℝ)≤2 by norm_num),Real.sqrt_nonneg (2:ℝ)] : (1:ℝ)≤Real.sqrt 2) (blockMask_card_le f hf) |
| calc |
| _ ≤ ∑f : Fin d → Fin p, ∑V∈S,ENNReal.ofReal K := by |
| apply (measure_mono hsub).trans |
| apply (measure_iUnion_fintype_le _ _).trans |
| apply sum_le_sum |
| intro f _ |
| by_cases hf : balancedBlocks f |
| · have he : (⋃ (_ : balancedBlocks f), ⋃ V∈S, {a : MatrixCoordinates (p+1) d | r-2*ε*M≤‖coordinateMask (blockMask f) (rotateCoordinates V a)‖}) = ⋃ V∈S, {a : MatrixCoordinates (p+1) d | r-2*ε*M≤‖coordinateMask (blockMask f) (rotateCoordinates V a)‖} := by ext a; simp [hf] |
| rw [he] |
| exact (measure_biUnion_finset_le S _).trans (sum_le_sum (fun V _ => hbound f hf V)) |
| · simp [hf] |
| _ = ENNReal.ofReal ((p:ℝ)^d*(S.card:ℝ)*K) := by |
| simp only [sum_const,nsmul_eq_mul,card_univ,Fintype.card_fun,Fintype.card_fin] |
| simp only [ENNReal.ofReal_mul (show 0≤(p:ℝ)^d by positivity), ENNReal.ofReal_pow (show (0:ℝ)≤p by positivity), ENNReal.ofReal_natCast, Nat.cast_pow, ENNReal.ofReal_mul (show (0:ℝ)≤S.card by positivity), mul_assoc] |
| _ ≤ _ := by |
| apply ENNReal.ofReal_le_ofReal |
| have hh := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_left hcard (by positivity : 0≤(p:ℝ)^d)) hK |
| simpa [K,mul_assoc] using hh |
| end FiniteEntropySeparation |
| |
| end |
| |
| end OAI |
| |
