File size: 7,622 Bytes
f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 | import OAI.Probability.CubeShuffle.SchattenBounds
namespace OAI
noncomputable section
open scoped BigOperators Classical
namespace CubeShuffle.WeightedSweep
open Specht UnitaryFinite
lemma finite_positive_lower_bound {ι : Type*} [Fintype ι] (f : ι → ℝ)
(hf : ∀ i, 0 < f i) : ∃ a : ℝ, 0 < a ∧ ∀ i, a ≤ f i := by
have hs (s : Finset ι) : ∃ a : ℝ, 0 < a ∧ ∀ i ∈ s, a ≤ f i := by
induction s using Finset.induction_on with
| empty => exact ⟨1, zero_lt_one, by simp⟩
| @insert i s hi ih =>
obtain ⟨a, ha, h⟩ := ih
refine ⟨min a (f i), lt_min ha (hf i), ?_⟩
intro j hj
rcases Finset.mem_insert.mp hj with rfl | hj
· exact min_le_right _ _
· exact (min_le_left _ _).trans (h j hj)
obtain ⟨a, ha, hs⟩ := hs Finset.univ
exact ⟨a, ha, fun i => hs i (Finset.mem_univ i)⟩
lemma uniformOperator_zero_of_dimension_gt_one {G V : Type*} [Group G]
[Fintype G] [NormedAddCommGroup V] [InnerProductSpace ℂ V]
[FiniteDimensional ℂ V] (ρ : Representation ℂ G V)
[Representation.IsIrreducible ρ] (hdim : 1 < Module.finrank ℂ V) :
uniformOperator ρ = 0 := by
apply ContinuousLinearMap.ext
intro v
change uniformOperator ρ v = 0
let u := uniformOperator ρ v
have hfixed (g : G) : ρ g u = u :=
congrArg (fun A : V →L[ℂ] V => A v) (uniformOperator_left ρ g)
let T : Representation.IntertwiningMap (Representation.trivial ℂ G ℂ) ρ :=
{ toLinearMap := LinearMap.toSpanSingleton ℂ V u
isIntertwining' g := by
apply LinearMap.ext
intro z
change z • u = ρ g (z • u)
rw [map_smul, hfixed] }
rcases Representation.IsIrreducible.surjective_or_eq_zero T with hs | hz
· have hd := LinearMap.finrank_le_finrank_of_surjective (f := T.toLinearMap) hs
simp only [Module.finrank_self] at hd
omega
· have he := congrArg (fun S : Representation.IntertwiningMap
(Representation.trivial ℂ G ℂ) ρ => S 1) hz
simpa [T] using he
lemma dimension_full_row (μ : YoungDiagram) (hμ : μ.rowLen 0 = μ.card) :
dimension μ = 1 := by
let := tabloid_subsingleton_of_full_row μ hμ
have h := (space μ).finrank_le
rw [Module.finrank_fintype_fun_eq_card] at h
have hcard : Fintype.card (Tabloid μ) ≤ 1 :=
Fintype.card_le_one_iff.mpr (fun _ _ => Subsingleton.elim _ _)
have he : Module.finrank ℂ (space μ) = 1 := by
have := Specht.dimension_pos μ
omega
simp [dimension, he]
def localExponent (d : ℕ) (μ : Shapes (2^d)) : ℝ :=
(1 - blockGap d) / (Real.log (dimension μ.1) + 1)
lemma localExponent_pos (d : ℕ) (μ : Shapes (2^d)) : 0 < localExponent d μ := by
unfold localExponent
exact div_pos (sub_pos.mpr (blockGap_lt_one d))
(by have := log_dimension_nonneg μ.1; linarith)
lemma palindrome_local_bound (d : ℕ) (μ : Shapes (2^d)) :
‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)‖ ≤
Real.exp (-localExponent d μ * Real.log (dimension μ.1)) := by
let : NormedAddCommGroup (hilbertSpace μ.1) := inferInstance
let : InnerProductSpace ℂ (hilbertSpace μ.1) := inferInstance
let : FiniteDimensional ℂ (hilbertSpace μ.1) := inferInstance
by_cases hdim : dimension μ.1 = 1
· rw [hdim, Real.log_one, mul_zero, Real.exp_zero]
exact sampleOperator_norm_le _ (cardRepresentation_unitary d μ) _
have hgt : 1 < dimension μ.1 :=
lt_of_le_of_ne (dimension_one_le μ.1) (Ne.symm hdim)
have hgt' : 1 < Module.finrank ℂ (hilbertSpace μ.1) := by
rw [finrank_hilbertSpace]
change 1 < (Module.finrank ℂ (space μ.1) : ℝ) at hgt
exact_mod_cast hgt
have hz := uniformOperator_zero_of_dimension_gt_one (V := hilbertSpace μ.1)
(cardRepresentation d μ) hgt'
have hgap := palindrome_off_invariants_gap d (V := hilbertSpace μ.1) (cardRepresentation d μ)
(cardRepresentation_unitary d μ)
rw [hz, sub_zero, ← sampleOperator_weighted] at hgap
have he : localExponent d μ * (Real.log (dimension μ.1) + 1) = 1 - blockGap d := by
exact div_mul_cancel₀ _ (ne_of_gt (by have := log_dimension_nonneg μ.1; linarith))
have hnum : blockGap d - 1 ≤ -localExponent d μ * Real.log (dimension μ.1) := by
have := localExponent_pos d μ
nlinarith
have hexp : blockGap d ≤ Real.exp (blockGap d - 1) := by
simpa using Real.add_one_le_exp (blockGap d - 1)
exact hgap.trans (hexp.trans (Real.exp_le_exp.mpr hnum))
theorem uniform_palindrome_dimension_decay :
∃ a : ℝ, 0 < a ∧ a ≤ 1 ∧ ∀ d : ℕ, 1 ≤ d → ∀ μ : Shapes (2^d),
‖sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ) (palindromePerm d)‖ ≤
Real.exp (-a * Real.log (dimension μ.1)) := by
obtain ⟨a₀, h, ha₀, hh, hevent⟩ := spectral_regimes
obtain ⟨d₀, hlarge⟩ := Filter.eventually_atTop.mp hevent
let ι := (d : Fin d₀) × Shapes (2^d.1)
obtain ⟨a₁, ha₁, hsmall⟩ := finite_positive_lower_bound
(fun i : ι => localExponent i.1.1 i.2) (fun i => localExponent_pos _ _)
let a := min 1 (min a₀ a₁)
have ha : 0 < a := lt_min zero_lt_one (lt_min ha₀ ha₁)
have ha₀' : a ≤ a₀ := (min_le_right _ _).trans (min_le_left _ _)
have ha₁' : a ≤ a₁ := (min_le_right _ _).trans (min_le_right _ _)
refine ⟨a, ha, min_le_left _ _, ?_⟩
intro d hd μ
by_cases hdim : dimension μ.1 = 1
· rw [hdim, Real.log_one, mul_zero, Real.exp_zero]
exact sampleOperator_norm_le _ (cardRepresentation_unitary d μ) _
by_cases hd₀ : d₀ ≤ d
· by_cases hc : (μ.1.card : ℝ) / 2 < μ.1.colLen 0
· have hz := palindrome_tall_zero_general d (by omega) μ.1 (cardLabels d μ) hc
change sampleOperator (V := hilbertSpace μ.1) (cardRepresentation d μ)
(palindromePerm d) = 0 at hz
rw [hz, norm_zero (E := (hilbertSpace μ.1 →L[ℂ] hilbertSpace μ.1))]
exact (Real.exp_pos _).le
· have hj : 0 < μ.1.card - μ.1.rowLen 0 := by
by_contra hn
have hrow : μ.1.rowLen 0 = μ.1.card := by
have := rowLen_zero_le_card μ.1
omega
exact hdim (dimension_full_row μ.1 hrow)
have hspec := (hlarge d hd₀ μ.1 (cardLabels d μ) hj (not_lt.mp hc)).1
apply hspec.trans (Real.exp_le_exp.mpr ?_)
have hmul := mul_le_mul_of_nonneg_right ha₀' (log_dimension_nonneg μ.1)
simpa only [neg_mul, dimension] using neg_le_neg hmul
· have hbase := palindrome_local_bound d μ
apply hbase.trans (Real.exp_le_exp.mpr ?_)
have hs := ha₁'.trans (hsmall ⟨⟨d, by omega⟩, μ⟩)
have hmul := mul_le_mul_of_nonneg_right hs (log_dimension_nonneg μ.1)
nlinarith
theorem uniform_sweep_dimension_decay :
∃ b : ℝ, 0 < b ∧ b ≤ 1 ∧ ∀ d : ℕ, 1 ≤ d → ∀ μ : Shapes (2^d),
‖fourierSweep d μ‖ ≤ dimension μ.1 ^ (-b) := by
obtain ⟨a, ha, ha1, h⟩ := uniform_palindrome_dimension_decay
refine ⟨a / 2, by positivity, by linarith, ?_⟩
intro d hd μ
have hs := h d hd μ
rw [← physical_palindrome_norm d _ (cardRepresentation_unitary d μ)] at hs
have he : (Real.exp (-a * Real.log (dimension μ.1) / 2)) ^ 2 =
Real.exp (-a * Real.log (dimension μ.1)) := by
rw [pow_two, ← Real.exp_add]
congr 1
ring
have hnorm : ‖fourierSweep d μ‖ ≤ Real.exp (-a * Real.log (dimension μ.1) / 2) := by
apply le_of_sq_le_sq ?_ (Real.exp_pos _).le
exact hs.trans_eq he.symm
convert hnorm using 1
rw [Real.rpow_def_of_pos (dimension_pos μ.1)]
congr 1
ring
end CubeShuffle.WeightedSweep
end
end OAI
|