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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Matrix.GOESubmatrix
import OAI.Probability.SKGap.Entropy.SparseSubsetEntropy
namespace OAI
section
noncomputable section
namespace SKGap
open MeasureTheory ProbabilityTheory Matrix Real Set
open scoped BigOperators RealInnerProductSpace
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
def principalOperatorNorm (M : Matrix ι ι ℝ) (S : Finset ι) : ℝ :=
‖matrixOperator (M.submatrix (fun i : S=>i.1) (fun i : S=>i.1))‖
omit [Fintype ι] in
lemma principalOperatorNorm_empty (M : Matrix ι ι ℝ) : principalOperatorNorm M ∅=0 := by
unfold principalOperatorNorm
have hz : matrixOperator (M.submatrix (fun i : (∅ : Finset ι)=>i.1) (fun i : (∅ : Finset ι)=>i.1))=0 :=
Subsingleton.elim _ _
rw [hz,norm_zero]
lemma goe_sparse_one_tail [Nonempty ι] (S : Finset ι) {j α ε : ℝ}
(hj : 0 < j) (_hα : 0 ≤ α) (hε : 0 < ε) (hS : (S.card:ℝ) ≤ α*(Fintype.card ι:ℝ))
(hedge : 2*sqrt (j*α) ≤ ε/2) :
(gaussianCoordinates (MatrixCoordinates ι)).real
{g | ε < principalOperatorNorm (goeMatrix (j/(Fintype.card ι:ℝ)) g) S} ≤
2*exp (-(Fintype.card ι:ℝ)*ε^2/(4*π^2*j)) := by
by_cases hne : S.Nonempty
· let : Nonempty S := Finset.Nonempty.to_subtype hne
have hn : (0:ℝ) < Fintype.card ι := Nat.cast_pos.mpr Fintype.card_pos
let e : S ↪ ι := ⟨Subtype.val,Subtype.val_injective⟩
have hsqrt : 2*sqrt (j/(Fintype.card ι:ℝ)*(Fintype.card S:ℝ)) ≤ ε/2 := by
apply le_trans _ hedge
apply mul_le_mul_of_nonneg_left (sqrt_le_sqrt _) (by norm_num)
rw [Fintype.card_coe]
have hm := mul_le_mul_of_nonneg_left hS hj.le
rw [div_mul_eq_mul_div]
apply (div_le_iff₀ hn).mpr
convert hm using 1; ring
have ht := goe_restricted_norm_tail e (div_pos hj hn) (by positivity : 0 ≤ ε/2)
apply (measureReal_mono (μ:=gaussianCoordinates (MatrixCoordinates ι)) ?_).trans
(ht.trans_eq ?_)
· intro g hg
change ε < principalOperatorNorm (goeMatrix (j/(Fintype.card ι:ℝ)) g) S at hg
change 2*sqrt (j/(Fintype.card ι:ℝ)*(Fintype.card S:ℝ))+ε/2 <
principalOperatorNorm (goeMatrix (j/(Fintype.card ι:ℝ)) g) S
linarith only [hsqrt,hg]
· congr 2
field_simp
ring
· have he : S=∅ := Finset.not_nonempty_iff_eq_empty.mp hne
subst S
simp only [principalOperatorNorm_empty,not_lt_of_ge hε.le,Set.ofPred_false,measureReal_empty]
positivity
theorem goe_sparse_union_tail [Nonempty ι] {j α ε l : ℝ}
(hj : 0 < j) (hα : 0 ≤ α) (hε : 0 < ε) (hl : 0 ≤ l)
(hedge : 2*sqrt (j*α) ≤ ε/2) :
(gaussianCoordinates (MatrixCoordinates ι)).real
{g | ∃ S : Finset ι,(S.card:ℝ) ≤ α*(Fintype.card ι:ℝ) ∧
ε < principalOperatorNorm (goeMatrix (j/(Fintype.card ι:ℝ)) g) S} ≤
2*exp (-(Fintype.card ι:ℝ)*(ε^2/(4*π^2*j)-(l*α+log (1+exp (-l))))) := by
classical
have he : {g : MatrixCoordinates ι→ℝ | ∃ S : Finset ι,(S.card:ℝ) ≤ α*(Fintype.card ι:ℝ) ∧
ε < principalOperatorNorm (goeMatrix (j/(Fintype.card ι:ℝ)) g) S} =
⋃ S∈smallSupports (ι:=ι) α,{g | ε < principalOperatorNorm (goeMatrix (j/(Fintype.card ι:ℝ)) g) S} := by
ext g
simp [smallSupports]
rw [he]
apply sparse_union_bound _ _ hl (by norm_num : (0:ℝ) ≤ 2)
intro S hS
have hc := (Finset.mem_filter.mp hS).2
convert goe_sparse_one_tail S hj hα hε hc hedge using 1
congr 2
ring
end SKGap
end
end
section
noncomputable section
namespace SKGap
open Matrix Real
open scoped BigOperators RealInnerProductSpace Matrix.Norms.L2Operator
variable {ι : Type*} [Fintype ι] [DecidableEq ι]
omit [DecidableEq ι] in
lemma supported_quad_eq (M : Matrix ι ι ℝ) (S : Finset ι)
(w : EuclideanSpace ℝ ι) (hw : ∀ i,i ∉ S → w i=0) :
matrixQuad M w=matrixQuad (M.submatrix (fun i : S=>i.1) (fun i : S=>i.1))
(WithLp.toLp 2 (fun i : S=>w i.1)) := by
change (∑ i,∑ k,w i*M i k*w k)=(∑ i : S,∑ k : S,w i.1*M i.1 k.1*w k.1)
calc
_ = ∑ i∈S,∑ k,w i*M i k*w k := (Finset.sum_subset (Finset.subset_univ S) (by
intro i _ hi
simp [hw i hi])).symm
_ = ∑ i : S,∑ k,w i.1*M i.1 k*w k :=
(Finset.sum_coe_sort S (fun i=>∑ k,w i*M i k*w k)).symm
_ = _ := by
apply Finset.sum_congr rfl
intro i _
rw [Finset.sum_coe_sort S (fun k=>w i.1*M i.1 k*w k)]
exact (Finset.sum_subset (Finset.subset_univ S) (by
intro k _ hk
simp [hw k hk])).symm
omit [DecidableEq ι] in
lemma supported_norm_sq_eq (S : Finset ι) (w : EuclideanSpace ℝ ι)
(hw : ∀ i,i ∉ S → w i=0) :
‖WithLp.toLp 2 (fun i : S=>w i.1)‖^2=‖w‖^2 := by
rw [EuclideanSpace.real_norm_sq_eq,EuclideanSpace.real_norm_sq_eq]
change (∑ i : S,w i.1^2)=(∑ i,w i^2)
rw [Finset.sum_coe_sort S (fun i=>w i^2)]
exact Finset.sum_subset (Finset.subset_univ S) (by intro i _ hi;simp [hw i hi])
lemma supported_quad_le_principal (M : Matrix ι ι ℝ) (S : Finset ι) {ε : ℝ}
(hM : principalOperatorNorm M S ≤ ε) (w : EuclideanSpace ℝ ι)
(hw : ∀ i,i ∉ S → w i=0) :
|⟪w,Matrix.toEuclideanCLM (𝕜:=ℝ) (n:=ι) M w⟫| ≤ ε*‖w‖^2 := by
rw [← matrixOperator_eq_toEuclideanCLM,← matrixQuad_inner,supported_quad_eq M S w hw,matrixQuad_inner]
let v := WithLp.toLp 2 (fun i : S=>w i.1)
calc
_ ≤ ‖v‖*‖matrixOperator (M.submatrix (fun i : S=>i.1) (fun i : S=>i.1)) v‖ :=
abs_real_inner_le_norm _ _
_ ≤ ‖v‖*(principalOperatorNorm M S*‖v‖) :=
mul_le_mul_of_nonneg_left ((matrixOperator _).le_opNorm v) (norm_nonneg v)
_ ≤ ‖v‖*(ε*‖v‖) := mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_right hM (norm_nonneg v)) (norm_nonneg v)
_ = ε*‖w‖^2 := by
rw [show ‖v‖*(ε*‖v‖)=ε*‖v‖^2 by ring,supported_norm_sq_eq S w hw]
end SKGap
end
end
end OAI
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