| 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 |
|
|
