| import Mathlib |
| import OAI.Probability.SKGap.Matrix.ConditionalEmpiricalNorm |
| import OAI.Probability.SKGap.Localization.SparseShrink |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Real Matrix MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| theorem empirical_conditional_wellDefined {j R t0 T : ℝ} |
| (hj : 0 < j) (hj1 : j < 1) (hR : 0 ≤ R) (ht0 : 0 < t0) : |
| ∃ U : Set (ScalarTimeDomain R t0 T), IsOpen U ∧ scalarEqualitySet j R t0 T ⊆ U ∧ |
| ∃ σ0 ε0 : ℝ, 0 < σ0 ∧ 0 < ε0 ∧ ∀ x ∈ U,∀ σ : ℝ, |σ| ≤ σ0 → |
| ∀ (n : ℕ) [NeZero n] (y a : Fin n → ℝ), empiricalLaw y=(x.1:ProbabilityMeasure ℝ) → |
| (∑ i,(a i-(1-tanh (y i)^2))^2) ≤ ε0^2*(n:ℝ) → |
| 0 < scalarQMoment (empiricalLaw y) ∧ 0 < scalarS j (empiricalLaw y,(x.2:ℝ),σ) ∧ |
| 0 < j/(n:ℝ)*∑ i,a i := by |
| obtain ⟨L,hL,U,hU,hKU,δ,hδ,hnb⟩ := scalar_conditional_neighborhood (R := R) (T := T) |
| hj hj1 hR ht0 (by norm_num : (0:ℝ)<1) |
| let ε := δ/(j+1) |
| have hε : 0 < ε := by dsimp [ε]; positivity |
| have hεδ : j*ε ≤ δ := by |
| have hh : (j+1)*ε=δ := by dsimp [ε]; field_simp |
| nlinarith only [hh,hε.le] |
| refine ⟨U,hU,hKU,δ,ε,hδ,hε,?_⟩ |
| intro x hx σ hσ n _ y a hP hclose |
| let r := j/(n:ℝ)*∑ i,a i-j*scalarBMoment (empiricalLaw y) |
| have hr : |r| ≤ δ := (empirical_mass_close a y hj.le hε.le hclose).trans hεδ |
| have hh := hnb x hx σ r hσ hr |
| rw [← hP] at hh |
| refine ⟨hh.1,hh.2.1,?_⟩ |
| have hB : j*scalarBMoment (empiricalLaw y)+r=j/(n:ℝ)*∑ i,a i := by dsimp [r]; ring |
| simpa only [hB] using hh.2.2.1 |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| lemma four_radius_sq_le {ε δ : ℝ} (hε : 0 ≤ ε) (hεδ : ε ≤ δ/2) : |
| 4*ε^2 ≤ δ^2 := by |
| have hh := mul_self_le_mul_self (by linarith : 0 ≤ 2*ε) (by linarith : 2*ε ≤ δ) |
| nlinarith only [hh] |
|
|
| theorem conditional_empirical_simultaneous {j A R t0 T : ℝ} |
| (hj : 0 < j) (hj1 : j < 1) (hA : 1 ≤ A) (hsub : sqrt j*A < 1) |
| (hR : 0 ≤ R) (ht0 : 0 < t0) : |
| ∃ m ν σ0 : ℝ, 0 < m ∧ 0 < ν ∧ 0 < σ0 ∧ |
| ∃ U : Set (ScalarTimeDomain R t0 T), IsOpen U ∧ scalarEqualitySet j R t0 T ⊆ U ∧ |
| ∀ R1 : ℝ, 1 ≤ R1 → ∃ ε c : ℝ, ∃ N : ℕ, 0 < ε ∧ 0 < c ∧ 0 < N ∧ |
| ∀ (n : ℕ) [NeZero n], N ≤ n → ∀ x ∈ U, ∀ y : Fin n → ℝ, |
| empiricalLaw y=(x.1 : ProbabilityMeasure ℝ) → |
| (∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ) → |
| ∀ σ : ℝ, |σ| ≤ σ0 → |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n)=>gaussianReal 0 1)).real |
| {g | ∃ a, (∀ i,a i ∈ Icc 0 A) ∧ |
| (∑ i,(a i-(1-tanh (y i)^2))^2) ≤ ε^2*(n:ℝ) ∧ |
| ¬(diagonalHessian a (empiricalConditionalMiddle j x.2 σ y g)- |
| m • (1 : Matrix (Fin n) (Fin n) ℝ)).PosSemidef} ≤ |
| 103*exp (-c*(n:ℝ)) := by |
| obtain ⟨m,ν,σf,hm,hν,hσf,Uf,hUf,hKUf,hfixed⟩ := |
| conditional_empirical_fixed_tail (T := T) hj hj1 hA hsub hR ht0 |
| obtain ⟨Uw,hUw,hKUw,σw,εw,hσw,hεw,hwell⟩ := |
| empirical_conditional_wellDefined (T := T) hj hj1 hR ht0 |
| obtain ⟨L,hL,hnorm⟩ := empiricalConditionalMiddle_uniform_norm (T := T) hj.le hR ht0 (2*sqrt j+1) |
| let σ0 := min 1 (min σf σw) |
| have hσ0 : 0 < σ0 := by dsimp [σ0]; positivity |
| refine ⟨m/2,ν,σ0,half_pos hm,hν,hσ0,Uf ∩ Uw,hUf.inter hUw,?_,?_⟩ |
| · exact fun x hx=>⟨hKUf hx,hKUw hx⟩ |
| intro R1 hR1 |
| obtain ⟨εf,c,N,hεf,hc,hN,hfix⟩ := hfixed R1 hR1 |
| obtain ⟨εs,hεs,hsparse⟩ := conditional_sparse_uniform hj (by linarith : 0 < A) |
| (by linarith : 0 < L) hm hc |
| let ε := min εs (min (εf/2) (εw/2)) |
| have hε : 0 < ε := by dsimp [ε]; positivity |
| have hεs' : ε ≤ εs := min_le_left _ _ |
| have hεf' : ε ≤ εf/2 := (min_le_right _ _).trans (min_le_left _ _) |
| have hεw' : ε ≤ εw/2 := (min_le_right _ _).trans (min_le_right _ _) |
| let c0 := min (1/(π^2*j)) (c/2) |
| have hc0 : 0 < c0 := by dsimp [c0]; positivity |
| refine ⟨ε,c0,N,hε,hc0,hN,?_⟩ |
| intro n _ hn x hx y hP hy σ hσ |
| have hσ1 : |σ| ≤ 1 := hσ.trans (min_le_left _ _) |
| have hσf' : |σ| ≤ σf := hσ.trans ((min_le_right _ _).trans (min_le_left _ _)) |
| have hσw' : |σ| ≤ σw := hσ.trans ((min_le_right _ _).trans (min_le_right _ _)) |
| let v : Fin n → ℝ := fun i=>1-tanh (y i)^2 |
| have hv (i : Fin n) : v i ∈ Icc 0 A := ⟨by dsimp [v]; linarith [tanh_sq_lt_one (y i)], |
| by dsimp [v]; linarith [sq_nonneg (tanh (y i))]⟩ |
| have hzero : (∑ i,(v i-(1-tanh (y i)^2))^2) ≤ εw^2*(n:ℝ) := by |
| simp only [v,sub_self,zero_pow (by norm_num : 2 ≠ 0),Finset.sum_const_zero] |
| positivity |
| have hpos := hwell x hx.2 σ hσw' n y v hP hzero |
| have hnorm' : ∀ g,opNorm (goeMatrix (j/(n:ℝ)) g) ≤ 2*sqrt j+1 → |
| opNorm (empiricalConditionalMiddle j x.2 σ y g) ≤ L := |
| hnorm n y (by rw [hP]; exact x.1.2) hpos.1 x.2 x.2.2 σ hσ1 |
| apply hsparse ε hε hεs' n v hv (empiricalConditionalMiddle j x.2 σ y) |
| (empiricalConditionalMiddle_hermitian j x.2 σ y) hnorm' |
| intro b hb hbclose |
| have hclosef : (∑ i,(b i-(1-tanh (y i)^2))^2) ≤ εf^2*(n:ℝ) := |
| hbclose.trans (mul_le_mul_of_nonneg_right (four_radius_sq_le hε.le hεf') (Nat.cast_nonneg n)) |
| have hclosew : (∑ i,(b i-(1-tanh (y i)^2))^2) ≤ εw^2*(n:ℝ) := |
| hbclose.trans (mul_le_mul_of_nonneg_right (four_radius_sq_le hε.le hεw') (Nat.cast_nonneg n)) |
| have hposb := hwell x hx.2 σ hσw' n y b hP hclosew |
| have heq (g : MatrixCoordinates (Fin n) → ℝ) := conditionalEmpiricalCore_eq_hessian j x.2 σ b y g |
| (fun i=>(hb i).1) hposb.1 hposb.2.1 hposb.2.2 |
| have hh := hfix n hn x hx.1 y hP hy σ hσf' b (fun i=>(hb i).1) (fun i=>(hb i).2) hclosef |
| simp only [heq] at hh |
| exact hh |
| end SKGap |
| end |
| end |
|
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| end OAI |
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