| import Mathlib |
| import OAI.Probability.GaussianInformation.Dyadic.LabelBounds |
|
|
| namespace OAI |
| open MeasureTheory ProbabilityTheory Set |
| open scoped BigOperators ENNReal |
| open MeasureTheory Metric Set |
| open scoped ENNReal NNReal |
| open MeasureTheory Set Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal Topology |
| open InnerProductSpace Matrix MeasureTheory Set |
| open scoped RealInnerProductSpace ENNReal |
| open MeasureTheory ProbabilityTheory Set |
| open scoped ENNReal NNReal |
| namespace RepeatedLocalization |
| lemma normalized_block_information {n k m : ℕ} (hn : 0 < n) (hk : 0 < k) (hm : 1 < m) |
| (hdim : m-2+2*(k+1) ≤ n/2) (hkn : k ≤ n) (Q : Cell n) |
| (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] |
| (hsupport : μ (cellSet Q)ᶜ = 0) {B : ℝ} (hB : 1 ≤ B) (hspread : DyadicSpread μ Q B) |
| {ι : Type*} [Fintype ι] [Nonempty ι] |
| (g : (Fin k → Fin (n+1) → ℝ) × (Fin k → ℝ) → ι → ℝ) |
| (hg : ∀ i,Measurable (fun q => g q i)) (hg0 : ∀ q i,0 ≤ g q i) |
| (hgs : ∀ q,∑ i,g q i = 1) : |
| finiteInformation μ (fun i z => ∫ X,g (X,projectVector X (normalizedChart Q z)) i ∂gaussianRows k (n+1)) ≤ |
| blockInformationConstant*(n+1)+Real.log B+ |
| (2/(m:ℝ)+Real.exp (-(k:ℝ)/2))*Real.log (Fintype.card ι) := by |
| classical |
| let ν := gaussianRows k (n+1) |
| let S := goodRegion k (n+1) |
| have hS : MeasurableSet S := measurableSet_goodRegion _ _ |
| have : IsProbabilityMeasure ν := gaussianRows_probability _ _ |
| have hg1 : ∀ q i,g q i ≤ 1 := by |
| intro q i |
| rw [← hgs q] |
| exact Finset.single_le_sum (fun j _ => hg0 q j) (Finset.mem_univ i) |
| let a := fun q : (Fin k → Fin (n+1) → ℝ) × Coordinate n => (q.1,projectVector q.1 (normalizedChart Q q.2)) |
| have ha : Measurable a := measurable_fst.prodMk (continuous_projectVector.measurable.comp |
| (measurable_fst.prodMk ((normalizedChart_measurable Q).comp measurable_snd))) |
| let gp := fun q i => S.indicator (fun q => g q i) q |
| let gq := fun q i => Sᶜ.indicator (fun q => g q i) q |
| have hgp : ∀ i,Measurable (fun q => gp q i) := fun i => (hg i).indicator hS |
| have hgq : ∀ i,Measurable (fun q => gq q i) := fun i => (hg i).indicator hS.compl |
| have hgp0 : ∀ q i,0 ≤ gp q i := fun q _ => Set.indicator_nonneg (fun _ _ => hg0 _ _) q |
| have hgq0 : ∀ q i,0 ≤ gq q i := fun q _ => Set.indicator_nonneg (fun _ _ => hg0 _ _) q |
| have hgp1 : ∀ q i,gp q i ≤ 1 := by intro q i; by_cases h : q ∈ S <;> simp [gp,h,hg1] |
| have hgq1 : ∀ q i,gq q i ≤ 1 := by intro q i; by_cases h : q ∈ S <;> simp [gq,h,hg1] |
| let p := averagedScores ν (fun q i => gp (a q) i) |
| let q := averagedScores ν (fun q i => gq (a q) i) |
| have hp : ∀ i,Measurable (p i) := measurable_averagedScores _ _ (fun i => (hgp i).comp ha) |
| have hq : ∀ i,Measurable (q i) := measurable_averagedScores _ _ (fun i => (hgq i).comp ha) |
| have hpb := averagedScores_bounds ν (fun q i => gp (a q) i) (fun i => (hgp i).comp ha) (fun _ _ => hgp0 _ _) (fun _ _ => hgp1 _ _) |
| have hqb := averagedScores_bounds ν (fun q i => gq (a q) i) (fun i => (hgq i).comp ha) (fun _ _ => hgq0 _ _) (fun _ _ => hgq1 _ _) |
| have hpq : ∀ i z,p i z+q i z = ∫ X,g (X,projectVector X (normalizedChart Q z)) i ∂ν := by |
| intro i z |
| dsimp only [p,q,averagedScores] |
| rw [← integral_add] |
| rotate_left |
| · exact integrable_score_section ν _ ((hgp i).comp ha) (fun _ => hgp0 _ _) (fun _ => hgp1 _ _) z |
| · exact integrable_score_section ν _ ((hgq i).comp ha) (fun _ => hgq0 _ _) (fun _ => hgq1 _ _) z |
| apply integral_congr_ae |
| filter_upwards [] with X |
| have he : gp (a (X,z)) i+gq (a (X,z)) i = g (a (X,z)) i := by |
| by_cases h : a (X,z) ∈ S <;> simp [gp,gq,h] |
| exact he |
| have hpqs : ∀ z,∑ i,(p i z+q i z) = 1 := by |
| intro z |
| simp_rw [hpq] |
| exact averagedScores_sum ν (fun q i => g (a q) i) (fun i => (hg i).comp ha) (fun _ _ => hg0 _ _) (fun _ _ => hg1 _ _) (fun _ => hgs _) z |
| have hvol : ν.prod volume S = volume (goodLabels k) := reference_goodRegion_volume _ _ |
| let v := (ν.prod volume).real S |
| have hvp : 0 < v := ENNReal.toReal_pos (by rw [hvol]; exact (volume_goodLabels_pos hk).ne') (by rw [hvol]; exact volume_goodLabels_ne_top _) |
| let b := fun i => (∫ q in S,g q i ∂ν.prod volume)/v |
| obtain ⟨hb0,hbs,hbi⟩ := finite_reference_scores (ν.prod volume) S hS |
| (by rw [hvol]; exact volume_goodLabels_ne_top _) (by rw [hvol]; exact volume_goodLabels_pos hk) g hg hg0 hg1 hgs |
| change (∀ i,0 ≤ b i) at hb0 |
| change (∑ i,b i = 1) at hbs |
| change ∀ i,(∫⁻ q,ENNReal.ofReal (gp q i) ∂ν.prod volume) = ENNReal.ofReal (v*b i) at hbi |
| let A := (4*tubeConstant*labelVolumeConstant)^(n+1)*Real.sqrt B |
| let r : ℝ := 1-1/(m:ℝ) |
| have hm0 : (0:ℝ) < m := by exact_mod_cast lt_trans Nat.zero_lt_one hm |
| have hm1 : (1:ℝ) < m := by exact_mod_cast hm |
| have hr : 0 < r := by dsimp [r]; rw [sub_pos]; exact (div_lt_one hm0).mpr hm1 |
| have hr1 : r ≤ 1 := by dsimp [r]; have := one_div_nonneg.mpr hm0.le; linarith |
| have hbase : 1 ≤ 4*tubeConstant*labelVolumeConstant := by nlinarith [tubeConstant_ge_one,labelVolumeConstant_ge_one] |
| have hA : 1 ≤ A := one_le_mul_of_one_le_of_one_le (one_le_pow₀ hbase) ((Real.le_sqrt (by norm_num) (by linarith : 0 ≤ B)).mpr (by simpa using hB)) |
| have hbound : ∀ i,(∫ z,(p i z)^2 ∂μ) ≤ (A*(b i)^r)^2 := by |
| intro i |
| have H := actual_good_score_l2bound hn hk hm hdim Q μ hsupport (by linarith : 0 ≤ B) hspread |
| (fun q => gp q i) (hgp i) (fun _ => hgp0 _ _) (fun _ => hgp1 _ _) (by rw [hbi i]; exact ENNReal.ofReal_ne_top) |
| apply H.trans |
| apply pow_le_pow_left₀ ENNReal.toReal_nonneg |
| rw [ENNReal.toReal_mul,← ENNReal.toReal_rpow,hbi i,← ENNReal.toReal_rpow,ENNReal.toReal_ofReal (mul_nonneg hvp.le (hb0 i)),weightedProjectionMoment_toReal (by linarith)] |
| rw [Real.mul_rpow hvp.le (hb0 i),← mul_assoc] |
| apply mul_le_mul_of_nonneg_right _ (Real.rpow_nonneg (hb0 i) _) |
| exact projection_constant_bound hm hkn hB tubeConstant_ge_one labelVolumeConstant_ge_one hvp.le |
| (by simpa only [v,measureReal_def,hvol] using volume_goodLabels_le hk) |
| have hbad : ∑ i,scoreMass μ (q i) ≤ Real.exp (-(k:ℝ)/2) := by |
| have hqi (i : ι) : Integrable (q i) μ := integrable_bounded_measurable μ _ (hq i) (C := 1) |
| (fun z => by rw [abs_of_nonneg (hqb.1 i z)]; exact hqb.2 i z) |
| unfold scoreMass |
| rw [← integral_finsetSum _ (fun i _ => hqi i)] |
| apply le_trans (integral_mono_ae (integrable_finsetSum _ (fun i _ => hqi i)) (integrable_const (Real.exp (-(k:ℝ)/2))) ?_) (by simp) |
| have hsupp : ∀ᵐ z ∂μ,z ∈ cellSet Q := ae_iff.mpr hsupport |
| filter_upwards [hsupp] with z hz |
| change ∑ i,(∫ X,gq (a (X,z)) i ∂ν) ≤ Real.exp (-(k:ℝ)/2) |
| rw [← integral_finsetSum] |
| rotate_left |
| · intro i _ |
| exact integrable_score_section ν _ ((hgq i).comp ha) (fun _ => hgq0 _ _) (fun _ => hgq1 _ _) z |
| have he : (fun X => ∑ i,gq (a (X,z)) i) = {X | projectVector X (normalizedChart Q z) ∉ goodLabels k}.indicator (fun _ => (1:ℝ)) := by |
| funext X |
| by_cases h : projectVector X (normalizedChart Q z) ∈ goodLabels k |
| · simp [gq,a,S,goodRegion,h] |
| · simp [gq,a,S,goodRegion,h,hgs] |
| rw [he,integral_indicator] |
| · simp only [integral_const,smul_eq_mul,mul_one] |
| apply ENNReal.toReal_le_of_le_ofReal (Real.exp_pos _).le |
| simpa only [Measure.restrict_apply_univ] using (normalized_label_tail (k := k) hn Q hz) |
| · exact ((measurableSet_goodLabels k).preimage (continuous_projectVector.measurable.comp (measurable_id.prodMk measurable_const))).compl |
| have H := finiteInformation_good_bad_bound μ p q hp hq hpb.1 hqb.1 hpqs b hb0 hbs hA hr hr1 hbound hbad |
| have hlogA : 2*Real.log A = 2*(n+1)*Real.log (4*tubeConstant*labelVolumeConstant)+Real.log B := by |
| rw [Real.log_mul (by positivity) (Real.sqrt_pos.mpr (by linarith)).ne',Real.log_pow,Real.log_sqrt (by linarith)] |
| push_cast |
| ring |
| simp_rw [hpq] at H |
| rw [hlogA] at H |
| dsimp only [r] at H |
| dsimp only [ν] at H |
| have he : 2-2*(1-1/(m:ℝ)) = 2/(m:ℝ) := by ring |
| rw [he] at H |
| unfold blockInformationConstant |
| have hnn : (0:ℝ) ≤ n := Nat.cast_nonneg _ |
| nlinarith |
|
|
| lemma exactLabel_normalization {n : ℕ} (Q : Cell n) (X : GaussianBlock n) (z : Coordinate n) : |
| (fun r => (2:ℝ)^(- (Q.1:ℤ))*projectVector X (normalizedChart Q z) r+exactLabel X (cellCenter Q) r) = |
| exactLabel X z := by |
| funext r |
| have hp : ((2:ℝ)^Q.1) ≠ 0 := by positivity |
| simp only [projectVector,normalizedChart,PiLp.smul_apply,smul_eq_mul,PiLp.sub_apply,eChart,exactLabel] |
| have he : ∑ j,X r j*((2:ℝ)^Q.1*(chart z j-chart (cellCenter Q) j)) = |
| (2:ℝ)^Q.1*((∑ j,X r j*chart z j)-(∑ j,X r j*chart (cellCenter Q) j)) := by |
| rw [mul_sub,Finset.mul_sum,Finset.mul_sum,← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro j _ |
| ring |
| rw [he] |
| simp only [_root_.zpow_neg,zpow_natCast] |
| field_simp |
| ring |
|
|
| lemma actual_block_information {n N : ℕ} (hn : 32 ≤ n) (hN : 1 ≤ N) |
| (Q : Cell n) (μ : Measure (Coordinate n)) [IsProbabilityMeasure μ] |
| (hsupport : μ (cellSet Q)ᶜ = 0) {B : ℝ} (hB : 1 ≤ B) (hspread : DyadicSpread μ Q B) |
| (rule : BlockRule n N) : |
| finiteInformation μ (fun w z => blockProbability rule z w) ≤ |
| blockInformationConstant*(n+1)+Real.log B+ |
| (2/(n/8:ℕ)+Real.exp (-(n/16:ℕ)/2))*Real.log N := by |
| classical |
| have : Nonempty (Fin N) := ⟨⟨0,by omega⟩⟩ |
| let a : GaussianBlock n × Label n → GaussianBlock n × Label n := |
| fun q => (q.1,fun r => (2:ℝ)^(- (Q.1:ℤ))*q.2 r+exactLabel q.1 (cellCenter Q) r) |
| have ha : Measurable a := by |
| apply measurable_fst.prodMk |
| apply Measurable.of_eval |
| intro r |
| exact (measurable_const.mul (measurable_pi_apply r |>.comp measurable_snd)).add |
| ((measurable_pi_apply r).comp ((continuous_exactLabel n).measurable.comp |
| (measurable_fst.prodMk measurable_const))) |
| let g := fun q w => rule.prob (a q) w |
| have h := normalized_block_information (n := n) (k := n/16) (m := n/8) |
| (by omega) (by omega) (by omega) (by omega) (by omega) Q μ hsupport hB hspread |
| g (fun w => (rule.measurable_prob w).comp ha) (fun q w => rule.nonneg (a q) w) |
| (fun q => rule.sum_one (a q)) |
| have heg (w : Fin N) (z : Coordinate n) (X : GaussianBlock n) : |
| g (X,projectVector X (normalizedChart Q z)) w = rule.prob (X,exactLabel X z) w := by |
| dsimp [g,a] |
| rw [exactLabel_normalization] |
| simp_rw [heg] at h |
| simpa only [gaussianRows,gaussianBlockLaw,blockProbability,Fintype.card_fin] using h |
|
|
| end RepeatedLocalization |
|
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| end OAI |
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