import OAI.Probability.GaussianRegression.Nets import OAI.Probability.GaussianRegression.Projection namespace OAI /-! Finite-volume enclosures for Gaussian matrix observations. -/ open MeasureTheory ProbabilityTheory Set Filter open scoped ENNReal NNReal RealInnerProductSpace Topology open InnerProductSpace noncomputable section universe u namespace NoiselessRegression open scoped Pointwise def coordinateNorm {m : ℕ} (y : Fin m → ℝ) : ℝ := ‖WithLp.toLp 2 y‖ @[fun_prop] lemma continuous_coordinateNorm {m : ℕ} : Continuous (coordinateNorm (m := m)) := by unfold coordinateNorm fun_prop lemma coordinateNorm_sq {m : ℕ} (y : Fin m → ℝ) : coordinateNorm y ^ 2 = ∑ j, (y j)^2 := by exact EuclideanSpace.real_norm_sq_eq (WithLp.toLp 2 y) lemma coordinateNorm_smul {m : ℕ} (c : ℝ) (y : Fin m → ℝ) : coordinateNorm (c • y) = |c| * coordinateNorm y := by simp [coordinateNorm, WithLp.toLp_smul, norm_smul, Real.norm_eq_abs] def coordinateBall (m : ℕ) (r : ℝ) : Set (Fin m → ℝ) := {y | coordinateNorm y ≤ r} lemma measurableSet_coordinateBall (m : ℕ) (r : ℝ) : MeasurableSet (coordinateBall m r) := measurableSet_le continuous_coordinateNorm.measurable measurable_const lemma coordinateBall_smul (m : ℕ) (c r : ℝ) (hc : 0 < c) : coordinateBall m (c * r) = c • coordinateBall m r := by ext y constructor · intro hy refine ⟨c⁻¹ • y, ?_, by simp [hc.ne']⟩ change coordinateNorm (c⁻¹ • y) ≤ r rw [coordinateNorm_smul, abs_of_pos (inv_pos.mpr hc)] exact (inv_mul_le_iff₀ hc).mpr hy · rintro ⟨y, hy, rfl⟩ change coordinateNorm (c • y) ≤ c * r rw [coordinateNorm_smul, abs_of_pos hc] exact mul_le_mul_of_nonneg_left hy hc.le lemma volume_coordinateBall_smul (m : ℕ) (c r : ℝ) (hc : 0 < c) : volume (coordinateBall m (c * r)) = ENNReal.ofReal (c^m) * volume (coordinateBall m r) := by rw [coordinateBall_smul m c r hc, Measure.addHaar_smul_of_nonneg volume hc.le] simp lemma gaussianPDF_le_exp (x : ℝ) : gaussianPDF 0 1 x ≤ ENNReal.ofReal (Real.exp (-(x^2)/2)) := by apply ENNReal.ofReal_le_ofReal unfold gaussianPDFReal simp only [NNReal.coe_one, mul_one, sub_zero] apply mul_le_of_le_one_left (Real.exp_nonneg _) apply inv_le_one_of_one_le₀ apply (Real.le_sqrt (by norm_num) (by positivity)).mpr nlinarith [Real.pi_gt_three] lemma lintegral_quarterGaussian_le : (∫⁻ x : ℝ, ENNReal.ofReal (Real.exp (-(x^2)/4))) ≤ 4 := by have he : (fun x : ℝ => Real.exp (-(x^2)/4)) = (fun x : ℝ => Real.exp (-(1/4 : ℝ) * x^2)) := by ext x congr 1 ring simp_rw [show ∀ x : ℝ, Real.exp (-(x^2)/4) = Real.exp (-(1/4 : ℝ) * x^2) from congrFun he] rw [← ofReal_integral_eq_lintegral_ofReal (integrable_exp_neg_mul_sq (by norm_num)) (by filter_upwards [] with x; positivity), integral_gaussian] have hs : Real.sqrt (Real.pi / (1/4 : ℝ)) ≤ 4 := by apply (Real.sqrt_le_iff).mpr constructor · norm_num · nlinarith [Real.pi_lt_four] exact (ENNReal.ofReal_le_ofReal hs).trans_eq (by norm_num) lemma gaussian_density_pair_bound {m : ℕ} (x y : Fin m → ℝ) (h : coordinateNorm y ≤ coordinateNorm x) : (∏ j, gaussianPDF 0 1 (x j)) ≤ (∏ j, ENNReal.ofReal (Real.exp (-(x j)^2/4))) * ∏ j, ENNReal.ofReal (Real.exp (-(y j)^2/4)) := by calc _ ≤ ∏ j, ENNReal.ofReal (Real.exp (-(x j)^2/2)) := Finset.prod_le_prod fun j _ => gaussianPDF_le_exp (x j) _ = ENNReal.ofReal (Real.exp (- (∑ j, (x j)^2) / 2)) := by rw [← ENNReal.ofReal_prod_of_nonneg (fun j _ => Real.exp_nonneg _), ← Real.exp_sum] congr 2 rw [← Finset.sum_div, Finset.sum_neg_distrib] _ ≤ ENNReal.ofReal (Real.exp (- (∑ j, (x j)^2)/4 - (∑ j, (y j)^2)/4)) := by apply ENNReal.ofReal_le_ofReal apply Real.exp_le_exp.mpr have hn : 0 ≤ coordinateNorm y := norm_nonneg _ have hs := pow_le_pow_left₀ hn h 2 rw [coordinateNorm_sq, coordinateNorm_sq] at hs linarith _ = _ := by rw [← ENNReal.ofReal_prod_of_nonneg (fun j _ => Real.exp_nonneg _), ← ENNReal.ofReal_prod_of_nonneg (fun j _ => Real.exp_nonneg _), ← Real.exp_sum, ← Real.exp_sum, ← ENNReal.ofReal_mul (Real.exp_nonneg _), ← Real.exp_add] congr 2 simp only [← Finset.sum_div, Finset.sum_neg_distrib] ring lemma measurable_coordinateVolume (m : ℕ) : Measurable (fun x : Fin m → ℝ => volume (coordinateBall m (coordinateNorm x))) := by have : Measurable (fun z : (Fin m → ℝ) × (Fin m → ℝ) => ({z | coordinateNorm z.2 ≤ coordinateNorm z.1} : Set _).indicator (fun _ => (1 : ℝ≥0∞)) z) := measurable_const.indicator (measurableSet_le (by fun_prop) (by fun_prop)) change Measurable (Function.uncurry (fun x y : Fin m → ℝ => (coordinateBall m (coordinateNorm x)).indicator (fun _ => (1 : ℝ≥0∞)) y)) at this convert this.lintegral_prod_right (ν := (volume : Measure (Fin m → ℝ))) using 1 ext x rw [lintegral_indicator (measurableSet_coordinateBall m _)] simp lemma gaussian_expected_coordinate_volume (m : ℕ) : (∫⁻ x : Fin m → ℝ, volume (coordinateBall m (coordinateNorm x)) ∂Measure.pi (fun _ => gaussianReal 0 1)) ≤ (16 : ℝ≥0∞)^m := by have hB := measurable_coordinateVolume m rw [pi_gaussian_withDensity m 1 (by norm_num), lintegral_withDensity_eq_lintegral_mul _ (by fun_prop) hB] have hi (x : Fin m → ℝ) : (∏ j, gaussianPDF 0 1 (x j)) * volume (coordinateBall m (coordinateNorm x)) = ∫⁻ y : Fin m → ℝ, (coordinateBall m (coordinateNorm x)).indicator (fun _ => ∏ j, gaussianPDF 0 1 (x j)) y := by rw [lintegral_indicator (measurableSet_coordinateBall m _)] simp simp only [Pi.mul_apply] simp_rw [hi] calc _ ≤ ∫⁻ x : Fin m → ℝ, ∫⁻ y : Fin m → ℝ, (∏ j, ENNReal.ofReal (Real.exp (-(x j)^2/4))) * ∏ j, ENNReal.ofReal (Real.exp (-(y j)^2/4)) := by apply lintegral_mono intro x apply lintegral_mono intro y by_cases hy : y ∈ coordinateBall m (coordinateNorm x) · rw [Set.indicator_of_mem hy] exact gaussian_density_pair_bound x y hy · rw [Set.indicator_of_notMem hy] exact zero_le _ = ((∫⁻ z : ℝ, ENNReal.ofReal (Real.exp (-(z^2)/4)))^m)^2 := by have hJ : (∫⁻ y : Fin m → ℝ, ∏ j, ENNReal.ofReal (Real.exp (-(y j)^2/4))) = (∫⁻ z : ℝ, ENNReal.ofReal (Real.exp (-(z^2)/4)))^m := by rw [volume_pi, lintegral_fin_prod volume m (fun _ => fun z : ℝ => ENNReal.ofReal (Real.exp (-(z^2)/4))) (fun _ => by fun_prop)] simp have hj (x : Fin m → ℝ) := lintegral_const_mul (μ := volume) (∏ j, ENNReal.ofReal (Real.exp (-(x j)^2/4))) (by fun_prop : Measurable (fun y : Fin m → ℝ => ∏ j, ENNReal.ofReal (Real.exp (-(y j)^2/4)))) simp_rw [hj, hJ] rw [lintegral_mul_const _ (by fun_prop), hJ, pow_two] _ ≤ ((4 : ℝ≥0∞)^m)^2 := pow_le_pow_left₀ (by positivity) (pow_le_pow_left₀ (by positivity) lintegral_quarterGaussian_le m) 2 _ = _ := by rw [← pow_mul, Nat.mul_comm, pow_mul]; norm_num section Rows variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] lemma stdGaussian_inner_map (u : E) : (stdGaussian E).map (fun x => ⟪u, x⟫) = gaussianReal 0 (NNReal.mk (‖u‖^2) (sq_nonneg ‖u‖)) := by have h := IsGaussian.map_eq_gaussianReal (μ := stdGaussian E) (innerSL ℝ u) have he : (⇑(innerSL ℝ u)) = (fun x => ⟪u, x⟫) := by ext x exact innerSL_apply_apply ℝ u x simp only [integral_strongDual_stdGaussian, variance_dual_stdGaussian, innerSL_apply_norm] at h rw [he] at h have hh : (‖u‖^2).toNNReal = NNReal.mk (‖u‖^2) (sq_nonneg ‖u‖) := by ext simp simpa only [integral_strongDual_stdGaussian, variance_dual_stdGaussian, innerSL_apply_norm, hh] using h lemma rows_inner_map (u : E) (m : ℕ) : (Measure.pi (fun _ : Fin m => stdGaussian E)).map (fun A => fun j => ⟪u, A j⟫) = (Measure.pi (fun _ : Fin m => gaussianReal 0 1)).map (fun x => ‖u‖ • x) := by rw [Measure.pi_map_pi (f := fun _ : Fin m => fun x : E => ⟪u, x⟫) (fun _ => (by fun_prop : Measurable (fun x : E => ⟪u, x⟫)).aemeasurable)] simp_rw [stdGaussian_inner_map] exact (pi_gaussian_map_scale m ‖u‖).symm lemma rows_expected_coordinate_volume (u : E) (hu : ‖u‖ ≤ 1) (m : ℕ) : (∫⁻ A : Fin m → E, volume (coordinateBall m (coordinateNorm (fun j => ⟪u, A j⟫))) ∂Measure.pi (fun _ => stdGaussian E)) ≤ (16 : ℝ≥0∞)^m := by have h := congrArg (fun μ : Measure (Fin m → ℝ) => ∫⁻ y, volume (coordinateBall m (coordinateNorm y)) ∂μ) (rows_inner_map u m) rw [lintegral_map (measurable_coordinateVolume m) (by fun_prop), lintegral_map (measurable_coordinateVolume m) (by fun_prop)] at h rw [h] apply le_trans _ (gaussian_expected_coordinate_volume m) apply lintegral_mono intro x apply measure_mono intro y hy change coordinateNorm y ≤ coordinateNorm x apply le_trans hy rw [coordinateNorm_smul, abs_of_nonneg (norm_nonneg _)] exact mul_le_of_le_one_left (norm_nonneg _) hu def rowOperator {m : ℕ} (A : Fin m → E) : E →L[ℝ] EuclideanSpace ℝ (Fin m) := (PiLp.continuousLinearEquiv 2 ℝ (fun _ : Fin m => ℝ)).symm.toContinuousLinearMap.comp (ContinuousLinearMap.pi (fun j => innerSL ℝ (A j))) omit [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] in lemma rowOperator_apply {m : ℕ} (A : Fin m → E) (t : E) : rowOperator A t = WithLp.toLp 2 (fun j => ⟪t, A j⟫) := by ext j exact real_inner_comm _ _ def netProjectionSupport (s : Finset E) (m : ℕ) : Set ((Fin m → E) × (Fin m → ℝ)) := ⋃ u ∈ s, {z | coordinateNorm z.2 ≤ 2 * coordinateNorm (fun j => ⟪u, z.1 j⟫)} lemma measurableSet_netProjectionSupport (s : Finset E) (m : ℕ) : MeasurableSet (netProjectionSupport s m) := by apply MeasurableSet.biUnion s.countable_toSet intro u _ exact measurableSet_le (by fun_prop) (by fun_prop) lemma netProjectionSupport_volume (s : Finset E) (hs : ∀ u ∈ s, ‖u‖ ≤ 1) (m : ℕ) : matrixLabelReference (E := E) m (netProjectionSupport s m) ≤ (s.card : ℝ≥0∞) * (32 : ℝ≥0∞)^m := by classical apply (measure_biUnion_finset_le s _).trans calc _ ≤ ∑ _u ∈ s, (32 : ℝ≥0∞)^m := by apply Finset.sum_le_sum intro u hu rw [matrixLabelReference, Measure.prod_apply (measurableSet_le (by fun_prop) (by fun_prop))] change (∫⁻ A : Fin m → E, volume (coordinateBall m (2 * coordinateNorm (fun j => ⟪u, A j⟫))) ∂Measure.pi (fun _ => stdGaussian E)) ≤ _ simp_rw [volume_coordinateBall_smul m 2 _ (by norm_num)] have hme : Measurable (fun A : Fin m → E => volume (coordinateBall m (coordinateNorm (fun j => ⟪u, A j⟫)))) := (measurable_coordinateVolume m).comp (by fun_prop) rw [lintegral_const_mul _ hme] calc _ ≤ ENNReal.ofReal ((2 : ℝ)^m) * (16 : ℝ≥0∞)^m := mul_le_mul' le_rfl (rows_expected_coordinate_volume u (hs u hu) m) _ = _ := by rw [ENNReal.ofReal_pow (by norm_num)]; norm_num [← mul_pow] _ = _ := by simp end Rows section NetNorm variable {E F : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] lemma opNorm_le_twice_net (f : E →L[ℝ] F) (s : Finset E) (hnet : ∀ x : E, ‖x‖ ≤ 1 → ∃ y ∈ s, dist x y ≤ 1/2) : ‖f‖ ≤ 2 * (s.sup (fun u => ‖f u‖₊) : ℝ≥0) := by classical let K : ℝ := (s.sup (fun u => ‖f u‖₊) : ℝ≥0) have hK : 0 ≤ K := NNReal.coe_nonneg _ have hbound : ‖f‖ ≤ K + ‖f‖/2 := by apply ContinuousLinearMap.opNorm_le_of_unit_norm (by positivity) intro x hx obtain ⟨y, hy, hxy⟩ := hnet x hx.le have hfy : ‖f y‖ ≤ K := by exact_mod_cast (Finset.le_sup (f := fun u => ‖f u‖₊) hy) calc ‖f x‖ = ‖f y + f (x-y)‖ := by rw [map_sub]; congr 1; abel _ ≤ ‖f y‖ + ‖f (x-y)‖ := norm_add_le _ _ _ ≤ K + ‖f‖ * (1/2) := add_le_add hfy ((f.le_opNorm (x-y)).trans (mul_le_mul_of_nonneg_left (by simpa [dist_eq_norm] using hxy) (norm_nonneg _))) _ = _ := by ring dsimp [K] at hbound linarith end NetNorm variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] theorem exists_projection_support (m : ℕ) (hm : m ≤ Module.finrank ℝ E) : ∃ D : Set ((Fin m → E) × (Fin m → ℝ)), MeasurableSet D ∧ (∀ A t, ‖t‖ ≤ 1 → (A, fun j => ⟪t, A j⟫) ∈ D) ∧ matrixLabelReference (E := E) m D ≤ (192 : ℝ≥0∞)^(Module.finrank ℝ E) := by classical obtain ⟨s, hs, hc, hnet⟩ := exists_unit_ball_net (E := E) (by norm_num : (0 : ℝ) < 1/2) (by norm_num : (1/2 : ℝ) ≤ 1) have hsne : s.Nonempty := by obtain ⟨u, hu, _⟩ := hnet 0 (by simp) exact ⟨u, hu⟩ refine ⟨netProjectionSupport s m, measurableSet_netProjectionSupport s m, ?_, ?_⟩ · intro A t ht obtain ⟨u, hu, he⟩ := s.exists_mem_eq_sup hsne (fun u => ‖rowOperator A u‖₊) refine Set.mem_iUnion.mpr ⟨u, Set.mem_iUnion.mpr ⟨hu, ?_⟩⟩ change coordinateNorm (fun j => ⟪t, A j⟫) ≤ 2 * coordinateNorm (fun j => ⟪u, A j⟫) change ‖WithLp.toLp 2 (fun j => ⟪t, A j⟫)‖ ≤ 2 * ‖WithLp.toLp 2 (fun j => ⟪u, A j⟫)‖ rw [← rowOperator_apply, ← rowOperator_apply] calc _ ≤ ‖rowOperator A‖ * ‖t‖ := (rowOperator A).le_opNorm t _ ≤ ‖rowOperator A‖ := mul_le_of_le_one_right (norm_nonneg _) ht _ ≤ _ := by simpa [he] using opNorm_le_twice_net (rowOperator A) s hnet · apply (netProjectionSupport_volume s hs m).trans have hcard : (s.card : ℝ≥0∞) ≤ (6 : ℝ≥0∞)^(Module.finrank ℝ E) := by have h : (s.card : ℝ) ≤ (6 : ℝ)^(Module.finrank ℝ E) := by norm_num at hc ⊢; exact hc simpa using ENNReal.ofReal_le_ofReal h calc _ ≤ (6 : ℝ≥0∞)^Module.finrank ℝ E * (32 : ℝ≥0∞)^Module.finrank ℝ E := mul_le_mul' hcard (pow_le_pow_right₀ (by norm_num) hm) _ = _ := by rw [← mul_pow]; norm_num end NoiselessRegression end end OAI