| import OAI.Probability.GaussianRegression.Nets |
| import OAI.Probability.GaussianRegression.InverseMoments |
|
|
| namespace OAI |
|
|
| /-! |
| Affine tube covers and inverse-distance bounds. |
| -/ |
|
|
| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
|
|
| noncomputable section |
|
|
| universe u |
|
|
| namespace NoiselessRegression |
|
|
| section AffineTubes |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] |
|
|
| open EuclideanGeometry |
|
|
| theorem exists_affine_tube_cover (L : AffineSubspace ℝ E) [Nonempty L] |
| {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ∃ s : Finset L.direction, |
| (s.card : ℝ) ≤ (3 / r) ^ Module.finrank ℝ L.direction ∧ |
| ∀ t : E, ‖t‖ ≤ 1 → Metric.infDist t L ≤ r → |
| ∃ y ∈ s, dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ 2 * r := by |
| classical |
| obtain ⟨s, _, hcard, hnet⟩ := exists_unit_ball_net (E := L.direction) hr hr1 |
| refine ⟨s, hcard, ?_⟩ |
| intro t ht hdist |
| obtain ⟨y, hy, hty⟩ := hnet (L.direction.orthogonalProjectionOnto t) |
| ((L.direction.norm_orthogonalProjectionOnto_apply_le t).trans ht) |
| refine ⟨y, hy, ?_⟩ |
| have hproj : (orthogonalProjection L t : E) = |
| (L.direction.orthogonalProjectionOnto t : E) + (orthogonalProjection L 0 : E) := by |
| have h := (orthogonalProjection L).map_vadd (0 : E) t |
| simpa only [vadd_eq_add, add_zero, orthogonalProjection_contLinear, |
| AffineSubspace.coe_vadd] using congrArg (fun x : L => (x : E)) h |
| have hpdist : dist (orthogonalProjection L t : E) |
| ((y : E) + (orthogonalProjection L 0 : E)) ≤ r := by |
| rw [hproj, dist_add_right] |
| exact hty |
| have hd : dist t (orthogonalProjection L t : E) ≤ r := by |
| rwa [dist_orthogonalProjection_eq_infDist] |
| calc |
| dist t ((y : E) + (orthogonalProjection L 0 : E)) ≤ |
| dist t (orthogonalProjection L t : E) + |
| dist (orthogonalProjection L t : E) ((y : E) + (orthogonalProjection L 0 : E)) := |
| dist_triangle _ _ _ |
| _ ≤ r + r := add_le_add hd hpdist |
| _ = 2 * r := by ring |
|
|
| theorem HasLocalMassBound.affine_tube {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ν {t | Metric.infDist t L ≤ r} ≤ |
| ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction * (4 * r) ^ a) := by |
| classical |
| obtain ⟨s, hcard, hcover⟩ := exists_affine_tube_cover L hr hr1 |
| let c : L.direction → E := fun y => (y : E) + (orthogonalProjection L 0 : E) |
| have hsub : {t | Metric.infDist t L ≤ r} ≤ᵐ[ν] |
| ⋃ y ∈ s, Metric.closedBall (c y) (2 * r) := by |
| filter_upwards [ν.support_mem_ae] with t ht |
| intro hdist |
| have htunit : ‖t‖ ≤ 1 := by simpa using hsupport ht |
| obtain ⟨y, hy, hty⟩ := hcover t htunit hdist |
| exact mem_iUnion₂.mpr ⟨y, hy, hty⟩ |
| calc |
| ν {t | Metric.infDist t L ≤ r} ≤ ν (⋃ y ∈ s, Metric.closedBall (c y) (2 * r)) := |
| measure_mono_ae hsub |
| _ ≤ ∑ y ∈ s, ν (Metric.closedBall (c y) (2 * r)) := measure_biUnion_finset_le _ _ |
| _ ≤ ∑ _y ∈ s, ENNReal.ofReal ((4 * r) ^ a) := by |
| refine Finset.sum_le_sum fun y _ => ?_ |
| simpa only [show (2 : ℝ) * (2 * r) = 4 * r by ring] using |
| hν.any_center (c y) (r := 2 * r) (by positivity) |
| _ = (s.card : ℝ≥0∞) * ENNReal.ofReal ((4 * r) ^ a) := by simp |
| _ ≤ ENNReal.ofReal ((3 / r) ^ Module.finrank ℝ L.direction) * |
| ENNReal.ofReal ((4 * r) ^ a) := by |
| gcongr |
| exact_mod_cast ENNReal.ofReal_le_ofReal hcard |
| _ = _ := (ENNReal.ofReal_mul (by positivity)).symm |
|
|
| lemma tube_radius_factors (ℓ : ℕ) (a : ℝ) {r : ℝ} (hr : 0 < r) : |
| (3 / r) ^ ℓ * (4 * r) ^ a = |
| (3 : ℝ) ^ ℓ * (4 : ℝ) ^ a * r ^ (a - ℓ) := by |
| rw [div_pow, Real.mul_rpow (by norm_num) hr.le, |
| Real.rpow_sub_natCast hr.ne'] |
| ring |
|
|
| theorem HasLocalMassBound.affine_tube_pow {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] (k : ℕ) |
| (hmargin : (Module.finrank ℝ L.direction : ℝ) + k ≤ a) |
| {r : ℝ} (hr : 0 < r) (hr1 : r ≤ 1) : |
| ν {t | Metric.infDist t L ≤ r} ≤ |
| ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) * |
| ENNReal.ofReal (r ^ k) := by |
| refine (hν.affine_tube hsupport L hr hr1).trans ?_ |
| rw [tube_radius_factors _ _ hr, ← ENNReal.ofReal_mul (by positivity)] |
| apply ENNReal.ofReal_le_ofReal |
| apply mul_le_mul_of_nonneg_left _ (by positivity) |
| rw [← Real.rpow_natCast] |
| exact Real.rpow_le_rpow_of_exponent_ge hr hr1 (by linarith) |
|
|
| theorem HasLocalMassBound.affine_inverse_moment {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] (m : ℕ) |
| (hmargin : (Module.finrank ℝ L.direction : ℝ) + (m + 1 : ℕ) ≤ a) : |
| ∫⁻ t, ENNReal.ofReal ((Metric.infDist t L)⁻¹ ^ m) ∂ν ≤ |
| ν Set.univ + |
| ENNReal.ofReal ((3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a) * |
| (2 : ℝ≥0∞) ^ (m + 1) := by |
| apply inverse_moment_of_tube_bound ν _ m (Metric.continuous_infDist_pt _).measurable |
| (fun _ => Metric.infDist_nonneg) |
| intro r hr hr1 |
| exact hν.affine_tube_pow hsupport L (m + 1) hmargin hr hr1 |
|
|
| theorem HasLocalMassBound.affine_null {ν : Measure E} {a : ℝ} |
| (hν : HasLocalMassBound ν a) (hsupport : ν.support ⊆ Metric.closedBall 0 1) |
| (L : AffineSubspace ℝ E) [Nonempty L] |
| (hmargin : (Module.finrank ℝ L.direction : ℝ) + 1 ≤ a) : ν L = 0 := by |
| let C : ℝ := (3 : ℝ) ^ Module.finrank ℝ L.direction * (4 : ℝ) ^ a |
| have hC : 0 ≤ C := by dsimp [C]; positivity |
| have hb (n : ℕ) : ν L ≤ ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n) := by |
| have hn : 0 < (1 / 2 : ℝ) ^ n := by positivity |
| have hn1 : (1 / 2 : ℝ) ^ n ≤ 1 := pow_le_one₀ (by norm_num) (by norm_num) |
| calc |
| ν L ≤ ν {t | Metric.infDist t L ≤ (1 / 2 : ℝ) ^ n} := by |
| apply measure_mono |
| intro t ht |
| simp only [mem_ofPred_eq, Metric.infDist_zero_of_mem ht] |
| exact hn.le |
| _ ≤ ENNReal.ofReal C * ENNReal.ofReal (((1 / 2 : ℝ) ^ n) ^ 1) := |
| hν.affine_tube_pow hsupport L 1 (by simpa using hmargin) hn hn1 |
| _ = _ := by rw [pow_one, ENNReal.ofReal_mul hC] |
| have hlim : Tendsto (fun n : ℕ => ENNReal.ofReal (C * (1 / 2 : ℝ) ^ n)) |
| atTop (𝓝 0) := by |
| convert ENNReal.tendsto_ofReal ((tendsto_const_nhds (x := C)).mul |
| (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : (0 : ℝ) ≤ 1 / 2) |
| (by norm_num : (1 / 2 : ℝ) < 1))) using 1 |
| simp |
| exact le_antisymm (ge_of_tendsto' hlim hb) (by positivity) |
|
|
| end AffineTubes |
|
|
| end NoiselessRegression |
|
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| end |
|
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| end OAI |
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