| import Mathlib |
| import OAI.Probability.GaussianRegression.Main |
| import OAI.Probability.MemoryPrecision.CubeGeometry |
| import OAI.Probability.MemoryPrecision.WeightedImage |
| import OAI.Probability.MemoryPrecision.WeightedSelection |
|
|
| namespace OAI |
|
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| open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
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| noncomputable section |
| universe u v |
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| open MeasureTheory ProbabilityTheory Set Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise |
| namespace MemoryPrecision |
| open NoiselessRegression |
|
|
| lemma cube_J_bound {d k : ℕ} (hd : 16 ≤ d) (hk : k ≤ (d/2)/4) : |
| ∀ j : ℕ, j ≤ (d/2)/4-2 → J (3*(d/2:ℕ)/4:ℝ) k j ≤ (7:ℝ)^d := by |
| have hp2 : 2 ≤ (d/2)/4 := by omega |
| have hn4 : (((d/2)/4:ℕ):ℝ)*4 ≤ (d/2:ℕ) := by exact_mod_cast Nat.div_mul_le_self (d/2) 4 |
| have hn : ((d/2:ℕ):ℝ) ≤ d := by exact_mod_cast Nat.div_le_self d 2 |
| have hm : (k:ℝ) ≤ ((d/2)/4:ℕ) := by exact_mod_cast hk |
| intro j hj |
| have hj' : (j:ℝ)+2 ≤ ((d/2)/4:ℕ) := by exact_mod_cast (show j+2 ≤ (d/2)/4 by omega) |
| have hgap : 0 < (3*(d/2:ℕ)/4:ℝ)-(j:ℝ)-(k:ℝ) := by linarith |
| have hquot : (k:ℝ)/((3*(d/2:ℕ)/4:ℝ)-(j:ℝ)-(k:ℝ)) ≤ 1 := by |
| rw [div_le_one hgap] |
| linarith |
| have h3 : (3:ℝ)^j ≤ (3:ℝ)^d := pow_le_pow_right₀ (by norm_num) (by omega) |
| have h2 : (2:ℝ)^(3*(d/2:ℕ)/4:ℝ) ≤ (2:ℝ)^d := by |
| rw [← Real.rpow_natCast] |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num) |
| linarith |
| calc |
| J (3*(d/2:ℕ)/4:ℝ) k j = 1+(3:ℝ)^j*(2:ℝ)^(3*(d/2:ℕ)/4:ℝ)* |
| ((k:ℝ)/((3*(d/2:ℕ)/4:ℝ)-(j:ℝ)-(k:ℝ))) := by unfold J; ring |
| _ ≤ 1+(3:ℝ)^j*(2:ℝ)^(3*(d/2:ℕ)/4:ℝ)*1 := by gcongr |
| _ ≤ 1+(3:ℝ)^d*(2:ℝ)^d := by rw [mul_one]; gcongr |
| _ = 1+(6:ℝ)^d := by rw [← mul_pow]; norm_num |
| _ ≤ (7:ℝ)^d := by |
| simpa only [one_pow, show (1:ℝ)+6=7 by norm_num] using |
| pow_add_pow_le (show (0:ℝ) ≤ 1 by norm_num) (show (0:ℝ) ≤ 6 by norm_num) (show d ≠ 0 by omega) |
| |
| lemma cube_weighted_ambient {d : ℕ} (hd : 2 ≤ d) {f : Cube d → ℝ≥0∞} |
| {L : ℝ} (hf : CubeLocal f L) : |
| AmbientBallBound (((CubeLaw d).withDensity f).map (CubeEmbedding d)) L (3*(d/2:ℕ)/4:ℝ) := by |
| intro x r hr |
| rw [weighted_image_ball _ (cubeEmbedding_measurable d)] |
| apply (lintegral_mono_set (show {u | dist (CubeEmbedding d u) x ≤ r} ⊆ |
| CubeBall d (Real.sqrt (d/2:ℕ) • cubeProject d x) r from ?_)).trans |
| (hf _ _ hr) |
| intro u hu |
| exact (cubeProject_parameter_dist hd u x).trans |
| (mul_le_mul_of_nonneg_left hu (Real.sqrt_nonneg _)) |
| |
| lemma cubeBall_embedding_subset {d : ℕ} (hd : 2 ≤ d) (z : Space (d/2)) {r : ℝ} |
| (u₀ : Cube d) (hu₀ : u₀ ∈ CubeBall d z r) : |
| CubeBall d z r ⊆ CubeEmbedding d ⁻¹' Metric.closedBall (CubeEmbedding d u₀) (4*r) := by |
| have hs : 0 < Real.sqrt (d/2:ℕ) := Real.sqrt_pos.2 (by exact_mod_cast (show 0 < d/2 by omega)) |
| intro u hu |
| have hu' : dist u.val z ≤ Real.sqrt (d/2:ℕ)*r := hu |
| have hu₀' : dist u₀.val z ≤ Real.sqrt (d/2:ℕ)*r := hu₀ |
| have hab : dist u.val u₀.val ≤ 2*Real.sqrt (d/2:ℕ)*r := by |
| have := dist_triangle u.val z u₀.val |
| rw [dist_comm z u₀.val] at this |
| linarith |
| apply (cubeEmbedding_dist hd u u₀).trans |
| calc |
| _ ≤ 2/Real.sqrt (d/2:ℕ)*(2*Real.sqrt (d/2:ℕ)*r) := |
| mul_le_mul_of_nonneg_left hab (by positivity) |
| _ = 4*r := by field_simp [ne_of_gt hs]; ring |
|
|
| lemma weightedBlock_le_sum {X : Type*} {d m N : ℕ} |
| (ι : X → Space d) (f : Fin N → X → ℝ≥0∞) |
| {g : Fin N → ((Fin m → Space d) × (Fin m → ℝ)) → ℝ≥0∞} |
| (hg : ∀ x, ∑ w, g w x ≤ 1) (x : X) : WeightedBlock ι f g x ≤ ∑ w, f w x := by |
| unfold WeightedBlock |
| calc |
| _ ≤ ∫⁻ _A : Fin m → Space d, (∑ w, f w x) ∂Measure.pi (fun _ => stdGaussian (Space d)) := by |
| apply lintegral_mono |
| intro A |
| apply Finset.sum_le_sum |
| intro w _ |
| exact mul_le_of_le_one_left' ((Finset.single_le_sum (fun _ _ => zero_le) (Finset.mem_univ w)).trans (hg _)) |
| _ = _ := by simp |
| |
| def CubeBlockConstant : ℝ := 4*1344^3 |
| lemma cubeBlockConstant_one : 1 ≤ CubeBlockConstant := by norm_num [CubeBlockConstant] |
| |
| lemma cube_constant_scale {d : ℕ} (hd : 2 ≤ d) {r : ℝ} (hr : 0 < r) : |
| (1344:ℝ)^d * (4*r)^(3*(d/2:ℕ)/4:ℝ) ≤ CubeBlockConstant^(d/2) * r^(3*(d/2:ℕ)/4:ℝ) := by |
| rw [Real.mul_rpow (by norm_num) hr.le] |
| have hp : (1344:ℝ)^d ≤ (1344^3:ℝ)^(d/2) := by |
| rw [← pow_mul] |
| apply pow_le_pow_right₀ (by norm_num) |
| omega |
| have h4 : (4:ℝ)^(3*(d/2:ℕ)/4:ℝ) ≤ (4:ℝ)^(d/2) := by |
| rw [← Real.rpow_natCast] |
| apply Real.rpow_le_rpow_of_exponent_le (by norm_num) |
| have := Nat.cast_nonneg (α := ℝ) (d/2) |
| linarith |
| calc |
| _ = ((1344:ℝ)^d*(4:ℝ)^(3*(d/2:ℕ)/4:ℝ))*r^(3*(d/2:ℕ)/4:ℝ) := by ring |
| _ ≤ ((1344^3:ℝ)^(d/2)*(4:ℝ)^(d/2))*r^(3*(d/2:ℕ)/4:ℝ) := by gcongr |
| _ = _ := by rw [← mul_pow]; congr 2; norm_num [CubeBlockConstant] |
| |
| def CubeBlockStatement : Prop := |
| ∃ C : ℝ, 1 ≤ C ∧ ∀ d k N : ℕ, 16 ≤ d → 1 ≤ k → k ≤ (d/2)/4 → 1 ≤ N → |
| ∀ L : ℝ, 0 ≤ L → ∀ f : Fin N → Cube d → ℝ≥0∞, |
| (∀ v, Measurable (f v)) → (∀ v u, f v u ≤ 1) → (∀ v, CubeLocal (f v) L) → |
| ∀ g : Fin N → ((Fin k → Space d) × (Fin k → ℝ)) → ℝ≥0∞, |
| (∀ v, Measurable (g v)) → (∀ x, ∑ v, g v x ≤ 1) → |
| CubeLocal (WeightedBlock (CubeEmbedding d) f g) |
| (C^(d/2) * (N:ℝ)^(1/(((d/2)/4:ℕ):ℝ)) * L) |
| theorem cube_block : CubeBlockStatement := by |
| refine ⟨CubeBlockConstant, cubeBlockConstant_one, ?_⟩ |
| intro d k N hd hk hkp hN L hL f hf hf1 hloc g hg hgs z r hr |
| by_cases hL0 : L = 0 |
| · simp only [hL0, mul_zero, zero_mul, ENNReal.ofReal_zero] |
| calc |
| _ ≤ ∫⁻ u in CubeBall d z r, ∑ w, f w u ∂CubeLaw d := |
| lintegral_mono (weightedBlock_le_sum _ _ hgs) |
| _ = ∑ w, ∫⁻ u in CubeBall d z r, f w u ∂CubeLaw d := |
| lintegral_finsetSum _ (fun w _ => hf w) |
| _ ≤ ∑ _w : Fin N, ENNReal.ofReal (L*r^(3*(d/2:ℕ)/4:ℝ)) := |
| Finset.sum_le_sum (fun w _ => hloc w z r hr) |
| _ = 0 := by simp [hL0] |
| have hLpos : 0 < L := lt_of_le_of_ne hL (Ne.symm hL0) |
| by_cases hempty : CubeBall d z r = ∅ |
| · simp [hempty] |
| obtain ⟨u₀,hu₀⟩ := Set.nonempty_iff_ne_empty.mpr hempty |
| let ν (w : Fin N) := ((CubeLaw d).withDensity (f w)).map (CubeEmbedding d) |
| let (w : Fin N) : IsFiniteMeasure (ν w) := weighted_image_finite (CubeLaw d) (CubeEmbedding d) (f w) (hf1 w) |
| have hν w : AmbientBallBound (ν w) L (3*(d/2:ℕ)/4:ℝ) := cube_weighted_ambient (by omega) (hloc w) |
| have hp : 2 ≤ (d/2)/4 := by omega |
| have hkd : k ≤ d := by omega |
| have hgap : (k:ℝ)+(((d/2)/4:ℕ):ℝ)-2 < (3*(d/2:ℕ)/4:ℝ) := by |
| have hp' : (((d/2)/4:ℕ):ℝ)*4 ≤ (d/2:ℕ) := by exact_mod_cast Nat.div_mul_le_self (d/2) 4 |
| have hk' : (k:ℝ) ≤ ((d/2)/4:ℕ) := by exact_mod_cast hkp |
| have hn' : (8:ℝ) ≤ (d/2:ℕ) := by exact_mod_cast (show 8 ≤ d/2 by omega) |
| linarith |
| have hsel := localized_weighted_selection ν hp hk hkd (by norm_num : (1:ℝ) ≤ 7) |
| hLpos hgap (cube_J_bound hd hkp) hν g hg hgs (CubeEmbedding d u₀) (by positivity : 0 < 4*r) |
| have he := weightedBlock_mass_identity (CubeLaw d) (cubeEmbedding_measurable d) hf hf1 hg |
| (show MeasurableSet (Metric.closedBall (CubeEmbedding d u₀) (4*r)) from Metric.isClosed_closedBall.measurableSet) |
| apply (lintegral_mono_set (cubeBall_embedding_subset (by omega) z u₀ hu₀)).trans |
| rw [he] |
| apply hsel.trans (ENNReal.ofReal_le_ofReal ?_) |
| have hscale := cube_constant_scale (d := d) (by omega) hr |
| norm_num only [show (7:ℝ)*192=1344 by norm_num] |
| calc |
| _ = ((1344:ℝ)^d*(4*r)^(3*(d/2:ℕ)/4:ℝ))*((N:ℝ)^(1/(((d/2)/4:ℕ):ℝ))*L) := by ring |
| _ ≤ (CubeBlockConstant^(d/2)*r^(3*(d/2:ℕ)/4:ℝ))*((N:ℝ)^(1/(((d/2)/4:ℕ):ℝ))*L) := |
| mul_le_mul_of_nonneg_right hscale (by positivity) |
| _ = _ := by ring |
| end MemoryPrecision |
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| end |
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| end OAI |
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