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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 | import Mathlib
import OAI.Probability.GaussianRegression.Main
import OAI.Probability.MemoryPrecision.CubeBlock
import OAI.Probability.MemoryPrecision.CubeCaps
import OAI.Probability.MemoryPrecision.RealContinuation
namespace OAI
open MeasureTheory ProbabilityTheory Set Filter EuclideanGeometry
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
noncomputable section
universe u v
open MeasureTheory ProbabilityTheory Set Filter
open scoped ENNReal NNReal RealInnerProductSpace Topology Pointwise
namespace MemoryPrecision
open NoiselessRegression
def CubeStepConstant : ℝ := Classical.choose cube_block
lemma cubeStepConstant_one : 1 ≤ CubeStepConstant := (Classical.choose_spec cube_block).1
lemma cubeStepConstant_pos : 0 < CubeStepConstant := lt_of_lt_of_le zero_lt_one cubeStepConstant_one
def CubeStep (d N : ℕ) : ℝ := CubeStepConstant^(d/2) * (N:ℝ)^(1/(((d/2)/4:ℕ):ℝ))
lemma cubeStep_one (d : ℕ) {N : ℕ} (hN : 1 ≤ N) : 1 ≤ CubeStep d N := by
apply one_le_mul_of_one_le_of_one_le
· exact one_le_pow₀ cubeStepConstant_one
· exact Real.one_le_rpow (by exact_mod_cast hN) (by positivity)
lemma cubeStep_pos (d : ℕ) {N : ℕ} (hN : 1 ≤ N) : 0 < CubeStep d N :=
lt_of_lt_of_le zero_lt_one (cubeStep_one d hN)
lemma cube_deterministic_step {d m N : ℕ} (hd : 16 ≤ d) (hm : 1 ≤ m)
(hmp : m ≤ (d/2)/4) (hN : 1 ≤ N) {H : ℝ} (hH : 0 < H)
(h : Fin N → Space d → ℝ≥0∞) (hh : ∀ w, Measurable (h w))
(hh1 : ∀ w s, h w s ≤ 1)
(hb : ∀ w, CubeLocal (fun u => h w (CubeEmbedding d u)) H)
(φ : ((Fin m → Space d) × (Fin m → ℝ)) → Fin N) (hφ : Measurable φ) :
CubeLocal (fun u => blockValue h φ (CubeEmbedding d u)) (CubeStep d N * H) := by
classical
let g (w : Fin N) (x : (Fin m → Space d) × (Fin m → ℝ)) : ℝ≥0∞ :=
if φ x = w then 1 else 0
have hg (w) : Measurable (g w) := by
exact Measurable.ite (measurableSet_eq_fun hφ measurable_const) measurable_const measurable_const
have hg1 (x) : ∑ w, g w x ≤ 1 := by simp [g]
have hb' := (Classical.choose_spec cube_block).2 d m N hd hm hmp hN H hH.le
(fun w u => h w (CubeEmbedding d u)) (fun w => (hh w).comp (cubeEmbedding_measurable d))
(fun w u => hh1 w (CubeEmbedding d u)) hb g hg hg1
convert hb' using 1
· funext u
unfold blockValue WeightedBlock
congr 1
funext A
simp [g]
· rfl
lemma CubeLocal.mono {d : ℕ} {h : Cube d → ℝ≥0∞} {H K : ℝ}
(hh : CubeLocal h H) (hHK : H ≤ K) : CubeLocal h K := by
intro z r hr
exact (hh z r hr).trans (ENNReal.ofReal_le_ofReal (mul_le_mul_of_nonneg_right hHK (by positivity)))
lemma cube_continuation_mass {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N)
{δ : ℕ → Fin N → (Space d × ℝ) → Fin N}
{w : ℕ → Fin N → Space d → ℝ≥0∞}
(hδ : ∀ n j, Measurable (δ n j)) (hw : ∀ n j, Measurable (w n j))
(hw1 : ∀ n j s, w n j s ≤ 1) {H : ℝ} (hH : 0 < H)
(hlocal : ∀ n j, CubeLocal (fun s => w n j (CubeEmbedding d s)) H)
(b k : ℕ) (hk : k ≤ b*((d/2)/4)) (n : ℕ) (j : Fin N) :
CubeLocal (fun s => continuation δ w n k j (CubeEmbedding d s))
(H * CubeStep d N ^ b) := by
have hp : 0 < (d/2)/4 := by omega
have hK := cubeStep_one d hN
have hK0 := cubeStep_pos d hN
induction b generalizing k n j with
| zero =>
have : k = 0 := by omega
subst k
simpa [continuation] using hlocal n j
| succ b ih =>
by_cases hk0 : k = 0
· subst k
apply (hlocal n j).mono
simpa [continuation] using le_mul_of_one_le_right hH.le (one_le_pow₀ hK)
· let m := min k ((d/2)/4)
have hm : 1 ≤ m := by dsimp [m]; omega
have hmp : m ≤ (d/2)/4 := min_le_right _ _
have hmk : m ≤ k := min_le_left _ _
have hk' : k-m ≤ b*((d/2)/4) := by
dsimp [m]
rw [Nat.succ_mul] at hk
omega
have hb := cube_deterministic_step hd hm hmp hN
(mul_pos hH (pow_pos hK0 b))
(fun a => continuation δ w (n+m) (k-m) a)
(fun a => measurable_continuation hδ hw _ _ a)
(fun a s => continuation_le_one hw1 _ _ a s)
(fun a => ih (k-m) hk' (n+m) a) (streamRoute δ n m j)
(measurable_streamRoute hδ n m j)
have he : (fun s : Cube d => continuation δ w n k j (CubeEmbedding d s)) =
(fun s : Cube d => blockValue (fun a => continuation δ w (n+m) (k-m) a)
(streamRoute δ n m j) (CubeEmbedding d s)) := by
funext s
simpa [Nat.add_sub_of_le hmk] using continuation_unroll hδ n m (k-m) j (CubeEmbedding d s)
rw [he]
convert hb using 1
rw [pow_succ]; ring
lemma cube_continuation_cap_integral {d N : ℕ} (hd : 16 ≤ d) (hN : 1 ≤ N)
{δ : ℕ → Fin N → (Space d × ℝ) → Fin N} (hδ : ∀ n j, Measurable (δ n j))
(y : ℕ → Fin N → Space d) {ε : ℝ} (hε : 0 < ε) (n k : ℕ) (j : Fin N) :
(∫⁻ u, continuation δ (fun n j => capWeight (y n j) ε) n k j (CubeEmbedding d u) ∂CubeLaw d) ≤
ENNReal.ofReal ((12:ℝ)^(d/2)*ε^((d/2:ℕ)/4:ℝ) * CubeStep d N ^ BlockCount k ((d/2)/4)) := by
apply cube_mass_bound
apply cube_continuation_mass hd hN hδ (fun n j => measurable_capWeight _ _)
(fun n j s => capWeight_le_one _ _ _) (by positivity)
(fun n j => cube_cap_local (by omega) (y n j) hε)
simpa [mul_comm] using blockCount_mul_ge k (by omega : 0 < (d/2)/4)
end MemoryPrecision
end
end OAI
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