| import Mathlib |
| import OAI.Probability.SKGap.Localization.DimensionDecay |
| |
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open scoped BigOperators |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory Metric Set |
| open scoped BigOperators NNReal ENNReal |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Matrix |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators Matrix Matrix.Norms.Elementwise NNReal ENNReal |
| open scoped BigOperators |
| open Filter Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped BigOperators NNReal ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal Topology |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal Topology BigOperators |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Set Matrix |
| open scoped BigOperators NNReal ENNReal Matrix.Norms.L2Operator |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open MeasureTheory ProbabilityTheory Filter Matrix |
| open scoped NNReal ENNReal BigOperators Topology Matrix Matrix.Norms.Elementwise |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter |
| open scoped Topology |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter ProbabilityTheory ContinuousLinearMap |
| open scoped Topology NNReal ENNReal |
| open Filter MeasureTheory ProbabilityTheory |
| open scoped Topology NNReal ENNReal |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| namespace SKGapCutoff |
|
|
| lemma dimensionDecay_pos (a : ℝ) (n : ℕ) : 0 < dimensionDecay a n := Real.exp_pos _ |
|
|
| lemma dimensionDecay_le_one {a : ℝ} (ha : 0 ≤ a) {n : ℕ} (hn : 0 < n) : |
| dimensionDecay a n ≤ 1 := by |
| apply Real.exp_le_one_iff.mpr |
| have hlog := Real.log_nonneg (show (1:ℝ) ≤ n by exact_mod_cast hn) |
| nlinarith |
|
|
| lemma dimensionDecay_mul (a b : ℝ) (n : ℕ) : |
| dimensionDecay a n * dimensionDecay b n = dimensionDecay (a+b) n := by |
| unfold dimensionDecay |
| rw [← Real.exp_add] |
| congr 1 |
| ring |
|
|
| lemma nat_mul_dimensionDecay {n : ℕ} (hn : 0 < n) (a : ℝ) : |
| (n:ℝ)*dimensionDecay a n = dimensionDecay (a-1) n := by |
| simpa only [show 1+(a-1) = a by ring, dimensionDecay] using mul_exp_log_identity hn (a-1) |
|
|
| lemma nat_sq_mul_dimensionDecay {n : ℕ} (hn : 0 < n) (a : ℝ) : |
| (n:ℝ)^2*dimensionDecay a n = dimensionDecay (a-2) n := by |
| rw [pow_two, mul_assoc, nat_mul_dimensionDecay hn, nat_mul_dimensionDecay hn] |
| congr 1 |
| ring |
|
|
| lemma exp_cutoffTime_eq_dimensionDecay {rate : ℝ} (hrate : rate ≠ 0) (ρ c : ℝ) (n : ℕ) : |
| Real.exp (-2*ρ*(c*cutoffTime rate n)) = dimensionDecay (c*ρ/rate) n := by |
| unfold cutoffTime dimensionDecay |
| congr 1 |
| field_simp [hrate] |
|
|
| lemma dimensionDecay_mul_cutoffTime_tendsto {q rate : ℝ} (hq : 0 < q) : |
| Tendsto (fun n => dimensionDecay q n*cutoffTime rate n) atTop (𝓝 0) := by |
| have hh := (tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero 1 q hq).comp |
| (Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop) |
| simp only [Function.comp_def, Real.rpow_one] at hh |
| have hh' := hh.div_const (2*rate) |
| convert hh' using 1 |
| · funext n |
| unfold dimensionDecay cutoffTime |
| ring |
| · simp |
|
|
| lemma linear_mean_error_tendsto {C α q : ℝ} (hα : 1 < α) (hq : 1 < q) : |
| Tendsto (fun n => 2*Real.sqrt (C*dimensionDecay (α-1) n+C*dimensionDecay (q-1) n)) |
| atTop (𝓝 0) := by |
| have hh := ((dimensionDecay_tendsto (α-1) (by linarith)).const_mul C).add |
| ((dimensionDecay_tendsto (q-1) (by linarith)).const_mul C) |
| simpa using hh.sqrt.const_mul 2 |
|
|
| lemma upper_polynomial_error_tendsto {C M θ m α δ q : ℝ} |
| (hexp : 1+2*δ < α) (hα : 1 < α) (htail : 2 < θ*δ) (hq : 1 < q) : |
| Tendsto (fun n => Real.sqrt (C/4*dimensionDecay (α-1) n + |
| C/4*dimensionDecay (α-1-2*δ) n + |
| 2*C*M*Real.exp (θ*m)*dimensionDecay (θ*δ-2) n + |
| C*dimensionDecay (q-1) n)) atTop (𝓝 0) := by |
| have h1 := (dimensionDecay_tendsto (α-1) (by linarith)).const_mul (C/4) |
| have h2 := (dimensionDecay_tendsto (α-1-2*δ) (by linarith)).const_mul (C/4) |
| have h3 := (dimensionDecay_tendsto (θ*δ-2) (by linarith)).const_mul (2*C*M*Real.exp (θ*m)) |
| have h4 := (dimensionDecay_tendsto (q-1) (by linarith)).const_mul C |
| simpa using ((h1.add h2).add h3 |>.add h4).sqrt |
| |
| end SKGapCutoff |
| |
| open MeasureTheory Filter |
| open scoped BigOperators Topology |
| |
| end |
| |
| end OAI |
| |
