| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Reduction.ArbitraryRealJunta |
|
|
| namespace OAI |
|
|
| namespace LargeIndependentSets.ProductAveraging |
| open MeasureTheory Set |
| open scoped BigOperators Classical |
| noncomputable def squareNorm {X : Type*} [MeasurableSpace X] (μ : Measure X) (f : X → ℝ) : ℝ := |
| Real.sqrt (∫ x, (f x)^2 ∂μ) |
| lemma squareNorm_eq_L2 {X : Type*} [MeasurableSpace X] (μ : Measure X) |
| {f : X → ℝ} (hf : MemLp f 2 μ) : squareNorm μ f = ‖hf.toLp f‖ := by |
| have hn : ‖hf.toLp f‖^2 = ∫ x, (f x)^2 ∂μ := by |
| calc |
| _ = inner ℝ (hf.toLp f) (hf.toLp f) := (real_inner_self_eq_norm_sq _).symm |
| _ = ∫ x, ((hf.toLp f) x)^2 ∂μ := by |
| rw [L2.inner_def] |
| simp only [inner_self_eq_norm_sq_to_K, RCLike.ofReal_real_eq_id, id_eq, Real.norm_eq_abs, sq_abs] |
| _ = _ := by |
| apply integral_congr_ae |
| filter_upwards [hf.coeFn_toLp] with x hx |
| rw [hx] |
| rw [squareNorm, ← hn, Real.sqrt_sq_eq_abs, abs_of_nonneg (norm_nonneg _)] |
| lemma squareNorm_sub_norm_le {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsFiniteMeasure μ] |
| {f g : X → ℝ} {B C : ℝ} (hf : Measurable f) (hg : Measurable g) |
| (hb : ∀ x, |f x| ≤ B) (hc : ∀ x, |g x| ≤ C) : |
| |squareNorm μ f-squareNorm μ g| ≤ squareNorm μ (f-g) := by |
| have hfl : MemLp f 2 μ := MemLp.of_bound hf.aestronglyMeasurable B |
| (Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hb x)) |
| have hgl : MemLp g 2 μ := MemLp.of_bound hg.aestronglyMeasurable C |
| (Filter.Eventually.of_forall (fun x => by simpa only [Real.norm_eq_abs] using hc x)) |
| rw [squareNorm_eq_L2 μ hfl, squareNorm_eq_L2 μ hgl, squareNorm_eq_L2 μ (hfl.sub hgl), MemLp.toLp_sub] |
| exact abs_norm_sub_norm_le _ _ |
| lemma squareNorm_le_bound {X : Type*} [MeasurableSpace X] (μ : Measure X) [IsProbabilityMeasure μ] |
| {f : X → ℝ} {B : ℝ} (hf : Measurable f) (hB : 0 ≤ B) (hb : ∀ x, |f x| ≤ B) : |
| squareNorm μ f ≤ B := by |
| apply (Real.sqrt_le_iff).mpr |
| refine ⟨hB,?_⟩ |
| have hi := bounded_sq_integrable (μ := μ) hf hb |
| calc |
| _ ≤ ∫ _ : X, B^2 ∂μ := integral_mono hi (integrable_const _) (fun x => by |
| have h := hb x |
| nlinarith [sq_abs (f x), abs_nonneg (f x)]) |
| _ = _ := by simp |
|
|
| noncomputable def influenceNorm {ι α : Type*} [Fintype ι] [DecidableEq ι] [MeasurableSpace α] |
| (μ : Measure α) (i : ι) (f : (ι → α) → ℝ) : ℝ := Real.sqrt (influenceSq μ i f) |
| lemma influenceNorm_difference {ι α : Type*} [Fintype ι] [DecidableEq ι] [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (i : ι) {f g : (ι → α) → ℝ} {B C : ℝ} |
| (hf : Measurable f) (hg : Measurable g) (hb : ∀ x, |f x| ≤ B) (hc : ∀ x, |g x| ≤ C) : |
| |influenceNorm μ i f-influenceNorm μ i g| ≤ squareNorm (Measure.pi (fun _ : ι => μ)) (f-g) := by |
| have hfm := coordinateMean_measurable μ i hf |
| have hgm := coordinateMean_measurable μ i hg |
| have hfb (x : ι → α) : |f x-coordinateMean μ i f x| ≤ B+B := |
| (abs_sub _ _).trans (add_le_add (hb x) (coordinateMean_bound μ i hb x)) |
| have hgb (x : ι → α) : |g x-coordinateMean μ i g x| ≤ C+C := |
| (abs_sub _ _).trans (add_le_add (hc x) (coordinateMean_bound μ i hc x)) |
| have h := squareNorm_sub_norm_le (Measure.pi (fun _ : ι => μ)) (hf.sub hfm) (hg.sub hgm) hfb hgb |
| change |influenceNorm μ i f-influenceNorm μ i g| ≤ _ at h |
| have hlin : (f-coordinateMean μ i f)-(g-coordinateMean μ i g) = |
| (f-g)-coordinateMean μ i (f-g) := by |
| funext x |
| rw [Pi.sub_apply, Pi.sub_apply, Pi.sub_apply, Pi.sub_apply, coordinateMean_sub μ i hf hg hb hc] |
| simp only [Pi.sub_apply] |
| ring |
| rw [hlin] at h |
| exact h.trans (Real.sqrt_le_sqrt (influenceSq_le μ i (hf.sub hg) |
| (fun x => (abs_sub _ _).trans (add_le_add (hb x) (hc x))))) |
| lemma influenceNorm_difference_le {ι α : Type*} [Fintype ι] [DecidableEq ι] [MeasurableSpace α] |
| (μ : Measure α) [IsProbabilityMeasure μ] (i : ι) {f g : (ι → α) → ℝ} {B C a : ℝ} |
| (hf : Measurable f) (hg : Measurable g) (hb : ∀ x, |f x| ≤ B) (hc : ∀ x, |g x| ≤ C) |
| (ha : 0 ≤ a) (hdiff : ∀ x, |f x-g x| ≤ a) : |
| |influenceNorm μ i f-influenceNorm μ i g| ≤ a := |
| (influenceNorm_difference μ i hf hg hb hc).trans |
| (squareNorm_le_bound _ (hf.sub hg) ha hdiff) |
| end LargeIndependentSets.ProductAveraging |
|
|
| namespace LargeIndependentSets.PhaseTest |
| open MeasureTheory Set |
| open scoped BigOperators Classical NNReal |
| lemma quotient_dist_le (x y : ℝ) : dist (x : Circle) (y : Circle) ≤ |x-y| := by |
| rw [dist_eq_norm, ← AddCircle.coe_sub] |
| exact QuotientAddGroup.norm_mk_le_norm |
|
|
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| variable {M : ι → Type*} [∀ i, Fintype (M i)] |
| variable {κ τ : Type*} [Fintype κ] |
|
|
| omit [DecidableEq ι] [∀ index, Fintype (M index)] [Fintype κ] in |
| lemma phase_measurable (p : ∀ j, κ → M j) (t : ι → ℝ) : Measurable (phase p t) := by |
| apply Measurable.of_eval |
| intro k |
| exact Finset.measurable_sum _ (fun l _ => quotient_preserving.measurable.comp |
| (measurable_const.mul (measurable_pi_apply (⟨l,p l k⟩ : Sigma M)))) |
|
|
| omit [DecidableEq ι] [∀ index, Fintype (M index)] in |
| lemma phase_dist_le (p : ∀ j, κ → M j) (t : ι → ℝ) (ht : ∀ j, |t j| ≤ 1) |
| (θ φ : Sigma M → ℝ) {e : ℝ} (he : 0 ≤ e) (hθ : ∀ c, |θ c-φ c| ≤ e) : |
| dist (phase p t θ) (phase p t φ) ≤ (Fintype.card ι : ℝ)*e := by |
| apply (dist_pi_le_iff (by positivity)).mpr |
| intro k |
| calc |
| _ ≤ ∑ j, dist (((t j*θ ⟨j,p j k⟩ : ℝ) : Circle)) (((t j*φ ⟨j,p j k⟩ : ℝ) : Circle)) := |
| dist_sum_sum_le _ _ _ |
| _ ≤ ∑ _j : ι, e := by |
| apply Finset.sum_le_sum |
| intro j _ |
| calc |
| _ ≤ |t j*θ ⟨j,p j k⟩-t j*φ ⟨j,p j k⟩| := quotient_dist_le _ _ |
| _ = |t j| * |θ ⟨j,p j k⟩-φ ⟨j,p j k⟩| := by rw [← mul_sub, abs_mul] |
| _ ≤ 1*e := mul_le_mul (ht j) (hθ _) (abs_nonneg _) (by norm_num) |
| _ = e := one_mul _ |
| _ = _ := by simp |
|
|
| omit [DecidableEq ι] in |
| lemma phase_lipschitz (p : ∀ j, κ → M j) (t : ι → ℝ) (ht : ∀ j, |t j| ≤ 1) : |
| LipschitzWith (Fintype.card ι : ℝ≥0) (phase p t) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro θ φ |
| exact phase_dist_le p t ht θ φ dist_nonneg (fun c => by simpa only [Real.dist_eq] using dist_le_pi_dist θ φ c) |
|
|
| omit [∀ index, Fintype (M index)] in |
| lemma phase_change_coefficient (p : ∀ j, κ → M j) (t t' : ι → ℝ) (j : ι) |
| (hc : ∀ l, l ≠ j → t l = t' l) (θ : Sigma M → ℝ) (hθ : ∀ c, |θ c| ≤ 1) |
| {u : ℝ} (hu : 0 ≤ u) (hd : |t j-t' j| ≤ u) : |
| dist (phase p t θ) (phase p t' θ) ≤ u := by |
| apply (dist_pi_le_iff hu).mpr |
| intro k |
| calc |
| _ ≤ ∑ l, dist (((t l*θ ⟨l,p l k⟩ : ℝ) : Circle)) (((t' l*θ ⟨l,p l k⟩ : ℝ) : Circle)) := |
| dist_sum_sum_le _ _ _ |
| _ ≤ ∑ l, if l=j then u else 0 := by |
| apply Finset.sum_le_sum |
| intro l _ |
| by_cases hl : l = j |
| · subst l |
| rw [ite_eq_left rfl] |
| calc |
| _ ≤ |t j*θ ⟨j,p j k⟩-t' j*θ ⟨j,p j k⟩| := quotient_dist_le _ _ |
| _ = |t j-t' j| * |θ ⟨j,p j k⟩| := by rw [← sub_mul, abs_mul] |
| _ ≤ u*1 := mul_le_mul hd (hθ _) (abs_nonneg _) hu |
| _ = u := mul_one _ |
| · simp only [hc l hl, hl, ↓reduceIte, dist_self, le_refl] |
| _ = _ := by simp |
| |
| omit [∀ index, Fintype (M index)] in |
| lemma auxiliary_change_coefficient (p : ∀ j, κ → M j) (r : κ → τ) (a : τ → ι) |
| (t t' : ι → ℝ) (j : ι) (hc : ∀ l, l ≠ j → t l = t' l) |
| (θ : Sigma M → ℝ) (hθ : ∀ c, |θ c| ≤ 1) (z : ι → Circle) |
| {u : ℝ} (hu : 0 ≤ u) (hd : |t j-t' j| ≤ u) : |
| dist (auxiliary p r a t θ z) (auxiliary p r a t' θ z) ≤ u := by |
| apply (dist_pi_le_iff hu).mpr |
| intro k |
| simpa only [auxiliary, dist_add_left] using |
| (dist_le_pi_dist (phase p t θ) (phase p t' θ) k).trans |
| (phase_change_coefficient p t t' j hc θ hθ hu hd) |
| |
| omit [∀ index, Fintype (M index)] in |
| lemma auxiliary_influence_step (p : ∀ j, κ → M j) (r : κ → τ) (a : τ → ι) |
| (t t' : ι → ℝ) (j : ι) (hc : ∀ l, l ≠ j → t l = t' l) |
| (θ : Sigma M → ℝ) (hθ : ∀ c, |θ c| ≤ 1) |
| {u : ℝ} (hu : 0 ≤ u) (hd : |t j-t' j| ≤ u) |
| {L : ℝ≥0} (F : (κ → Circle) → ℝ) (hF : LipschitzWith L F) (hb : ∀ x, |F x| ≤ 1) : |
| |ProductAveraging.influenceNorm volume j (fun z => F (auxiliary p r a t θ z)) - |
| ProductAveraging.influenceNorm volume j (fun z => F (auxiliary p r a t' θ z))| ≤ L*u := by |
| apply ProductAveraging.influenceNorm_difference_le volume j |
| ((hF.comp (auxiliary_lipschitz p r a t θ)).continuous.measurable) |
| ((hF.comp (auxiliary_lipschitz p r a t' θ)).continuous.measurable) |
| (fun z => hb _) (fun z => hb _) (by positivity) |
| intro z |
| calc |
| _ = dist (F (auxiliary p r a t θ z)) (F (auxiliary p r a t' θ z)) := (Real.dist_eq _ _).symm |
| _ ≤ L*dist (auxiliary p r a t θ z) (auxiliary p r a t' θ z) := hF.dist_le_mul _ _ |
| _ ≤ _ := mul_le_mul_of_nonneg_left |
| (auxiliary_change_coefficient p r a t t' j hc θ hθ z hu hd) (by positivity) |
| |
| lemma squared_value_distance {X : Type*} [PseudoMetricSpace X] {L : ℝ≥0} |
| {F : X → ℝ} (hF : LipschitzWith L F) (hb : ∀ x, |F x| ≤ 1) (x y : X) : |
| |(F x)^2-(F y)^2| ≤ 2*L*dist x y := by |
| calc |
| _ = |F x-F y| * |F x+F y| := by rw [← abs_mul]; congr 1; ring |
| _ ≤ (L*dist x y)*2 := mul_le_mul (by simpa only [Real.dist_eq] using hF.dist_le_mul x y) |
| ((abs_add_le _ _).trans (by linarith [hb x,hb y])) (abs_nonneg _) (by positivity) |
| _ = _ := by ring |
| |
| omit [DecidableEq ι] [∀ index, Fintype (M index)] in |
| lemma grid_square_error (p : ∀ j, κ → M j) (t : ι → ℝ) (ht : ∀ j, |t j| ≤ 1) |
| (θ φ : Sigma M → ℝ) {e : ℝ} (he : 0 ≤ e) (hθ : ∀ c, |θ c-φ c| ≤ e) |
| {L : ℝ≥0} {F : (κ → Circle) → ℝ} (hF : LipschitzWith L F) (hb : ∀ x, |F x| ≤ 1) : |
| |(F (phase p t θ))^2-(F (phase p t φ))^2| ≤ 2*L*Fintype.card ι*e := by |
| calc |
| _ ≤ 2*L*dist (phase p t θ) (phase p t φ) := squared_value_distance hF hb _ _ |
| _ ≤ 2*L*((Fintype.card ι : ℝ)*e) := mul_le_mul_of_nonneg_left (phase_dist_le p t ht θ φ he hθ) (by positivity) |
| _ = _ := by ring |
| end LargeIndependentSets.PhaseTest |
| |
| end OAI |
| |
