| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Polynomial Filter |
| open scoped BigOperators Topology |
| namespace MinUncut.GaussianHermite |
| variable {ι : Type*} [Fintype ι] |
|
|
| omit [Fintype ι] in |
| lemma mem_range_bump [DecidableEq ι] (i : ι) (a : ι → ℕ) : |
| a ∈ Set.range (bump i) ↔ 0 < a i := by |
| constructor |
| · rintro ⟨b,rfl⟩ |
| simp [bump] |
| · intro ha |
| refine ⟨Function.update a i (a i-1), ?_⟩ |
| funext j |
| by_cases hj : j=i |
| · subst j |
| simp only [bump, Function.update_self] |
| omega |
| · simp [bump, Function.update_of_ne hj] |
|
|
| lemma productHermiteBasis_apply (a : ι → ℕ) : |
| productHermiteBasis a = psiVec a := by |
| simp only [productHermiteBasis, HilbertBasis.coe_mkOfOrthogonalEqBot] |
|
|
| lemma coeff_eq_inner {f : (ι → ℝ) → ℝ} (hf : MemLp f 2 (γpi ι)) (a : ι → ℕ) : |
| coeff f a = inner ℝ (productHermiteBasis a) (hf.toLp f) := by |
| rw [productHermiteBasis_apply] |
| exact (integral_inner_toLp (memLp_psi a) hf).symm |
|
|
| lemma hasSum_coeff_sq {f : (ι → ℝ) → ℝ} (hf : MemLp f 2 (γpi ι)) : |
| HasSum (fun a => (coeff f a)^2) (∫ x, (f x)^2 ∂γpi ι) := by |
| have hs := productHermiteBasis.hasSum_inner_mul_inner (hf.toLp f) (hf.toLp f) |
| have hc (a : ι → ℕ) : inner ℝ (hf.toLp f) (productHermiteBasis a) = coeff f a := by |
| rw [real_inner_comm] |
| exact (coeff_eq_inner hf a).symm |
| simp only [hc, ← coeff_eq_inner hf, ← pow_two] at hs |
| simpa only [integral_inner_toLp hf hf, ← pow_two] using hs |
|
|
| lemma parseval {f : (ι → ℝ) → ℝ} (hf : MemLp f 2 (γpi ι)) : |
| (∑' a, (coeff f a)^2) = ∫ x, (f x)^2 ∂γpi ι := (hasSum_coeff_sq hf).tsum_eq |
|
|
| lemma coeff_const_mul (r : ℝ) (f : (ι → ℝ) → ℝ) (a : ι → ℕ) : |
| coeff (fun x => r*f x) a = r*coeff f a := by |
| unfold coeff |
| rw [← integral_const_mul] |
| apply integral_congr_ae |
| filter_upwards [] with x |
| ring |
|
|
| def gradient (ρ s : ℝ) (Q : (ι → ℝ) → ℝ) (i : ι) (x : ι → ℝ) : ℝ := |
| (ρ/s)*score ρ s Q i x |
|
|
| lemma memLp_gradient (ρ s : ℝ) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) |
| (hb : ∀ x, |Q x| ≤ 1) (i : ι) : MemLp (gradient ρ s Q i) 2 (γpi ι) := |
| (memLp_score ρ s hQ hb i).const_mul (ρ/s) |
|
|
| lemma gradient_coefficient [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) (hs : s ≠ 0) |
| {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) |
| (a : ι → ℕ) (i : ι) : |
| coeff (gradient ρ s Q i) a = |
| Real.sqrt (a i+1:ℝ)*ρ^(degree a+1)*coeff Q (bump i a) := by |
| change coeff (fun x => (ρ/s)*score ρ s Q i x) a = _ |
| rw [coeff_const_mul] |
| unfold coeff |
| rw [score_coefficient ρ s h hQ hb] |
| dsimp [degree, bump] |
| rw [pow_succ] |
| field_simp |
|
|
| lemma gradient_coefficient_sq [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) (hs : s ≠ 0) |
| {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) |
| (a : ι → ℕ) (i : ι) : |
| (coeff (gradient ρ s Q i) a)^2 = |
| (bump i a i : ℝ)*ρ^(2*degree (bump i a))*(coeff Q (bump i a))^2 := by |
| rw [gradient_coefficient ρ s h hs hQ hb, mul_pow, mul_pow, |
| Real.sq_sqrt (by positivity), ← pow_mul, degree_bump] |
| simp only [bump, Function.update_self, Nat.cast_add, Nat.cast_one] |
| congr 2 |
| ring |
|
|
| lemma hasSum_gradient_energy_coord [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) |
| (hs : s ≠ 0) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) (i : ι) : |
| HasSum (fun a => (a i : ℝ)*ρ^(2*degree a)*(coeff Q a)^2) |
| (∫ x, (gradient ρ s Q i x)^2 ∂γpi ι) := by |
| apply (bump_injective i).hasSum_iff (fun a ha => ?_) |>.mp |
| · simpa only [Function.comp_def, ← gradient_coefficient_sq ρ s h hs hQ hb] using |
| hasSum_coeff_sq (memLp_gradient ρ s hQ hb i) |
| · have hz : a i = 0 := by |
| rw [mem_range_bump] at ha |
| omega |
| simp [hz] |
|
|
| lemma hasSum_gradient_energy [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) |
| (hs : s ≠ 0) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) : |
| HasSum (fun a => (degree a : ℝ)*ρ^(2*degree a)*(coeff Q a)^2) |
| (∑ i, ∫ x, (gradient ρ s Q i x)^2 ∂γpi ι) := by |
| have hh := hasSum_sum (s := Finset.univ) (fun i _ => |
| hasSum_gradient_energy_coord ρ s h hs hQ hb i) |
| simpa only [← Finset.sum_mul, ← Nat.cast_sum, degree] using hh |
|
|
| lemma memLp_noise (ρ s : ℝ) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) |
| (hb : ∀ x, |Q x| ≤ 1) : MemLp (noise ρ s Q) 2 (γpi ι) := |
| MemLp.of_bound (measurable_noise ρ s hQ).aestronglyMeasurable 1 |
| (ae_of_all _ fun x => by simpa only [Real.norm_eq_abs] using noise_bound ρ s hb x) |
|
|
| lemma coeff_noise (ρ s : ℝ) (h : ρ^2+s^2=1) {Q : (ι → ℝ) → ℝ} |
| (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) (a : ι → ℕ) : |
| coeff (noise ρ s Q) a = ρ^(degree a)*coeff Q a := noise_coefficient ρ s h hQ hb a |
|
|
| lemma coeff_noise_sq (ρ s : ℝ) (h : ρ^2+s^2=1) {Q : (ι → ℝ) → ℝ} |
| (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) (a : ι → ℕ) : |
| (coeff (noise ρ s Q) a)^2 = ρ^(2*degree a)*(coeff Q a)^2 := by |
| rw [coeff_noise ρ s h hQ hb, mul_pow, ← pow_mul, mul_comm (degree a) 2] |
|
|
| @[simp] lemma psi_zero (x : ι → ℝ) : psi 0 x = 1 := by |
| simp [psi, H] |
|
|
| @[simp] lemma coeff_zero (f : (ι → ℝ) → ℝ) : coeff f 0 = ∫ x, f x ∂γpi ι := by |
| simp [coeff] |
|
|
| lemma degree_eq_zero_iff (a : ι → ℕ) : degree a = 0 ↔ a = 0 := by |
| simp only [degree, Finset.sum_eq_zero_iff, Finset.mem_univ, true_implies, |
| funext_iff, Pi.zero_apply] |
|
|
| lemma poincare [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) (hs : s ≠ 0) |
| {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) |
| (hmean : (∫ x, noise ρ s Q x ∂γpi ι) = 0) : |
| (∫ x, (noise ρ s Q x)^2 ∂γpi ι) ≤ |
| ∑ i, ∫ x, (gradient ρ s Q i x)^2 ∂γpi ι := by |
| apply hasSum_le (fun a => ?_) (hasSum_coeff_sq (memLp_noise ρ s hQ hb)) |
| (hasSum_gradient_energy ρ s h hs hQ hb) |
| by_cases ha : degree a = 0 |
| · have haz := (degree_eq_zero_iff a).mp ha |
| subst a |
| simp only [coeff_zero, hmean, zero_pow (by omega : 2 ≠ 0), degree] |
| simp |
| · rw [coeff_noise_sq ρ s h hQ hb] |
| have hd : (1:ℝ) ≤ degree a := by exact_mod_cast Nat.one_le_iff_ne_zero.mpr ha |
| calc |
| _ = 1*(ρ^(2*degree a)*(coeff Q a)^2) := by ring |
| _ ≤ (degree a : ℝ)*(ρ^(2*degree a)*(coeff Q a)^2) := by |
| apply mul_le_mul_of_nonneg_right hd |
| rw [pow_mul] |
| exact mul_nonneg (pow_nonneg (sq_nonneg ρ) _) (sq_nonneg _) |
| _ = _ := by ring |
|
|
| def low [DecidableEq ι] (d : ℕ) : Finset (ι → ℕ) := |
| (Fintype.piFinset (fun _ : ι => Finset.range (d+1))).filter (fun a => degree a ≤ d) |
|
|
| @[simp] lemma mem_low [DecidableEq ι] (d : ℕ) (a : ι → ℕ) : |
| a ∈ low d ↔ degree a ≤ d := by |
| simp only [low, Finset.mem_filter, Fintype.mem_piFinset, Finset.mem_range] |
| constructor |
| · exact And.right |
| · intro h |
| refine ⟨fun i => ?_, h⟩ |
| have hi : a i ≤ degree a := Finset.single_le_sum (fun j _ => Nat.zero_le (a j)) |
| (Finset.mem_univ i) |
| omega |
|
|
| def project [DecidableEq ι] (d : ℕ) (f : (ι → ℝ) → ℝ) (x : ι → ℝ) : ℝ := |
| ∑ a ∈ low d, coeff f a * psi a x |
|
|
| lemma memLp_project [DecidableEq ι] (d : ℕ) (f : (ι → ℝ) → ℝ) : |
| MemLp (project d f) 2 (γpi ι) := by |
| exact memLp_finsetSum _ (fun a _ => (memLp_psi a).const_mul _) |
|
|
| lemma coeff_project [DecidableEq ι] (d : ℕ) (f : (ι → ℝ) → ℝ) (a : ι → ℕ) : |
| coeff (project d f) a = if degree a ≤ d then coeff f a else 0 := by |
| unfold coeff project |
| simp_rw [Finset.mul_sum] |
| rw [integral_finsetSum _ (fun b _ => |
| ((memLp_psi a).integrable_mul (memLp_psi b)).const_mul (coeff f b) |>.congr |
| (ae_of_all _ fun x => by dsimp; ring))] |
| simp_rw [show ∀ b : ι → ℕ, (fun x => psi a x * (coeff f b * psi b x)) = |
| (fun x => coeff f b * (psi a x * psi b x)) by intro b; funext x; ring, |
| integral_const_mul, psi_product] |
| simp [coeff, mem_low] |
|
|
| lemma coeff_sub {f g : (ι → ℝ) → ℝ} (hf : MemLp f 2 (γpi ι)) |
| (hg : MemLp g 2 (γpi ι)) (a : ι → ℕ) : |
| coeff (fun x => f x-g x) a = coeff f a-coeff g a := by |
| unfold coeff |
| simp_rw [mul_sub] |
| exact integral_sub ((memLp_psi a).integrable_mul hf) ((memLp_psi a).integrable_mul hg) |
|
|
| lemma coeff_residual [DecidableEq ι] {f : (ι → ℝ) → ℝ} (hf : MemLp f 2 (γpi ι)) |
| (d : ℕ) (a : ι → ℕ) : |
| coeff (fun x => f x-project d f x) a = if d < degree a then coeff f a else 0 := by |
| rw [coeff_sub hf (memLp_project d f), coeff_project] |
| split_ifs <;> simp_all; omega |
|
|
| lemma hasSum_gradient_tail_coord [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) |
| (hs : s ≠ 0) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) |
| (d : ℕ) (i : ι) : |
| HasSum (fun a => if d+2 ≤ degree a then |
| (a i : ℝ)*ρ^(2*degree a)*(coeff Q a)^2 else 0) |
| (∫ x, (gradient ρ s Q i x-project d (gradient ρ s Q i) x)^2 ∂γpi ι) := by |
| apply (bump_injective i).hasSum_iff (fun a ha => ?_) |>.mp |
| · convert hasSum_coeff_sq ((memLp_gradient ρ s hQ hb i).sub |
| (memLp_project d (gradient ρ s Q i))) using 1 |
| · funext a |
| change (if d+2 ≤ degree (bump i a) then |
| (bump i a i : ℝ)*ρ^(2*degree (bump i a))*(coeff Q (bump i a))^2 else 0) = |
| (coeff (fun x => gradient ρ s Q i x-project d (gradient ρ s Q i) x) a)^2 |
| rw [coeff_residual (memLp_gradient ρ s hQ hb i)] |
| split_ifs with hd he he |
| · exact (gradient_coefficient_sq ρ s h hs hQ hb a i).symm |
| · rw [degree_bump] at hd; omega |
| · rw [degree_bump] at hd; omega |
| · simp |
| · have hz : a i = 0 := by rw [mem_range_bump] at ha; omega |
| simp [hz] |
|
|
| lemma hasSum_gradient_tail [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) |
| (hs : s ≠ 0) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) |
| (d : ℕ) : |
| HasSum (fun a => if d+2 ≤ degree a then |
| (degree a : ℝ)*ρ^(2*degree a)*(coeff Q a)^2 else 0) |
| (∑ i, ∫ x, (gradient ρ s Q i x-project d (gradient ρ s Q i) x)^2 ∂γpi ι) := by |
| have hh := hasSum_sum (s := Finset.univ) (fun i _ => |
| hasSum_gradient_tail_coord ρ s h hs hQ hb d i) |
| convert hh using 1 |
| funext a |
| split_ifs <;> simp [← Finset.sum_mul, degree, Nat.cast_sum] |
|
|
| lemma gradient_tail_le [DecidableEq ι] (ρ s : ℝ) (h : ρ^2+s^2=1) |
| (hs : s ≠ 0) {Q : (ι → ℝ) → ℝ} (hQ : Measurable Q) (hb : ∀ x, |Q x| ≤ 1) |
| (d : ℕ) {ε : ℝ} (hε : 0 ≤ ε) (hd : ∀ k : ℕ, d+2 ≤ k → (k:ℝ)*ρ^(2*k) ≤ ε) : |
| (∑ i, ∫ x, (gradient ρ s Q i x-project d (gradient ρ s Q i) x)^2 ∂γpi ι) ≤ ε := by |
| have hq : MemLp Q 2 (γpi ι) := MemLp.of_bound hQ.aestronglyMeasurable 1 |
| (ae_of_all _ fun x => by simpa only [Real.norm_eq_abs] using hb x) |
| have he := hasSum_le (f := fun a => if d+2 ≤ degree a then |
| (degree a : ℝ)*ρ^(2*degree a)*(coeff Q a)^2 else 0) |
| (g := fun a => ε*(coeff Q a)^2) (fun a => ?_) |
| (hasSum_gradient_tail ρ s h hs hQ hb d) ((hasSum_coeff_sq hq).mul_left ε) |
| · have hi : (∫ x, Q x^2 ∂γpi ι) ≤ 1 := by |
| calc |
| _ ≤ ∫ _ : ι → ℝ, (1:ℝ) ∂γpi ι := integral_mono |
| (hq.integrable_sq) (integrable_const 1) (fun x => by |
| have ha := hb x; nlinarith [sq_abs (Q x), abs_nonneg (Q x)]) |
| _ = 1 := by simp |
| exact he.trans (by nlinarith) |
| · split_ifs with ha |
| · exact mul_le_mul_of_nonneg_right (hd _ ha) (sq_nonneg _) |
| · positivity |
|
|
| lemma exists_cutoff {ρ ε : ℝ} (hρ : |ρ| < 1) (hε : 0 < ε) : |
| ∃ d : ℕ, ∀ k : ℕ, d+2 ≤ k → (k:ℝ)*ρ^(2*k) ≤ ε := by |
| have hh : ρ^2 < 1 := by nlinarith [sq_abs ρ, abs_nonneg ρ] |
| have ht := tendsto_self_mul_const_pow_of_lt_one (sq_nonneg ρ) hh |
| obtain ⟨d, hd⟩ := (eventually_atTop.1 (ht.eventually (gt_mem_nhds hε))) |
| refine ⟨d, fun k hk => ?_⟩ |
| have he := hd k (by omega) |
| simpa only [pow_mul] using he.le |
|
|
| end MinUncut.GaussianHermite |
|
|
| end |
|
|
| end OAI |
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|
