| import OAI.Combinatorics.CliqueFree.Model |
|
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| namespace OAI |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
| open Finset |
|
|
| attribute [local instance] Classical.propDecidable |
|
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| namespace CliqueFreeIndependence |
|
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| universe u v |
|
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| namespace FiniteEntropy |
|
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| variable {I : Type u} [Fintype I] |
|
|
| / |
| def IsProb (p : I → ℝ) : Prop := (∀ i, 0 ≤ p i) ∧ ∑ i, p i = 1 |
|
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| / |
| def entropy (w p : I → ℝ) : ℝ := ∑ i, p i * Real.log (w i / p i) |
|
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| / |
| def divergence (p q : I → ℝ) : ℝ := ∑ i, p i * Real.log (p i / q i) |
|
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| / |
| def tv (p q : I → ℝ) : ℝ := (∑ i, |p i - q i|) / 2 |
|
|
| lemma relTerm_nonneg {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) |
| (hs : 0 < p → 0 < q) : 0 ≤ p * Real.log (p / q) - p + q := by |
| rcases hp.eq_or_lt with hp | hp |
| · simp [← hp, hq] |
| have hq' := hs hp |
| have h := Real.log_le_sub_one_of_pos (div_pos hq' hp) |
| have hl : Real.log (q / p) = -Real.log (p / q) := by |
| rw [Real.log_div hq'.ne' hp.ne', Real.log_div hp.ne' hq'.ne'] |
| ring |
| rw [hl] at h |
| have hm := mul_le_mul_of_nonneg_left h hp.le |
| have hc : p * (q / p - 1) = q - p := by field_simp |
| rw [hc] at hm |
| linarith |
|
|
| lemma sqrt_sub_sq_le_relTerm {p q : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) |
| (hs : 0 < p → 0 < q) : |
| (Real.sqrt p - Real.sqrt q) ^ 2 ≤ p * Real.log (p / q) - p + q := by |
| rcases hp.eq_or_lt with hp | hp |
| · simp [← hp, Real.sq_sqrt hq] |
| have hq' := hs hp |
| have hsp := Real.sqrt_pos.2 hp |
| have hsq := Real.sqrt_pos.2 hq' |
| have h := Real.log_le_sub_one_of_pos (div_pos hsq hsp) |
| have hl : Real.log (Real.sqrt q / Real.sqrt p) = -(Real.log (p / q)) / 2 := by |
| rw [Real.log_div hsq.ne' hsp.ne', Real.log_sqrt hq, Real.log_sqrt hp.le, |
| Real.log_div hp.ne' hq'.ne'] |
| ring |
| rw [hl] at h |
| have hm := mul_le_mul_of_nonneg_left h (show 0 ≤ 2 * p by positivity) |
| have hc : p * (Real.sqrt q / Real.sqrt p) = Real.sqrt p * Real.sqrt q := by |
| calc |
| _ = (Real.sqrt p) ^ 2 * (Real.sqrt q / Real.sqrt p) := by rw [Real.sq_sqrt hp.le] |
| _ = Real.sqrt p * Real.sqrt q := by field_simp |
| have hp2 := Real.sq_sqrt hp.le |
| have hq2 := Real.sq_sqrt hq |
| nlinarith |
| |
| lemma divergence_nonneg {p q : I → ℝ} (hp : IsProb p) (hq : IsProb q) |
| (hs : ∀ i, 0 < p i → 0 < q i) : 0 ≤ divergence p q := by |
| have h := sum_nonneg (s := univ) fun i _ ↦ relTerm_nonneg (hp.1 i) (hq.1 i) (hs i) |
| simpa [divergence, sum_add_distrib, sum_sub_distrib, hp.2, hq.2] using h |
| |
| lemma hellinger_le_divergence {p q : I → ℝ} (hp : IsProb p) (hq : IsProb q) |
| (hs : ∀ i, 0 < p i → 0 < q i) : |
| (∑ i, (Real.sqrt (p i) - Real.sqrt (q i)) ^ 2) ≤ divergence p q := by |
| have h := sum_le_sum (s := univ) fun i _ ↦ |
| sqrt_sub_sq_le_relTerm (hp.1 i) (hq.1 i) (hs i) |
| simpa [divergence, sum_add_distrib, sum_sub_distrib, hp.2, hq.2] using h |
| |
| lemma tv_nonneg (p q : I → ℝ) : 0 ≤ tv p q := by |
| exact div_nonneg (sum_nonneg fun _ _ ↦ abs_nonneg _) (by norm_num) |
| |
| lemma tv_sq_le_hellinger {p q : I → ℝ} (hp : IsProb p) (hq : IsProb q) : |
| tv p q ^ 2 ≤ ∑ i, (Real.sqrt (p i) - Real.sqrt (q i)) ^ 2 := by |
| let f := fun i ↦ |Real.sqrt (p i) - Real.sqrt (q i)| |
| let g := fun i ↦ Real.sqrt (p i) + Real.sqrt (q i) |
| have hfactor (i : I) : |p i - q i| = f i * g i := by |
| have he : p i - q i = |
| (Real.sqrt (p i) - Real.sqrt (q i)) * |
| (Real.sqrt (p i) + Real.sqrt (q i)) := by |
| nlinarith [Real.sq_sqrt (hp.1 i), Real.sq_sqrt (hq.1 i)] |
| rw [he, abs_mul, abs_of_nonneg (by positivity : 0 ≤ Real.sqrt (p i) + Real.sqrt (q i))] |
| have hsum : (∑ i, f i ^ 2) + (∑ i, g i ^ 2) = 4 := by |
| rw [← sum_add_distrib] |
| calc |
| _ = (∑ i, (2 * p i + 2 * q i)) := by |
| apply sum_congr rfl |
| intro i _ |
| dsimp [f, g] |
| rw [sq_abs] |
| nlinarith [Real.sq_sqrt (hp.1 i), Real.sq_sqrt (hq.1 i)] |
| _ = 4 := by norm_num [sum_add_distrib, ← mul_sum, hp.2, hq.2] |
| have hf0 : 0 ≤ ∑ i, f i ^ 2 := sum_nonneg fun _ _ ↦ sq_nonneg _ |
| have hg4 : ∑ i, g i ^ 2 ≤ 4 := by linarith |
| have hc := sum_mul_sq_le_sq_mul_sq univ f g |
| have hle := mul_le_mul_of_nonneg_left hg4 hf0 |
| have he : (∑ i, f i ^ 2) = ∑ i, (Real.sqrt (p i) - Real.sqrt (q i)) ^ 2 := by |
| simp [f] |
| rw [tv, show (∑ i, |p i - q i|) = ∑ i, f i * g i from sum_congr rfl (fun i _ ↦ hfactor i)] |
| rw [← he] |
| nlinarith |
| |
| lemma entropy_eq_sum {w p : I → ℝ} (hw : ∀ i, 0 < w i) (hp : ∀ i, 0 ≤ p i) : |
| entropy w p = ∑ i, (p i * Real.log (w i) - p i * Real.log (p i)) := by |
| apply sum_congr rfl |
| intro i _ |
| rcases (hp i).eq_or_lt with hi | hi |
| · simp [← hi] |
| · rw [Real.log_div (hw i).ne' hi.ne', mul_sub] |
| |
| lemma divergence_eq_sum {p q : I → ℝ} (hp : ∀ i, 0 ≤ p i) |
| (hs : ∀ i, 0 < p i → 0 < q i) : |
| divergence p q = ∑ i, (p i * Real.log (p i) - p i * Real.log (q i)) := by |
| apply sum_congr rfl |
| intro i _ |
| rcases (hp i).eq_or_lt with hi | hi |
| · simp [← hi] |
| · rw [Real.log_div hi.ne' (hs i hi).ne', mul_sub] |
| |
| lemma entropy_ge_neg_log_of_density {w p : I → ℝ} (_hw : ∀ i, 0 < w i) |
| (hp : IsProb p) {b : ℝ} (hb : 0 < b) (hden : ∀ i, p i ≤ b * w i) : |
| -Real.log b ≤ entropy w p := by |
| have hi (i : I) : p i * (-Real.log b) ≤ p i * Real.log (w i / p i) := by |
| rcases (hp.1 i).eq_or_lt with hi | hi |
| · simp [← hi] |
| apply mul_le_mul_of_nonneg_left _ hi.le |
| rw [← Real.log_inv] |
| apply Real.log_le_log (inv_pos.2 hb) |
| apply (le_div_iff₀ hi).2 |
| calc |
| b⁻¹ * p i = p i / b := by ring |
| _ ≤ w i := (div_le_iff₀ hb).2 (by simpa [mul_comm] using hden i) |
| have h := sum_le_sum (s := univ) fun i _ ↦ hi i |
| simpa [← sum_mul, hp.2, entropy] using h |
| |
| lemma tv_comm (p q : I → ℝ) : tv p q = tv q p := by |
| simp only [tv, abs_sub_comm] |
| |
| lemma tv_triangle (p q r : I → ℝ) : tv p r ≤ tv p q + tv q r := by |
| have h := sum_le_sum (s := univ) fun i _ ↦ abs_sub_le (p i) (q i) (r i) |
| simpa only [sum_add_distrib, tv, ← add_div] using div_le_div_of_nonneg_right h (by norm_num : (0 : ℝ) ≤ 2) |
| |
| lemma weighted_sq_le {a f : I → ℝ} (ha : IsProb a) : |
| (∑ i, a i * f i) ^ 2 ≤ ∑ i, a i * (f i) ^ 2 := by |
| let m := ∑ i, a i * f i |
| have h := sum_nonneg (s := univ) fun i _ ↦ |
| mul_nonneg (ha.1 i) (sq_nonneg (f i - m)) |
| have he : (∑ i, a i * (f i - m) ^ 2) = (∑ i, a i * (f i) ^ 2) - m ^ 2 := by |
| calc |
| _ = ∑ i, (a i * (f i) ^ 2 - 2 * (a i * f i) * m + a i * m ^ 2) := by |
| apply sum_congr rfl |
| intro i _ |
| ring |
| _ = _ := by |
| rw [sum_add_distrib, sum_sub_distrib, ← sum_mul, ← mul_sum, ← sum_mul, ha.2] |
| dsimp [m] |
| ring |
| rw [he] at h |
| dsimp [m] at h |
| linarith |
| |
| variable {J : Type v} [Fintype J] |
| |
| /-- Mixing finitely many probability vectors. -/ |
| lemma mix_isProb {a : J → ℝ} {p : J → I → ℝ} (ha : IsProb a) |
| (hp : ∀ j, IsProb (p j)) : IsProb (fun i ↦ ∑ j, a j * p j i) := by |
| refine ⟨fun i ↦ sum_nonneg fun j _ ↦ mul_nonneg (ha.1 j) ((hp j).1 i), ?_⟩ |
| rw [sum_comm] |
| simp_rw [← mul_sum, fun j ↦ (hp j).2, mul_one] |
| exact ha.2 |
| |
| omit [Fintype I] in |
| lemma mix_support {a : J → ℝ} {p : J → I → ℝ} (ha : ∀ j, 0 ≤ a j) |
| (hp : ∀ j i, 0 ≤ p j i) {j : J} (hj : 0 < a j) {i : I} (hi : 0 < p j i) : |
| 0 < ∑ k, a k * p k i := by |
| classical |
| exact (mul_pos hj hi).trans_le (single_le_sum |
| (fun k _ ↦ mul_nonneg (ha k) (hp k i)) (mem_univ j)) |
| |
| /-- The finite entropy-mixture identity for the averaged increment. -/ |
| lemma entropy_mix_gap {w : I → ℝ} (hw : ∀ i, 0 < w i) |
| {a : J → ℝ} {p : J → I → ℝ} (ha : IsProb a) (hp : ∀ j, IsProb (p j)) : |
| entropy w (fun i ↦ ∑ j, a j * p j i) - ∑ j, a j * entropy w (p j) = |
| ∑ j, a j * divergence (p j) (fun i ↦ ∑ k, a k * p k i) := by |
| let b := fun i ↦ ∑ j, a j * p j i |
| have hb := mix_isProb ha hp |
| have hd (j : J) : a j * divergence (p j) b = |
| a j * (∑ i, (p j i * Real.log (p j i) - p j i * Real.log (b i))) := by |
| rcases (ha.1 j).eq_or_lt with hj | hj |
| · simp [← hj] |
| · rw [divergence_eq_sum (hp j).1 |
| (fun i hi ↦ mix_support ha.1 (fun k ↦ (hp k).1) hj hi)] |
| change entropy w b - ∑ j, a j * entropy w (p j) = ∑ j, a j * divergence (p j) b |
| rw [entropy_eq_sum hw hb.1] |
| simp_rw [entropy_eq_sum hw (hp _).1, hd, mul_sum, mul_sub, sum_sub_distrib] |
| have he (f : I → ℝ) : (∑ i, b i * f i) = ∑ j, ∑ i, a j * (p j i * f i) := by |
| dsimp [b] |
| simp_rw [sum_mul, mul_assoc] |
| rw [sum_comm] |
| rw [he, he] |
| ring |
| |
| lemma average_tv_sq_le_average_divergence {a : J → ℝ} (ha : IsProb a) |
| {p q : J → I → ℝ} (hp : ∀ j, IsProb (p j)) (hq : ∀ j, IsProb (q j)) |
| (hs : ∀ j, 0 < a j → ∀ i, 0 < p j i → 0 < q j i) : |
| (∑ j, a j * tv (p j) (q j)) ^ 2 ≤ ∑ j, a j * divergence (p j) (q j) := by |
| apply (weighted_sq_le ha).trans |
| apply sum_le_sum |
| intro j _ |
| rcases (ha.1 j).eq_or_lt with hj | hj |
| · simp [← hj] |
| apply mul_le_mul_of_nonneg_left _ hj.le |
| exact (tv_sq_le_hellinger (hp j) (hq j)).trans |
| (hellinger_le_divergence (hp j) (hq j) (hs j hj)) |
| |
| omit [Fintype I] in |
| lemma mul_log_div {p w : ℝ} (hp : 0 ≤ p) (hw : 0 < w) : |
| p * Real.log (w / p) = p * Real.log w - p * Real.log p := by |
| rcases hp.eq_or_lt with hp | hp |
| · simp [← hp] |
| · rw [Real.log_div hw.ne' hp.ne', mul_sub] |
| |
| omit [Fintype I] in |
| lemma partial_entropy_le (s : Finset I) {w p : I → ℝ} |
| (hw : ∀ i ∈ s, 0 < w i) (hp : ∀ i ∈ s, 0 ≤ p i) |
| {m : ℝ} (hm : 0 < m) (hwm : ∑ i ∈ s, w i ≤ m) : |
| (∑ i ∈ s, p i * Real.log (w i / p i)) ≤ |
| (∑ i ∈ s, p i) * Real.log (m / (∑ i ∈ s, p i)) := by |
| let t := ∑ i ∈ s, p i |
| have ht : 0 ≤ t := sum_nonneg hp |
| rcases ht.eq_or_lt with ht | ht |
| · have hz : ∀ i ∈ s, p i = 0 := (sum_eq_zero_iff_of_nonneg hp).1 ht.symm |
| simp only [show (∑ i ∈ s, p i) = 0 from ht.symm, zero_mul] |
| apply le_of_eq |
| exact sum_eq_zero (fun i hi ↦ by rw [hz i hi, zero_mul]) |
| have hi (i : I) (his : i ∈ s) : |
| p i * Real.log (w i / p i) ≤ p i * Real.log (m / t) - p i + t * w i / m := by |
| rcases (hp i his).eq_or_lt with hpi | hpi |
| · simp only [← hpi, zero_mul, sub_self, zero_add] |
| exact (div_pos (mul_pos ht (hw i his)) hm).le |
| have hq : 0 < t * w i / m := div_pos (mul_pos ht (hw i his)) hm |
| have hr := relTerm_nonneg hpi.le hq.le (fun _ ↦ hq) |
| have hl : Real.log (p i / (t * w i / m)) = |
| Real.log (m / t) - Real.log (w i / p i) := by |
| rw [Real.log_div hpi.ne' hq.ne', |
| Real.log_div (mul_pos ht (hw i his)).ne' hm.ne', |
| Real.log_mul ht.ne' (hw i his).ne', |
| Real.log_div hm.ne' ht.ne', Real.log_div (hw i his).ne' hpi.ne'] |
| ring |
| rw [hl] at hr |
| nlinarith |
| have h := sum_le_sum hi |
| simp only [sum_add_distrib, sum_sub_distrib, ← sum_mul, ← sum_div, ← mul_sum] at h |
| have hq : t * (∑ i ∈ s, w i) / m ≤ t := by |
| apply (div_le_iff₀ hm).2 |
| exact mul_le_mul_of_nonneg_left hwm ht.le |
| change (∑ i ∈ s, p i * Real.log (w i / p i)) ≤ t * Real.log (m / t) |
| change (∑ i ∈ s, p i * Real.log (w i / p i)) ≤ |
| t * Real.log (m / t) - t + t * (∑ i ∈ s, w i) / m at h |
| linarith |
| |
| lemma prob_sum_on {p : I → ℝ} (hp : IsProb p) (D : Finset I) |
| (hs : ∀ i, i ∉ D → p i = 0) : ∑ i ∈ D, p i = 1 := by |
| rw [← hp.2] |
| exact sum_subset (subset_univ _) (fun i _ hi ↦ hs i hi) |
| |
| lemma entropy_sum_on {w p : I → ℝ} (D : Finset I) |
| (hs : ∀ i, i ∉ D → p i = 0) : entropy w p = ∑ i ∈ D, p i * Real.log (w i / p i) := by |
| symm |
| apply sum_subset (subset_univ _) |
| intro i _ hi |
| rw [hs i hi, zero_mul] |
| |
| lemma entropy_supported_le_log (D : Finset I) {w p : I → ℝ} |
| (hw : ∀ i ∈ D, 0 < w i) (hp : IsProb p) |
| (hs : ∀ i, i ∉ D → p i = 0) {m : ℝ} (hm : 0 < m) (hwm : ∑ i ∈ D, w i ≤ m) : |
| entropy w p ≤ Real.log m := by |
| have h := partial_entropy_le D hw (fun i _ ↦ hp.1 i) hm hwm |
| simpa only [prob_sum_on hp D hs, div_one, one_mul, ← entropy_sum_on D hs] using h |
| |
| lemma entropy_supported_concentration (D s : Finset I) (hsD : s ⊆ D) {w p : I → ℝ} |
| (hw : ∀ i ∈ D, 0 < w i) (hp : IsProb p) (hs : ∀ i, i ∉ D → p i = 0) |
| {m M : ℝ} (hm : 1 ≤ m) (hM : 0 < M) |
| (hwsmall : ∑ i ∈ s, w i ≤ m) (hwall : ∑ i ∈ D, w i ≤ M) : |
| entropy w p ≤ Real.log 2 + Real.log m + (1 - ∑ i ∈ s, p i) * Real.log M := by |
| classical |
| let t := ∑ i ∈ s, p i |
| let d := ∑ i ∈ D \ s, p i |
| have ht : 0 ≤ t := sum_nonneg fun i _ ↦ hp.1 i |
| have hd : 0 ≤ d := sum_nonneg fun i _ ↦ hp.1 i |
| have htd : d + t = 1 := by |
| dsimp [t, d] |
| rw [sum_sdiff hsD, prob_sum_on hp D hs] |
| have hm0 : 0 < m := lt_of_lt_of_le zero_lt_one hm |
| have hwc : (∑ i ∈ D \ s, w i) ≤ M := by |
| apply le_trans _ hwall |
| exact sum_le_sum_of_subset_of_nonneg sdiff_subset (fun i hi _ ↦ (hw i hi).le) |
| have ha := partial_entropy_le s (fun i hi ↦ hw i (hsD hi)) (fun i _ ↦ hp.1 i) hm0 hwsmall |
| have hc := partial_entropy_le (D \ s) (fun i hi ↦ hw i (mem_sdiff.1 hi).1) |
| (fun i _ ↦ hp.1 i) hM hwc |
| have he : entropy w p ≤ d * Real.log (M / d) + t * Real.log (m / t) := by |
| rw [entropy_sum_on D hs, ← sum_sdiff hsD] |
| exact add_le_add hc ha |
| rw [mul_log_div hd hM, mul_log_div ht hm0] at he |
| have hbin : -t * Real.log t - d * Real.log d ≤ Real.log 2 := by |
| have hb := Real.binEntropy_le_log_two (p := t) |
| have hd' : d = 1 - t := by linarith |
| rw [hd'] |
| simpa [Real.binEntropy, Real.log_inv] using hb |
| have hl := Real.log_nonneg hm |
| have ht1 : t ≤ 1 := by linarith |
| have hmterm : t * Real.log m ≤ Real.log m := by nlinarith |
| change entropy w p ≤ Real.log 2 + Real.log m + (1 - t) * Real.log M |
| have hd' : 1 - t = d := by linarith |
| rw [hd'] |
| linarith |
| |
| lemma entropy_tail_term {p w x a : ℝ} (hp : 0 ≤ p) (hw : 0 < w) |
| (hx : 0 < x) (hd : p ≤ x * w) : |
| (a + Real.log x) * (if p < w * Real.exp (-a) then p else 0) ≤ |
| p * (Real.log (w / p) + Real.log x) := by |
| rcases hp.eq_or_lt with hp | hp |
| · simp [← hp] |
| have hlog := Real.log_le_log hp hd |
| rw [Real.log_mul hx.ne' hw.ne'] at hlog |
| have hlower : 0 ≤ Real.log (w / p) + Real.log x := by |
| rw [Real.log_div hw.ne' hp.ne'] |
| linarith |
| split_ifs with hsmall |
| · have h := Real.log_lt_log hp hsmall |
| rw [Real.log_mul hw.ne' (Real.exp_ne_zero _), Real.log_exp] at h |
| have ha : a ≤ Real.log (w / p) := by |
| rw [Real.log_div hw.ne' hp.ne'] |
| linarith |
| nlinarith |
| · simpa using mul_nonneg hp.le hlower |
|
|
| lemma entropy_tail {p w : I → ℝ} (hp : IsProb p) (hw : ∀ i, 0 < w i) |
| {x a : ℝ} (hx : 0 < x) (hd : ∀ i, p i ≤ x * w i) : |
| (a + Real.log x) * (∑ i, if p i < w i * Real.exp (-a) then p i else 0) ≤ |
| entropy w p + Real.log x := by |
| have h := sum_le_sum (s := univ) fun i _ ↦ |
| entropy_tail_term (a := a) (hp.1 i) (hw i) hx (hd i) |
| simpa only [mul_sum, mul_add, sum_add_distrib, ← sum_mul, hp.2, one_mul, |
| entropy] using h |
|
|
| end FiniteEntropy |
|
|
| end CliqueFreeIndependence |
|
|
| end |
|
|
| end OAI |
|
|
