planted cliques iterative thresholding and message
play

Planted Cliques, Iterative Thresholding and Message Passing - PowerPoint PPT Presentation

Sep 09, 2022 •24 likes •1.15k views

Planted Cliques, Iterative Thresholding and Message Passing Algorithms Yash Deshpande and Andrea Montanari Stanford University November 5, 2013 Deshpande, Montanari Planted Cliques November 5, 2013 1 / 49 Problem Definition Given

  1. Seems much harder than it looks! ◮ “Statistical algorithms” fail if k = n 1 ❂ 2 � ✍ : [Feldman et al., 2012] ♣ ◮ r -Lovász-Schrijver fails for k ✔ n ❂ 2 r : [Feige, Krauthgamer, 2002] ♣ n ◮ r -Lasserre fails for k ✔ ( log n ) r 2 : [Widgerson and Meka, 2013] Deshpande, Montanari Planted Cliques November 5, 2013 21 / 49

  2. Our result Theorem (Deshpande, Montanari, 2013) ♣ n ❂ e , there exists an O ( n 2 log n ) time algorithm that If ❥ S ❥ = k ✕ ( 1 + ✧ ) identifies S with high probability. I will present: 1 A (wrong) heuristic analysis 2 How to fix the heuristic 3 Lower bounds Deshpande, Montanari Planted Cliques November 5, 2013 22 / 49

  3. Our result Theorem (Deshpande, Montanari, 2013) ♣ n ❂ e , there exists an O ( n 2 log n ) time algorithm that If ❥ S ❥ = k ✕ ( 1 + ✧ ) identifies S with high probability. I will present: 1 A (wrong) heuristic analysis 2 How to fix the heuristic 3 Lower bounds Deshpande, Montanari Planted Cliques November 5, 2013 22 / 49

  4. Our result Theorem (Deshpande, Montanari, 2013) ♣ n ❂ e , there exists an O ( n 2 log n ) time algorithm that If ❥ S ❥ = k ✕ ( 1 + ✧ ) identifies S with high probability. I will present: 1 A (wrong) heuristic analysis 2 How to fix the heuristic 3 Lower bounds Deshpande, Montanari Planted Cliques November 5, 2013 22 / 49

  5. Iterative Thresholding The power iteration: v t + 1 = A v t ✿ Improvement: v t + 1 = AF t ( v t ) ✿ where F t ( v ) = ( f t ( v 1 ) ❀ f t ( v 2 ) ❀ ✁ ✁ ✁ ❀ f t ( v n )) T Choose f t ( ✁ ) to exploit sparsity of e S Deshpande, Montanari Planted Cliques November 5, 2013 23 / 49

  6. Iterative Thresholding The power iteration: v t + 1 = A v t ✿ Improvement: v t + 1 = AF t ( v t ) ✿ where F t ( v ) = ( f t ( v 1 ) ❀ f t ( v 2 ) ❀ ✁ ✁ ✁ ❀ f t ( v n )) T Choose f t ( ✁ ) to exploit sparsity of e S Deshpande, Montanari Planted Cliques November 5, 2013 23 / 49

  7. Iterative Thresholding The power iteration: v t + 1 = A v t ✿ Improvement: v t + 1 = AF t ( v t ) ✿ where F t ( v ) = ( f t ( v 1 ) ❀ f t ( v 2 ) ❀ ✁ ✁ ✁ ❀ f t ( v n )) T Choose f t ( ✁ ) to exploit sparsity of e S Deshpande, Montanari Planted Cliques November 5, 2013 23 / 49

  8. Iterative Thresholding The power iteration: v t + 1 = A v t ✿ Improvement: v t + 1 = AF t ( v t ) ✿ where F t ( v ) = ( f t ( v 1 ) ❀ f t ( v 2 ) ❀ ✁ ✁ ✁ ❀ f t ( v n )) T Choose f t ( ✁ ) to exploit sparsity of e S Deshpande, Montanari Planted Cliques November 5, 2013 23 / 49

  9. Iterative Thresholding The power iteration: v t + 1 = A v t ✿ Improvement: v t + 1 = AF t ( v t ) ✿ where F t ( v ) = ( f t ( v 1 ) ❀ f t ( v 2 ) ❀ ✁ ✁ ✁ ❀ f t ( v n )) T Choose f t ( ✁ ) to exploit sparsity of e S Deshpande, Montanari Planted Cliques November 5, 2013 23 / 49

  10. (Wrong) Analysis ❳ 1 v t + 1 A ij f t ( v t ♣ n = j ) ✿ i j A ij are random ✝ 1 r.v. Use Central Limit Theorem for v t + 1 i Deshpande, Montanari Planted Cliques November 5, 2013 24 / 49

  11. (Wrong) Analysis ❳ 1 v t + 1 A ij f t ( v t ♣ n = j ) ✿ i j A ij are random ✝ 1 r.v. Use Central Limit Theorem for v t + 1 i Deshpande, Montanari Planted Cliques November 5, 2013 24 / 49

  12. (Wrong) Analysis If i ❂ ✷ S : ❳ 1 v t + 1 A ij f t ( v t ♣ n = j ) i j ✒ ✓ ❳ 0 ❀ 1 f t ( v t j ) 2 ✙ N n j Letting v t i ✙ N ( 0 ❀ ✛ 2 t ) . . . ❳ t + 1 = 1 ✛ 2 f t ( v t j ) 2 n j = E ❢ f t ( ✛ t ✘ ) 2 ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 25 / 49

  13. (Wrong) Analysis If i ❂ ✷ S : ❳ 1 v t + 1 A ij f t ( v t ♣ n = j ) i j ✒ ✓ ❳ 0 ❀ 1 f t ( v t j ) 2 ✙ N n j Letting v t i ✙ N ( 0 ❀ ✛ 2 t ) . . . ❳ t + 1 = 1 ✛ 2 f t ( v t j ) 2 n j = E ❢ f t ( ✛ t ✘ ) 2 ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 25 / 49

  14. (Wrong) Analysis If i ❂ ✷ S : ❳ 1 v t + 1 A ij f t ( v t ♣ n = j ) i j ✒ ✓ ❳ 0 ❀ 1 f t ( v t j ) 2 ✙ N n j Letting v t i ✙ N ( 0 ❀ ✛ 2 t ) . . . ❳ t + 1 = 1 ✛ 2 f t ( v t j ) 2 n j = E ❢ f t ( ✛ t ✘ ) 2 ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 25 / 49

  15. (Wrong) Analysis If i ❂ ✷ S : ❳ 1 v t + 1 A ij f t ( v t ♣ n = j ) i j ✒ ✓ ❳ 0 ❀ 1 f t ( v t j ) 2 ✙ N n j Letting v t i ✙ N ( 0 ❀ ✛ 2 t ) . . . ❳ t + 1 = 1 ✛ 2 f t ( v t j ) 2 n j = E ❢ f t ( ✛ t ✘ ) 2 ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 25 / 49

  16. (Wrong) Analysis If i ✷ S : ❳ ❳ 1 + 1 v t + 1 f t ( v t A ij f t ( v t = ♣ n j ) ♣ n j ) i j ✷ S j ❂ ✷ S ⑤ ④③ ⑥ ⑤ ④③ ⑥ ✖ t + 1 ✙ N ( 0 ❀✛ 2 t + 1 ) where ❳ 1 f t ( v t ♣ n ✖ t + 1 = j ) j ✷ S ✒ k ✓ ✒ 1 ✓ ❳ f t ( v t ♣ n = j ) k j ✷ S = ✔ E ❢ f t ( ✖ t + ✛ t ✘ ) ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 26 / 49

  17. (Wrong) Analysis If i ✷ S : ❳ ❳ 1 + 1 v t + 1 f t ( v t A ij f t ( v t = ♣ n j ) ♣ n j ) i j ✷ S j ❂ ✷ S ⑤ ④③ ⑥ ⑤ ④③ ⑥ ✖ t + 1 ✙ N ( 0 ❀✛ 2 t + 1 ) where ❳ 1 f t ( v t ♣ n ✖ t + 1 = j ) j ✷ S ✒ k ✓ ✒ 1 ✓ ❳ f t ( v t ♣ n = j ) k j ✷ S = ✔ E ❢ f t ( ✖ t + ✛ t ✘ ) ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 26 / 49

  18. (Wrong) Analysis If i ✷ S : ❳ ❳ 1 + 1 v t + 1 f t ( v t A ij f t ( v t = ♣ n j ) ♣ n j ) i j ✷ S j ❂ ✷ S ⑤ ④③ ⑥ ⑤ ④③ ⑥ ✖ t + 1 ✙ N ( 0 ❀✛ 2 t + 1 ) where ❳ 1 f t ( v t ♣ n ✖ t + 1 = j ) j ✷ S ✒ k ✓ ✒ 1 ✓ ❳ f t ( v t ♣ n = j ) k j ✷ S = ✔ E ❢ f t ( ✖ t + ✛ t ✘ ) ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 26 / 49

  19. (Wrong) Analysis If i ✷ S : ❳ ❳ 1 + 1 v t + 1 f t ( v t A ij f t ( v t = ♣ n j ) ♣ n j ) i j ✷ S j ❂ ✷ S ⑤ ④③ ⑥ ⑤ ④③ ⑥ ✖ t + 1 ✙ N ( 0 ❀✛ 2 t + 1 ) where ❳ 1 f t ( v t ♣ n ✖ t + 1 = j ) j ✷ S ✒ k ✓ ✒ 1 ✓ ❳ f t ( v t ♣ n = j ) k j ✷ S = ✔ E ❢ f t ( ✖ t + ✛ t ✘ ) ❣ ❀ ✘ ✘ N ( 0 ❀ 1 ) Deshpande, Montanari Planted Cliques November 5, 2013 26 / 49

  20. Summarizing . . . v t i , i / ∈ S 0 . 3 0 . 2 0 . 1 v t i , i ∈ S 0 − 4 − 2 0 2 4 Deshpande, Montanari Planted Cliques November 5, 2013 27 / 49

  21. State Evolution ✖ t + 1 = ✔ E ❢ f t ( ✖ t + ✛ t ✘ ) ❣ ✛ 2 t + 1 = E ❢ f t ( ✛ t ✘ ) 2 ❣ ✿ Using the optimal function f t ( x ) = e ✖ t x � ✖ 2 t ✖ t + 1 = ✔ e ✖ 2 t ❂ 2 ✛ 2 t + 1 = 1 Deshpande, Montanari Planted Cliques November 5, 2013 28 / 49

  22. State Evolution ✖ t + 1 = ✔ E ❢ f t ( ✖ t + ✛ t ✘ ) ❣ ✛ 2 t + 1 = E ❢ f t ( ✛ t ✘ ) 2 ❣ ✿ Using the optimal function f t ( x ) = e ✖ t x � ✖ 2 t ✖ t + 1 = ✔ e ✖ 2 t ❂ 2 ✛ 2 t + 1 = 1 Deshpande, Montanari Planted Cliques November 5, 2013 28 / 49

  23. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  24. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  25. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  26. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  27. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  28. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  29. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  30. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  31. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  32. Fixed points develop below threshold! 1 If ✔ ❃ ♣ e µ t +1 µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  33. Fixed points develop below threshold! 1 1 If ✔ ❃ If ✔ ❁ ♣ e ♣ e µ t +1 µ t +1 µ t µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  34. Fixed points develop below threshold! 1 1 If ✔ ❃ If ✔ ❁ ♣ e ♣ e µ t +1 µ t +1 µ t µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  35. Fixed points develop below threshold! 1 1 If ✔ ❃ If ✔ ❁ ♣ e ♣ e µ t +1 µ t +1 µ t µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  36. Fixed points develop below threshold! 1 1 If ✔ ❃ If ✔ ❁ ♣ e ♣ e µ t +1 µ t +1 µ t µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  37. Fixed points develop below threshold! 1 1 If ✔ ❃ If ✔ ❁ ♣ e ♣ e µ t +1 µ t +1 µ t µ t Deshpande, Montanari Planted Cliques November 5, 2013 29 / 49

  38. Analysis is wrong but. . . Theorem (Deshpande, Montanari, 2013) ♣ n ❂ e , there exists an O ( n 2 log n ) time algorithm that If ❥ S ❥ = k ✕ ( 1 + ✧ ) identifies S with high probability. . . . so we modify the algorithm. Deshpande, Montanari Planted Cliques November 5, 2013 30 / 49

  39. What algorithm? Slight modification to iterative scheme: ( v t i ) i ✷ [ n ] ✦ ( v t i ✦ j ) i ❀ j ✷ [ n ] ❳ 1 v t + 1 A i ❵ f t ( v t i ✦ j = ♣ n ❵ ✦ i ) ✿ ❵ ✻ = i ❀ j Analysis is exact as n ✦ ✶ . Deshpande, Montanari Planted Cliques November 5, 2013 31 / 49

  40. What algorithm? Slight modification to iterative scheme: ( v t i ) i ✷ [ n ] ✦ ( v t i ✦ j ) i ❀ j ✷ [ n ] ❳ 1 v t + 1 A i ❵ f t ( v t i ✦ j = ♣ n ❵ ✦ i ) ✿ ❵ ✻ = i ❀ j Analysis is exact as n ✦ ✶ . Deshpande, Montanari Planted Cliques November 5, 2013 31 / 49

  41. What algorithm? Slight modification to iterative scheme: ( v t i ) i ✷ [ n ] ✦ ( v t i ✦ j ) i ❀ j ✷ [ n ] ❳ 1 v t + 1 A i ❵ f t ( v t i ✦ j = ♣ n ❵ ✦ i ) ✿ ❵ ✻ = i ❀ j Analysis is exact as n ✦ ✶ . Deshpande, Montanari Planted Cliques November 5, 2013 31 / 49

  42. What algorithm? Slight modification to iterative scheme: ( v t i ) i ✷ [ n ] ✦ ( v t i ✦ j ) i ❀ j ✷ [ n ] ❳ 1 v t + 1 A i ❵ f t ( v t i ✦ j = ♣ n ❵ ✦ i ) ✿ ❵ ✻ = i ❀ j Analysis is exact as n ✦ ✶ . Deshpande, Montanari Planted Cliques November 5, 2013 31 / 49

  43. Fixing the heuristic Lemma Let ( f t ( z )) t ✕ 0 be a sequence of polynomials. Then, for every fixed t , and bounded, continuous function ✥ : R ✦ R the following limit holds in probability: ❳ 1 ✥ ( v t ♣ n lim i ✦ j ) = ✔ E ❢ ✥ ( ✖ t + ✛ t ✘ ) ❣ ❀ n ✦✶ i ✷ S ❳ 1 ✥ ( v t lim i ✦ j ) = E ❢ ✥ ( ✛ t ✘ ) ❣ ❀ n ✦✶ n i ✷ [ n ] ♥ S where ✘ ✘ N ( 0 ❀ 1 ) . Deshpande, Montanari Planted Cliques November 5, 2013 32 / 49

  44. Proof Technique Key ideas: Expand v t i ✦ j for polynomial f t ( ✁ ) Wrong analysis works if A ✦ A t Deshpande, Montanari Planted Cliques November 5, 2013 33 / 49

  45. Proof Technique Key ideas: Expand v t i ✦ j for polynomial f t ( ✁ ) Wrong analysis works if A ✦ A t Deshpande, Montanari Planted Cliques November 5, 2013 33 / 49

  46. Proof Technique Key ideas: Expand v t i ✦ j for polynomial f t ( ✁ ) Wrong analysis works if A ✦ A t Deshpande, Montanari Planted Cliques November 5, 2013 33 / 49

  47. Proof Technique - Expanding v t Let f t ( x ) = x 2 ❀ v 0 i ✦ j = 1 j ❳ i v 1 i ✦ j = A ik ✿ k ✻ = j k Deshpande, Montanari Planted Cliques November 5, 2013 34 / 49

  48. Proof Technique - Expanding v t j ❳ v 2 A ik ( v 1 k ✦ i ) 2 i ✦ j = i k ✻ = j ✒ ❳ ✓✒ ❳ ✓ ❳ = A ik A k ❵ A km k ✻ = j ❵ ✻ = i m ✻ = i k ❳ ❳ ❳ = A ik A k ❵ A km ✿ k ✻ = j ❵ ✻ = i m ✻ = i m ℓ Deshpande, Montanari Planted Cliques November 5, 2013 35 / 49

  49. Proof Technique - Expanding v t j ❳ v 2 A ik ( v 1 k ✦ i ) 2 i ✦ j = i k ✻ = j ✒ ❳ ✓✒ ❳ ✓ ❳ = A ik A k ❵ A km k ✻ = j ❵ ✻ = i m ✻ = i k ❳ ❳ ❳ = A ik A k ❵ A km ✿ k ✻ = j ❵ ✻ = i m ✻ = i m ℓ Deshpande, Montanari Planted Cliques November 5, 2013 35 / 49

  50. Proof Technique - Expanding v t j ❳ v 2 A ik ( v 1 k ✦ i ) 2 i ✦ j = i k ✻ = j ✒ ❳ ✓✒ ❳ ✓ ❳ = A ik A k ❵ A km k ✻ = j ❵ ✻ = i m ✻ = i k ❳ ❳ ❳ = A ik A k ❵ A km ✿ k ✻ = j ❵ ✻ = i m ✻ = i m ℓ Deshpande, Montanari Planted Cliques November 5, 2013 35 / 49

  51. Proof Technique ❳ ❳ v t + 1 A ik f t ( v t ✘ t + 1 A t ik f t ( ✘ t i ✦ j = k ✦ i ) ✿ i ✦ j = k ✦ i ) ✿ k ✻ = i k ✻ = i i i k k o o ℓ ℓ m m p p r r q q s w s w Deshpande, Montanari Planted Cliques November 5, 2013 36 / 49

  52. Proof Technique - a Combinatorial Lemma Lemma ❳ v t A ( T )Γ( T ) v 0 ( T ) i ✦ j = T ✷❚ t i ✦ j where ❚ t i ✦ j consists rooted, labeled trees that: 1 have maximum depth t . 2 do not backtrack. (Similarly for the ✘ t i ✦ j ) Deshpande, Montanari Planted Cliques November 5, 2013 37 / 49

  53. Proof Technique - Moment Method ❳ ❳ v t + 1 A ik f t ( v t ✘ t + 1 A t ik f t ( ✘ t i ✦ j = k ✦ i ) ✿ i ✦ j = k ✦ i ) ✿ k ✻ = i k ✻ = i i i k k o o ℓ ℓ m m p p r r q q s w s w lim n ✦✶ Moments of v t + 1 = Moments of ✘ t + 1 . Deshpande, Montanari Planted Cliques November 5, 2013 38 / 49

  54. Proof Technique - Moment Method ❳ ❳ v t + 1 A ik f t ( v t ✘ t + 1 A t ik f t ( ✘ t i ✦ j = k ✦ i ) ✿ i ✦ j = k ✦ i ) ✿ k ✻ = i k ✻ = i i i k k o o ℓ ℓ m m p p r r q q s w s w lim n ✦✶ Moments of v t + 1 = Moments of ✘ t + 1 . Deshpande, Montanari Planted Cliques November 5, 2013 38 / 49

  55. Progress(4) Complexity Spectral threshold = √ n n 2 log n n 2 log n n 2 k 1 . 261 √ n C √ n 2 √ n log n � n 2 log 2 n e Deshpande, Montanari Planted Cliques November 5, 2013 39 / 49

  56. Is this threshold fundamental? Rest of the talk: perhaps Deshpande, Montanari Planted Cliques November 5, 2013 40 / 49

  57. The “Hidden Set” Problem A ij ∼ Q 1 Given G n = ([ n ] ❀ E n ) A Set ✦ S ✚ [ n ] ✭ Q 1 if i ❀ j ✷ S ❀ Data ✦ A ij ✘ otherwise. Q 0 A ij ∼ Q 0 Problem: Given edge labels ( A ij ) ( i ❀ j ) ✷ E n , identify S Deshpande, Montanari Planted Cliques November 5, 2013 41 / 49

  58. The “Hidden Set” Problem A ij ∼ Q 1 Given G n = ([ n ] ❀ E n ) A Set ✦ S ✚ [ n ] ✭ Q 1 if i ❀ j ✷ S ❀ Data ✦ A ij ✘ otherwise. Q 0 A ij ∼ Q 0 Problem: Given edge labels ( A ij ) ( i ❀ j ) ✷ E n , identify S Deshpande, Montanari Planted Cliques November 5, 2013 41 / 49

  59. The “Hidden Set” Problem S Given G n = ([ n ] ❀ E n ) S A Set ✦ S ✚ [ n ] ✭ A = if i ❀ j ✷ S ❀ Q 1 Data ✦ A ij ✘ Q 0 otherwise. Problem: Given edge labels ( A ij ) ( i ❀ j ) ✷ E n , identify S Deshpande, Montanari Planted Cliques November 5, 2013 42 / 49

  60. “Local” Algorithms A t -local algorithm computes: Estimate at i : u ( i ) = F ( A Ball ( i ❀ t ) ) ❜ Deshpande, Montanari Planted Cliques November 5, 2013 43 / 49

  61. The Sparse Graph Analogue A ij ∼ Q 1 G n = ([ n ] ❀ E n ) ❀ n ✕ 1 satisfies: ◮ locally tree-like ◮ regular degree ∆ Further ◮ Q 1 = ✍ + 1 , Q 0 = 1 2 ✍ + 1 + 1 2 ✍ � 1 A ij ∼ Q 0 What can local algorithms achieve? Deshpande, Montanari Planted Cliques November 5, 2013 44 / 49

  62. The Sparse Graph Analogue A ij ∼ Q 1 G n = ([ n ] ❀ E n ) ❀ n ✕ 1 satisfies: ◮ locally tree-like ◮ regular degree ∆ Further ◮ Q 1 = ✍ + 1 , Q 0 = 1 2 ✍ + 1 + 1 2 ✍ � 1 A ij ∼ Q 0 What can local algorithms achieve? Deshpande, Montanari Planted Cliques November 5, 2013 44 / 49

Recommend

Convergence of Iterative Hard Thresholding Variants with Application to Asynchronous Parallel

Convergence of Iterative Hard Thresholding Variants with Application to Asynchronous Parallel

Convergence of Iterative Hard Thresholding Variants with Application to Asynchronous Parallel Methods for Sparse Recovery Jamie Haddock Asilomar Conference on Signals, Systems, and Computers, November 4, 2019 Computational and Applied

1.21k views • 31 slides
Stochastic Iterative Hard Thresholding for Graph-Structured Sparsity Optimization Baojian Zhou 1 ,

Stochastic Iterative Hard Thresholding for Graph-Structured Sparsity Optimization Baojian Zhou 1 ,

Stochastic Iterative Hard Thresholding for Graph-Structured Sparsity Optimization Baojian Zhou 1 , Feng Chen 1 , and Yiming Ying 2 1 Department of Computer Science, 2 Department of Mathematics and Statistics, University at Albany, NY, USA

289 views • 7 slides
Cliques & communities Network Analysis in Python I Cliques Social cliques:

Cliques & communities Network Analysis in Python I Cliques Social cliques:

NETWORK ANALYSIS IN PYTHON I Cliques & communities Network Analysis in Python I Cliques Social cliques: tightly-knit groups Network cliques: completely connected graphs Network Analysis in Python I Cliques Simplest

244 views • 20 slides
Thresholding of Text Documents Oliver A Nina William A Barrett Thresholding or Binarization

Thresholding of Text Documents Oliver A Nina William A Barrett Thresholding or Binarization

Thresholding of Text Documents Oliver A Nina William A Barrett Thresholding or Binarization Simple method of image segmentation The image is separated in two parts: object of interest background Thresholding Important

782 views • 45 slides
CLIQUES: Security for Dynamic Peer Groups CLIQUES: Gene Tsudik Yongdae Kim Formation Member

CLIQUES: Security for Dynamic Peer Groups CLIQUES: Gene Tsudik Yongdae Kim Formation Member

CLIQUES: Security for Dynamic Peer Groups CLIQUES: Gene Tsudik Yongdae Kim Formation Member add Member leave Group fusion Giuseppe Ateniese Damian Group fission Hasse 4/ 11/ 99 1 Background Targeted environment

442 views • 11 slides
Inventor Natalie Heckert SEEDS PLANTED = HARVEST? SEEDS PLANTED. LETS GROW! 1986

Inventor Natalie Heckert SEEDS PLANTED = HARVEST? SEEDS PLANTED. LETS GROW! 1986

STRUGGLES = SUCCESS! Inventor Natalie Heckert SEEDS PLANTED = HARVEST? SEEDS PLANTED. LETS GROW! 1986 Cruised to LA 1990 Cruised back to MN 1992 Graduated 4 degree from SCSU 2002 Proposed ATCC Degree Program & Tastefully

415 views • 27 slides
From Cliques to Equilibria: From Cliques to Equilibria: The Dominant- -Set Framework for

From Cliques to Equilibria: From Cliques to Equilibria: The Dominant- -Set Framework for

From Cliques to Equilibria: From Cliques to Equilibria: The Dominant- -Set Framework for Pairwise Data Clustering Set Framework for Pairwise Data Clustering The Dominant Marcello Pelillo Marcello Pelillo Department of Computer Science

1.17k views • 99 slides
for Planted Clique Part I Lecture Outline Part I: Planted Clique and the Meka-Wigderson

for Planted Clique Part I Lecture Outline Part I: Planted Clique and the Meka-Wigderson

Lecture 12: SOS Lower Bounds for Planted Clique Part I Lecture Outline Part I: Planted Clique and the Meka-Wigderson Moments Part II: MPW Analysis Preprocessing Part III: MPW Analysis with Graph Matrices Part IV: The Pessimist

714 views • 46 slides
Score Distribution Based Term Specific Thresholding for Spoken Term Detection D. Can M. Sarac

Score Distribution Based Term Specific Thresholding for Spoken Term Detection D. Can M. Sarac

Score Distribution Based Term Specific Thresholding for Spoken Term Detection D. Can M. Sarac lar Bo gazic i University Department of Electrical & Electronics Engineering BUSIM Lab Introduction Thresholding for Spoken Term

654 views • 30 slides
1 Message Encryption- see Table 11.1 in the book Message Encryption if public-key encryption

1 Message Encryption- see Table 11.1 in the book Message Encryption if public-key encryption

Message Authentication CPE 542: CRYPTOGRAPHY & NETWORK SECURITY message authentication is concerned with: protecting the integrity of a message Chapter 11 Message Authentication and validating identity of originator Hash

462 views • 6 slides
Iterative Techniques in Matrix Algebra Jacobi & Gauss-Seidel Iterative Techniques II

Iterative Techniques in Matrix Algebra Jacobi & Gauss-Seidel Iterative Techniques II

Iterative Techniques in Matrix Algebra Jacobi & Gauss-Seidel Iterative Techniques II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University 2011

1.51k views • 107 slides
Lecture 14: Planted Sparse Vector Lecture Outline Part I: Planted Sparse Vector and 2 to 4

Lecture 14: Planted Sparse Vector Lecture Outline Part I: Planted Sparse Vector and 2 to 4

Lecture 14: Planted Sparse Vector Lecture Outline Part I: Planted Sparse Vector and 2 to 4 Norm Part II: SOS and 2 to 4 Norm on Random Subspaces Part III: Warmup: Showing 1 Part IV: 4-Norm Analysis Part V:

566 views • 54 slides
Development Figures are from : Agile and Iterative Development: A Manager's Guide, Craig

Development Figures are from : Agile and Iterative Development: A Manager's Guide, Craig

Incremental & Iterative Development Figures are from : Agile and Iterative Development: A Manager's Guide, Craig Larman Addison Wesley, 2004 Incremental & Iterative Development Iterative development: an approach to building

385 views • 10 slides
On cliques in edge-regular graphs Leonard Soicher Queen Mary University of London Modern Trends

On cliques in edge-regular graphs Leonard Soicher Queen Mary University of London Modern Trends

On cliques in edge-regular graphs Leonard Soicher Queen Mary University of London Modern Trends in Algebraic Graph Theory, Villanova, 2014 Leonard Soicher (QMUL) Cliques in ERGs Villanova, 2014 1 / 10 The setup All graphs in this talk are

504 views • 19 slides
Lecture Notes: Message Management 1 Slide 1: Message Management Message Management A critical

Lecture Notes: Message Management 1 Slide 1: Message Management Message Management A critical

Lecture Notes: Message Management 1 Slide 1: Message Management Message Management A critical aspect of successful advocacy is the ability to create and deliver a message. While your passion matters greatly, your coalitions make the work

487 views • 15 slides
Graphs with = have big cliques Daniel W. Cranston Virginia Commonwealth University

Graphs with = have big cliques Daniel W. Cranston Virginia Commonwealth University

Graphs with = have big cliques Daniel W. Cranston Virginia Commonwealth University dcranston@vcu.edu Joint with Landon Rabern Slides available on my webpage Discrete Math Days of the Northeast Wesleyan University, 5 October 2013

988 views • 84 slides
Iterative Methods Mostly for SPD systems Iterative Linear conjugate gradient and its variants

Iterative Methods Mostly for SPD systems Iterative Linear conjugate gradient and its variants

Iterative Methods Mostly for SPD systems Iterative Linear conjugate gradient and its variants System Solvers for Large, Sparse Matrices minimize through successive line searches CS 176 Spring 2011 CS 176 Spring 2011 1 2 Steepest

432 views • 3 slides
Graphs with = have big cliques Daniel W. Cranston Virginia Commonwealth University

Graphs with = have big cliques Daniel W. Cranston Virginia Commonwealth University

Graphs with = have big cliques Daniel W. Cranston Virginia Commonwealth University dcranston@vcu.edu Joint with Landon Rabern Slides available on my webpage West Fest A celebration of Dougs 60th birthday! 20 June 2014 Dougs big

1.05k views • 83 slides
Basic Techniques II: Iterative Compression Marek Cygan Institute of Informatics University of

Basic Techniques II: Iterative Compression Marek Cygan Institute of Informatics University of

Basic Techniques II: Iterative Compression Marek Cygan Institute of Informatics University of Warsaw 18th August 2014, Bdlewo Marek Cygan Iterative compression 1/36 What iterative compression is? Iterative compression - jist Recursive

1.47k views • 105 slides
Topic 11: A message What Is a Message in Communication? In rhetorical and communication studies,

Topic 11: A message What Is a Message in Communication? In rhetorical and communication studies,

Topic 11: A message What Is a Message in Communication? In rhetorical and communication studies, a message is defined as information conveyed by words (in speech or writing), and/or other signs and symbols. A message (verbal or nonverbal, or

485 views • 17 slides
Automatic Scaling Iterative Computations Guozhang Wang Cornell University Aug. 7 th , 2012

Automatic Scaling Iterative Computations Guozhang Wang Cornell University Aug. 7 th , 2012

Automatic Scaling Iterative Computations Guozhang Wang Cornell University Aug. 7 th , 2012 1 What are Non-Iterative Computations? Non-iterative computation flow Input Data Directed Acyclic Operator 1 Examples Batch style

664 views • 44 slides
GSM Short Message Service GSM Short Message Service GSM Short Message Service GSM Short Message

GSM Short Message Service GSM Short Message Service GSM Short Message Service GSM Short Message

National Taiwan University Department of Computer Science and Information Engineering GSM Short Message Service GSM Short Message Service GSM Short Message Service GSM Short Message Service Phone Lin Phone Lin Ph.D. Ph.D. Email:

360 views • 13 slides
Message Passing Algorithms for Compressed Sensing DLD, Arian Maleki, Andrea Montanari Stanford

Message Passing Algorithms for Compressed Sensing DLD, Arian Maleki, Andrea Montanari Stanford

Compressed Sensing Phase Transitions Simple Iterative Algorithms Heuristics Message Passing Algorithms Message Passing Algorithms for Compressed Sensing DLD, Arian Maleki, Andrea Montanari Stanford September 2, 2009 DLD, Arian Maleki,

688 views • 36 slides
Thresholding and Learning theory Dominique Picard Laboratoire Probabilit es et Mod` eles Al

Thresholding and Learning theory Dominique Picard Laboratoire Probabilit es et Mod` eles Al

Thresholding and Learning theory Dominique Picard Laboratoire Probabilit es et Mod` eles Al eatoires Universit es Paris VII Joint work with G. Kerkyacharian (LPMA) Columbia- SC May 2008. http

604 views • 41 slides

More recommend