Dimension Reduction Methods with Application to ... - Rice University

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46 / 86 L 2 -L 1 : ACHLIOPTAS RANDOM MATRIX Gaussian ◮ same lower bound as in L 2 -L 1 case using Gaussian Random Matrix (NR 2 bound) ◮ bounding the mgf of Achlioptas-typed r.v. by that of a standard Gaussian ◮ 36%-40% improvement
SUMMARY Random Projection ◮ Gaussian random matrix ◮ ◮ L 2 -L 2 projection: improvement of 12%-34% on Dasgupta and Gupta bound L 2 -L 1 projection: improvement of 36%-40% on Chazelle & Ailon bound ◮ Achlioptas-typed random matrix ◮ ◮ L2 -L 2 projection: improvement of 20%-40% on Achlioptas bound (Rademacher random matrix) L 2 -L 1 projection: improvement of 36%-40% on Ma<strong>to</strong>usek bound ◮ lower bound for k is still large for practical purposes: active research 47 / 86
Page 1 and 2: 1 / 86 Dimension Reduction Methods
Page 3 and 4: OUTLINE 3 / 86 ◮ Microarray Data
Page 5 and 6: OUTLINE ◮ Microarray Data ◮ Dim
Page 7 and 8: WHAT IS DNA MICROARRAY? 7 / 86 ◮
Page 9 and 10: OLIGONUCLEOTIDE MICROARRAY 9 / 86
Page 11 and 12: APPLICATIONS OF MICROARRAY 11 / 86
Page 13 and 14: ANALYZING MICROARRAY DATA 13 / 86 2
Page 15 and 16: R k R d 15 / 86 RANDOM PROJECTION (
Page 17 and 18: JL LEMMA: IMPROVEMENT ON LOWER BOUN
Page 19 and 20: JL LEMMA: DASGUPTA AND GUPTA 19 / 8
Page 21 and 22: 21 / 86 JL LEMMA: DASGUPTA AND GUPT
Page 23 and 24: JL LEMMA Projection of all ( n 2) p
Page 25 and 26: 25 / 86 IMPROVEMENT ON DASGUPTA AND
Page 27 and 28: OUTLINE OF PROOF 27 / 86 Theorem 1:
Page 29 and 30: COMPARISON OF THE LOWER BOUNDS ON k
Page 31 and 32: OUTLINE: OUR CONTRIBUTIONS 31 / 86
Page 33 and 34: L 2 -L 1 RP: IMPROVEMENT TO AILON &
Page 35 and 36: COMPARISON OF THE LOWER BOUNDS ON k
Page 37 and 38: ACHLIOPTAS-TYPED RANDOM MATRICES 37
Page 39 and 40: 39 / 86 L 2 -L 2 RP: IMPROVEMENT ON
Page 41 and 42: 41 / 86 OUTLINE OF PROOF ◮ Right
Page 43 and 44: 43 / 86 L 2 -L 2 RP: IMPROVEMENT ON
Page 45: COMPARISON OF THE LOWER BOUNDS ON k
Page 49 and 50: PRINCIPAL COMPONENT ANALYSIS (PCA)
Page 51 and 52: PCA 51 / 86 How to Select K: number
Page 53 and 54: INCORPORATING CENSORING IN PLS 53 /
Page 55 and 56: OUTLINE: OUR CONTRIBUTIONS Dimensio
Page 57 and 58: NGUYEN AND ROJO (2009A, B): RANK-BA
Page 59 and 60: OUTLINE ◮ Microarray Data ◮ Dim
Page 61 and 62: SURVIVAL ANALYSIS 61 / 86 Cox Propo
Page 63 and 64: ASSESSMENT OF THE METHODS Selection
Page 65 and 66: ASSESSMENT OF THE METHODS 65 / 86 S
Page 67 and 68: 67 / 86 UNIV, SPCR, CPCR ◮ Univar
Page 69 and 70: 69 / 86 REAL DATASETS Table: AFT Lo
Page 71 and 72: GENE RANKING 71 / 86 Table: Number
Page 73 and 74: SIMULATION SETUP 73 / 86 Simulation
Page 75 and 76: SIMULATION SETUP 75 / 86 Simulation
Page 77 and 78: PERFORMANCE MEASURES Performance Me
Page 79 and 80: 79 / 86 SIMULATION RESULTS: COX MOD
Page 81 and 82: SIMULATION RESULTS: AFT MODEL, CV m
Page 83 and 84: SUMMARY 83 / 86 ◮ Rank-based Part
Page 85 and 86: THANK YOU! 85 / 86 Special Thanks:
Page 87 and 88: DATTA: REWEIGHTING (RWPLS) 87 / 86
Page 89 and 90: 89 / 86 BIBLIOGRAPHY ◮ Achlioptas

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