Structured Matrix Completion with Applications to Genomic Data Integration

Transcription

1 Structured Matrix Completion with Applications to Genomic Data Integration Aaron Jones Duke University BIOSTAT 900 October 14, 2016 Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

2 Reference Tianxi Cai, T. Tony Cai & Anru Zhang (2016) Structured Matrix Completion with Applications to Genomic Data Integration, Journal of the American Statistical Association, 111:514, , DOI: / Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

3 Overview 1 Introduction Genomic Data Integration Structured Matrix Completion 2 Methodology Exact Low-Rank Matrix Approximate Low-Rank Matrix Known Rank r Unknown Rank r 3 Theoretical Analysis 4 Simulation 5 Application Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

4 Genomic Data Integration In genomics, often analyze data drawn from multiple studies/sources E.g., combine separate studies conducted using different architecture E.g., funding for NGS in a subset of patients, but SNP chip for the rest E.g., may have other data for some patients (mirna, methylation) Complete case analysis reduces power, and may bias associations The observed data are full rows (patients) and columns (loci) of the data matrix A; the missing data form a rectangular submatrix of A Take advantage of the missingness structure to impute missing values Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

5 Structured Matrix Completion For p 1 p 2 matrix A, observe m 1 < p 1 rows and m 2 < p 2 columns: A = m 2 p 2 m 2 A11 A 12 A 21 (A 22 ) m 1 p 1 m 1 Goal: fill in the missing block A 22, given fully observed A 11, A 12, A 21 Problem: A 22 could be anything, without some assumptions about A Solution: Assume A is approximately low-rank sensible in genomics Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

6 Exact Low-Rank Matrix Proposition 1: Suppose A is of rank r, the SVD of A 11 is A 11 = UΣV T, where U R p1 r, Σ R r r, and V R p2 r. If ( ) A11 rank( A 11 A 12 ) = rank = rank(a) = r, A 21 then rank(a 11 ) = r and A 22 is exactly given by A 22 = A 21 (A 11 ) A 12 = A 21 V (Σ) 1 U T A 12. Simple, analytic solution, but (A 11 ) is not continuous in A 11, so this method does not give approximate A 22 for approximately low-rank A Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

7 Approximate Low-Rank Matrix Definition: A is approximately rank r if there is a significant gap between the rth and (r + 1)th singular values, σ r (A) and σ r+1 (A), ( ) 1/q and the tail k r+1 σq k (A) is small. Let A = UΣV be the SVD of an approximately low-rank matrix A and partition U R p 1 p 1, Σ R p 1 p 2, V R p 2 p 2 into blocks as U = Σ = V = r p 1 r U11 U 12 U 21 U 22 r p 2 r Σ1 0 0 Σ 2 r p 2 r V11 V 12 V 21 V 22 m 1 p 1 m 1 r p 1 r m 2 p 2 m 2 Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

8 Approximate Low-Rank Matrix A = UΣV T U11 U = 12 Σ1 0 V T 11 V T 21 U 21 U 22 0 Σ 2 V12 T V22 T U11 = Σ U 1 V T 11 V21 T U U 22 U11 Σ = 1 V11 T U 11 Σ 1 V21 T U 21 Σ 1 V11 T U 21 Σ 1 V21 T = A max(r) + A max(r), Σ 2 V T 12 V T 22 + U12 Σ 2 V T 12 U 12 Σ 2 V T 22 U 22 Σ 2 V T 12 U 22 Σ 2 V T 22 where A max(r) is a rank-r approximation to A with the largest r singular values, and A max(r) has small singular values. Then by Proposition 1: U 21 Σ 1 V T 21 = (U 21 Σ 1 V T 11)(U 11 Σ 1 V T 11) 1 (U 11 Σ 1 V T 21) Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

9 Known Rank r Define the notation M k := M1k M 2k and M k := M k1 M k2 When r is known, we can estimate A 22 by estimating U 1 and V 1 using the r principal components of A 1 and A 1 as described below: Algorithm 1 Structured Matrix Completion with a Known Rank r 1 Input: A 11 R m 1 m 2, A 12 R (p 1 m 1 ) m 2, A 21 R m 1 (p 2 m 2 ) 2 Calculate the SVD of A 1 = U (1) Σ (1) V (1)T, A 1 = U (2) Σ (2) V (2)T 3 Estimate the column space of U 11 and V 11 by ˆM = U (2),1:r, ˆN = V (1),1:r 4 Estimate A 22 as  22 = A 21 ˆN( ˆM T A 11 ˆN) 1 ˆM T A 12 Problem: Algorithm 1 assumes r is known, but r is generally unknown Solution: First estimate r with some ˆr, then run Algorithm 1 using ˆr Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

10 Unknown Rank r The algorithm to recover A 22 when r is unknown has three steps: 1 Rotate A 1 and A 1 by SVD to move significant factors to the front: A 1 = U (1) Σ (1) V (1)T, A 1 = U (2) Σ (2) V (2)T Z11 Z = Z = 12 U = (2)T A 11 V (1) U (2)T A 12 Z 21 Z 22 A 21 V (1) A 22 Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

11 Unknown Rank r 2 If A were exactly rank-r, the r + 1,..., m 1 rows and r + 1,..., m 2 columns of Z would be zero, but they are nonzero (yet small) due to the perturbation A max(r). So, since we want A max(r), the best rank-r approximation to A, ignore these rows/columns and use the first r. However, r is unknown, so estimate it by the largest ˆr such that Z 11,1:ˆr,1:ˆr is nonsingular and σ 1 (Z 21,1:ˆr,1:ˆr Z 1 11,1:ˆr,1:ˆr ) 2 p1 m 1. 3 As before, estimate A 22 as Â22 = Ẑ22 = Z 21,,1:ˆr Z 1 11,1:ˆr,1:ˆr Z 12,1:ˆr, Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

12 Unknown Rank r Algorithm 2 Structured Matrix Completion with an Unknown Rank r 1 Input: A 11 R m 1 m 2, A 12 R (p 1 m 1 ) m 2, A 21 R m 1 (p 2 m 2 ), thresholding level T R (or T C ) 2 Calculate the SVD of A 1 = U (1) Σ (1) V (1)T, A 1 = U (2) Σ (2) V (2)T 3 Calculate Z 11 = U (2)T A 11 V (1), Z 12 = U (2)T A 12, Z 21 = A 21 V (1) 4 Estimate the column space of U 11 and V 11 by ˆM = U (2),1:r, ˆN = V (1),1:r 5 For s = min(m 1, m 2 ),..., 2, 1 : Calculate D R,s = Z 21,,1:s Z 1 11,1:s,1:s (or D C,s = Z 1 11,1:s,1:s Z 12,1:s,) If Z 11,1:s,1:s is not singular and σ 1 (D R,s ) T R (or σ 1 (D C,s ) T C ): ˆr = s 6 If ˆr is still unassigned, then ˆr = 0 7 Estimate A 22 as Â22 = Ẑ22 = Z 21,,1:ˆr Z 1 11,1:ˆr,1:ˆr Z 12,1:ˆr, Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

13 Theoretical Analysis The paper presents upper and lower bounds for the estimation errors of Algorithms 1 & 2, so the optimal rate of recovery can be given for certain classes of approximately low-rank matrices There are also probability bounds on the estimation errors for fixed A and random rows/columns observed, and also for random A Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

14 Simulation Fix p 1 = p 2 = 1000, m 1 = m 2 = 50 Choose singular values as {1, r 2..., 1, g 1 1 1, g 1 2 1,...} Vary gap ratio g = 1, 2,..., 10, rank r = 4, 12, 20 Algorithm improves as r gets smaller and g = σr (A) σ r+1 (A) gets larger Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

15 Simulation Fix p 1 = p 2 = 1000, m 1 = m 2 = 50 Choose singular values as {j α : j = 1, 2,..., min(p 1, p 2 )} Vary α between 0.3 and 2, and T R = c p1 m1 for c between 1 and 6 Algorithm does well if α is not too small and improves as α gets larger The paper identifies c = 2 as the recommended optimal value Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

16 Simulation Fix p 1 = p 2 = 1000 Choose singular values as {j α : j = 1, 2,..., min(p 1, p 2 )} Vary α between 0.6 and 2, and m 1 = m 2 = 50 or 100 Compare SMC to constrained nuclear norm minimization (NNM) SMC outperforms NNM in approximately low-rank matrices with rectangular missingness Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17

17 Application Imputing the missing mirna expression levels reduces the standard errors and increases power Adding the imputed mirna significantly improves the predictive ability of the model Aaron Jones (BIOSTAT 900) Structured Matrix Completion October 14, / 17