UC San Diego

We consider the problem of detecting and localizing an submatrix with larger-than-usual

entries inside a large, noisy matrix. This problem arises from analysis of data in

genetics, bioinformatics, and social sciences. We consider that entries of the data matrix are

independently following distributions from a natural exponential family, which generalizes

the common Gaussian assumptions in the literature. In Chapter 2 a permutation test for

testing the existence of the elevated submatrix is studied. The test's asymptotic power is

illustrated, and its robust variation (rank method) is also studied. In The latter part of

Chapter 2 and Chapter 3 we remove the prior knowledge of the submatrix size, aiming

to develop adaptive methods for detection and localization. Latter part of Chapter 2

proposes a Bonferroni testing framework based on the permutation scan test, to solve

the detection problem. An accelerating framework is also developed without sacricing

asymptotic power. In Chapter 3, a new size-adaptive estimator is proposed to solve the

localization problem. Its asymptotic performance is studied, and two fast algorithms to

approximate the estimator are developed.