**High-dimensional low-rank matrix and tensor completion with applications in genomic data integration and brain imaging**

**Anru Zhang, PhD**

**Universty of Wisconsin at Madison**

### Location and Address

1811 Posvar Hall

adjacent to the elvators on the first floor

### Schedule of Events

##### High-dimensional low-rank matrix and tensor completion with applications in genomic data integration and brain imaging

**January 27, 2017 - 2:00pm**

High-dimensional low-rank structure arises in many applications including genomics, brain imaging, signal processing, and social science. In this talk, we discuss some recent results on high-dimensional low-rank matrix and tensor completion. First, motivated by applications in genomic data integration, we propose a new framework of structured matrix completion (SMC) to treat structured missingness in matrix by design. Specifically, our proposed method aims at efficient matrix recovery when a subset of the rows and columns of an approximately low-rank matrix are observed. The method is applied to integrate several ovarian cancer genomic studies with different extent of genomic measurements, which enables us to construct more accurate prediction rules for ovarian cancer survival. Next, we propose a framework for low-rank tensor completion via a novel tensor measurement scheme we name Cross. The proposed procedure is efficient and easy to implement. In particular, we show that a third order tensor of Tucker rank-$(r_1, r_2, r_3)$ in $p_1$-by-$p_2$-by-$p_3$ dimensional space can be recovered from as few as $r_1r_2r_3 + r_1(p_1-r_1) + r_2(p_2-r_2) + r_3(p_3-r_3)$ noiseless measurements, which matches the sample complexity lower-bound. The results can be further extended to fourth or higher-order tensors. Finally, the procedure is illustrated through a real dataset in neuroimaging.