##### Department of Mathematics,

University of California San Diego

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### Math 278C - Optimization Seminar

## Ke Ye

#### University of Chicago

## Tensor Network Ranks

##### Abstract:

At the beginning of this talk, we will introduce the background of tensor network states (TNS) in various areas such as quantum physics, quantum chemistry and numerical partial differential equations. Famous TNS includes tensor trains (TT), matrix product states (MPS), projected entangled pair states (PEPS) and multi-scale entanglement renormalization ansatz (MERA). Then we will explain how to define TNS by graphs and we will define tensor network ranks which can be used to measure the complexity of TNS. We will see that the notion of tensor network ranks is an analogue of tensor rank and multilinear rank. We will discuss basic properties of tensor network ranks and the comparison among tensor network ranks, tensors rank and multilinear rank. If time permits, we will also discuss the dimension of tensor networks and the geometry of TNS. This talk is based on papers joined with Lek-Heng Lim.

Host: Jiawang Nie

### January 25, 2017

### 2:00 PM

### AP&M 7321

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