[Ml-stat-talks] Fwd: Wilks Statistics Seminar: Anru Zhang, Today, April 28, 2017 12:30 PM, Sherrerd Hall 101

Barbara Engelhardt bee at princeton.edu
Fri Apr 28 09:35:18 EDT 2017

Talk of interest today.

***   Wilks Statistics Seminar   ***

DATE:  Today, April 28, 2017

TIME:   12:30 pm

LOCATION:   Sherrerd Hall 101

SPEAKER:  Anru Zhang, University of Wisconsin

TITLE:   Inference for High-dimensional Tensor Data

ABSTRACT:  High-dimensional high-order arrays, or tensors, arise in many
modern scientific applications including genomics, brain imaging, and
social science. In this talk, we consider two specific problems in tensor
data analysis: low-rank tensor completion and tensor PCA.

We first 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 the tensors of
Tucker low-rank can be recovered from a limited number of noiseless
measurements, which matches the sample complexity lower-bound. In the case
of noisy measurements, we also develop a theoretical upper bound and the
matching minimax lower bound for recovery error over certain classes of
low-rank tensors for the proposed procedure.

Then we propose a general framework of tensor principal component analysis,
which focuses on the methodology and theory for extracting the hidden
low-rank structure from the high-dimensional tensor data. A unified
solution is provided for tensor PCA with considerations in both statistical
limits and computational costs. The problem exhibits three different phases
according to the signal-noise-ratio (SNR). In particular, with strong SNR,
the fast spectral power iteration method that achieves the minimax optimal
rate of convergence in estimation; with weak SNR, the
information-theoretical lower bound shows that it is impossible to have
consistent estimation in general; with moderate SNR, we show that the
non-convex maximum likelihood estimation provides optimal solution, but
with NP-hard computational cost; moreover, under the hardness hypothesis of
hypergraphic planted clique detection, there are no polynomial-time
algorithms performing consistently in general.
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