2 research outputs found
Structure Learning in Graphical Modeling
A graphical model is a statistical model that is associated to a graph whose
nodes correspond to variables of interest. The edges of the graph reflect
allowed conditional dependencies among the variables. Graphical models admit
computationally convenient factorization properties and have long been a
valuable tool for tractable modeling of multivariate distributions. More
recently, applications such as reconstructing gene regulatory networks from
gene expression data have driven major advances in structure learning, that is,
estimating the graph underlying a model. We review some of these advances and
discuss methods such as the graphical lasso and neighborhood selection for
undirected graphical models (or Markov random fields), and the PC algorithm and
score-based search methods for directed graphical models (or Bayesian
networks). We further review extensions that account for effects of latent
variables and heterogeneous data sources
ISLET: Fast and Optimal Low-rank Tensor Regression via Importance Sketching
In this paper, we develop a novel procedure for low-rank tensor regression,
namely \emph{\underline{I}mportance \underline{S}ketching \underline{L}ow-rank
\underline{E}stimation for \underline{T}ensors} (ISLET). The central idea
behind ISLET is \emph{importance sketching}, i.e., carefully designed sketches
based on both the responses and low-dimensional structure of the parameter of
interest. We show that the proposed method is sharply minimax optimal in terms
of the mean-squared error under low-rank Tucker assumptions and under
randomized Gaussian ensemble design. In addition, if a tensor is low-rank with
group sparsity, our procedure also achieves minimax optimality. Further, we
show through numerical study that ISLET achieves comparable or better
mean-squared error performance to existing state-of-the-art methods while
having substantial storage and run-time advantages including capabilities for
parallel and distributed computing. In particular, our procedure performs
reliable estimation with tensors of dimension and is or
orders of magnitude faster than baseline methods