Tensor Biclustering
Consider a dataset where data is collected on multiple features of multiple individuals over multiple times. This type of data can be represented as a three dimensional individual/feature/time tensor and has become increasingly prominent in various areas of science. The tensor biclustering problem computes a subset of individuals and a subset of features whose signal trajectories over time lie in a low-dimensional subspace, modeling similarity among the signal trajectories while allowing different scalings across different individuals or different features. We study the information-theoretic limit of this problem under a generative model. Moreover, we propose an efficient spectral algorithm to solve the tensor biclustering problem and analyze its achievability bound in an asymptotic regime. Finally, we show the efficiency of our proposed method in several synthetic and real datasets.
This paper studies the tensor biclustering problem, which is
	defined as follows: given an order-3 tensor that is assumed to
	be of rank 1 with additive Gaussian noise, we want to find
	index sets along with mode 1 and mode 2 where the factor
	vectors have non-zero entries. The paper considers several
	methods and analyzes their recovery performance. The
	statistical lower bound (i.e. the best achievable performance)
	is also evaluated. The performance is also evaluated by experiments. 
	The paper is clearly written, self-contained, and easy to follow.
	Overall, this paper aims to develop a relatively minor
	problem, tensor biclustering, theoretically in depth, and it
	seems to be succeeded. For computationally efficient methods
	(the proposed algorithms), the asymptotic order of variance
	where the non-zero indices are correctly recovered is
	identified. In addition, the optimal bound of performance is
	evaluated, which is helpful as a baseline. 
	Still, I feel there is room for improvement. 
	
	1. The model formulation (1) looks less general. The strongest
	restriction may be the setting q=1. Although the paper says
	the analysis can be generalized into the case q>1 (Line 83),
	it is not discussed in the rest of paper. For me, the analysis
	for q>1 is not trivial. It is very helpful if the paper
	contains discussion on, for q>1, how we can adapt the
	analysis, how the theoretical results change, etc.
	2. Synthetic experiments can be elaborate. I expected to see
	the experiments that support the theoretical results
	(e.g. Theorems 1 and 2). For example, for various n and
	\sigma_1^2, you can plot monochrome tiles where each tile
	represents the recovery rate at corresponding n and \sigma_1^2
	(e.g. the tile is black if the rate is 0 and is white if the
	rate is 1), which ideally shows the phase transition that the
	theorems foreshow.