8 research outputs found
Statistical and computational rates in high rank tensor estimation
Higher-order tensor datasets arise commonly in recommendation systems,
neuroimaging, and social networks. Here we develop probable methods for
estimating a possibly high rank signal tensor from noisy observations. We
consider a generative latent variable tensor model that incorporates both high
rank and low rank models, including but not limited to, simple hypergraphon
models, single index models, low-rank CP models, and low-rank Tucker models.
Comprehensive results are developed on both the statistical and computational
limits for the signal tensor estimation. We find that high-dimensional latent
variable tensors are of log-rank; the fact explains the pervasiveness of
low-rank tensors in applications. Furthermore, we propose a polynomial-time
spectral algorithm that achieves the computationally optimal rate. We show that
the statistical-computational gap emerges only for latent variable tensors of
order 3 or higher. Numerical experiments and two real data applications are
presented to demonstrate the practical merits of our methods.Comment: 38 pages, 8 figure
Computational Barriers to Estimation from Low-Degree Polynomials
One fundamental goal of high-dimensional statistics is to detect or recover
structure from noisy data. In many cases, the data can be faithfully modeled by
a planted structure (such as a low-rank matrix) perturbed by random noise. But
even for these simple models, the computational complexity of estimation is
sometimes poorly understood. A growing body of work studies low-degree
polynomials as a proxy for computational complexity: it has been demonstrated
in various settings that low-degree polynomials of the data can match the
statistical performance of the best known polynomial-time algorithms for
detection. While prior work has studied the power of low-degree polynomials for
the task of detecting the presence of hidden structures, it has failed to
address the estimation problem in settings where detection is qualitatively
easier than estimation.
In this work, we extend the method of low-degree polynomials to address
problems of estimation and recovery. For a large class of "signal plus noise"
problems, we give a user-friendly lower bound for the best possible mean
squared error achievable by any degree-D polynomial. To our knowledge, this is
the first instance in which the low-degree polynomial method can establish
low-degree hardness of recovery problems where the associated detection problem
is easy. As applications, we give a tight characterization of the low-degree
minimum mean squared error for the planted submatrix and planted dense subgraph
problems, resolving (in the low-degree framework) open problems about the
computational complexity of recovery in both cases.Comment: 38 page