The (efficient and parsimonious) decomposition of higher-order tensors is a
fundamental problem with numerous applications in a variety of fields. Several
methods have been proposed in the literature to that end, with the Tucker and
PARAFAC decompositions being the most prominent ones. Inspired by the latter,
in this work we propose a multi-resolution low-rank tensor decomposition to
describe (approximate) a tensor in a hierarchical fashion. The central idea of
the decomposition is to recast the tensor into \emph{multiple}
lower-dimensional tensors to exploit the structure at different levels of
resolution. The method is first explained, an alternating least squares
algorithm is discussed, and preliminary simulations illustrating the potential
practical relevance are provided.

通过多分辨率低秩张量分解以层次化方式描述张量，这种方法能够利用不同层次分辨率上的结构，创造性地解决了高阶张量分解这一基础问题。

多分辨率低秩张量分解

A Multi-resolution Low-rank Tensor Decomposition

Value-function (VF) approximation is a central problem in Reinforcement
Learning (RL). Classical non-parametric VF estimation suffers from the curse of
dimensionality. As a result, parsimonious parametric models have been adopted
to approximate VFs in high-dimensional spaces, with most efforts being focused
on linear and neural-network-based approaches. Differently, this paper puts
forth a a parsimonious non-parametric approach, where we use stochastic
low-rank algorithms to estimate the VF matrix in an online and model-free
fashion. Furthermore, as VFs tend to be multi-dimensional, we propose replacing
the classical VF matrix representation with a tensor (multi-way array)
representation and, then, use the PARAFAC decomposition to design an online
model-free tensor low-rank algorithm. Different versions of the algorithms are
proposed, their complexity is analyzed, and their performance is assessed
numerically using standardized RL environments.

论文提出了一种基于随机低秩算法和张量拟合方法的、无模型且在线的环境中的 VF 矩阵估计算法，有效地解决了高维状态空间下基于线性或神经网络的 VF 估计方法的维度灾难问题，得到了满意的性能评估效果。

强化学习中的张量和矩阵低秩值函数逼近

Tensor and Matrix Low-Rank Value-Function Approximation in Reinforcement Learning

Intelligent reflective surface (IRS) is an emergent technology for future
wireless communications. It consists of a large 2D array of passive scattering
elements that control the electromagnetic properties of radio-frequency waves
so that the reflected signals add coherently at the intended receiver or
destructively to reduce co-channel interference. The promised gains of
IRS-assisted communications depend on the accuracy of the channel state
information. In this paper, we propose two novel channel estimation methods for
an IRS-assisted multiple-input multiple-output (MIMO) communication system.
Assuming a structured time-domain pattern of pilots and IRS phase shifts, we
show that the received signal follows a parallel factor (PARAFAC) tensor model
that can be exploited to estimate the involved communication channels in
closed-form or iteratively. Numerical results corroborate the effectiveness of
the proposed channel estimation methods and highlight the involved tradeoffs.

本文提出了两种新的信道估计方法，利用平行因子分解张量模型，闭合形式或迭代估计涉及通讯通道，证实了智能反射表面 (IRS) 辅助通信的有效性。