We study offline multitask representation learning in reinforcement learning
(RL), where a learner is provided with an offline dataset from different tasks
that share a common representation and is asked to learn the shared
representation. We theoretically investigate offline multitask low-rank RL, and
propose a new algorithm called MORL for offline multitask representation
learning. Furthermore, we examine downstream RL in reward-free, offline and
online scenarios, where a new task is introduced to the agent that shares the
same representation as the upstream offline tasks. Our theoretical results
demonstrate the benefits of using the learned representation from the upstream
offline task instead of directly learning the representation of the low-rank
model.

我们研究了强化学习中的离线多任务表示学习，理论上研究了离线多任务低秩强化学习，并提出了一种名为 MORL 的用于离线多任务表示学习的新算法。此外，我们还研究了奖励缺失、离线和在线情景下的下游强化学习，在其中引入了一个与上游离线任务共享相同表示的新任务。我们的理论结果证明了使用上游离线任务学到的表示而不是直接学习低秩模型的表示的好处。

强化学习的离线多任务表示学习

Offline Multitask Representation Learning for Reinforcement Learning

Let us consider a case where all of the elements in some continuous slices
are missing in tensor data.
In this case, the nuclear-norm and total variation regularization methods
usually fail to recover the missing elements.
The key problem is capturing some delay/shift-invariant structure.
In this study, we consider a low-rank model in an embedded space of a tensor.
For this purpose, we extend a delay embedding for a time series to a
"multi-way delay-embedding transform" for a tensor, which takes a given
incomplete tensor as the input and outputs a higher-order incomplete Hankel
tensor.
The higher-order tensor is then recovered by Tucker-based low-rank tensor
factorization.
Finally, an estimated tensor can be obtained by using the inverse multi-way
delay embedding transform of the recovered higher-order tensor.
Our experiments showed that the proposed method successfully recovered
missing slices for some color images and functional magnetic resonance images.

本研究提出了一种用于多维张量数据的低秩模型和一种基于 Tucker 分解的低秩张量分解方法，该方法能够成功地恢复一些彩色图像和功能磁共振成像中的丢失部分。

嵌入空间低秩模型的张量缺失切片恢复

Missing Slice Recovery for Tensors Using a Low-rank Model in Embedded  Space

The performance of principal component analysis (PCA) suffers badly in the
presence of outliers. This paper proposes two novel approaches for robust PCA
based on semidefinite programming. The first method, maximum mean absolute
deviation rounding (MDR), seeks directions of large spread in the data while
damping the effect of outliers. The second method produces a low-leverage
decomposition (LLD) of the data that attempts to form a low-rank model for the
data by separating out corrupted observations. This paper also presents
efficient computational methods for solving these SDPs. Numerical experiments
confirm the value of these new techniques.

本文提出了两种基于半定规划的鲁棒主成分分析方法，一种是寻找数据中大波动方向的最大平均绝对偏差舍入方法 (MDR)，另一种是通过分离出受损观测值的低杠杆分解 (LLD) 来尝试建立数据的低秩模型。通过数值实验可以证明这些新技术的价值。