Motion planning and control problems are embedded and essential in almost all
robotics applications. These problems are often formulated as stochastic
optimal control problems and solved using dynamic programming algorithms.
Unfortunately, most existing algorithms that guarantee convergence to optimal
solutions suffer from the curse of dimensionality: the run time of the
algorithm grows exponentially with the dimension of the state space of the
system. We propose novel dynamic programming algorithms that alleviate the
curse of dimensionality in problems that exhibit certain low-rank structure.
The proposed algorithms are based on continuous tensor decompositions recently
developed by the authors. Essentially, the algorithms represent
high-dimensional functions (e.g., the value function) in a compressed format,
and directly perform dynamic programming computations (e.g., value iteration,
policy iteration) in this format. Under certain technical assumptions, the new
algorithms guarantee convergence towards optimal solutions with arbitrary
precision. Furthermore, the run times of the new algorithms scale polynomially
with the state dimension and polynomially with the ranks of the value function.
This approach realizes substantial computational savings in "compressible"
problem instances, where value functions admit low-rank approximations. We
demonstrate the new algorithms in a wide range of problems, including a
simulated six-dimensional agile quadcopter maneuvering example and a
seven-dimensional aircraft perching example. In some of these examples, we
estimate computational savings of up to ten orders of magnitude over standard
value iteration algorithms. We further demonstrate the algorithms running in
real time on board a quadcopter during a flight experiment under motion
capture.

本文提出一种新的基于张量分解的动态规划算法，通过压缩高维函数并在其上直接执行动态规划计算，使其在具有低秩结构的问题实例中能够缓解维数灾难，并对最终的解提供任意精度的保证，同时可实现巨大的计算节省。

使用连续张量分解的高维随机最优控制

High-Dimensional Stochastic Optimal Control using Continuous Tensor  Decompositions

We analyze the local Rademacher complexity of empirical risk minimization
(ERM)-based multi-label learning algorithms, and in doing so propose a new
algorithm for multi-label learning. Rather than using the trace norm to
regularize the multi-label predictor, we instead minimize the tail sum of the
singular values of the predictor in multi-label learning. Benefiting from the
use of the local Rademacher complexity, our algorithm, therefore, has a sharper
generalization error bound and a faster convergence rate. Compared to methods
that minimize over all singular values, concentrating on the tail singular
values results in better recovery of the low-rank structure of the multi-label
predictor, which plays an import role in exploiting label correlations. We
propose a new conditional singular value thresholding algorithm to solve the
resulting objective function. Empirical studies on real-world datasets validate
our theoretical results and demonstrate the effectiveness of the proposed
algorithm.

本文研究了基于 ERM 的多标签学习算法的本地 Rademacher 复杂度，并提出一种新算法，通过集中在尾奇异值上而不是所有奇异值上，更好地恢复多标签预测器的低秩结构，从而利用标签之间的相关性。我们提出了一种新的条件奇异值阈值算法来解决本文提出的目标函数。实证研究验证了我们的理论结果，并证明了所提出算法的有效性。

多标记学习的本地 Rademacher 复杂度

Local Rademacher Complexity for Multi-label Learning

Higher-order tensors are becoming prevalent in many scientific areas such as
computer vision, social network analysis, data mining and neuroscience.
Traditional tensor decomposition approaches face three major challenges: model
selecting, gross corruptions and computational efficiency. To address these
problems, we first propose a parallel trace norm regularized tensor
decomposition method, and formulate it as a convex optimization problem. This
method does not require the rank of each mode to be specified beforehand, and
can automatically determine the number of factors in each mode through our
optimization scheme. By considering the low-rank structure of the observed
tensor, we analyze the equivalent relationship of the trace norm between a
low-rank tensor and its core tensor. Then, we cast a non-convex tensor
decomposition model into a weighted combination of multiple much smaller-scale
matrix trace norm minimization. Finally, we develop two parallel alternating
direction methods of multipliers (ADMM) to solve our problems. Experimental
results verify that our regularized formulation is effective, and our methods
are robust to noise or outliers.

针对高阶张量分解中的模型选择、大规模数据和计算效率等难点，本文提出了一种基于迹范数的正则化并可并行计算的分解方法，能够有效分解低秩结构的张量，并具有较强的鲁棒性。