In this work, we establish the first separation between computation with bounded and unbounded space, for problems with short outputs (i.e., working memory can be exponentially larger than output size), both in the classical and the quantum setting. Towards that, we introduce a problem called nested collision finding, and show that optimal query complexity can not be achieved without exponential memory.

Millimeter-wave massive multiple-input multiple-output employing a large-scale antenna array is a promising technology for 5G and 6G cellular networks. It also provides strong support for high-speed, low-latency communications and diverse applications. In order to enhance the accuracy and efficiency of channel estimation, in this paper, we formulate a tensor-based channel estimation model with sparse regularization aiming at characterizing the sparse structure of a large-scale channel in the delay-angular domain. An efficient subspace Newton least squares algorithm is designed to solve the nonconvex discontinuous tensor-based model, operating in restricted subspaces and being capable of handling singularities of the Hessian matrix. Our proposed algorithm is also proved to enjoy global and linear (or sublinear) convergence. Some numerical simulations are performed, which demonstrate the feasibility of our proposed model and the time validity of our algorithm.

Time series is a very common but important data type. A large number of time series data are generated in various professional research fields and daily life. Although there are many models being developed to deal with time series, the cluster methods for time series are insufficient and need to improve. This paper is focused on time series clustering, which uses deep learning approach to discover the shape characteristics of time series. We establish a new neural network model of time series clustering to jointly optimize the representation learning and clustering tasks of time series. Focusing on shape features with time series, we built the Soft-DTW layer into the neural network to learn the interpretable time series representation. Maximized regularization mutual information is used to jointly optimize representation learning and clustering tasks. Experiments show that this model can help obtain an excellent representation of time series. In comparison with the benchmark model, the best clustering effect is achieved in the proposed model on multiple data sets. This model has broad applicability in time series data.