T. Luo

Yaoyu Zhang, Zhongwang Zhang, Leyang Zhang et al.

Models with nonlinear architectures/parameterizations such as deep neural networks (DNNs) are well known for their mysteriously good generalization performance at overparameterization. In this work, we tackle this mystery from a novel perspective focusing on the transition of the target recovery/fitting accuracy as a function of the training data size. We propose a rank stratification for general nonlinear models to uncover a model rank as an "effective size of parameters" for each function in the function space of the corresponding model. Moreover, we establish a linear stability theory proving that a target function almost surely becomes linearly stable when the training data size equals its model rank. Supported by our experiments, we propose a linear stability hypothesis that linearly stable functions are preferred by nonlinear training. By these results, model rank of a target function predicts a minimal training data size for its successful recovery. Specifically for the matrix factorization model and DNNs of fully-connected or convolutional architectures, our rank stratification shows that the model rank for specific target functions can be much lower than the size of model parameters. This result predicts the target recovery capability even at heavy overparameterization for these nonlinear models as demonstrated quantitatively by our experiments. Overall, our work provides a unified framework with quantitative prediction power to understand the mysterious target recovery behavior at overparameterization for general nonlinear models.

Yaoyu Zhang, Zhongwang Zhang, Leyang Zhang et al.

We propose an optimistic estimate to evaluate the best possible fitting performance of nonlinear models. It yields an optimistic sample size that quantifies the smallest possible sample size to fit/recover a target function using a nonlinear model. We estimate the optimistic sample sizes for matrix factorization models, deep models, and deep neural networks (DNNs) with fully-connected or convolutional architecture. For each nonlinear model, our estimates predict a specific subset of targets that can be fitted at overparameterization, which are confirmed by our experiments. Our optimistic estimate reveals two special properties of the DNN models -- free expressiveness in width and costly expressiveness in connection. These properties suggest the following architecture design principles of DNNs: (i) feel free to add neurons/kernels; (ii) restrain from connecting neurons. Overall, our optimistic estimate theoretically unveils the vast potential of nonlinear models in fitting at overparameterization. Based on this framework, we anticipate gaining a deeper understanding of how and why numerous nonlinear models such as DNNs can effectively realize their potential in practice in the near future.

Ouwen Huan, Yang Yang, Tao Luo et al.

In this paper, a multi-modal data based semi-supervised learning (SSL) framework that jointly use channel state information (CSI) data and RGB images for vehicle positioning is designed. In particular, an outdoor positioning system where the vehicle locations are determined by a base station (BS) is considered. The BS equipped with several cameras can collect a large amount of unlabeled CSI data and a small number of labeled CSI data of vehicles, and the images taken by cameras. Although the collected images contain partial information of vehicles (i.e. azimuth angles of vehicles), the relationship between the unlabeled CSI data and its azimuth angle, and the distances between the BS and the vehicles captured by images are both unknown. Therefore, the images cannot be directly used as the labels of unlabeled CSI data to train a positioning model. To exploit unlabeled CSI data and images, a SSL framework that consists of a pretraining stage and a downstream training stage is proposed. In the pretraining stage, the azimuth angles obtained from the images are considered as the labels of unlabeled CSI data to pretrain the positioning model. In the downstream training stage, a small sized labeled dataset in which the accurate vehicle positions are considered as labels is used to retrain the model. Simulation results show that the proposed method can reduce the positioning error by up to 30% compared to a baseline where the model is not pretrained.