On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action $\Gamma$, and obtain an asymptotic estimate for the $\Gamma$-dimension of the harmonic space with respect to the tensor times $k$ in the holomorphic line bundle $L^{k}\otimes E$ and the type $(n,q)$ of the differential form, when $L$ is semipositive. In particular, we estimate the $\Gamma$-dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. Essentially, we obtain a local estimate of the pointwise norm of harmonic forms with valued in semipositive line bundles over Hermitian manifolds

Stability of matrix factorization for collaborative filtering

We study the stability vis a vis adversarial noise of matrix factorization algorithm for matrix completion. In particular, our results include: (I) we bound the gap between the solution matrix of the factorization method and the ground truth in terms of root mean square error; (II) we treat the matrix factorization as a subspace fitting problem and analyze the difference between the solution subspace and the ground truth; (III) we analyze the prediction error of individual users based on the subspace stability. We apply these results to the problem of collaborative filtering under manipulator attack, which leads to useful insights and guidelines for collaborative filtering system design.Comment: ICML201

Data Dropout: Optimizing Training Data for Convolutional Neural Networks

Deep learning models learn to fit training data while they are highly expected to generalize well to testing data. Most works aim at finding such models by creatively designing architectures and fine-tuning parameters. To adapt to particular tasks, hand-crafted information such as image prior has also been incorporated into end-to-end learning. However, very little progress has been made on investigating how an individual training sample will influence the generalization ability of a model. In other words, to achieve high generalization accuracy, do we really need all the samples in a training dataset? In this paper, we demonstrate that deep learning models such as convolutional neural networks may not favor all training samples, and generalization accuracy can be further improved by dropping those unfavorable samples. Specifically, the influence of removing a training sample is quantifiable, and we propose a Two-Round Training approach, aiming to achieve higher generalization accuracy. We locate unfavorable samples after the first round of training, and then retrain the model from scratch with the reduced training dataset in the second round. Since our approach is essentially different from fine-tuning or further training, the computational cost should not be a concern. Our extensive experimental results indicate that, with identical settings, the proposed approach can boost performance of the well-known networks on both high-level computer vision problems such as image classification, and low-level vision problems such as image denoising