Convergence Analysis for Learning Orthonormal Deep Linear Neural Networks

Enforcing orthonormal or isometric property for the weight matrices has been shown to enhance the training of deep neural networks by mitigating gradient exploding/vanishing and increasing the robustness of the learned networks. However, despite its practical performance, the theoretical analysis of orthonormality in neural networks is still lacking; for example, how orthonormality affects the convergence of the training process. In this letter, we aim to bridge this gap by providing convergence analysis for training orthonormal deep linear neural networks. Specifically, we show that Riemannian gradient descent with an appropriate initialization converges at a linear rate for training orthonormal deep linear neural networks with a class of loss functions. Unlike existing works that enforce orthonormal weight matrices for all the layers, our approach excludes this requirement for one layer, which is crucial to establish the convergence guarantee. Our results shed light on how increasing the number of hidden layers can impact the convergence speed. Experimental results validate our theoretical analysis.

Prototype Fission: Closing Set for Robust Open-set Semi-supervised Learning

Semi-supervised Learning (SSL) has been proven vulnerable to out-of-distribution (OOD) samples in realistic large-scale unsupervised datasets due to over-confident pseudo-labeling OODs as in-distribution (ID). A key underlying problem is class-wise latent space spreading from closed seen space to open unseen space, and the bias is further magnified in SSL's self-training loops. To close the ID distribution set so that OODs are better rejected for safe SSL, we propose Prototype Fission(PF) to divide class-wise latent spaces into compact sub-spaces by automatic fine-grained latent space mining, driven by coarse-grained labels only. Specifically, we form multiple unique learnable sub-class prototypes for each class, optimized towards both diversity and consistency. The Diversity Modeling term encourages samples to be clustered by one of the multiple sub-class prototypes, while the Consistency Modeling term clusters all samples of the same class to a global prototype. Instead of "opening set", i.e., modeling OOD distribution, Prototype Fission "closes set" and makes it hard for OOD samples to fit in sub-class latent space. Therefore, PF is compatible with existing methods for further performance gains. Extensive experiments validate the effectiveness of our method in open-set SSL settings in terms of successfully forming sub-classes, discriminating OODs from IDs and improving overall accuracy. Codes will be released.

Feature Importance-aware Graph Attention Network and Dueling Double Deep Q-Network Combined Approach for Critical Node Detection Problems

Detecting critical nodes in sparse networks is important in a variety of application domains. A Critical Node Problem (CNP) aims to find a set of critical nodes from a network whose deletion maximally degrades the pairwise connectivity of the residual network. Due to its general NP-hard nature, state-of-the-art CNP solutions are based on heuristic approaches. Domain knowledge and trial-and-error are usually required when designing such approaches, thus consuming considerable effort and time. This work proposes a feature importance-aware graph attention network for node representation and combines it with dueling double deep Q-network to create an end-to-end algorithm to solve CNP for the first time. It does not need any problem-specific knowledge or labeled datasets as required by most of existing methods. Once the model is trained, it can be generalized to cope with various types of CNPs (with different sizes and topological structures) without re-training. Extensive experiments on 28 real-world networks show that the proposed method is highly comparable to state-of-the-art methods. It does not require any problem-specific knowledge and, hence, can be applicable to many applications including those impossible ones by using the existing approaches. It can be combined with some local search methods to further improve its solution quality. Extensive comparison results are given to show its effectiveness in solving CNP.