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Implicit deep learning has received increasing attention recently due to the fact that it generalizes the recursive prediction rules of many commonly used neural network architectures. Its prediction rule is provided implicitly based on the solution of an equilibrium equation. Although a line of recent empirical studies has demonstrated its superior performances, the theoretical understanding of implicit neural networks is limited. In general, the equilibrium equation may not be well-posed during the training. As a result, there is no guarantee that a vanilla (stochastic) gradient descent (SGD) training nonlinear implicit neural networks can converge. This paper fills the gap by analyzing the gradient flow of Rectified Linear Unit (ReLU) activated implicit neural networks. For an $m$-width implicit neural network with ReLU activation and $n$ training samples, we show that a randomly initialized gradient descent converges to a global minimum at a linear rate for the square loss function if the implicit neural network is \textit{over-parameterized}. It is worth noting that, unlike existing works on the convergence of (S)GD on finite-layer over-parameterized neural networks, our convergence results hold for implicit neural networks, where the number of layers is \textit{infinite}.

This paper develops a robust dynamic mode decomposition (RDMD) method endowed with statistical and numerical robustness. Statistical robustness ensures estimation efficiency at the Gaussian and non-Gaussian probability distributions, including heavy-tailed distributions. The proposed RDMD is statistically robust because the outliers in the data set are flagged via projection statistics and suppressed using a Schweppe-type Huber generalized maximum-likelihood estimator that minimizes a convex Huber cost function. The latter is solved using the iteratively reweighted least-squares algorithm that is known to exhibit a better convergence property and numerical stability than the Newton algorithms. Several numerical simulations using canonical models of dynamical systems demonstrate the excellent performance of the proposed RDMD method. The results reveal that it outperforms several other methods proposed in the literature.

We address a three-tier numerical framework based on manifold learning for the forecasting of high-dimensional time series. At the first step, we embed the time series into a reduced low-dimensional space using a nonlinear manifold learning algorithm such as Locally Linear Embedding and Diffusion Maps. At the second step, we construct reduced-order regression models on the manifold, in particular Multivariate Autoregressive (MVAR) and Gaussian Process Regression (GPR) models, to forecast the embedded dynamics. At the final step, we lift the embedded time series back to the original high-dimensional space using Radial Basis Functions interpolation and Geometric Harmonics. For our illustrations, we test the forecasting performance of the proposed numerical scheme with four sets of time series: three synthetic stochastic ones resembling EEG signals produced from linear and nonlinear stochastic models with different model orders, and one real-world data set containing daily time series of 10 key foreign exchange rates (FOREX) spanning the time period 03/09/2001-29/10/2020. The forecasting performance of the proposed numerical scheme is assessed using the combinations of manifold learning, modelling and lifting approaches. We also provide a comparison with the Principal Component Analysis algorithm as well as with the naive random walk model and the MVAR and GPR models trained and implemented directly in the high-dimensional space.

In this paper, we consider the problem of partitioning a polygon into a set of connected disjoint sub-polygons, each of which covers an area of a specific size. The work is motivated by terrain covering applications in robotics, where the goal is to find a set of efficient plans for a team of heterogeneous robots to cover a given area. Within this application, solving a polygon partitioning problem is an essential stepping stone. Unlike previous work, the problem formulation proposed in this paper also considers a compactness metric of the generated sub-polygons, in addition to the area size constraints. Maximizing the compactness of sub-polygons directly influences the optimality of any generated motion plans. Consequently, this increases the efficiency with which robotic tasks can be performed within each sub-region. The proposed problem representation is based on grid cell decomposition and a potential field model that allows for the use of standard optimization techniques. A new algorithm, the AreaDecompose algorithm, is proposed to solve this problem. The algorithm includes a number of existing and new optimization techniques combined with two post-processing methods. The approach has been evaluated on a set of randomly generated polygons which are then divided using different criteria and the results have been compared with a state-of-the-art algorithm. Results show that the proposed algorithm can efficiently divide polygon regions maximizing compactness of the resulting partitions, where the sub-polygon regions are on average up to 73% more compact in comparison to existing techniques.

Learning disentanglement aims at finding a low dimensional representation which consists of multiple explanatory and generative factors of the observational data. The framework of variational autoencoder (VAE) is commonly used to disentangle independent factors from observations. However, in real scenarios, factors with semantics are not necessarily independent. Instead, there might be an underlying causal structure which renders these factors dependent. We thus propose a new VAE based framework named CausalVAE, which includes a Causal Layer to transform independent exogenous factors into causal endogenous ones that correspond to causally related concepts in data. We further analyze the model identifiabitily, showing that the proposed model learned from observations recovers the true one up to a certain degree. Experiments are conducted on various datasets, including synthetic and real word benchmark CelebA. Results show that the causal representations learned by CausalVAE are semantically interpretable, and their causal relationship as a Directed Acyclic Graph (DAG) is identified with good accuracy. Furthermore, we demonstrate that the proposed CausalVAE model is able to generate counterfactual data through "do-operation" to the causal factors.

Recently, numerous handcrafted and searched networks have been applied for semantic segmentation. However, previous works intend to handle inputs with various scales in pre-defined static architectures, such as FCN, U-Net, and DeepLab series. This paper studies a conceptually new method to alleviate the scale variance in semantic representation, named dynamic routing. The proposed framework generates data-dependent routes, adapting to the scale distribution of each image. To this end, a differentiable gating function, called soft conditional gate, is proposed to select scale transform paths on the fly. In addition, the computational cost can be further reduced in an end-to-end manner by giving budget constraints to the gating function. We further relax the network level routing space to support multi-path propagations and skip-connections in each forward, bringing substantial network capacity. To demonstrate the superiority of the dynamic property, we compare with several static architectures, which can be modeled as special cases in the routing space. Extensive experiments are conducted on Cityscapes and PASCAL VOC 2012 to illustrate the effectiveness of the dynamic framework. Code is available at //github.com/yanwei-li/DynamicRouting.

Graph convolution is the core of most Graph Neural Networks (GNNs) and usually approximated by message passing between direct (one-hop) neighbors. In this work, we remove the restriction of using only the direct neighbors by introducing a powerful, yet spatially localized graph convolution: Graph diffusion convolution (GDC). GDC leverages generalized graph diffusion, examples of which are the heat kernel and personalized PageRank. It alleviates the problem of noisy and often arbitrarily defined edges in real graphs. We show that GDC is closely related to spectral-based models and thus combines the strengths of both spatial (message passing) and spectral methods. We demonstrate that replacing message passing with graph diffusion convolution consistently leads to significant performance improvements across a wide range of models on both supervised and unsupervised tasks and a variety of datasets. Furthermore, GDC is not limited to GNNs but can trivially be combined with any graph-based model or algorithm (e.g. spectral clustering) without requiring any changes to the latter or affecting its computational complexity. Our implementation is available online.

Inferencing with network data necessitates the mapping of its nodes into a vector space, where the relationships are preserved. However, with multi-layered networks, where multiple types of relationships exist for the same set of nodes, it is crucial to exploit the information shared between layers, in addition to the distinct aspects of each layer. In this paper, we propose a novel approach that first obtains node embeddings in all layers jointly via DeepWalk on a \textit{supra} graph, which allows interactions between layers, and then fine-tunes the embeddings to encourage cohesive structure in the latent space. With empirical studies in node classification, link prediction and multi-layered community detection, we show that the proposed approach outperforms existing single- and multi-layered network embedding algorithms on several benchmarks. In addition to effectively scaling to a large number of layers (tested up to $37$), our approach consistently produces highly modular community structure, even when compared to methods that directly optimize for the modularity function.

Meta-learning has been proposed as a framework to address the challenging few-shot learning setting. The key idea is to leverage a large number of similar few-shot tasks in order to learn how to adapt a base-learner to a new task for which only a few labeled samples are available. As deep neural networks (DNNs) tend to overfit using a few samples only, meta-learning typically uses shallow neural networks (SNNs), thus limiting its effectiveness. In this paper we propose a novel few-shot learning method called meta-transfer learning (MTL) which learns to adapt a deep NN for few shot learning tasks. Specifically, "meta" refers to training multiple tasks, and "transfer" is achieved by learning scaling and shifting functions of DNN weights for each task. In addition, we introduce the hard task (HT) meta-batch scheme as an effective learning curriculum for MTL. We conduct experiments using (5-class, 1-shot) and (5-class, 5-shot) recognition tasks on two challenging few-shot learning benchmarks: miniImageNet and Fewshot-CIFAR100. Extensive comparisons to related works validate that our meta-transfer learning approach trained with the proposed HT meta-batch scheme achieves top performance. An ablation study also shows that both components contribute to fast convergence and high accuracy.

Autonomous urban driving navigation with complex multi-agent dynamics is under-explored due to the difficulty of learning an optimal driving policy. The traditional modular pipeline heavily relies on hand-designed rules and the pre-processing perception system while the supervised learning-based models are limited by the accessibility of extensive human experience. We present a general and principled Controllable Imitative Reinforcement Learning (CIRL) approach which successfully makes the driving agent achieve higher success rates based on only vision inputs in a high-fidelity car simulator. To alleviate the low exploration efficiency for large continuous action space that often prohibits the use of classical RL on challenging real tasks, our CIRL explores over a reasonably constrained action space guided by encoded experiences that imitate human demonstrations, building upon Deep Deterministic Policy Gradient (DDPG). Moreover, we propose to specialize adaptive policies and steering-angle reward designs for different control signals (i.e. follow, straight, turn right, turn left) based on the shared representations to improve the model capability in tackling with diverse cases. Extensive experiments on CARLA driving benchmark demonstrate that CIRL substantially outperforms all previous methods in terms of the percentage of successfully completed episodes on a variety of goal-directed driving tasks. We also show its superior generalization capability in unseen environments. To our knowledge, this is the first successful case of the learned driving policy through reinforcement learning in the high-fidelity simulator, which performs better-than supervised imitation learning.