Hyperbolic spaces have recently gained momentum in the context of machine learning due to their high capacity and tree-likeliness properties. However, the representational power of hyperbolic geometry is not yet on par with Euclidean geometry, mostly because of the absence of corresponding hyperbolic neural networklayers.

The emergence of Deep Convolutional Neural Networks (DCNNs) has been a pervasive tool for accomplishing widespread applications in computer vision. Despite its potential capability to capture intricate patterns inside the data, the underlying embedding space remains Euclidean and primarily pursues contractive convolution. Several instances can serve as a precedent for the exacerbating performance of DCNNs. The recent advancement of neural networks in the hyperbolic spaces gained traction, incentivizing the development of convolutional deep neural networks in the hyperbolic space. In this work, we propose Hyperbolic DCNN based on the Poincar\'{e} Disc. The work predominantly revolves around analyzing the nature of expansive convolution in the context of the non-Euclidean domain. We further offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Several simulations were performed not only on the synthetic datasets but also on some real-world datasets. The experimental results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.

Hyperbolic representation learning is well known for its ability to capture hierarchical information. However, the distance between samples from different levels of hierarchical classes can be required large. We reveal that the hyperbolic discriminant objective forces the backbone to capture this hierarchical information, which may inevitably increase the Lipschitz constant of the backbone. This can hinder the full utilization of the backbone's generalization ability. To address this issue, we introduce second-order pooling into hyperbolic representation learning, as it naturally increases the distance between samples without compromising the generalization ability of the input features. In this way, the Lipschitz constant of the backbone does not necessarily need to be large. However, current off-the-shelf low-dimensional bilinear pooling methods cannot be directly employed in hyperbolic representation learning because they inevitably reduce the distance expansion capability. To solve this problem, we propose a kernel approximation regularization, which enables the low-dimensional bilinear features to approximate the kernel function well in low-dimensional space. Finally, we conduct extensive experiments on graph-structured datasets to demonstrate the effectiveness of the proposed method.

Existing data-driven and feedback traffic control strategies do not consider the heterogeneity of real-time data measurements. Besides, traditional reinforcement learning (RL) methods for traffic control usually converge slowly for lacking data efficiency. Moreover, conventional optimal perimeter control schemes require exact knowledge of the system dynamics and thus would be fragile to endogenous uncertainties. To handle these challenges, this work proposes an integral reinforcement learning (IRL) based approach to learning the macroscopic traffic dynamics for adaptive optimal perimeter control. This work makes the following primary contributions to the transportation literature: (a) A continuous-time control is developed with discrete gain updates to adapt to the discrete-time sensor data. (b) To reduce the sampling complexity and use the available data more efficiently, the experience replay (ER) technique is introduced to the IRL algorithm. (c) The proposed method relaxes the requirement on model calibration in a "model-free" manner that enables robustness against modeling uncertainty and enhances the real-time performance via a data-driven RL algorithm. (d) The convergence of the IRL-based algorithms and the stability of the controlled traffic dynamics are proven via the Lyapunov theory. The optimal control law is parameterized and then approximated by neural networks (NN), which moderates the computational complexity. Both state and input constraints are considered while no model linearization is required. Numerical examples and simulation experiments are presented to verify the effectiveness and efficiency of the proposed method.

Hyperbolic embeddings are state-of-the-art models to learn representations of data with an underlying hierarchical structure [18]. The hyperbolic space serves as a geometric prior to hierarchical structures, tree graphs, heavy-tailed distributions, e.g., scale-free, powerlaw [45]. A relevant tool to implement hyperbolic space algorithms is the Möbius gyrovector spaces or Gyrovector spaces [66]. Gyrovector spaces are an algebraic formalism, which leads to vector-like operations, i.e., gyrovector, in the Poincaré model of the hyperbolic space. Thanks to this formalism, we can quickly build estimators that are well-suited to perform end-to-end optimization [6]. Gyrovector spaces are essential to design the hyperbolic version of several machine learning algorithms, like Hyperbolic Neural Networks (HNN) [24], Hyperbolic Graph NN [36], Hyperbolic Graph Convolutional NN [12], learning latent feature representations [41, 46], word embeddings [62, 25], and image embeddings [29]. Modern machine learning algorithms rely on the availability to accumulate large volumes of data, often coming from various sources, e.g., acquisition devices or languages. However, these massive amounts of heterogeneous data can entangle downstream learning tasks since the data may follow different distributions. Alignment aims at building connections between two or more disparate data sets by aligning their underlying manifolds.

Graphical models are powerful tools for modeling high-dimensional data, but learning graphical models in the presence of latent variables is well-known to be difficult. In this work we give new results for learning Restricted Boltzmann Machines, probably the most well-studied class of latent variable models. Our results are based on new connections to learning two-layer neural networks under $\ell_{\infty}$ bounded input; for both problems, we give nearly optimal results under the conjectured hardness of sparse parity with noise. Using the connection between RBMs and feedforward networks, we also initiate the theoretical study of $supervised~RBMs$ [Hinton, 2012], a version of neural-network learning that couples distributional assumptions induced from the underlying graphical model with the architecture of the unknown function class. We then give an algorithm for learning a natural class of supervised RBMs with better runtime than what is possible for its related class of networks without distributional assumptions.

In artificial neural networks, learning from data is a computationally demanding task in which a large number of connection weights are iteratively tuned through stochastic-gradient-based heuristic processes over a cost-function. It is not well understood how learning occurs in these systems, in particular how they avoid getting trapped in configurations with poor computational performance. Here we study the difficult case of networks with discrete weights, where the optimization landscape is very rough even for simple architectures, and provide theoretical and numerical evidence of the existence of rare - but extremely dense and accessible - regions of configurations in the network weight space. We define a novel measure, which we call the "robust ensemble" (RE), which suppresses trapping by isolated configurations and amplifies the role of these dense regions. We analytically compute the RE in some exactly solvable models, and also provide a general algorithmic scheme which is straightforward to implement: define a cost-function given by a sum of a finite number of replicas of the original cost-function, with a constraint centering the replicas around a driving assignment. To illustrate this, we derive several powerful new algorithms, ranging from Markov Chains to message passing to gradient descent processes, where the algorithms target the robust dense states, resulting in substantial improvements in performance. The weak dependence on the number of precision bits of the weights leads us to conjecture that very similar reasoning applies to more conventional neural networks. Analogous algorithmic schemes can also be applied to other optimization problems.

We introduce a useful tool for analyzing boosting algorithms called the ``smooth margin function,'' a differentiable approximation of the usual margin for boosting algorithms. We present two boosting algorithms based on this smooth margin, ``coordinate ascent boosting'' and ``approximate coordinate ascent boosting,'' which are similar to Freund and Schapire's AdaBoost algorithm and Breiman's arc-gv algorithm. We give convergence rates to the maximum margin solution for both of our algorithms and for arc-gv. We then study AdaBoost's convergence properties using the smooth margin function. We precisely bound the margin attained by AdaBoost when the edges of the weak classifiers fall within a specified range. This shows that a previous bound proved by R\"{a}tsch and Warmuth is exactly tight. Furthermore, we use the smooth margin to capture explicit properties of AdaBoost in cases where cyclic behavior occurs.

This paper provides a system-level analysis of a scalable distributed sensing model for networked sensors. In our system model, a data center acquires data from a bunch of L sensors which each independently encode their noisy observations of an original binary sequence, and transmit their encoded data sequences to the data center at a combined rate R, which is limited. Supposing that the sensors use independent LDGM rate distortion codes, we show that the system performance can be evaluated for any given finite R when the number of sensors L goes to infinity . The analysis shows how the optimal strategy for the distributed sensing problem changes at critical values of the data rate R or the noise level.

This paper provides a system-level analysis of a scalable distributed sensing model for networked sensors. In our system model, a data center acquires data from a bunch of L sensors which each independently encode their noisy observations of an original binary sequence, and transmit their encoded data sequences to the data center at a combined rate R, which is limited. Supposing that the sensors use independent LDGM rate distortion codes, we show that the system performance can be evaluated for any given finite R when the number of sensors L goes to infinity . The analysis shows how the optimal strategy for the distributed sensing problem changes at critical values of the data rate R or the noise level.