On the Homology of the Space of Curves Immersed in The Sphere with Curvature Constrained to a Prescribed Interval

While the topology of the space of all smooth immersed curves on the $2$-sphere $\mathbb{S}^2$ that start and end at given points in given directions is well known, it is an open problem to understand the homotopy type of its subspaces consisting of the curves whose geodesic curvatures are constrained to a prescribed proper open interval. In this article we prove that, under certain circumstances for endpoints and end directions, these subspaces are not homotopically equivalent to the whole space. Moreover, we give an explicit construction of exotic generators for some homotopy and cohomology groups. It turns out that the dimensions of these generators depend on endpoints and end directions. A version of the h-principle is used to prove these results.Comment: 62 Pages, 19 figures. This is the article version of author's PhD Thesis advised by Nicolau C. Saldanh

Event-Trigger Based Robust-Optimal Control for Energy Harvesting Transmitter

This paper studies an online algorithm for an energy harvesting transmitter, where the transmission (completion) time is considered as the system performance. Unlike the existing online algorithms which more or less require the knowledge on the future behavior of the energy-harvesting rate, we consider a practical but significantly more challenging scenario where the energy-harvesting rate is assumed to be totally unknown. Our design is formulated as a robust-optimal control problem which aims to optimize the worst-case performance. The transmit power is designed only based on the current battery energy level and the data queue length directly monitored by the transmitter itself. Specifically, we apply an event-trigger approach in which the transmitter continuously monitors the battery energy and triggers an event when a significant change occurs. Once an event is triggered, the transmit power is updated according to the solution to the robust-optimal control problem, which is given in a simple analytic form. We present numerical results on the transmission time achieved by the proposed design and demonstrate its robust-optimality.Comment: The paper is accepted for publication in IEEE Transactions on Wireless Communication

A Distributed Hierarchical SGD Algorithm with Sparse Global Reduction

Reducing communication in training large-scale machine learning applications on distributed platform is still a big challenge. To address this issue, we propose a distributed hierarchical averaging stochastic gradient descent (Hier-AVG) algorithm with infrequent global reduction by introducing local reduction. As a general type of parallel SGD, Hier-AVG can reproduce several popular synchronous parallel SGD variants by adjusting its parameters. We show that Hier-AVG with infrequent global reduction can still achieve standard convergence rate for non-convex optimization problems. In addition, we show that more frequent local averaging with more participants involved can lead to faster training convergence. By comparing Hier-AVG with another popular distributed training algorithm K-AVG, we show that through deploying local averaging with fewer number of global averaging, Hier-AVG can still achieve comparable training speed while frequently get better test accuracy. This indicates that local averaging can serve as an alternative remedy to effectively reduce communication overhead when the number of learners is large. Experimental results of Hier-AVG with several state-of-the-art deep neural nets on CIFAR-10 and IMAGENET-1K are presented to validate our analysis and show its superiority.Comment: 38 page

Stabilization and Consensus of Linear Systems with Multiple Input Delays by Truncated Pseudo-Predictor Feedback

This paper provides an alternative approach referred to as pseudo-predictor feedback (PPF) for stabilization of linear systems with multiple input delays. Differently from the traditional predictor feedback which is from the model reduction appoint of view, the proposed PPF utilizes the idea of prediction by generalizing the corresponding results for linear systems with a single input delay to the case of multiple input delays. Since the PPF will generally lead to distributed controllers, a truncated pseudopredictor feedback (TPPF) approach is established instead which gives finite dimensional controllers. It is shown that the TPPF can compensate arbitrarily large yet bounded delays as long as the open-loop system is only polynomially unstable. The proposed TPPF approach is then used to solve the consensus problems for multi-agent systems characterized by linear systems with multiple input delays. Numerical examples show the effectiveness of the proposed approach.Comment: 19pages, 4 figures. submitted to a journal for publication consideratio

Probing the linear polarization of photons in ultraperipheral heavy ion collisions

We propose to measure the linear polarization of the external electromagnetic fields of a relativistic heavy ion through azimuthal asymmetries in dilepton production in ultraperipheral collisions. The asymmetries estimated with the equivalent photon approximation are shown to be sizable.Comment: The version accepted by the journal, 7 pages, 4 figure

Fast Simulation of Hyperplane-Truncated Multivariate Normal Distributions

We introduce a fast and easy-to-implement simulation algorithm for a multivariate normal distribution truncated on the intersection of a set of hyperplanes, and further generalize it to efficiently simulate random variables from a multivariate normal distribution whose covariance (precision) matrix can be decomposed as a positive-definite matrix minus (plus) a low-rank symmetric matrix. Example results illustrate the correctness and efficiency of the proposed simulation algorithms.Comment: To appear in Bayesian Analysi

Gamma Belief Networks

To infer multilayer deep representations of high-dimensional discrete and nonnegative real vectors, we propose an augmentable gamma belief network (GBN) that factorizes each of its hidden layers into the product of a sparse connection weight matrix and the nonnegative real hidden units of the next layer. The GBN's hidden layers are jointly trained with an upward-downward Gibbs sampler that solves each layer with the same subroutine. The gamma-negative binomial process combined with a layer-wise training strategy allows inferring the width of each layer given a fixed budget on the width of the first layer. Example results illustrate interesting relationships between the width of the first layer and the inferred network structure, and demonstrate that the GBN can add more layers to improve its performance in both unsupervisedly extracting features and predicting heldout data. For exploratory data analysis, we extract trees and subnetworks from the learned deep network to visualize how the very specific factors discovered at the first hidden layer and the increasingly more general factors discovered at deeper hidden layers are related to each other, and we generate synthetic data by propagating random variables through the deep network from the top hidden layer back to the bottom data layer.Comment: 44 pages, 24 figure

Towards Efficient Scheduling of Federated Mobile Devices under Computational and Statistical Heterogeneity

Originated from distributed learning, federated learning enables privacy-preserved collaboration on a new abstracted level by sharing the model parameters only. While the current research mainly focuses on optimizing learning algorithms and minimizing communication overhead left by distributed learning, there is still a considerable gap when it comes to the real implementation on mobile devices. In this paper, we start with an empirical experiment to demonstrate computation heterogeneity is a more pronounced bottleneck than communication on the current generation of battery-powered mobile devices, and the existing methods are haunted by mobile stragglers. Further, non-identically distributed data across the mobile users makes the selection of participants critical to the accuracy and convergence. To tackle the computational and statistical heterogeneity, we utilize data as a tuning knob and propose two efficient polynomial-time algorithms to schedule different workloads on various mobile devices, when data is identically or non-identically distributed. For identically distributed data, we combine partitioning and linear bottleneck assignment to achieve near-optimal training time without accuracy loss. For non-identically distributed data, we convert it into an average cost minimization problem and propose a greedy algorithm to find a reasonable balance between computation time and accuracy. We also establish an offline profiler to quantify the runtime behavior of different devices, which serves as the input to the scheduling algorithms. We conduct extensive experiments on a mobile testbed with two datasets and up to 20 devices. Compared with the common benchmarks, the proposed algorithms achieve 2-100x speedup epoch-wise, 2-7% accuracy gain and boost the convergence rate by more than 100% on CIFAR10

Regional Multi-Armed Bandits

We consider a variant of the classic multi-armed bandit problem where the expected reward of each arm is a function of an unknown parameter. The arms are divided into different groups, each of which has a common parameter. Therefore, when the player selects an arm at each time slot, information of other arms in the same group is also revealed. This regional bandit model naturally bridges the non-informative bandit setting where the player can only learn the chosen arm, and the global bandit model where sampling one arms reveals information of all arms. We propose an efficient algorithm, UCB-g, that solves the regional bandit problem by combining the Upper Confidence Bound (UCB) and greedy principles. Both parameter-dependent and parameter-free regret upper bounds are derived. We also establish a matching lower bound, which proves the order-optimality of UCB-g. Moreover, we propose SW-UCB-g, which is an extension of UCB-g for a non-stationary environment where the parameters slowly vary over time.Comment: AISTATS 201

Scaling parameters in anomalous and nonlinear Hall effects depend on temperature

In the study of the anomalous Hall effect, the scaling relations between the anomalous Hall and longitudinal resistivities play the central role. The scaling parameters by definition are fixed as the scaling variable (longitudinal resistivity) changes. Contrary to this paradigm, we unveil that the electron-phonon scattering can result in apparent temperature-dependence of scaling parameters when the longitudinal resistivity is tuned through temperature. An experimental approach is proposed to observe this hitherto unexpected temperature-dependence. We further show that this phenomenon also exists in the nonlinear Hall effect in nonmagnetic inversion-breaking materials and may help identify experimentally the presence of the side-jump contribution besides the Berry-curvature dipole.Comment: 4 pages, 2 figures, considerable change of the presentation in this versio