arXiv subjects

Yi Zhou

Publications and source records attributed to Yi Zhou.

At least 523 records · Page 29Linked to original sources

arXiv2018

Convergence of Cubic Regularization for Nonconvex Optimization under KL Property

Cubic-regularized Newton's method (CR) is a popular algorithm that guarantees to produce a second-order stationary solution for solving nonconvex optimization problems. However, existing understandings of the convergence rate of CR are conditioned on special types of geometrical properties of the objective function. In this paper, we explore the asymptotic convergence rate of CR by exploiting the ubiquitous Kurdyka-Lojasiewicz (KL) property of nonconvex objective functions. In specific, we characterize the asymptotic convergence rate of various types of optimality measures for CR including function value gap, variable distance gap, gradient norm and least eigenvalue of the Hessian matrix. Our results fully characterize the diverse convergence behaviors of these optimality measures in the full parameter regime of the KL property. Moreover, we show that the obtained asymptotic convergence rates of CR are order-wise faster than those of first-order gradient descent algorithms under the KL property.

math.OC↗

arXiv2018

A Note on Inexact Condition for Cubic Regularized Newton's Method

This note considers the inexact cubic-regularized Newton's method (CR), which has been shown in \cite{Cartis2011a} to achieve the same order-level convergence rate to a secondary stationary point as the exact CR \citep{Nesterov2006}. However, the inexactness condition in \cite{Cartis2011a} is not implementable due to its dependence on future iterates variable. This note fixes such an issue by proving the same convergence rate for nonconvex optimization under an inexact adaptive condition that depends on only the current iterate. Our proof controls the sufficient decrease of the function value over the total iterations rather than each iteration as used in the previous studies, which can be of independent interest in other contexts.

math.OC↗

arXiv2018

Global Instability of the Multi-dimensional Plane Shocks for the isothermal flow

In this paper, we are concerned with the long time behavior of the piecewise smooth solutions to the generalized Riemann problem governed by the compressible isothermal Euler equations in two and three dimensions. Non-existence result is established for the fan-shaped wave structure solution, including two shocks and one contact discontinuity and which is a perturbation of plane waves. Therefore, unlike the one-dimensional case, the multi-dimensional plane shocks are not stable globally. What is more, the sharp lifespan estimate is established which is the same as the lifespan estimate for the nonlinear wave equations in both two and three space dimensions.

math.AP↗

arXiv2018

Semi-Dense 3D Reconstruction with a Stereo Event Camera

Event cameras are bio-inspired sensors that offer several advantages, such as low latency, high-speed and high dynamic range, to tackle challenging scenarios in computer vision. This paper presents a solution to the problem of 3D reconstruction from data captured by a stereo event-camera rig moving in a static scene, such as in the context of stereo Simultaneous Localization and Mapping. The proposed method consists of the optimization of an energy function designed to exploit small-baseline spatio-temporal consistency of events triggered across both stereo image planes. To improve the density of the reconstruction and to reduce the uncertainty of the estimation, a probabilistic depth-fusion strategy is also developed. The resulting method has no special requirements on either the motion of the stereo event-camera rig or on prior knowledge about the scene. Experiments demonstrate our method can deal with both texture-rich scenes as well as sparse scenes, outperforming state-of-the-art stereo methods based on event data image representations.

cs.CV↗

arXiv2018

HairNet: Single-View Hair Reconstruction using Convolutional Neural Networks

We introduce a deep learning-based method to generate full 3D hair geometry from an unconstrained image. Our method can recover local strand details and has real-time performance. State-of-the-art hair modeling techniques rely on large hairstyle collections for nearest neighbor retrieval and then perform ad-hoc refinement. Our deep learning approach, in contrast, is highly efficient in storage and can run 1000 times faster while generating hair with 30K strands. The convolutional neural network takes the 2D orientation field of a hair image as input and generates strand features that are evenly distributed on the parameterized 2D scalp. We introduce a collision loss to synthesize more plausible hairstyles, and the visibility of each strand is also used as a weight term to improve the reconstruction accuracy. The encoder-decoder architecture of our network naturally provides a compact and continuous representation for hairstyles, which allows us to interpolate naturally between hairstyles. We use a large set of rendered synthetic hair models to train our network. Our method scales to real images because an intermediate 2D orientation field, automatically calculated from the real image, factors out the difference between synthetic and real hairs. We demonstrate the effectiveness and robustness of our method on a wide range of challenging real Internet pictures and show reconstructed hair sequences from videos.

cs.GR↗

arXiv2018

Auto-Conditioned Recurrent Networks for Extended Complex Human Motion Synthesis

We present a real-time method for synthesizing highly complex human motions using a novel training regime we call the auto-conditioned Recurrent Neural Network (acRNN). Recently, researchers have attempted to synthesize new motion by using autoregressive techniques, but existing methods tend to freeze or diverge after a couple of seconds due to an accumulation of errors that are fed back into the network. Furthermore, such methods have only been shown to be reliable for relatively simple human motions, such as walking or running. In contrast, our approach can synthesize arbitrary motions with highly complex styles, including dances or martial arts in addition to locomotion. The acRNN is able to accomplish this by explicitly accommodating for autoregressive noise accumulation during training. Our work is the first to our knowledge that demonstrates the ability to generate over 18,000 continuous frames (300 seconds) of new complex human motion w.r.t. different styles.

cs.LG↗

arXiv2018

Exact Solution to a Class of Generalized Kitaev Spin-$1/2$ Models in Arbitrary Dimensions

We construct a class of exactly solvable generalized Kitaev spin-$1/2$ models in arbitrary dimensions, which is beyond the category of quantum compass models. The Jordan-Wigner transformation is employed to prove the exact solvability. An exactly solvable quantum spin-$1/2$ models can be mapped to a gas of free Majorana fermions coupled to static $Z_2$ gauge fields. We classify these exactly solvable models according to their parent models. Any model belonging to this class can be generated by one of the parent models. For illustration, a two dimensional ($2D$) tetragon-octagon model and a three dimensional ($3D$) $xy$ bond model are studied.

cond-mat.str-el↗

arXiv2018

Non-Abelian Chiral Spin Liquid on the Kagome Lattice

We study $S=1$ spin liquid states on the kagome lattice constructed by Gutzwiller-projected $p_x+ip_y$ superconductors. We show that the obtained spin liquids are either non-Abelian or Abelian topological phases, depending on the topology of the fermionic mean-field state. By calculating the modular matrices $S$ and $T$, we confirm that projected topological superconductors are non-Abelian chiral spin liquid (NACSL). The chiral central charge and the spin Hall conductance we obtained agree very well with the $SO(3)_1$ (or, equivalently, $SU(2)_2$) field theory predictions. We propose a local Hamiltonian which may stabilize the NACSL. From a variational study we observe a topological phase transition from the NACSL to the $Z_2$ Abelian spin liquid.

cond-mat.str-el↗

arXiv2018

The sharp lifespan estimate for semilinear damped wave equation with Fujita critical power in high dimensions

This paper is concerned about the lifespan estimate to the Cauchy problem of semilinear damped wave equations with the Fujita critical exponent in high dimensions$(n\geq 4)$. We establish the sharp upper bound of the lifespan in the following form \begin{equation}\nonumber\\ \begin{aligned} T(\varepsilon)\leq \exp(C\varepsilon^{-\frac 2n}), \end{aligned} \end{equation} by using the heat kernel as the test function.

math.AP↗

arXiv2018

Pressure induced superconductivity bordering a charge-density-wave state in NbTe4 with strong spinorbit coupling

Transition-metal chalcogenides host various phases of matter, such as charge-density wave (CDW), superconductors, and topological insulators or semimetals. Superconductivity and its competition with CDW in low-dimensional compounds have attracted much interest and stimulated considerable research. Here we report pressure induced superconductivity in a strong spin-orbit (SO) coupled quasi-one-dimensional (1D) transition-metal chalcogenide NbTe$_4$, which is a CDW material under ambient pressure. With increasing pressure, the CDW transition temperature is gradually suppressed, and superconducting transition, which is fingerprinted by a steep resistivity drop, emerges at pressures above 12.4 GPa. Under pressure $p$ = 69 GPa, zero resistance is detected with a transition temperature $T_c$ = 2.2 K and an upper critical field $H_{c2}$= 2 T. We also find large magnetoresistance (MR) up to 102\% at low temperatures, which is a distinct feature differentiating NbTe$_4$ from other conventional CDW materials.

cond-mat.supr-con↗

arXiv2018

Air-Ground Integrated Mobile Edge Networks: Architecture, Challenges and Opportunities

The ever-increasing mobile data demands have posed significant challenges in the current radio access networks, while the emerging computation-heavy Internet of things (IoT) applications with varied requirements demand more flexibility and resilience from the cloud/edge computing architecture. In this article, to address the issues, we propose a novel air-ground integrated mobile edge network (AGMEN), where UAVs are flexibly deployed and scheduled, and assist the communication, caching, and computing of the edge network. In specific, we present the detailed architecture of AGMEN, and investigate the benefits and application scenarios of drone-cells, and UAV-assisted edge caching and computing. Furthermore, the challenging issues in AGMEN are discussed, and potential research directions are highlighted.

eess.SP↗

arXiv2018

Global classical solutions of 3D viscous MHD system without magnetic diffusion on periodic boxes

In this paper, we study the global existence of classical solutions to the three dimensional incompressible viscous magneto-hydrodynamical system without magnetic diffusion on periodic boxes, i.e., with periodic boundary conditions. We work in Eulerian coordinate and employ a time-weighted energy estimate to prove the global existence result, under the assumptions that the initial magnetic field is close enough to an equilibrium state and the initial data have some symmetries.

math.AP↗

arXiv2018

Conditional Accelerated Lazy Stochastic Gradient Descent

In this work we introduce a conditional accelerated lazy stochastic gradient descent algorithm with optimal number of calls to a stochastic first-order oracle and convergence rate $O\left(\frac{1}{\varepsilon^2}\right)$ improving over the projection-free, Online Frank-Wolfe based stochastic gradient descent of Hazan and Kale [2012] with convergence rate $O\left(\frac{1}{\varepsilon^4}\right)$.

cs.LG↗

arXiv2017

A Simple Convergence Analysis of Bregman Proximal Gradient Algorithm

In this paper, we provide a simple convergence analysis of proximal gradient algorithm with Bregman distance, which provides a tighter bound than existing result. In particular, for the problem of minimizing a class of convex objective functions, we show that proximal gradient algorithm with Bregman distance can be viewed as proximal point algorithm that incorporates another Bregman distance. Consequently, the convergence result of the proximal gradient algorithm with Bregman distance follows directly from that of the proximal point algorithm with Bregman distance, and this leads to a simpler convergence analysis with a tighter convergence bound than existing ones. We further propose and analyze the backtracking line search variant of the proximal gradient algorithm with Bregman distance. Simulation results show that the line search method significantly improves the convergence performance of the algorithm.

math.OC↗

arXiv2017

Random gradient extrapolation for distributed and stochastic optimization

In this paper, we consider a class of finite-sum convex optimization problems defined over a distributed multiagent network with $m$ agents connected to a central server. In particular, the objective function consists of the average of $m$ ($\ge 1$) smooth components associated with each network agent together with a strongly convex term. Our major contribution is to develop a new randomized incremental gradient algorithm, namely random gradient extrapolation method (RGEM), which does not require any exact gradient evaluation even for the initial point, but can achieve the optimal ${\cal O}(\log(1/ε))$ complexity bound in terms of the total number of gradient evaluations of component functions to solve the finite-sum problems. Furthermore, we demonstrate that for stochastic finite-sum optimization problems, RGEM maintains the optimal ${\cal O}(1/ε)$ complexity (up to a certain logarithmic factor) in terms of the number of stochastic gradient computations, but attains an ${\cal O}(\log(1/ε))$ complexity in terms of communication rounds (each round involves only one agent). It is worth noting that the former bound is independent of the number of agents $m$, while the latter one only linearly depends on $m$ or even $\sqrt m$ for ill-conditioned problems. To the best of our knowledge, this is the first time that these complexity bounds have been obtained for distributed and stochastic optimization problems. Moreover, our algorithms were developed based on a novel dual perspective of Nesterov's accelerated gradient method.

math.OC↗

arXiv2017

Critical Points of Neural Networks: Analytical Forms and Landscape Properties

Due to the success of deep learning to solving a variety of challenging machine learning tasks, there is a rising interest in understanding loss functions for training neural networks from a theoretical aspect. Particularly, the properties of critical points and the landscape around them are of importance to determine the convergence performance of optimization algorithms. In this paper, we provide full (necessary and sufficient) characterization of the analytical forms for the critical points (as well as global minimizers) of the square loss functions for various neural networks. We show that the analytical forms of the critical points characterize the values of the corresponding loss functions as well as the necessary and sufficient conditions to achieve global minimum. Furthermore, we exploit the analytical forms of the critical points to characterize the landscape properties for the loss functions of these neural networks. One particular conclusion is that: The loss function of linear networks has no spurious local minimum, while the loss function of one-hidden-layer nonlinear networks with ReLU activation function does have local minimum that is not global minimum.

stat.ML↗

arXiv2017

Characterization of Gradient Dominance and Regularity Conditions for Neural Networks

The past decade has witnessed a successful application of deep learning to solving many challenging problems in machine learning and artificial intelligence. However, the loss functions of deep neural networks (especially nonlinear networks) are still far from being well understood from a theoretical aspect. In this paper, we enrich the current understanding of the landscape of the square loss functions for three types of neural networks. Specifically, when the parameter matrices are square, we provide an explicit characterization of the global minimizers for linear networks, linear residual networks, and nonlinear networks with one hidden layer. Then, we establish two quadratic types of landscape properties for the square loss of these neural networks, i.e., the gradient dominance condition within the neighborhood of their full rank global minimizers, and the regularity condition along certain directions and within the neighborhood of their global minimizers. These two landscape properties are desirable for the optimization around the global minimizers of the loss function for these neural networks.

stat.ML↗

arXiv2017

Selective equal spin Andreev reflection at vortex core center in magnetic semiconductor-superconductor heterostructure

Sau, Lutchyn, Tewari and Das Sarma (SLTD) proposed a heterostructure consisting of a semiconducting thin film sandwiched between an s-wave superconductor and a magnetic insulator and showed possible Majorana zero mode. Here we study spin polarization of the vortex core states and spin selective Andreev reflection at the vortex center of the SLTD model. In the topological phase, the differential conductance at the vortex center contributed from the Andreev reflection, is spin selective and has a quantized value $(dI/dV)^{topo}_A =2e^2/h$ at zero bias. In the topological trivial phase, $(dI/dV)^{trivial}_A$ at the lowest quasiparticle energy of the vortex core is spin selective due to the spin-orbit coupling (SOC). Unlike in the topological phase, $(dI/dV)^{trivial}_A$ is suppressed in the Giaever limit and vanishes exactly at zero bias due to the quantum destruction interference.

cond-mat.supr-con↗