Simultaneous Learning of Nonlinear Manifold and Dynamical Models for High-dimensional Time Series

The goal of this work is to learn a parsimonious and informative representation for high-dimensional time series. Conceptually, this comprises two distinct yet tightly coupled tasks: learning a low-dimensional manifold and modeling the dynamical process. These two tasks have a complementary relationship as the temporal constraints provide valuable neighborhood information for dimensionality reduction and conversely, the low-dimensional space allows dynamics to be learnt efficiently. Solving these two tasks simultaneously allows important information to be exchanged mutually. If nonlinear models are required to capture the rich complexity of time series, then the learning problem becomes harder as the nonlinearities in both tasks are coupled. The proposed solution approximates the nonlinear manifold and dynamics using piecewise linear models. The interactions among the linear models are captured in a graphical model. By exploiting the model structure, efficient inference and learning algorithms are obtained without oversimplifying the model of the underlying dynamical process. Evaluation of the proposed framework with competing approaches is conducted in three sets of experiments: dimensionality reduction and reconstruction using synthetic time series, video synthesis using a dynamic texture database, and human motion synthesis, classification and tracking on a benchmark data set. In all experiments, the proposed approach provides superior performance.National Science Foundation (IIS 0308213, IIS 0329009, CNS 0202067

Efficient techniques for recovering 2D human body poses from images (PhD thesis)

Human parsing recovers the 2D spatial layout of a human figure in an image. First, patches in the image that resemble body parts, i.e., head, torso and limbs, are identified, then a coherent human figure is assembled from these candidate positions. The human model is represented as a graph where each vertex represents a body part and each edge represents a relationship between parts. If the graph is a tree, then the optimal solution can be recovered efficiently using the Min-Sum (MS) algorithm. Tree models often return incorrect solutions with the left and right legs stacked on top of one another. To overcome this problem, we add constraints to the tree model, yielding a graph that contains loops. Finding the optimal solution for a loopy graph is computationally intensive. We propose a Branch and Bound search algorithm to recover the optimal solution. Our algorithm converges quickly in practice due to a novel tree structured lower bound and a fast way for evaluating these lower bounds. Naively, evaluating each lower bound requires $O(nh)$ time for a graph with $n$ vertices and $h$ candidate body part locations. We develop an $O(1)$ time method for evaluating the lower bound (in most iterations of the algorithm) by reusing messages from the MS algorithm and using a Range Minimum Query data structure. We also propose a human parsing model that encodes the viewpoint and walking phase of the human figure using the Common Factor Model (CFM). The main computational bottleneck of the CFM human parsing algorithm involves message creation for each iteration of the MS algorithm. The original CFM inference requires $O(kn)$ messages to be created for $k$ iterations of the MS algorithm in a graph with $n$ vertices. Our new algorithm reduces this to $O(n)$ messages created. This speedup is based on the insight that the messages are shifted from one iteration to the next and, therefore, messages can be created once and then shifted in subsequent iterations (shifting is an efficient operation which requires $O(1)$ time). In our experiments, the two proposed algorithms yield an order of magnitude computational speedup over competing algorithms

Fast Globally Optimal 2D Human Detection with Loopy Graph Models

This paper presents an algorithm for recovering the globally optimal 2D human figure detection using a loopy graph model. This is computationally challenging because the time complexity scales exponentially in the size of the largest clique in the graph. The proposed algorithm uses Branch and Bound (BB) to search for the globally optimal solution. The algorithm converges rapidly in practice and this is due to a novel method for quickly computing tree based lower bounds. The key idea is to recycle the dynamic programming (DP) tables associated with the tree model to look up the tree based lower bound rather than recomputing the lower bound from scratch. This technique is further sped up using Range Minimum Query data structures to provide $O(1)$ cost for computing the lower bound for most iterations of the BB algorithm. The algorithm is evaluated on the Iterative Parsing dataset and it is shown to run fast empirically

Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations II

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies either $|v (x,t)| \le C_*{|t|^{-1/2}}$ or, for some \e > 0, $|v (x,t)| \le C_* r^{-1+\epsilon} |t|^{-\epsilon /2}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large. We prove that $v$ is regular at time zero.Comment: More explanations and a new appendi

Variations in photoprotective potential along gradients of leaf development and plant succession in subtropical forests under contrasting irradiances

The successful development of photosynthetic organs is the basis of plant growth and community development. To reveal photo-acclimation to high irradiance in tree species during the course of leaf development and plant succession of subtropical forests, photosynthetic efficiency and photoprotective compounds were analyzed in young and mature leaves of three mid-successional tree species (Castanopsis fissa, Castanopsis chinensis and Schima superba) and three late-successional tree species (Machilus chinensis, Cryptocarya chinensis and Cryptocarya concinna), grown in 100% full sunlight (FL) or 30% of FL (low light, LL). Young leaves of the two species groups exhibited lower chlorophyll (Chl) content, Rubisco content, net photosynthetic rate (Pn), carboxylation efficiency (CE), effective photochemical yield (ΦPSII), photorespiratory electron flow (JO), but higher dark respiration (Rd), and ratios of carotenoids/chlorophylls (Car/Chl), anthocyanins/chlorophylls (Anth/Chl), flavonoids/chlorophylls (Flav/Chl), phenols/chlorophylls (Phen/Chl) and total antioxidant capacity/chlorophylls (TAC/Chl) than those of mature leaves, regardless of growth irradiance. Young leaves of both species groups demonstrated a higher flexibility of Anth/Chl, Flav/Chl, Phen/Chl and TAC/Chl in response to different light conditions than mature leaves. Flav/Chl in young leaves of late-successional group was remarkably higher than that of mid-successional group under the same light conditions. There was a negative correlation between antioxidant-dependent photoprotective potential and photosynthetic efficiency in young and mature leaves of the six tree species grown under either FL or LL. Our results explain partial mechanisms that lie behind the replacement of communities in subtropical forests: highly integrated photoprotective potential allows young leaves of shade-tolerant late-successional species to develop smoothly into mature organs under high irradiance.This work was funded by the National Natural Science Foundation of China (31570398, 31270287). The study was also supported by the key programme of Guangdong Province Natural Science Foundation (2015A030311023)

Lower bound on the blow-up rate of the axisymmetric Navier-Stokes equations

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in $\R^3$ with non-trivial swirl. Such solutions are not known to be globally defined, but it is shown in \cite{MR673830} that they could only blow up on the axis of symmetry. Let $z$ denote the axis of symmetry and $r$ measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound $|v (x,t)| \le C_*{(r^2 -t)^{-1/2}}$ for $-T_0\le t < 0$ and $0<C_*<\infty$ allowed to be large, we then prove that $v$ is regular at time zero.Comment: 25 page

Attention Mechanisms in Computer Vision: A Survey

Humans can naturally and effectively find salient regions in complex scenes. Motivated by this observation, attention mechanisms were introduced into computer vision with the aim of imitating this aspect of the human visual system. Such an attention mechanism can be regarded as a dynamic weight adjustment process based on features of the input image. Attention mechanisms have achieved great success in many visual tasks, including image classification, object detection, semantic segmentation, video understanding, image generation, 3D vision, multi-modal tasks and self-supervised learning. In this survey, we provide a comprehensive review of various attention mechanisms in computer vision and categorize them according to approach, such as channel attention, spatial attention, temporal attention and branch attention; a related repository https://github.com/MenghaoGuo/Awesome-Vision-Attentions is dedicated to collecting related work. We also suggest future directions for attention mechanism research.Comment: 27 pages, 9 figure

Interior regularity criteria for suitable weak solutions of the Navier-Stokes equations

We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point $z$ if either the scaled $L^{p,q}_{x,t}$-norm of the velocity with $3/p+2/q\leq 2$, $1\leq q\leq \infty$, or the $L^{p,q}_{x,t}$-norm of the vorticity with $3/p+2/q\leq 3$, $1 \leq q < \infty$, or the $L^{p,q}_{x,t}$-norm of the gradient of the vorticity with $3/p+2/q\leq 4$, $1 \leq q$, $1 \leq p$, is sufficiently small near $z$