We introduce a method for learning minimal-dimensional dynamical models from high-dimensional time series data that lie on a low-dimensional manifold, as arises for many processes. For an arbitrary manifold, there is no smooth global coordinate representation, so following the formalism of differential topology we represent the manifold as an atlas of charts. We first partition the data into overlapping regions. Then undercomplete autoencoders are used to find low-dimensional coordinate representations for each region. We then use the data to learn dynamical models in each region, which together yield a global low-dimensional dynamical model. We apply this method to examples ranging from simple periodic dynamics to complex, nominally high-dimensional non-periodic bursting dynamics of the Kuramoto-Sivashinsky equation. We demonstrate that it: (1) can yield dynamical models of the lowest possible dimension, where previous methods generally cannot; (2) exhibits computational benefits including scalability, parallelizability, and adaptivity; and (3) separates state space into regions of distinct behaviours.
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Coulomb interaction, following an inverse-square force-law, quantifies the amount of force between two stationary and electrically charged particles. The long-range nature of Coulomb interactions poses a major challenge to molecular dynamics simulations which are major tools for problems at the nano-/micro- scale. Various algorithms are developed to calculate the pairwise Coulomb interactions to a linear scaling but the poor scalability limits the size of simulated systems. Here, we conduct an efficient molecular dynamics algorithm with the random batch Ewald method on all-atom systems where the complete Fourier components in the Coulomb interaction are replaced by randomly selected mini-batches. By simulating the $N$-body systems up to 100 million particles using $10$ thousand CPU cores, we show that this algorithm furnishes $O(N)$ complexity, almost perfect scalability and an order of magnitude faster computational speed when compared to the existing state-of-the-art algorithms. Further examinations of our algorithm on distinct systems, including pure water, micro-phase-separated electrolyte and protein solution demonstrate that the spatiotemporal information on all time and length scales investigated and thermodynamic quantities derived from our algorithm are in perfect agreement with those obtained from the existing algorithms. Therefore, our algorithm provides a breakthrough solution on scalability of computing the Coulomb interaction. It is particularly useful and cost-effective to simulate ultra-large systems, which was either impossible or very costing to conduct using existing algorithms, thus would benefit the broad community of sciences.
In this article, I introduce the differential equation model and review their frequentist and Bayesian computation methods. A numerical example of the FitzHugh-Nagumo model is given.
This chapter describes how gradient flows and nonlinear power methods in Banach spaces can be used to solve nonlinear eigenvector-dependent eigenvalue problems, and how convergence of (discretized) approximations can be verified. We review several flows from literature, which were proposed to compute nonlinear eigenfunctions, and show that they all relate to normalized gradient flows. Furthermore, we show that the implicit Euler discretization of gradient flows gives rise to a nonlinear power method of the proximal operator and prove their convergence to nonlinear eigenfunctions. Finally, we prove that $\Gamma$-convergence of functionals implies convergence of their ground states, which is important for discrete approximations.
We introduce kernel thinning, a new procedure for compressing a distribution $\mathbb{P}$ more effectively than i.i.d. sampling or standard thinning. Given a suitable reproducing kernel $\mathbf{k}$ and $\mathcal{O}(n^2)$ time, kernel thinning compresses an $n$-point approximation to $\mathbb{P}$ into a $\sqrt{n}$-point approximation with comparable worst-case integration error across the associated reproducing kernel Hilbert space. With high probability, the maximum discrepancy in integration error is $\mathcal{O}_d(n^{-\frac{1}{2}}\sqrt{\log n})$ for compactly supported $\mathbb{P}$ and $\mathcal{O}_d(n^{-\frac{1}{2}} \sqrt{(\log n)^{d+1}\log\log n})$ for sub-exponential $\mathbb{P}$ on $\mathbb{R}^d$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n^{-\frac14})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\mathbb{R}^d$ and a wide range of common kernels. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat\'ern, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning, in dimensions $d=2$ through $100$.
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.
With the advances of data-driven machine learning research, a wide variety of prediction problems have been tackled. It has become critical to explore how machine learning and specifically deep learning methods can be exploited to analyse healthcare data. A major limitation of existing methods has been the focus on grid-like data; however, the structure of physiological recordings are often irregular and unordered which makes it difficult to conceptualise them as a matrix. As such, graph neural networks have attracted significant attention by exploiting implicit information that resides in a biological system, with interactive nodes connected by edges whose weights can be either temporal associations or anatomical junctions. In this survey, we thoroughly review the different types of graph architectures and their applications in healthcare. We provide an overview of these methods in a systematic manner, organized by their domain of application including functional connectivity, anatomical structure and electrical-based analysis. We also outline the limitations of existing techniques and discuss potential directions for future research.
The use of orthogonal projections on high-dimensional input and target data in learning frameworks is studied. First, we investigate the relations between two standard objectives in dimension reduction, maximizing variance and preservation of pairwise relative distances. The derivation of their asymptotic correlation and numerical experiments tell that a projection usually cannot satisfy both objectives. In a standard classification problem we determine projections on the input data that balance them and compare subsequent results. Next, we extend our application of orthogonal projections to deep learning frameworks. We introduce new variational loss functions that enable integration of additional information via transformations and projections of the target data. In two supervised learning problems, clinical image segmentation and music information classification, the application of the proposed loss functions increase the accuracy.
In this work we present a novel unsupervised framework for hard training example mining. The only input to the method is a collection of images relevant to the target application and a meaningful initial representation, provided e.g. by pre-trained CNN. Positive examples are distant points on a single manifold, while negative examples are nearby points on different manifolds. Both types of examples are revealed by disagreements between Euclidean and manifold similarities. The discovered examples can be used in training with any discriminative loss. The method is applied to unsupervised fine-tuning of pre-trained networks for fine-grained classification and particular object retrieval. Our models are on par or are outperforming prior models that are fully or partially supervised.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
In this work, we present a method for tracking and learning the dynamics of all objects in a large scale robot environment. A mobile robot patrols the environment and visits the different locations one by one. Movable objects are discovered by change detection, and tracked throughout the robot deployment. For tracking, we extend the Rao-Blackwellized particle filter of previous work with birth and death processes, enabling the method to handle an arbitrary number of objects. Target births and associations are sampled using Gibbs sampling. The parameters of the system are then learnt using the Expectation Maximization algorithm in an unsupervised fashion. The system therefore enables learning of the dynamics of one particular environment, and of its objects. The algorithm is evaluated on data collected autonomously by a mobile robot in an office environment during a real-world deployment. We show that the algorithm automatically identifies and tracks the moving objects within 3D maps and infers plausible dynamics models, significantly decreasing the modeling bias of our previous work. The proposed method represents an improvement over previous methods for environment dynamics learning as it allows for learning of fine grained processes.
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