Data assimilation techniques are widely used to predict complex dynamical systems with uncertainties, based on time-series observation data. Error covariance matrices modelling is an important element in data assimilation algorithms which can considerably impact the forecasting accuracy. The estimation of these covariances, which usually relies on empirical assumptions and physical constraints, is often imprecise and computationally expensive especially for systems of large dimension. In this work, we propose a data-driven approach based on long short term memory (LSTM) recurrent neural networks (RNN) to improve both the accuracy and the efficiency of observation covariance specification in data assimilation for dynamical systems. Learning the covariance matrix from observed/simulated time-series data, the proposed approach does not require any knowledge or assumption about prior error distribution, unlike classical posterior tuning methods. We have compared the novel approach with two state-of-the-art covariance tuning algorithms, namely DI01 and D05, first in a Lorenz dynamical system and then in a 2D shallow water twin experiments framework with different covariance parameterization using ensemble assimilation. This novel method shows significant advantages in observation covariance specification, assimilation accuracy and computational efficiency.
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Graph analytics attract much attention from both research and industry communities. Due to the linear time complexity, the $k$-core decomposition is widely used in many real-world applications such as biology, social networks, community detection, ecology, and information spreading. In many such applications, the data graphs continuously change over time. The changes correspond to edge insertion and removal. Instead of recomputing the $k$-core, which is time-consuming, we study how to maintain the $k$-core efficiently. That is, when inserting or deleting an edge, we need to identify the affected vertices by searching for more vertices. The state-of-the-art order-based method maintains an order, the so-called $k$-order, among all vertices, which can significantly reduce the searching space. However, this order-based method is complicated for understanding and implementation, and its correctness is not formally discussed. In this work, we propose a simplified order-based approach by introducing the classical Order Data Structure to maintain the $k$-order, which significantly improves the worst-case time complexity for both edge insertion and removal algorithms. Also, our simplified method is intuitive to understand and implement; it is easy to argue the correctness formally. Additionally, we discuss a simplified batch insertion approach. The experiments evaluate our simplified method over 12 real and synthetic graphs with billions of vertices. Compared with the existing method, our simplified approach achieves high speedups up to 7.7x and 9.7x for edge insertion and removal, respectively.
Matrix Reordering for Noisy Disordered Matrices: Optimality and Computationally Efficient Algorithms
Motivated by applications in single-cell biology and metagenomics, we consider matrix reordering based on the noisy disordered matrix model. We first establish the fundamental statistical limit for the matrix reordering problem in a decision-theoretic framework and show that a constrained least square estimator is rate-optimal. Given the computational hardness of the optimal procedure, we analyze a popular polynomial-time algorithm, spectral seriation, and show that it is suboptimal. We then propose a novel polynomial-time adaptive sorting algorithm with guaranteed improvement on the performance. The superiority of the adaptive sorting algorithm over the existing methods is demonstrated in simulation studies and in the analysis of two real single-cell RNA sequencing datasets.
This study presents a policy optimisation framework for structured nonlinear control of continuous-time (deterministic) dynamic systems. The proposed approach prescribes a structure for the controller based on relevant scientific knowledge (such as Lyapunov stability theory or domain experiences) while considering the tunable elements inside the given structure as the point of parametrisation with neural networks. To optimise a cost represented as a function of the neural network weights, the proposed approach utilises the continuous-time policy gradient method based on adjoint sensitivity analysis as a means for correct and performant computation of cost gradient. This enables combining the stability, robustness, and physical interpretability of an analytically-derived structure for the feedback controller with the representational flexibility and optimised resulting performance provided by machine learning techniques. Such a hybrid paradigm for fixed-structure control synthesis is particularly useful for optimising adaptive nonlinear controllers to achieve improved performance in online operation, an area where the existing theory prevails the design of structure while lacking clear analytical understandings about tuning of the gains and the uncertainty model basis functions that govern the performance characteristics. Numerical experiments on aerospace applications illustrate the utility of the structured nonlinear controller optimisation framework.
This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI). The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system.
Weak lensing mass-mapping is a useful tool to access the full distribution of dark matter on the sky, but because of intrinsic galaxy ellipticies and finite fields/missing data, the recovery of dark matter maps constitutes a challenging ill-posed inverse problem. We introduce a novel methodology allowing for efficient sampling of the high-dimensional Bayesian posterior of the weak lensing mass-mapping problem, and relying on simulations for defining a fully non-Gaussian prior. We aim to demonstrate the accuracy of the method on simulations, and then proceed to applying it to the mass reconstruction of the HST/ACS COSMOS field. The proposed methodology combines elements of Bayesian statistics, analytic theory, and a recent class of Deep Generative Models based on Neural Score Matching. This approach allows us to do the following: 1) Make full use of analytic cosmological theory to constrain the 2pt statistics of the solution. 2) Learn from cosmological simulations any differences between this analytic prior and full simulations. 3) Obtain samples from the full Bayesian posterior of the problem for robust Uncertainty Quantification. We demonstrate the method on the $\kappa$TNG simulations and find that the posterior mean significantly outperfoms previous methods (Kaiser-Squires, Wiener filter, Sparsity priors) both on root-mean-square error and in terms of the Pearson correlation. We further illustrate the interpretability of the recovered posterior by establishing a close correlation between posterior convergence values and SNR of clusters artificially introduced into a field. Finally, we apply the method to the reconstruction of the HST/ACS COSMOS field and yield the highest quality convergence map of this field to date.
The principle of least action is one of the most fundamental physical principle. It says that among all possible motions connecting two points in a phase space, the system will exhibit those motions which extremise an action functional. Many qualitative features of dynamical systems, such as the presence of conservation laws and energy balance equations, are related to the existence of an action functional. Incorporating variational structure into learning algorithms for dynamical systems is, therefore, crucial in order to make sure that the learned model shares important features with the exact physical system. In this paper we show how to incorporate variational principles into trajectory predictions of learned dynamical systems. The novelty of this work is that (1) our technique relies only on discrete position data of observed trajectories. Velocities or conjugate momenta do {\em not} need to be observed or approximated and {\em no} prior knowledge about the form of the variational principle is assumed. Instead, they are recovered using backward error analysis. (2) Moreover, our technique compensates discretisation errors when trajectories are computed from the learned system. This is important when moderate to large step-sizes are used and high accuracy is required. For this, we introduce and rigorously analyse the concept of inverse modified Lagrangians by developing an inverse version of variational backward error analysis. (3) Finally, we introduce a method to perform system identification from position observations only, based on variational backward error analysis.
The essence of multivariate sequential learning is all about how to extract dependencies in data. These data sets, such as hourly medical records in intensive care units and multi-frequency phonetic time series, often time exhibit not only strong serial dependencies in the individual components (the "marginal" memory) but also non-negligible memories in the cross-sectional dependencies (the "joint" memory). Because of the multivariate complexity in the evolution of the joint distribution that underlies the data generating process, we take a data-driven approach and construct a novel recurrent network architecture, termed Memory-Gated Recurrent Networks (mGRN), with gates explicitly regulating two distinct types of memories: the marginal memory and the joint memory. Through a combination of comprehensive simulation studies and empirical experiments on a range of public datasets, we show that our proposed mGRN architecture consistently outperforms state-of-the-art architectures targeting multivariate time series.
Cognitive diagnosis is a fundamental issue in intelligent education, which aims to discover the proficiency level of students on specific knowledge concepts. Existing approaches usually mine linear interactions of student exercising process by manual-designed function (e.g., logistic function), which is not sufficient for capturing complex relations between students and exercises. In this paper, we propose a general Neural Cognitive Diagnosis (NeuralCD) framework, which incorporates neural networks to learn the complex exercising interactions, for getting both accurate and interpretable diagnosis results. Specifically, we project students and exercises to factor vectors and leverage multi neural layers for modeling their interactions, where the monotonicity assumption is applied to ensure the interpretability of both factors. Furthermore, we propose two implementations of NeuralCD by specializing the required concepts of each exercise, i.e., the NeuralCDM with traditional Q-matrix and the improved NeuralCDM+ exploring the rich text content. Extensive experimental results on real-world datasets show the effectiveness of NeuralCD framework with both accuracy and interpretability.
Recurrent neural networks (RNNs) provide state-of-the-art performance in processing sequential data but are memory intensive to train, limiting the flexibility of RNN models which can be trained. Reversible RNNs---RNNs for which the hidden-to-hidden transition can be reversed---offer a path to reduce the memory requirements of training, as hidden states need not be stored and instead can be recomputed during backpropagation. We first show that perfectly reversible RNNs, which require no storage of the hidden activations, are fundamentally limited because they cannot forget information from their hidden state. We then provide a scheme for storing a small number of bits in order to allow perfect reversal with forgetting. Our method achieves comparable performance to traditional models while reducing the activation memory cost by a factor of 10--15. We extend our technique to attention-based sequence-to-sequence models, where it maintains performance while reducing activation memory cost by a factor of 5--10 in the encoder, and a factor of 10--15 in the decoder.
In this paper we introduce a covariance framework for the analysis of EEG and MEG data that takes into account observed temporal stationarity on small time scales and trial-to-trial variations. We formulate a model for the covariance matrix, which is a Kronecker product of three components that correspond to space, time and epochs/trials, and consider maximum likelihood estimation of the unknown parameter values. An iterative algorithm that finds approximations of the maximum likelihood estimates is proposed. We perform a simulation study to assess the performance of the estimator and investigate the influence of different assumptions about the covariance factors on the estimated covariance matrix and on its components. Apart from that, we illustrate our method on real EEG and MEG data sets. The proposed covariance model is applicable in a variety of cases where spontaneous EEG or MEG acts as source of noise and realistic noise covariance estimates are needed for accurate dipole localization, such as in evoked activity studies, or where the properties of spontaneous EEG or MEG are themselves the topic of interest, such as in combined EEG/fMRI experiments in which the correlation between EEG and fMRI signals is investigated.
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