We investigate spatio-temporal prediction and introduce a novel prediction algorithm. Our approach is based on the point processes, which we use to model the event arrivals in both space and time. Although we specifically use the Hawkes process, other processes can be readily used as provided remarks in the paper. Moreover, we partition the given spatial region into subregions by an adaptive decision tree and model each subregion with individual and interacting point processes. With individual point processes for each subregion, we estimate the time and location of the events using the past event times and locations. Furthermore, thanks to the nonstationary and self-exciting point generation mechanism in the Hawkes process and the adaptive partitioning of the space, we model the data as nonstationary in both time and space. Finally, we provide a gradient based joint optimization algorithm for the adaptive tree parameter and the point process parameters. With the joint optimization, our algorithm can infer the source statistics and adaptive partitioning of the region. We also provide a training algorithm for the online setup, where we update the model parameters with newly arrived points. We provide experimental results on both simulated data and real-life data where we compare our approach with the standard approaches and demonstrate significant performance improvements thanks to the adaptive spatial partitioning mechanism and the joint optimization procedure.

Numerical integration is a computational procedure that is widely encountered across disciplines when reasoning about data. We derive a formula in closed form to calculate the multidimensional integral of functions fw that are representable using a shallow feed-forward neural network with weights w and a sigmoid activation function. We demonstrate its applicability in estimating numerical integration of arbitrary functions f over hyper-rectangular domains in the absence of a prior. To achieve this, we first train the network to learn $fw \approx f$ using point-samples of the integrand. We then use our formula to calculate the exact integral of the learned function fw. Our formula operates on the weights w of the trained approximator network. We show that this formula can itself be expressed as a shallow feed-forward network, which we call a Q-NET, with w as its inputs. Although the Q-NET does not have any learnable parameters, we use this abstraction to derive a family of elegant parametric formulae that represent the marginal distributions of the input function over arbitrary subsets of input dimensions in functional form. We perform empirical evaluations of Q-NETs for integrating smooth functions as well as functions with discontinuities.

Autoencoders have been widely used for dimensional reduction and feature extraction. Various types of autoencoders have been proposed by introducing regularization terms. Most of these regularizations improve representation learning by constraining the weights in the encoder part, which maps input into hidden nodes and affects the generation of features. In this study, we show that a constraint to the decoder can also significantly improve its performance because the decoder determines how the latent variables contribute to the reconstruction of input. Inspired by the structural modal analysis method in mechanical engineering, a new modal autoencoder (MAE) is proposed by othogonalising the columns of the readout weight matrix. The new regularization helps to disentangle explanatory factors of variation and forces the MAE to extract fundamental modes in data. The learned representations are functionally independent in the reconstruction of input and perform better in consecutive classification tasks. The results were validated on the MNIST variations and USPS classification benchmark suite. Comparative experiments clearly show that the new algorithm has a surprising advantage. The new MAE introduces a very simple training principle for autoencoders and could be promising for the pre-training of deep neural networks.

Many machine learning models require a training procedure based on running stochastic gradient descent. A key element for the efficiency of those algorithms is the choice of the learning rate schedule. While finding good learning rates schedules using Bayesian optimisation has been tackled by several authors, adapting it dynamically in a data-driven way is an open question. This is of high practical importance to users that need to train a single, expensive model. To tackle this problem, we introduce an original probabilistic model for traces of optimisers, based on latent Gaussian processes and an auto-/regressive formulation, that flexibly adjusts to abrupt changes of behaviours induced by new learning rate values. As illustrated, this model is well-suited to tackle a set of problems: first, for the on-line adaptation of the learning rate for a cold-started run; then, for tuning the schedule for a set of similar tasks (in a classical BO setup), as well as warm-starting it for a new task.

Private machine learning involves addition of noise while training, resulting in lower accuracy. Intuitively, greater stability can imply greater privacy and improve this privacy-utility tradeoff. We study this role of stability in private empirical risk minimization, where differential privacy is achieved by output perturbation, and establish a corresponding theoretical result showing that for strongly-convex loss functions, an algorithm with uniform stability of $\beta$ implies a bound of $O(\sqrt{\beta})$ on the scale of noise required for differential privacy. The result applies to both explicit regularization and to implicitly stabilized ERM, such as adaptations of Stochastic Gradient Descent that are known to be stable. Thus, it generalizes recent results that improve privacy through modifications to SGD, and establishes stability as the unifying perspective. It implies new privacy guarantees for optimizations with uniform stability guarantees, where a corresponding differential privacy guarantee was previously not known. Experimental results validate the utility of stability enhanced privacy in several problems, including application of elastic nets and feature selection.

In this paper, we propose a novel learning scheme called epoch-evolving Gaussian Process Guided Learning (GPGL), which aims at characterizing the correlation information between the batch-level distribution and the global data distribution. Such correlation information is encoded as context labels and needs renewal every epoch. With the guidance of the context label and ground truth label, GPGL scheme provides a more efficient optimization through updating the model parameters with a triangle consistency loss. Furthermore, our GPGL scheme can be further generalized and naturally applied to the current deep models, outperforming the existing batch-based state-of-the-art models on mainstream datasets (CIFAR-10, CIFAR-100, and Tiny-ImageNet) remarkably.

A great variety of natural and artificial systems can be represented as networks of elementary structural entities coupled by relations between them. The abstraction of such systems as networks helps us understand, predict and optimize their behaviour [1, 2]. In this sense, node and graph embeddings have been established as standard feature representations in many learning tasks for graphs and complex networks [3, 4]. Node embedding methods map each node of a graph into a low-dimensional vector, that can be then used to solve downstream tasks such as edge prediction, network reconstruction and node classification. Node embeddings have proven successful in achieving low-dimensional encoding of static network structures, but many real-world networks are inherently dynamic, with interactions among nodes changing over time [5]. Time-resolved networks are also the support of important dynamical processes, such as epidemic or rumor spreading, cascading failures, consensus formation, etc. [6] Time-resolved node embeddings have been shown to yield improved performance for predicting the outcome of dynamical processes over networks, such as information diffusion and disease spreading [7]. In this paper we propose a representation learning model that performs an implicit tensor factorization on different higher-order representations of time-varying graphs.

Quadratic discriminant analysis (QDA) is a widely used classification technique that generalizes the linear discriminant analysis (LDA) classifier to the case of distinct covariance matrices among classes. For the QDA classifier to yield high classification performance, an accurate estimation of the covariance matrices is required. Such a task becomes all the more challenging in high dimensional settings, wherein the number of observations is comparable with the feature dimension. A popular way to enhance the performance of QDA classifier under these circumstances is to regularize the covariance matrix, giving the name regularized QDA (R-QDA) to the corresponding classifier. In this work, we consider the case in which the population covariance matrix has a spiked covariance structure, a model that is often assumed in several applications. Building on the classical QDA, we propose a novel quadratic classification technique, the parameters of which are chosen such that the fisher-discriminant ratio is maximized. Numerical simulations show that the proposed classifier not only outperforms the classical R-QDA for both synthetic and real data but also requires lower computational complexity, making it suitable to high dimensional settings.

A new loss function is proposed for neural networks on classification tasks which extends the hinge loss by assigning gradients to its critical points. We will show that for a linear classifier on linearly separable data with fixed step size, the margin of this modified hinge loss converges to the $\ell_2$ max-margin at the rate of $\mathcal{O}( 1/t )$. This rate is fast when compared with the $\mathcal{O}(1/\log t)$ rate of exponential losses such as the logistic loss. Furthermore, empirical results suggest that this increased convergence speed carries over to ReLU networks.

Prediction of spatio-temporal chaotic systems is important in various fields, such as Numerical Weather Prediction (NWP). While data assimilation methods have been applied in NWP, machine learning techniques, such as Reservoir Computing (RC), are recently recognized as promising tools to predict spatio-temporal chaotic systems. However, the sensitivity of the skill of the machine learning based prediction to the imperfectness of observations is unclear. In this study, we evaluate the skill of RC with noisy and sparsely distributed observations. We intensively compare the performances of RC and Local Ensemble Transform Kalman Filter (LETKF) by applying them to the prediction of the Lorenz 96 system. Although RC can successfully predict the Lorenz 96 system if the system is perfectly observed, we find that RC is vulnerable to observation sparsity compared with LETKF. To overcome this limitation of RC, we propose to combine LETKF and RC. In our proposed method, the system is predicted by RC that learned the analysis time series estimated by LETKF. Our proposed method can successfully predict the Lorenz 96 system using noisy and sparsely distributed observations. Most importantly, our method can predict better than LETKF when the process-based model is imperfect.