Deep Koopman-layered Model with Universal Property Based on Toeplitz Matrices

We propose deep Koopman-layered models with learnable parameters in the form of Toeplitz matrices for analyzing the dynamics of time-series data. The proposed model has both theoretical solidness and flexibility. By virtue of the universal property of Toeplitz matrices and the reproducing property underlined in the model, we can show its universality and the generalization property. In addition, the flexibility of the proposed model enables the model to fit time-series data coming from nonautonomous dynamical systems. When training the model, we apply Krylov subspace methods for efficient computations. In addition, the proposed model can be regarded as a neural ODE-based model. In this sense, the proposed model establishes a new connection among Koopman operators, neural ODEs, and numerical linear algebraic methods.

Meta-learning for Positive-unlabeled Classification

We propose a meta-learning method for positive and unlabeled (PU) classification, which improves the performance of binary classifiers obtained from only PU data in unseen target tasks. PU learning is an important problem since PU data naturally arise in real-world applications such as outlier detection and information retrieval. Existing PU learning methods require many PU data, but sufficient data are often unavailable in practice. The proposed method minimizes the test classification risk after the model is adapted to PU data by using related tasks that consist of positive, negative, and unlabeled data. We formulate the adaptation as an estimation problem of the Bayes optimal classifier, which is an optimal classifier to minimize the classification risk. The proposed method embeds each instance into a task-specific space using neural networks. With the embedded PU data, the Bayes optimal classifier is estimated through density-ratio estimation of PU densities, whose solution is obtained as a closed-form solution. The closed-form solution enables us to efficiently and effectively minimize the test classification risk. We empirically show that the proposed method outperforms existing methods with one synthetic and three real-world datasets.

Deep Positive-Unlabeled Anomaly Detection for Contaminated Unlabeled Data

Semi-supervised anomaly detection, which aims to improve the performance of the anomaly detector by using a small amount of anomaly data in addition to unlabeled data, has attracted attention. Existing semi-supervised approaches assume that unlabeled data are mostly normal. They train the anomaly detector to minimize the anomaly scores for the unlabeled data, and to maximize those for the anomaly data. However, in practice, the unlabeled data are often contaminated with anomalies. This weakens the effect of maximizing the anomaly scores for anomalies, and prevents us from improving the detection performance. To solve this problem, we propose the positive-unlabeled autoencoder, which is based on positive-unlabeled learning and the anomaly detector such as the autoencoder. With our approach, we can approximate the anomaly scores for normal data using the unlabeled and anomaly data. Therefore, without the labeled normal data, we can train the anomaly detector to minimize the anomaly scores for normal data, and to maximize those for the anomaly data. In addition, our approach is applicable to various anomaly detectors such as the DeepSVDD. Experiments on various datasets show that our approach achieves better detection performance than existing approaches.

Neural Operators Meet Energy-based Theory: Operator Learning for Hamiltonian and Dissipative PDEs

The operator learning has received significant attention in recent years, with the aim of learning a mapping between function spaces. Prior works have proposed deep neural networks (DNNs) for learning such a mapping, enabling the learning of solution operators of partial differential equations (PDEs). However, these works still struggle to learn dynamics that obeys the laws of physics. This paper proposes Energy-consistent Neural Operators (ENOs), a general framework for learning solution operators of PDEs that follows the energy conservation or dissipation law from observed solution trajectories. We introduce a novel penalty function inspired by the energy-based theory of physics for training, in which the energy functional is modeled by another DNN, allowing one to bias the outputs of the DNN-based solution operators to ensure energetic consistency without explicit PDEs. Experiments on multiple physical systems show that ENO outperforms existing DNN models in predicting solutions from data, especially in super-resolution settings.

Meta-Learning for Neural Network-based Temporal Point Processes

Human activities generate various event sequences such as taxi trip records, bike-sharing pick-ups, crime occurrence, and infectious disease transmission. The point process is widely used in many applications to predict such events related to human activities. However, point processes present two problems in predicting events related to human activities. First, recent high-performance point process models require the input of sufficient numbers of events collected over a long period (i.e., long sequences) for training, which are often unavailable in realistic situations. Second, the long-term predictions required in real-world applications are difficult. To tackle these problems, we propose a novel meta-learning approach for periodicity-aware prediction of future events given short sequences. The proposed method first embeds short sequences into hidden representations (i.e., task representations) via recurrent neural networks for creating predictions from short sequences. It then models the intensity of the point process by monotonic neural networks (MNNs), with the input being the task representations. We transfer the prior knowledge learned from related tasks and can improve event prediction given short sequences of target tasks. We design the MNNs to explicitly take temporal periodic patterns into account, contributing to improved long-term prediction performance. Experiments on multiple real-world datasets demonstrate that the proposed method has higher prediction performance than existing alternatives.

Meta-learning to Calibrate Gaussian Processes with Deep Kernels for Regression Uncertainty Estimation

Although Gaussian processes (GPs) with deep kernels have been successfully used for meta-learning in regression tasks, its uncertainty estimation performance can be poor. We propose a meta-learning method for calibrating deep kernel GPs for improving regression uncertainty estimation performance with a limited number of training data. The proposed method meta-learns how to calibrate uncertainty using data from various tasks by minimizing the test expected calibration error, and uses the knowledge for unseen tasks. We design our model such that the adaptation and calibration for each task can be performed without iterative procedures, which enables effective meta-learning. In particular, a task-specific uncalibrated output distribution is modeled by a GP with a task-shared encoder network, and it is transformed to a calibrated one using a cumulative density function of a task-specific Gaussian mixture model (GMM). By integrating the GP and GMM into our neural network-based model, we can meta-learn model parameters in an end-to-end fashion. Our experiments demonstrate that the proposed method improves uncertainty estimation performance while keeping high regression performance compared with the existing methods using real-world datasets in few-shot settings.

Meta-learning of semi-supervised learning from tasks with heterogeneous attribute spaces

We propose a meta-learning method for semi-supervised learning that learns from multiple tasks with heterogeneous attribute spaces. The existing semi-supervised meta-learning methods assume that all tasks share the same attribute space, which prevents us from learning with a wide variety of tasks. With the proposed method, the expected test performance on tasks with a small amount of labeled data is improved with unlabeled data as well as data in various tasks, where the attribute spaces are different among tasks. The proposed method embeds labeled and unlabeled data simultaneously in a task-specific space using a neural network, and the unlabeled data's labels are estimated by adapting classification or regression models in the embedding space. For the neural network, we develop variable-feature self-attention layers, which enable us to find embeddings of data with different attribute spaces with a single neural network by considering interactions among examples, attributes, and labels. Our experiments on classification and regression datasets with heterogeneous attribute spaces demonstrate that our proposed method outperforms the existing meta-learning and semi-supervised learning methods.

Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs

We propose a neural network-based meta-learning method to efficiently solve partial differential equation (PDE) problems. The proposed method is designed to meta-learn how to solve a wide variety of PDE problems, and uses the knowledge for solving newly given PDE problems. We encode a PDE problem into a problem representation using neural networks, where governing equations are represented by coefficients of a polynomial function of partial derivatives, and boundary conditions are represented by a set of point-condition pairs. We use the problem representation as an input of a neural network for predicting solutions, which enables us to efficiently predict problem-specific solutions by the forwarding process of the neural network without updating model parameters. To train our model, we minimize the expected error when adapted to a PDE problem based on the physics-informed neural network framework, by which we can evaluate the error even when solutions are unknown. We demonstrate that our proposed method outperforms existing methods in predicting solutions of PDE problems.

Explanation-Based Training with Differentiable Insertion/Deletion Metric-Aware Regularizers

The quality of explanations for the predictions of complex machine learning predictors is often measured using insertion and deletion metrics, which assess the faithfulness of the explanations, i.e., how correctly the explanations reflect the predictor's behavior. To improve the faithfulness, we propose insertion/deletion metric-aware explanation-based optimization (ID-ExpO), which optimizes differentiable predictors to improve both insertion and deletion scores of the explanations while keeping their predictive accuracy. Since the original insertion and deletion metrics are indifferentiable with respect to the explanations and directly unavailable for gradient-based optimization, we extend the metrics to be differentiable and use them to formalize insertion and deletion metric-based regularizers. The experimental results on image and tabular datasets show that the deep neural networks-based predictors fine-tuned using ID-ExpO enable popular post-hoc explainers to produce more faithful and easy-to-interpret explanations while keeping high predictive accuracy.

Information-theoretic Analysis of Test Data Sensitivity in Uncertainty

Bayesian inference is often utilized for uncertainty quantification tasks. A recent analysis by Xu and Raginsky 2022 rigorously decomposed the predictive uncertainty in Bayesian inference into two uncertainties, called aleatoric and epistemic uncertainties, which represent the inherent randomness in the data-generating process and the variability due to insufficient data, respectively. They analyzed those uncertainties in an information-theoretic way, assuming that the model is well-specified and treating the model's parameters as latent variables. However, the existing information-theoretic analysis of uncertainty cannot explain the widely believed property of uncertainty, known as the sensitivity between the test and training data. It implies that when test data are similar to training data in some sense, the epistemic uncertainty should become small. In this work, we study such uncertainty sensitivity using our novel decomposition method for the predictive uncertainty. Our analysis successfully defines such sensitivity using information-theoretic quantities. Furthermore, we extend the existing analysis of Bayesian meta-learning and show the novel sensitivities among tasks for the first time.