Designing efficient and accurate numerical solvers for high-dimensional partial differential equations (PDEs) remains a challenging and important topic in computational science and engineering, mainly due to the "curse of dimensionality" in designing numerical schemes that scale in dimension. This paper introduces a new methodology that seeks an approximate PDE solution in the space of functions with finitely many analytic expressions and, hence, this methodology is named the finite expression method (FEX). It is proved in approximation theory that FEX can avoid the curse of dimensionality. As a proof of concept, a deep reinforcement learning method is proposed to implement FEX for various high-dimensional PDEs in different dimensions, achieving high and even machine accuracy with a memory complexity polynomial in dimension and an amenable time complexity. An approximate solution with finite analytic expressions also provides interpretable insights into the ground truth PDE solution, which can further help to advance the understanding of physical systems and design postprocessing techniques for a refined solution.
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To alleviate the cost of regression testing in continuous integration (CI), a large number of machine learning-based (ML-based) test case prioritization techniques have been proposed. However, it is yet unknown how they perform under the same experimental setup, because they are evaluated on different datasets with different metrics. To bridge this gap, we conduct the first comprehensive study on these ML-based techniques in this paper. We investigate the performance of 11 representative ML-based prioritization techniques for CI on 11 open-source subjects and obtain a series of findings. For example, the performance of the techniques changes across CI cycles, mainly resulting from the changing amount of training data, instead of code evolution and test removal/addition. Based on the findings, we give some actionable suggestions on enhancing the effectiveness of ML-based techniques, e.g., pretraining a prioritization technique with cross-subject data to get it thoroughly trained and then finetuning it with within-subject data dramatically improves its performance. In particular, the pretrained MART achieves state-of-the-art performance, producing the optimal sequence on 80% subjects, while the existing best technique, the original MART, only produces the optimal sequence on 50% subjects.
Most conventional crowd counting methods utilize a fully-supervised learning framework to establish a mapping between scene images and crowd density maps. They usually rely on a large quantity of costly and time-intensive pixel-level annotations for training supervision. One way to mitigate the intensive labeling effort and improve counting accuracy is to leverage large amounts of unlabeled images. This is attributed to the inherent self-structural information and rank consistency within a single image, offering additional qualitative relation supervision during training. Contrary to earlier methods that utilized the rank relations at the original image level, we explore such rank-consistency relation within the latent feature spaces. This approach enables the incorporation of numerous pyramid partial orders, strengthening the model representation capability. A notable advantage is that it can also increase the utilization ratio of unlabeled samples. Specifically, we propose a Deep Rank-consistEnt pyrAmid Model (DREAM), which makes full use of rank consistency across coarse-to-fine pyramid features in latent spaces for enhanced crowd counting with massive unlabeled images. In addition, we have collected a new unlabeled crowd counting dataset, FUDAN-UCC, comprising 4,000 images for training purposes. Extensive experiments on four benchmark datasets, namely UCF-QNRF, ShanghaiTech PartA and PartB, and UCF-CC-50, show the effectiveness of our method compared with previous semi-supervised methods. The codes are available at //github.com/bridgeqiqi/DREAM.
Matrix factorization (MF) is a simple collaborative filtering technique that achieves superior recommendation accuracy by decomposing the user-item rating matrix into user and item latent matrices. This approach relies on learning from user-item interactions, which may not effectively capture the underlying shared dependencies between users or items. Therefore, there is scope to explicitly capture shared dependencies to further improve recommendation accuracy and the interpretability of learning results by summarizing user-item interactions. Based on these insights, we propose "Hierarchical Matrix Factorization" (HMF), which incorporates clustering concepts to capture the hierarchy, where leaf nodes and other nodes correspond to users/items and clusters, respectively. Central to our approach, called hierarchical embeddings, is the additional decomposition of the user and item latent matrices (embeddings) into probabilistic connection matrices, which link the hierarchy, and a root cluster latent matrix. Thus, each node is represented by the weighted average of the embeddings of its parent clusters. The embeddings are differentiable, allowing simultaneous learning of interactions and clustering using a single gradient descent method. Furthermore, the obtained cluster-specific interactions naturally summarize user-item interactions and provide interpretability. Experimental results on rating and ranking predictions demonstrated the competitiveness of HMF over vanilla and hierarchical MF methods, especially its robustness in sparse interactions. Additionally, it was confirmed that the clustering integration of HMF has the potential for faster learning convergence and mitigation of overfitting compared to MF, and also provides interpretability through a cluster-centered case study.
The common spatial pattern analysis (CSP) is a widely used signal processing technique in brain-computer interface (BCI) systems to increase the signal-to-noise ratio in electroencephalogram (EEG) recordings. Despite its popularity, the CSP's performance is often hindered by the nonstationarity and artifacts in EEG signals. The minmax CSP improves the robustness of the CSP by using data-driven covariance matrices to accommodate the uncertainties. We show that by utilizing the optimality conditions, the minmax CSP can be recast as an eigenvector-dependent nonlinear eigenvalue problem (NEPv). We introduce a self-consistent field (SCF) iteration with line search that solves the NEPv of the minmax CSP. Local quadratic convergence of the SCF for solving the NEPv is illustrated using synthetic datasets. More importantly, experiments with real-world EEG datasets show the improved motor imagery classification rates and shorter running time of the proposed SCF-based solver compared to the existing algorithm for the minmax CSP.
Recent work in data-driven modeling has demonstrated that a weak formulation of model equations enhances the noise robustness of a wide range of computational methods. In this paper, we demonstrate the power of the weak form to enhance the LaSDI (Latent Space Dynamics Identification) algorithm, a recently developed data-driven reduced order modeling technique. We introduce a weak form-based version WLaSDI (Weak-form Latent Space Dynamics Identification). WLaSDI first compresses data, then projects onto the test functions and learns the local latent space models. Notably, WLaSDI demonstrates significantly enhanced robustness to noise. With WLaSDI, the local latent space is obtained using weak-form equation learning techniques. Compared to the standard sparse identification of nonlinear dynamics (SINDy) used in LaSDI, the variance reduction of the weak form guarantees a robust and precise latent space recovery, hence allowing for a fast, robust, and accurate simulation. We demonstrate the efficacy of WLaSDI vs. LaSDI on several common benchmark examples including viscid and inviscid Burgers', radial advection, and heat conduction. For instance, in the case of 1D inviscid Burgers' simulations with the addition of up to 100% Gaussian white noise, the relative error remains consistently below 6% for WLaSDI, while it can exceed 10,000% for LaSDI. Similarly, for radial advection simulations, the relative errors stay below 15% for WLaSDI, in stark contrast to the potential errors of up to 10,000% with LaSDI. Moreover, speedups of several orders of magnitude can be obtained with WLaSDI. For example applying WLaSDI to 1D Burgers' yields a 140X speedup compared to the corresponding full order model. Python code to reproduce the results in this work is available at (//github.com/MathBioCU/PyWSINDy_ODE) and (//github.com/MathBioCU/PyWLaSDI).
Recent contrastive representation learning methods rely on estimating mutual information (MI) between multiple views of an underlying context. E.g., we can derive multiple views of a given image by applying data augmentation, or we can split a sequence into views comprising the past and future of some step in the sequence. Contrastive lower bounds on MI are easy to optimize, but have a strong underestimation bias when estimating large amounts of MI. We propose decomposing the full MI estimation problem into a sum of smaller estimation problems by splitting one of the views into progressively more informed subviews and by applying the chain rule on MI between the decomposed views. This expression contains a sum of unconditional and conditional MI terms, each measuring modest chunks of the total MI, which facilitates approximation via contrastive bounds. To maximize the sum, we formulate a contrastive lower bound on the conditional MI which can be approximated efficiently. We refer to our general approach as Decomposed Estimation of Mutual Information (DEMI). We show that DEMI can capture a larger amount of MI than standard non-decomposed contrastive bounds in a synthetic setting, and learns better representations in a vision domain and for dialogue generation.
The accurate and interpretable prediction of future events in time-series data often requires the capturing of representative patterns (or referred to as states) underpinning the observed data. To this end, most existing studies focus on the representation and recognition of states, but ignore the changing transitional relations among them. In this paper, we present evolutionary state graph, a dynamic graph structure designed to systematically represent the evolving relations (edges) among states (nodes) along time. We conduct analysis on the dynamic graphs constructed from the time-series data and show that changes on the graph structures (e.g., edges connecting certain state nodes) can inform the occurrences of events (i.e., time-series fluctuation). Inspired by this, we propose a novel graph neural network model, Evolutionary State Graph Network (EvoNet), to encode the evolutionary state graph for accurate and interpretable time-series event prediction. Specifically, Evolutionary State Graph Network models both the node-level (state-to-state) and graph-level (segment-to-segment) propagation, and captures the node-graph (state-to-segment) interactions over time. Experimental results based on five real-world datasets show that our approach not only achieves clear improvements compared with 11 baselines, but also provides more insights towards explaining the results of event predictions.
Cold-start problems are long-standing challenges for practical recommendations. Most existing recommendation algorithms rely on extensive observed data and are brittle to recommendation scenarios with few interactions. This paper addresses such problems using few-shot learning and meta learning. Our approach is based on the insight that having a good generalization from a few examples relies on both a generic model initialization and an effective strategy for adapting this model to newly arising tasks. To accomplish this, we combine the scenario-specific learning with a model-agnostic sequential meta-learning and unify them into an integrated end-to-end framework, namely Scenario-specific Sequential Meta learner (or s^2 meta). By doing so, our meta-learner produces a generic initial model through aggregating contextual information from a variety of prediction tasks while effectively adapting to specific tasks by leveraging learning-to-learn knowledge. Extensive experiments on various real-world datasets demonstrate that our proposed model can achieve significant gains over the state-of-the-arts for cold-start problems in online recommendation. Deployment is at the Guess You Like session, the front page of the Mobile Taobao.
High spectral dimensionality and the shortage of annotations make hyperspectral image (HSI) classification a challenging problem. Recent studies suggest that convolutional neural networks can learn discriminative spatial features, which play a paramount role in HSI interpretation. However, most of these methods ignore the distinctive spectral-spatial characteristic of hyperspectral data. In addition, a large amount of unlabeled data remains an unexploited gold mine for efficient data use. Therefore, we proposed an integration of generative adversarial networks (GANs) and probabilistic graphical models for HSI classification. Specifically, we used a spectral-spatial generator and a discriminator to identify land cover categories of hyperspectral cubes. Moreover, to take advantage of a large amount of unlabeled data, we adopted a conditional random field to refine the preliminary classification results generated by GANs. Experimental results obtained using two commonly studied datasets demonstrate that the proposed framework achieved encouraging classification accuracy using a small number of data for training.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis.
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