Reducing operator complexity in Algebraic Multigrid with Machine Learning Approaches

We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in algebraic multigrid (AMG) methods, addressing the well-known issue of increasing operator complexity. Guided by the AMG theory on spectrally equivalent coarse-grid operators, we have developed novel ML algorithms that utilize neural networks (NNs) combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall AMG convergence for solving parametric partial differential equation (PDE) problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and compare with existing methods for computing non-Galerkin coarse-grid operators.

MedDiff: Generating Electronic Health Records using Accelerated Denoising Diffusion Model

Due to patient privacy protection concerns, machine learning research in healthcare has been undeniably slower and limited than in other application domains. High-quality, realistic, synthetic electronic health records (EHRs) can be leveraged to accelerate methodological developments for research purposes while mitigating privacy concerns associated with data sharing. The current state-of-the-art model for synthetic EHR generation is generative adversarial networks, which are notoriously difficult to train and can suffer from mode collapse. Denoising Diffusion Probabilistic Models, a class of generative models inspired by statistical thermodynamics, have recently been shown to generate high-quality synthetic samples in certain domains. It is unknown whether these can generalize to generation of large-scale, high-dimensional EHRs. In this paper, we present a novel generative model based on diffusion models that is the first successful application on electronic health records. Our model proposes a mechanism to perform class-conditional sampling to preserve label information. We also introduce a new sampling strategy to accelerate the inference speed. We empirically show that our model outperforms existing state-of-the-art synthetic EHR generation methods.

MuG: A Multimodal Classification Benchmark on Game Data with Tabular, Textual, and Visual Fields

Multimodal learning has attracted the interest of the machine learning community due to its great potential in a variety of applications. To help achieve this potential, we propose a multimodal benchmark MuG with eight datasets allowing researchers to test the multimodal perceptron capabilities of their models. These datasets are collected from four different genres of games that cover tabular, textual, and visual modalities. We conduct multi-aspect data analysis to provide insights into the benchmark, including label balance ratios, percentages of missing features, distributions of data within each modality, and the correlations between labels and input modalities. We further present experimental results obtained by several state-of-the-art unimodal classifiers and multimodal classifiers, which demonstrate the challenging and multimodal-dependent properties of the benchmark. MuG is released at https://github.com/lujiaying/MUG-Bench with the data, documents, tutorials, and implemented baselines. Extensions of MuG are welcomed to facilitate the progress of research in multimodal learning problems.

Data-Driven Linear Complexity Low-Rank Approximation of General Kernel Matrices: A Geometric Approach

A general, {\em rectangular} kernel matrix may be defined as $K_{ij} = \kappa(x_i,y_j)$ where $\kappa(x,y)$ is a kernel function and where $X=\{x_i\}_{i=1}^m$ and $Y=\{y_i\}_{i=1}^n$ are two sets of points. In this paper, we seek a low-rank approximation to a kernel matrix where the sets of points $X$ and $Y$ are large and are not well-separated (e.g., the points in $X$ and $Y$ may be ``intermingled''). Such rectangular kernel matrices may arise, for example, in Gaussian process regression where $X$ corresponds to the training data and $Y$ corresponds to the test data. In this case, the points are often high-dimensional. Since the point sets are large, we must exploit the fact that the matrix arises from a kernel function, and avoid forming the matrix, and thus ruling out most algebraic techniques. In particular, we seek methods that can scale linearly, i.e., with computational complexity $O(m)$ or $O(n)$ for a fixed accuracy or rank. The main idea in this paper is to {\em geometrically} select appropriate subsets of points to construct a low rank approximation. An analysis in this paper guides how this selection should be performed.

An Efficient Nonlinear Acceleration method that Exploits Symmetry of the Hessian

Nonlinear acceleration methods are powerful techniques to speed up fixed-point iterations. However, many acceleration methods require storing a large number of previous iterates and this can become impractical if computational resources are limited. In this paper, we propose a nonlinear Truncated Generalized Conjugate Residual method (nlTGCR) whose goal is to exploit the symmetry of the Hessian to reduce memory usage. The proposed method can be interpreted as either an inexact Newton or a quasi-Newton method. We show that, with the help of global strategies like residual check techniques, nlTGCR can converge globally for general nonlinear problems and that under mild conditions, nlTGCR is able to achieve superlinear convergence. We further analyze the convergence of nlTGCR in a stochastic setting. Numerical results demonstrate the superiority of nlTGCR when compared with several other competitive baseline approaches on a few problems. Our code will be available in the future.

AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows

Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting function classes or model parameters and the inverse transformation is often approximated by root-finding algorithms as a closed-form inverse is unavailable. In this paper, we introduce a new integral-based approach termed "Atomic Unrestricted Time Machine (AUTM)", equipped with unrestricted integrands and easy-to-compute explicit inverse. AUTM offers a versatile and efficient way to the design of normalizing flows with explicit inverse and unrestricted function classes or parameters. Theoretically, we present a constructive proof that AUTM is universal: all monotonic normalizing flows can be viewed as limits of AUTM flows. We provide a concrete example to show how to approximate any given monotonic normalizing flow using AUTM flows with guaranteed convergence. The result implies that AUTM can be used to transform an existing flow into a new one equipped with explicit inverse and unrestricted parameters. The performance of the new approach is evaluated on high dimensional density estimation, variational inference and image generation. Experiments demonstrate superior speed and memory efficiency of AUTM.

Solve Minimax Optimization by Anderson Acceleration

Many modern machine learning algorithms such as generative adversarial networks (GANs) and adversarial training can be formulated as minimax optimization. Gradient descent ascent (GDA) is the most commonly used algorithm due to its simplicity. However, GDA can converge to non-optimal minimax points. We propose a new minimax optimization framework, GDA-AM, that views the GDAdynamics as a fixed-point iteration and solves it using Anderson Mixing to con-verge to the local minimax. It addresses the diverging issue of simultaneous GDAand accelerates the convergence of alternating GDA. We show theoretically that the algorithm can achieve global convergence for bilinear problems under mild conditions. We also empirically show that GDA-AMsolves a variety of minimax problems and improves GAN training on several datasets

Learning optimal multigrid smoothers via neural networks

Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smoothing, which aims at reducing high-frequency errors on each grid level. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. In this paper, we propose an efficient adaptive framework for learning optimized smoothers from operator stencils in the form of convolutional neural networks (CNNs). The CNNs are trained on small-scale problems from a given type of PDEs based on a supervised loss function derived from multigrid convergence theories, and can be applied to large-scale problems of the same class of PDEs. Numerical results on anisotropic rotated Laplacian problems demonstrate improved convergence rates and solution time compared with classical hand-crafted relaxation methods.

Generating a Doppelganger Graph: Resembling but Distinct

Deep generative models, since their inception, have become increasingly more capable of generating novel and perceptually realistic signals (e.g., images and sound waves). With the emergence of deep models for graph structured data, natural interests seek extensions of these generative models for graphs. Successful extensions were seen recently in the case of learning from a collection of graphs (e.g., protein data banks), but the learning from a single graph has been largely under explored. The latter case, however, is important in practice. For example, graphs in financial and healthcare systems contain so much confidential information that their public accessibility is nearly impossible, but open science in these fields can only advance when similar data are available for benchmarking. In this work, we propose an approach to generating a doppelganger graph that resembles a given one in many graph properties but nonetheless can hardly be used to reverse engineer the original one, in the sense of a near zero edge overlap. The approach is an orchestration of graph representation learning, generative adversarial networks, and graph realization algorithms. Through comparison with several graph generative models (either parameterized by neural networks or not), we demonstrate that our result barely reproduces the given graph but closely matches its properties. We further show that downstream tasks, such as node classification, on the generated graphs reach similar performance to the use of the original ones.