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Approximating the Koopman operator from data is numerically challenging when many lifting functions are considered. Even low-dimensional systems can yield unstable or ill-conditioned results in a high-dimensional lifted space. In this paper, Extended Dynamic Mode Decomposition (DMD) and DMD with control, two popular methods for approximating the Koopman operator, are reformulated as convex optimization problems with linear matrix inequality constraints. Hard asymptotic stability constraints and system norm regularizers are considered as methods to improve the numerical conditioning of the approximate Koopman operator. In particular, the H-infinity norm is used as a regularizer to penalize the input-output gain of the linear system defined by the Koopman operator. Weighting functions are then applied to penalize the system gain at specific frequencies. Experimental results using data from an aircraft fatigue structural test rig and a soft robot arm highlight the advantages of the proposed regression methods.

We propose an optimization method for the automatic design of approximate multipliers, which minimizes the average error according to the operand distributions. Our multiplier achieves up to 50.24% higher accuracy than the best reproduced approximate multiplier in DNNs, with 15.76% smaller area, 25.05% less power consumption, and 3.50% shorter delay. Compared with an exact multiplier, our multiplier reduces the area, power consumption, and delay by 44.94%, 47.63%, and 16.78%, respectively, with negligible accuracy losses. The tested DNN accelerator modules with our multiplier obtain up to 18.70% smaller area and 9.99% less power consumption than the original modules.

We report on the potential for using algorithms for non-negative matrix factorization (NMF) to improve parameter estimation in topic models. While several papers have studied connections between NMF and topic models, none have suggested leveraging these connections to develop new algorithms for fitting topic models. NMF avoids the "sum-to-one" constraints on the topic model parameters, resulting in an optimization problem with simpler structure and more efficient computations. Building on recent advances in optimization algorithms for NMF, we show that first solving the NMF problem then recovering the topic model fit can produce remarkably better fits, and in less time, than standard algorithms for topic models. While we focus primarily on maximum likelihood estimation, we show that this approach also has the potential to improve variational inference for topic models. Our methods are implemented in the R package fastTopics.

Ensembles of networks arise in various fields where multiple independent networks are observed on the same set of nodes, for example, a collection of brain networks constructed on the same brain regions for different individuals. However, there are few models that describe both the variations and characteristics of networks in an ensemble at the same time. In this paper, we propose to model the ensemble of networks using a Dirichlet Process Mixture of Exponential Random Graph Models (DPM-ERGMs), which divides the ensemble into different clusters and models each cluster of networks using a separate Exponential Random Graph Model (ERGM). By employing a Dirichlet process mixture, the number of clusters can be determined automatically and changed adaptively with the data provided. Moreover, in order to perform full Bayesian inference for DPM-ERGMs, we employ the intermediate importance sampling technique inside the Metropolis-within-slice sampling scheme, which addressed the problem of sampling from the intractable ERGMs on an infinite sample space. We also demonstrate the performance of DPM-ERGMs with both simulated and real datasets.

We formulate an efficient approximation for multi-agent batch reinforcement learning, the approximated multi-agent fitted Q iteration (AMAFQI). We present a detailed derivation of our approach. We propose an iterative policy search and show that it yields a greedy policy with respect to multiple approximations of the centralized, learned Q-function. In each iteration and policy evaluation, AMAFQI requires a number of computations that scales linearly with the number of agents whereas the analogous number of computations increase exponentially for the fitted Q iteration (FQI), a commonly used approaches in batch reinforcement learning. This property of AMAFQI is fundamental for the design of a tractable multi-agent approach. We evaluate the performance of AMAFQI and compare it to FQI in numerical simulations. The simulations illustrate the significant computation time reduction when using AMAFQI instead of FQI in multi-agent problems and corroborate the similar performance of both approaches.

We introduce a novel method for clustering using a semidefinite programming (SDP) relaxation of the Max k-Cut problem. The approach is based on a new methodology for rounding the solution of an SDP relaxation using iterated linear optimization. We show the vertices of the Max k-Cut SDP relaxation correspond to partitions of the data into at most k sets. We also show the vertices are attractive fixed points of iterated linear optimization. Each step of this iterative procedure solves a relaxation of the closest vertex problem and leads to a new clustering problem where the underlying clusters are more clearly defined. Our experiments show that using fixed point iteration for rounding the Max k-Cut SDP relaxation leads to significantly better results when compared to randomized rounding.

We introduce PyTorch Geometric, a library for deep learning on irregularly structured input data such as graphs, point clouds and manifolds, built upon PyTorch. In addition to general graph data structures and processing methods, it contains a variety of recently published methods from the domains of relational learning and 3D data processing. PyTorch Geometric achieves high data throughput by leveraging sparse GPU acceleration, by providing dedicated CUDA kernels and by introducing efficient mini-batch handling for input examples of different size. In this work, we present the library in detail and perform a comprehensive comparative study of the implemented methods in homogeneous evaluation scenarios.

Proximal Policy Optimization (PPO) is a highly popular model-free reinforcement learning (RL) approach. However, in continuous state and actions spaces and a Gaussian policy -- common in computer animation and robotics -- PPO is prone to getting stuck in local optima. In this paper, we observe a tendency of PPO to prematurely shrink the exploration variance, which naturally leads to slow progress. Motivated by this, we borrow ideas from CMA-ES, a black-box optimization method designed for intelligent adaptive Gaussian exploration, to derive PPO-CMA, a novel proximal policy optimization approach that can expand the exploration variance on objective function slopes and shrink the variance when close to the optimum. This is implemented by using separate neural networks for policy mean and variance and training the mean and variance in separate passes. Our experiments demonstrate a clear improvement over vanilla PPO in many difficult OpenAI Gym MuJoCo tasks.

Clustering and classification critically rely on distance metrics that provide meaningful comparisons between data points. We present mixed-integer optimization approaches to find optimal distance metrics that generalize the Mahalanobis metric extensively studied in the literature. Additionally, we generalize and improve upon leading methods by removing reliance on pre-designated "target neighbors," "triplets," and "similarity pairs." Another salient feature of our method is its ability to enable active learning by recommending precise regions to sample after an optimal metric is computed to improve classification performance. This targeted acquisition can significantly reduce computational burden by ensuring training data completeness, representativeness, and economy. We demonstrate classification and computational performance of the algorithms through several simple and intuitive examples, followed by results on real image and medical datasets.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.