When, in terms of the number of data points, the size of a dataset exceeds available computing resources, or when labeling is expensive, an attractive solution consists of selecting only some of the data points (subdata) for further consideration. A central question for selecting subdata of size $n$ from $N$ available data points is which $n$ points to select. While an answer to this question depends on the objective, one approach for a parametric model and a focus on parameter estimation is to select subdata that retains maximal information. Identifying such subdata is a classical NP-hard problem due to its inherent discreteness. Based on optimal approximate design theory, we develop a new methodology for information-based subdata selection, resulting in subdata that approaches the optimal solution. To achieve this, we develop a novel algorithm that applies to a general model, accommodates arbitrary choices of $N$ and $n$, and supports multiple optimality criteria, and we prove its convergence. Moreover, the new methodology facilitates an assessment of the efficiency of subdata selected by any method by obtaining tight lower and upper bounds for the efficiency. We show that the subdata obtained through the new methodology is highly efficient and outperforms all existing methods.

The asymptotic behavior of the proposed estimation procedure is studied. Our theoretical results show that the proposed method can achieve a faster convergence rate than the optimal convergence rate for sampling methods.

We derive an estimator of the spectral density of a functional time series that is the output of a multilayer perceptron neural network. The estimator is motivated by difficulties with the computation of existing spectral density estimators for time series of functions defined on very large grids that arise, for example, in climate compute models and medical scans. Existing estimators use autocovariance kernels represented as large $G \times G$ matrices, where $G$ is the number of grid points on which the functions are evaluated. In many recent applications, functions are defined on 2D and 3D domains, and $G$ can be of the order $G \sim 10^5$, making the evaluation of the autocovariance kernels computationally intensive or even impossible. We use the theory of spectral functional principal components to derive our deep learning estimator and prove that it is a universal approximator to the spectral density under general assumptions. Our estimator can be trained without computing the autocovariance kernels and it can be parallelized to provide the estimates much faster than existing approaches. We validate its performance by simulations and an application to fMRI images.

We propose a novel randomized framework for the estimation problem of large-scale linear statistical models, namely Sequential Least-Squares Estimators with Fast Randomized Sketching (SLSE-FRS), which integrates Sketch-and-Solve and Iterative-Sketching methods for the first time. By iteratively constructing and solving sketched least-squares (LS) subproblems with increasing sketch sizes to achieve better precisions, SLSE-FRS gradually refines the estimators of the true parameter vector, ultimately producing high-precision estimators. We analyze the convergence properties of SLSE-FRS, and provide its efficient implementation. Numerical experiments show that SLSE-FRS outperforms the state-of-the-art methods, namely the Preconditioned Conjugate Gradient (PCG) method, and the Iterative Double Sketching (IDS) method.

An effective method for optimizing path planning for a specific model of a 6-degree-of-freedom (6-DOF) robot manipulator is presented as part of the motion planning of the manipulator using computer algebra. We assume that we are given a path in the form of a set of line segments that the end-effector should follow. We also assume that we have a method to solve the inverse kinematic problem of the manipulator at each via-point of the trajectory. The proposed method consists of three steps. First, we calculate the feasible region of the manipulator under a specific configuration of the end-effector. Next, we aim to find a trajectory on the line segments and a sequence of joint configurations the manipulator should follow to move the end-effector along the specified trajectory. Finally, we find the optimal combination of solutions to the inverse kinematic problem at each via-point along the trajectory by reducing the problem to a shortest-path problem of the graph and applying Dijkstra's algorithm. We show the effectiveness of the proposed method by experiments.

We consider the inverse problem of multi-armed bandits (IMAB) that are widely used in neuroscience and psychology research for behavior modelling. We first show that the IMAB problem is not convex in general, but can be relaxed to a convex problem via variable transformation. Based on this result, we propose a two-step sequential heuristic for (approximately) solving the IMAB problem. We discuss a condition where our method provides global solution to the IMAB problem with certificate, as well as approximations to further save computing time. Numerical experiments indicate that our heuristic method is more robust than directly solving the IMAB problem via repeated local optimization, and can achieve the performance of Monte Carlo methods within a significantly decreased running time. We provide the implementation of our method based on CVXPY, which allows straightforward application by users not well versed in convex optimization.

Given the popularity of decision trees, proposing an efficient parallel implementation of this method is of course very relevant. The proposed parallelization is original with respect to existing methods and it should indeed lead to less communications than other methods. The theoretical analysis is sound and I like the discussion of the impact of the main problem and method parameters that follows from the lower bound provided in theorem 4.1. Experiments are conducted on two very large problems, where, in the limit of the tested settings (see below), PV-tree is clearly shown to outperform other parallel implementations, in terms of both computing times to reach a given accuracy level and communication costs. I nevertheless have two major concerns with the proposed parallelization.

In the graph-based semi-supervised learning, the Green-function method is a classical method that works by computing the Green's function in the graph space. However, when applied to large graphs, especially those sparse ones, this method performs unstably and unsatisfactorily. We make a detailed analysis on it and propose a novel method from the perspective of optimization. On fully connected graphs, the method is equivalent to the Green-function method and can be seen as another interpretation with physical meanings, while on non-fully connected graphs, it helps to explain why the Green-function method causes a mess on large sparse graphs. To solve this dilemma, we propose a workable approach to improve our proposed method. Unlike the original method, our improved method can also apply two accelerating techniques, Gaussian Elimination, and Anchored Graphs to become more efficient on large graphs. Finally, the extensive experiments prove our conclusions and the efficiency, accuracy, and stability of our improved Green's function method.

The recent developments of adiabatic quantum machine learning (AQML) methods and applications based on the quadratic unconstrained binary optimization (QUBO) model have received attention from academics and practitioners. Traditional machine learning methods such as support vector machines, balanced k-means clustering, linear regression, Decision Tree Splitting, Restricted Boltzmann Machines, and Deep Belief Networks can be transformed into a QUBO model. The training of adiabatic quantum machine learning models is the bottleneck for computation. Heuristics-based quantum annealing solvers such as Simulated Annealing and Multiple Start Tabu Search (MSTS) are implemented to speed up the training of AQML based on the QUBO model. The main purpose of this paper is to present a hybrid heuristic embedding an r-flip strategy to solve large-scale QUBO with an improved solution and shorter computing time compared to the state-of-the-art MSTS method. The results of the substantial computational experiments are reported to compare an r-flip strategy embedded hybrid heuristic and a multiple start tabu search algorithm on a set of benchmark instances and three large-scale QUBO instances. The r-flip strategy embedded algorithm provides very high-quality solutions within the CPU time limits of 60 and 600 seconds.