Factor models have been widely used in economics and finance. However, the heavy-tailed nature of macroeconomic and financial data is often neglected in the existing literature. To address this issue and achieve robustness, we propose an approach to estimate factor loadings and scores by minimizing the Huber loss function, which is motivated by the equivalence of conventional Principal Component Analysis (PCA) and the constrained least squares method in the factor model. We provide two algorithms that use different penalty forms. The first algorithm, which we refer to as Huber PCA, minimizes the $\ell_2$-norm-type Huber loss and performs PCA on the weighted sample covariance matrix. The second algorithm involves an element-wise type Huber loss minimization, which can be solved by an iterative Huber regression algorithm. Our study examines the theoretical minimizer of the element-wise Huber loss function and demonstrates that it has the same convergence rate as conventional PCA when the idiosyncratic errors have bounded second moments. We also derive their asymptotic distributions under mild conditions. Moreover, we suggest a consistent model selection criterion that relies on rank minimization to estimate the number of factors robustly. We showcase the benefits of Huber PCA through extensive numerical experiments and a real financial portfolio selection example. An R package named ``HDRFA" has been developed to implement the proposed robust factor analysis.
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Robots with the ability to balance time against the thoroughness of search have the potential to provide time-critical assistance in applications such as search and rescue. Current advances in ergodic coverage-based search methods have enabled robots to completely explore and search an area in a fixed amount of time. However, optimizing time against the quality of autonomous ergodic search has yet to be demonstrated. In this paper, we investigate solutions to the time-optimal ergodic search problem for fast and adaptive robotic search and exploration. We pose the problem as a minimum time problem with an ergodic inequality constraint whose upper bound regulates and balances the granularity of search against time. Solutions to the problem are presented analytically using Pontryagin's conditions of optimality and demonstrated numerically through a direct transcription optimization approach. We show the efficacy of the approach in generating time-optimal ergodic search trajectories in simulation and with drone experiments in a cluttered environment. Obstacle avoidance is shown to be readily integrated into our formulation, and we perform ablation studies that investigate parameter dependence on optimized time and trajectory sensitivity for search.
Sequential transfer optimization (STO), which aims to improve optimization performance by exploiting knowledge captured from previously-solved optimization tasks stored in a database, has been gaining increasing research attention in recent years. However, despite significant advancements in algorithm design, the test problems in STO are not well designed. Oftentimes, they are either randomly assembled by other benchmark functions that have identical optima or are generated from practical problems that exhibit limited variations. The relationships between the optimal solutions of source and target tasks in these problems are manually configured and thus monotonous, limiting their ability to represent the diverse relationships of real-world problems. Consequently, the promising results achieved by many algorithms on these problems are highly biased and difficult to be generalized to other problems. In light of this, we first introduce a few rudimentary concepts for characterizing STO problems (STOPs) and present an important problem feature overlooked in previous studies, namely similarity distribution, which quantitatively delineates the relationship between the optima of source and target tasks. Then, we propose general design guidelines and a problem generator with superior extendibility. Specifically, the similarity distribution of a problem can be systematically customized by modifying a parameterized density function, enabling a broad spectrum of representation for the diverse similarity relationships of real-world problems. Lastly, a benchmark suite with 12 individual STOPs is developed using the proposed generator, which can serve as an arena for comparing different STO algorithms. The source code of the benchmark suite is available at //github.com/XmingHsueh/STOP.
The majority of work in privacy-preserving federated learning (FL) has been focusing on horizontally partitioned datasets where clients share the same sets of features and can train complete models independently. However, in many interesting problems, such as financial fraud detection and disease detection, individual data points are scattered across different clients/organizations in vertical federated learning. Solutions for this type of FL require the exchange of gradients between participants and rarely consider privacy and security concerns, posing a potential risk of privacy leakage. In this work, we present a novel design for training vertical FL securely and efficiently using state-of-the-art security modules for secure aggregation. We demonstrate empirically that our method does not impact training performance whilst obtaining 9.1e2 ~3.8e4 speedup compared to homomorphic encryption (HE).
We consider problems of minimizing functionals $\mathcal{F}$ of probability measures on the Euclidean space. To propose an accelerated gradient descent algorithm for such problems, we consider gradient flow of transport maps that give push-forward measures of an initial measure. Then we propose a deterministic accelerated algorithm by extending Nesterov's acceleration technique with momentum. This algorithm do not based on the Wasserstein geometry. Furthermore, to estimate the convergence rate of the accelerated algorithm, we introduce new convexity and smoothness for $\mathcal{F}$ based on transport maps. As a result, we can show that the accelerated algorithm converges faster than a normal gradient descent algorithm. Numerical experiments support this theoretical result.
Most machine learning methods require tuning of hyper-parameters. For kernel ridge regression with the Gaussian kernel, the hyper-parameter is the bandwidth. The bandwidth specifies the length-scale of the kernel and has to be carefully selected in order to obtain a model with good generalization. The default methods for bandwidth selection is cross-validation and marginal likelihood maximization, which often yields good results, albeit at high computational costs. Furthermore, the estimates provided by these methods tend to have very high variance, especially when training data are scarce. Inspired by Jacobian regularization, we formulate an approximate expression for how the derivatives of the functions inferred by kernel ridge regression with the Gaussian kernel depend on the kernel bandwidth. We then use this expression to propose a closed-form, computationally feather-light, bandwidth selection heuristic based on controlling the Jacobian. In addition, the Jacobian expression illuminates how the bandwidth selection is a trade-off between the smoothness of the inferred function, and the conditioning of the training data kernel matrix. We show on real and synthetic data that compared to cross-validation and marginal likelihood maximization, our method is considerably faster and considerably more stable in terms of bandwidth selection.
The problem of packing equal spheres in a spherical container is a classic global optimization problem, which has attracted enormous studies in academia and found various applications in industry. This problem is computationally challenging, and many efforts focus on small-scale instances with the number of spherical items less than 200 in the literature. In this work, we propose an efficient local search heuristic algorithm named solution space exploring and descent for solving this problem, which can quantify the solution's quality to determine the number of exploring actions and quickly discover a high-quality solution. Besides, we propose an adaptive neighbor object maintenance method to speed up the convergence of the continuous optimization process and reduce the time consumption. Computational experiments on a large number of benchmark instances with $5 \leq n \leq 400$ spherical items show that our algorithm significantly outperforms the state-of-the-art algorithm. In particular, it improves the 274 best-known results and matches the 84 best-known results out of the 396 well-known benchmark instances.
With the growing availability of large-scale biomedical data, it is often time-consuming or infeasible to directly perform traditional statistical analysis with relatively limited computing resources at hand. We propose a fast subsampling method to effectively approximate the full data maximum partial likelihood estimator in Cox's model, which largely reduces the computational burden when analyzing massive survival data. We establish consistency and asymptotic normality of a general subsample-based estimator. The optimal subsampling probabilities with explicit expressions are determined via minimizing the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator. We propose a two-step subsampling algorithm for practical implementation, which has a significant reduction in computing time compared to the full data method. The asymptotic properties of the resulting two-step subsample-based estimator is also established. Extensive numerical experiments and a real-world example are provided to assess our subsampling strategy.
Integrated visible light positioning and communication (VLPC), capable of combining advantages of visible light communications (VLC) and visible light positioning (VLP), is a promising key technology for the future Internet of Things. In VLPC networks, positioning and communications are inherently coupled, which has not been sufficiently explored in the literature. We propose a robust power allocation scheme for integrated VLPC Networks by exploiting the intrinsic relationship between positioning and communications. Specifically, we derive explicit relationships between random positioning errors, following both a Gaussian distribution and an arbitrary distribution, and channel state information errors. Then, we minimize the Cramer-Rao lower bound (CRLB) of positioning errors, subject to the rate outage constraint and the power constraints, which is a chance-constrained optimization problem and generally computationally intractable. To circumvent the nonconvex challenge, we conservatively transform the chance constraints to deterministic forms by using the Bernstein-type inequality and the conditional value-at-risk for the Gaussian and arbitrary distributed positioning errors, respectively, and then approximate them as convex semidefinite programs. Finally, simulation results verify the robustness and effectiveness of our proposed integrated VLPC design schemes.
We present an $\ell^2_2+\ell_1$-regularized discrete least squares approximation over general regions under assumptions of hyperinterpolation, named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft thresholding operator and a filter function to shrink the Fourier coefficients approximated by a high-order quadrature rule of a given continuous function with respect to some orthonormal basis, is a combination of Lasso and filtered hyperinterpolations. Hybrid hyperinterpolation inherits features of them to deal with noisy data once the regularization parameter and the filter function are chosen well. We not only provide $L_2$ errors in theoretical analysis for hybrid hyperinterpolation to approximate continuous functions with noise and noise-free, but also decompose $L_2$ errors into three exact computed terms with the aid of a prior regularization parameter choices rule. This rule, making fully use of coefficients of hyperinterpolation to choose a regularization parameter, reveals that $L_2$ errors for hybrid hyperinterpolation sharply decline and then slowly increase when the sparsity of coefficients ranges from one to large values. Numerical examples show the enhanced performance of hybrid hyperinterpolation when regularization parameters and noise vary. Theoretical $L_2$ errors bounds are verified in numerical examples on the interval, the unit-disk, the unit-sphere and the unit-cube, the union of disks.
Generative adversarial networks (GANs) have been extensively studied in the past few years. Arguably their most significant impact has been in the area of computer vision where great advances have been made in challenges such as plausible image generation, image-to-image translation, facial attribute manipulation and similar domains. Despite the significant successes achieved to date, applying GANs to real-world problems still poses significant challenges, three of which we focus on here. These are: (1) the generation of high quality images, (2) diversity of image generation, and (3) stable training. Focusing on the degree to which popular GAN technologies have made progress against these challenges, we provide a detailed review of the state of the art in GAN-related research in the published scientific literature. We further structure this review through a convenient taxonomy we have adopted based on variations in GAN architectures and loss functions. While several reviews for GANs have been presented to date, none have considered the status of this field based on their progress towards addressing practical challenges relevant to computer vision. Accordingly, we review and critically discuss the most popular architecture-variant, and loss-variant GANs, for tackling these challenges. Our objective is to provide an overview as well as a critical analysis of the status of GAN research in terms of relevant progress towards important computer vision application requirements. As we do this we also discuss the most compelling applications in computer vision in which GANs have demonstrated considerable success along with some suggestions for future research directions. Code related to GAN-variants studied in this work is summarized on //github.com/sheqi/GAN_Review.
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