Due to the curse of dimensionality, it is often prohibitively expensive to generate deterministic space-filling designs. On the other hand, when using na{\"i}ve uniform random sampling to generate designs cheaply, design points tend to concentrate in a small region of the design space. Although, it is preferable in these cases to utilize quasi-random techniques such as Sobol sequences and Latin hypercube designs over uniform random sampling in many settings, these methods have their own caveats especially in high-dimensional spaces. In this paper, we propose a technique that addresses the fundamental issue of measure concentration by updating high-dimensional distribution functions to produce better space-filling designs. Then, we show that our technique can outperform Latin hypercube sampling and Sobol sequences by the discrepancy metric while generating moderately-sized space-filling samples for high-dimensional problems.
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The theory of mixed finite element methods for solving different types of elliptic partial differential equations in saddle-point formulation is well established since many decades. However, this topic was mostly studied for variational formulations defined upon the same finite-element product spaces of both shape- and test-pairs of primal variable-multiplier. Whenever these two product spaces are different the saddle point problem is asymmetric. It turns out that the conditions to be satisfied by the finite elements product spaces stipulated in the few works on this case may be of limited use in practice. The purpose of this paper is to provide an in-depth analysis of the well-posedness and the uniform stability of asymmetric approximate saddle point problems, based on the theory of continuous linear operators on Hilbert spaces. Our approach leads to necessary and sufficient conditions for such properties to hold, expressed in a readily exploitable form with fine constants. In particular standard interpolation theory suffices to estimate the error of a conforming method.
We observe n possibly dependent random variables, the distribution of which is presumed to be stationary even though this might not be true, and we aim at estimating the stationary distribution. We establish a non-asymptotic deviation bound for the Hellinger distance between the target distribution and our estimator. If the dependence within the observations is small, the estimator performs as good as if the data were independent and identically distributed. In addition our estimator is robust to misspecification and contamination. If the dependence is too high but the observed process is mixing, we can select a subset of observations that is almost independent and retrieve results similar to what we have in the i.i.d. case. We apply our procedure to the estimation of the invariant distribution of a diffusion process and to finite state space hidden Markov models.
We define data transformations that leave certain classes of distributions invariant, while acting in a specific manner upon the parameters of the said distributions. It is shown that under such transformations the maximum likelihood estimators behave in exactly the same way as the parameters being estimated. As a consequence goodness--of--fit tests based on standardized data obtained through the inverse of this invariant data--transformation reduce to the case of testing a standard member of the family with fixed parameter values. While presenting our results, we also provide a selective review of the subject of equivariant estimators always in connection to invariant goodness--of--fit tests. A small Monte Carlo study is presented for the special case of testing for the Weibull distribution, along with real--data illustrations.
Visible-infrared person re-identification seeks to retrieve images of the same individual captured over a distributed network of RGB and IR sensors. Several V-I ReID approaches directly integrate both V and I modalities to discriminate persons within a shared representation space. However, given the significant gap in data distributions between V and I modalities, cross-modal V-I ReID remains challenging. Some recent approaches improve generalization by leveraging intermediate spaces that can bridge V and I modalities, yet effective methods are required to select or generate data for such informative domains. In this paper, the Adaptive Generation of Privileged Intermediate Information training approach is introduced to adapt and generate a virtual domain that bridges discriminant information between the V and I modalities. The key motivation behind AGPI^2 is to enhance the training of a deep V-I ReID backbone by generating privileged images that provide additional information. These privileged images capture shared discriminative features that are not easily accessible within the original V or I modalities alone. Towards this goal, a non-linear generative module is trained with an adversarial objective, translating V images into intermediate spaces with a smaller domain shift w.r.t. the I domain. Meanwhile, the embedding module within AGPI^2 aims to produce similar features for both V and generated images, encouraging the extraction of features that are common to all modalities. In addition to these contributions, AGPI^2 employs adversarial objectives for adapting the intermediate images, which play a crucial role in creating a non-modality-specific space to address the large domain shifts between V and I domains. Experimental results conducted on challenging V-I ReID datasets indicate that AGPI^2 increases matching accuracy without extra computational resources during inference.
Many stochastic processes in the physical and biological sciences can be modelled using Brownian dynamics with multiplicative noise. However, numerical integrators for these processes can lose accuracy or even fail to converge when the diffusion term is configuration-dependent. One remedy is to construct a transform to a constant-diffusion process and sample the transformed process instead. In this work, we explain how coordinate-based and time-rescaling-based transforms can be used either individually or in combination to map a general class of variable-diffusion Brownian motion processes into constant-diffusion ones. The transforms are invertible, thus allowing recovery of the original dynamics. We motivate our methodology using examples in one dimension before then considering multivariate diffusion processes. We illustrate the benefits of the transforms through numerical simulations, demonstrating how the right combination of integrator and transform can improve computational efficiency and the order of convergence to the invariant distribution. Notably, the transforms that we derive are applicable to a class of multibody, anisotropic Stokes-Einstein diffusion that has applications in biophysical modelling.
Robots are notoriously difficult to design because of complex interdependencies between their physical structure, sensory and motor layouts, and behavior. Despite this, almost every detail of every robot built to date has been manually determined by a human designer after several months or years of iterative ideation, prototyping, and testing. Inspired by evolutionary design in nature, the automated design of robots using evolutionary algorithms has been attempted for two decades, but it too remains inefficient: days of supercomputing are required to design robots in simulation that, when manufactured, exhibit desired behavior. Here we show for the first time de-novo optimization of a robot's structure to exhibit a desired behavior, within seconds on a single consumer-grade computer, and the manufactured robot's retention of that behavior. Unlike other gradient-based robot design methods, this algorithm does not presuppose any particular anatomical form; starting instead from a randomly-generated apodous body plan, it consistently discovers legged locomotion, the most efficient known form of terrestrial movement. If combined with automated fabrication and scaled up to more challenging tasks, this advance promises near instantaneous design, manufacture, and deployment of unique and useful machines for medical, environmental, vehicular, and space-based tasks.
Generative models can be categorized into two types: explicit generative models that define explicit density forms and allow exact likelihood inference, such as score-based diffusion models (SDMs) and normalizing flows; implicit generative models that directly learn a transformation from the prior to the data distribution, such as generative adversarial nets (GANs). While these two types of models have shown great success, they suffer from respective limitations that hinder them from achieving fast sampling and high sample quality simultaneously. In this paper, we propose a unified theoretic framework for SDMs and GANs. We shown that: i) the learning dynamics of both SDMs and GANs can be described as a novel SDE named Discriminator Denoising Diffusion Flow (DiffFlow) where the drift can be determined by some weighted combinations of scores of the real data and the generated data; ii) By adjusting the relative weights between different score terms, we can obtain a smooth transition between SDMs and GANs while the marginal distribution of the SDE remains invariant to the change of the weights; iii) we prove the asymptotic optimality and maximal likelihood training scheme of the DiffFlow dynamics; iv) under our unified theoretic framework, we introduce several instantiations of the DiffFLow that provide new algorithms beyond GANs and SDMs with exact likelihood inference and have potential to achieve flexible trade-off between high sample quality and fast sampling speed.
This paper examines the approximation of log-determinant for large-scale symmetric positive definite matrices. Inspired by the variance reduction technique, we split the approximation of $\log\det(A)$ into two parts. The first to compute is the trace of the projection of $\log(A)$ onto a suboptimal subspace, while the second is the trace of the projection on the corresponding orthogonal complementary space. For these two approximations, the stochastic Lanczos quadrature method is used. Furthermore, in the construction of the suboptimal subspace, we utilize a projection-cost-preserving sketch to bound the size of the Gaussian random matrix and the dimension of the suboptimal subspace. We provide a rigorous error analysis for our proposed method and explicit lower bounds for its design parameters, offering guidance for practitioners. We conduct numerical experiments to demonstrate our method's effectiveness and illustrate the quality of the derived bounds.
The multivariate adaptive regression spline (MARS) is one of the popular estimation methods for nonparametric multivariate regressions. However, as MARS is based on marginal splines, to incorporate interactions of covariates, products of the marginal splines must be used, which leads to an unmanageable number of basis functions when the order of interaction is high and results in low estimation efficiency. In this paper, we improve the performance of MARS by using linear combinations of the covariates which achieve sufficient dimension reduction. The special basis functions of MARS facilitate calculation of gradients of the regression function, and estimation of the linear combinations is obtained via eigen-analysis of the outer-product of the gradients. Under some technical conditions, the asymptotic theory is established for the proposed estimation method. Numerical studies including both simulation and empirical applications show its effectiveness in dimension reduction and improvement over MARS and other commonly-used nonparametric methods in regression estimation and prediction.
Mutual coherence is a measure of similarity between two opinions. Although the notion comes from philosophy, it is essential for a wide range of technologies, e.g., the Wahl-O-Mat system. In Germany, this system helps voters to find candidates that are the closest to their political preferences. The exact computation of mutual coherence is highly time-consuming due to the iteration over all subsets of an opinion. Moreover, for every subset, an instance of the SAT model counting problem has to be solved which is known to be a hard problem in computer science. This work is the first study to accelerate this computation. We model the distribution of the so-called confirmation values as a mixture of three Gaussians and present efficient heuristics to estimate its model parameters. The mutual coherence is then approximated with the expected value of the distribution. Some of the presented algorithms are fully polynomial-time, others only require solving a small number of instances of the SAT model counting problem. The average squared error of our best algorithm lies below 0.0035 which is insignificant if the efficiency is taken into account. Furthermore, the accuracy is precise enough to be used in Wahl-O-Mat-like systems.
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