As a regression technique in spatial statistics, the spatiotemporally varying coefficient model (STVC) is an important tool for discovering nonstationary and interpretable response-covariate associations over both space and time. However, it is difficult to apply STVC for large-scale spatiotemporal analyses due to the high computational cost. To address this challenge, we summarize the spatiotemporally varying coefficients using a third-order tensor structure and propose to reformulate the spatiotemporally varying coefficient model as a special low-rank tensor regression problem. The low-rank decomposition can effectively model the global patterns of the large data sets with a substantially reduced number of parameters. To further incorporate the local spatiotemporal dependencies, we use Gaussian process (GP) priors on the spatial and temporal factor matrices. We refer to the overall framework as Bayesian Kernelized Tensor Regression (BKTR). For model inference, we develop an efficient Markov chain Monte Carlo (MCMC) algorithm, which uses Gibbs sampling to update factor matrices and slice sampling to update kernel hyperparameters. We conduct extensive experiments on both synthetic and real-world data sets, and our results confirm the superior performance and efficiency of BKTR for model estimation and parameter inference.
It is well-known that classical optical cavities can exhibit localized phenomena associated to scattering resonances (using the Black Box Scattering Theory), leading to numerical instabilities in approximating the solution. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical propertie(s), for example made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and extending the Black Box Scattering Theory to problems with sign-changing coefficient. Our asymptotic analysis reveals that, depending on the metamaterial's properties, scattering resonances situated closed to the real axis are associated to surface plasmons. Examples for several metamaterial cavities are provided.
Video super-resolution (VSR) aims to restore a sequence of high-resolution (HR) frames from their low-resolution (LR) counterparts. Although some progress has been made, there are grand challenges to effectively utilize temporal dependency in entire video sequences. Existing approaches usually align and aggregate video frames from limited adjacent frames (e.g., 5 or 7 frames), which prevents these approaches from satisfactory results. In this paper, we take one step further to enable effective spatio-temporal learning in videos. We propose a novel Trajectory-aware Transformer for Video Super-Resolution (TTVSR). In particular, we formulate video frames into several pre-aligned trajectories which consist of continuous visual tokens. For a query token, self-attention is only learned on relevant visual tokens along spatio-temporal trajectories. Compared with vanilla vision Transformers, such a design significantly reduces the computational cost and enables Transformers to model long-range features. We further propose a cross-scale feature tokenization module to overcome scale-changing problems that often occur in long-range videos. Experimental results demonstrate the superiority of the proposed TTVSR over state-of-the-art models, by extensive quantitative and qualitative evaluations in four widely-used video super-resolution benchmarks. Both code and pre-trained models can be downloaded at //github.com/researchmm/TTVSR.
Super-Resolution is the technique to improve the quality of a low-resolution photo by boosting its plausible resolution. The computer vision community has extensively explored the area of Super-Resolution. However, previous Super-Resolution methods require vast amounts of data for training which becomes problematic in domains where very few low-resolution, high-resolution pairs might be available. One such area is statistical downscaling, where super-resolution is increasingly being used to obtain high-resolution climate information from low-resolution data. Acquiring high-resolution climate data is extremely expensive and challenging. To reduce the cost of generating high-resolution climate information, Super-Resolution algorithms should be able to train with a limited number of low-resolution, high-resolution pairs. This paper tries to solve the aforementioned problem by introducing a semi-supervised way to perform super-resolution that can generate sharp, high-resolution images with as few as 500 paired examples. The proposed semi-supervised technique can be used as a plug-and-play module with any supervised GAN-based Super-Resolution method to enhance its performance. We quantitatively and qualitatively analyze the performance of the proposed model and compare it with completely supervised methods as well as other unsupervised techniques. Comprehensive evaluations show the superiority of our method over other methods on different metrics. We also offer the applicability of our approach in statistical downscaling to obtain high-resolution climate images.
Many recent state-of-the-art (SOTA) optical flow models use finite-step recurrent update operations to emulate traditional algorithms by encouraging iterative refinements toward a stable flow estimation. However, these RNNs impose large computation and memory overheads, and are not directly trained to model such stable estimation. They can converge poorly and thereby suffer from performance degradation. To combat these drawbacks, we propose deep equilibrium (DEQ) flow estimators, an approach that directly solves for the flow as the infinite-level fixed point of an implicit layer (using any black-box solver), and differentiates through this fixed point analytically (thus requiring $O(1)$ training memory). This implicit-depth approach is not predicated on any specific model, and thus can be applied to a wide range of SOTA flow estimation model designs. The use of these DEQ flow estimators allows us to compute the flow faster using, e.g., fixed-point reuse and inexact gradients, consumes $4\sim6\times$ times less training memory than the recurrent counterpart, and achieves better results with the same computation budget. In addition, we propose a novel, sparse fixed-point correction scheme to stabilize our DEQ flow estimators, which addresses a longstanding challenge for DEQ models in general. We test our approach in various realistic settings and show that it improves SOTA methods on Sintel and KITTI datasets with substantially better computational and memory efficiency.
We investigate the feature compression of high-dimensional ridge regression using the optimal subsampling technique. Specifically, based on the basic framework of random sampling algorithm on feature for ridge regression and the A-optimal design criterion, we first obtain a set of optimal subsampling probabilities. Considering that the obtained probabilities are uneconomical, we then propose the nearly optimal ones. With these probabilities, a two step iterative algorithm is established which has lower computational cost and higher accuracy. We provide theoretical analysis and numerical experiments to support the proposed methods. Numerical results demonstrate the decent performance of our methods.
Given two strings $T$ and $S$ and a set of strings $P$, for each string $p \in P$, consider the unique substrings of $T$ that have $p$ as their prefix and $S$ as their suffix. Two problems then come to mind; the first problem being the counting of such substrings, and the second problem being the problem of listing all such substrings. In this paper, we describe linear-time, linear-space suffix tree-based algorithms for both problems. More specifically, we describe an $O(|T| + |P|)$ time algorithm for the counting problem, and an $O(|T| + |P| + \#(ans))$ time algorithm for the listing problem, where $\#(ans)$ refers to the number of strings being listed in total, and $|P|$ refers to the total length of the strings in $P$. We also consider the reversed version of the problems, where one prefix condition string and multiple suffix condition strings are given instead, and similarly describe linear-time, linear-space algorithms to solve them.
The four-parameter generalized beta distribution of the second kind (GBII) has been proposed for modelling insurance losses with heavy-tailed features. The aim of this paper is to present a parametric composite GBII regression modelling by splicing two GBII distributions using mode matching method. It is designed for simultaneous modeling of small and large claims and capturing the policyholder heterogeneity by introducing the covariates into the location parameter. In such cases, the threshold that splits two GBII distributions varies across individuals policyholders based on their risk features. The proposed regression modelling also contains a wide range of insurance loss distributions as the head and the tail respectively and provides the close-formed expressions for parameter estimation and model prediction. A simulation study is conducted to show the accuracy of the proposed estimation method and the flexibility of the regressions. Some illustrations of the applicability of the new class of distributions and regressions are provided with a Danish fire losses data set and a Chinese medical insurance claims data set, comparing with the results of competing models from the literature.
We study efficient estimation of an interventional mean associated with a point exposure treatment under a causal graphical model represented by a directed acyclic graph without hidden variables. Under such a model, it may happen that a subset of the variables are uninformative in that failure to measure them neither precludes identification of the interventional mean nor changes the semiparametric variance bound for regular estimators of it. We develop a set of graphical criteria that are sound and complete for eliminating all the uninformative variables so that the cost of measuring them can be saved without sacrificing estimation efficiency, which could be useful when designing a planned observational or randomized study. Further, we construct a reduced directed acyclic graph on the set of informative variables only. We show that the interventional mean is identified from the marginal law by the g-formula (Robins, 1986) associated with the reduced graph, and the semiparametric variance bounds for estimating the interventional mean under the original and the reduced graphical model agree. This g-formula is an irreducible, efficient identifying formula in the sense that the nonparametric estimator of the formula, under regularity conditions, is asymptotically efficient under the original causal graphical model, and no formula with such property exists that only depends on a strict subset of the variables.
A High-dimensional and sparse (HiDS) matrix is frequently encountered in a big data-related application like an e-commerce system or a social network services system. To perform highly accurate representation learning on it is of great significance owing to the great desire of extracting latent knowledge and patterns from it. Latent factor analysis (LFA), which represents an HiDS matrix by learning the low-rank embeddings based on its observed entries only, is one of the most effective and efficient approaches to this issue. However, most existing LFA-based models perform such embeddings on a HiDS matrix directly without exploiting its hidden graph structures, thereby resulting in accuracy loss. To address this issue, this paper proposes a graph-incorporated latent factor analysis (GLFA) model. It adopts two-fold ideas: 1) a graph is constructed for identifying the hidden high-order interaction (HOI) among nodes described by an HiDS matrix, and 2) a recurrent LFA structure is carefully designed with the incorporation of HOI, thereby improving the representa-tion learning ability of a resultant model. Experimental results on three real-world datasets demonstrate that GLFA outperforms six state-of-the-art models in predicting the missing data of an HiDS matrix, which evidently supports its strong representation learning ability to HiDS data.
Bayesian model selection provides a powerful framework for objectively comparing models directly from observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (model evidence), which is computationally challenging, prohibiting its use in many high-dimensional Bayesian inverse problems. With Bayesian imaging applications in mind, in this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models for applications that use images to inform decisions under uncertainty. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving l_1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices of dictionary and measurement model.