We consider the task of filtering a dynamic parameter evolving as a diffusion process, given data collected at discrete times from a likelihood which is conjugate to the marginal law of the diffusion, when a generic dual process on a discrete state space is available. Recently, it was shown that duality with respect to a death-like process implies that the filtering distributions are finite mixtures, making exact filtering and smoothing feasible through recursive algorithms with polynomial complexity in the number of observations. Here we provide general results for the case of duality between the diffusion and a regular jump continuous-time Markov chain on a discrete state space, which typically leads to filtering distribution given by countable mixtures indexed by the dual process state space. We investigate the performance of several approximation strategies on two hidden Markov models driven by Cox-Ingersoll-Ross and Wright-Fisher diffusions, which admit duals of birth-and-death type, and compare them with the available exact strategies based on death-type duals and with bootstrap particle filtering on the diffusion state space as a general benchmark.
相關內容
Numerical simulations of kinetic problems can become prohibitively expensive due to their large memory footprint and computational costs. A method that has proven to successfully reduce these costs is the dynamical low-rank approximation (DLRA). One key question when using DLRA methods is the construction of robust time integrators that preserve the invariances and associated conservation laws of the original problem. In this work, we demonstrate that the augmented basis update & Galerkin integrator (BUG) preserves solution invariances and the associated conservation laws when using a conservative truncation step and an appropriate time and space discretization. We present numerical comparisons to existing conservative integrators and discuss advantages and disadvantages
Approximate message passing (AMP) is a family of iterative algorithms that generalize matrix power iteration. AMP algorithms are known to optimally solve many average-case optimization problems. In this paper, we show that a large class of AMP algorithms can be simulated in polynomial time by \emph{local statistics hierarchy} semidefinite programs (SDPs), even when an unknown principal minor of measure $1/\mathrm{polylog}(\mathrm{dimension})$ is adversarially corrupted. Ours are the first robust guarantees for many of these problems. Further, our results offer an interesting counterpoint to strong lower bounds against less constrained SDP relaxations for average-case max-cut-gain (a.k.a. "optimizing the Sherrington-Kirkpatrick Hamiltonian") and other problems.
Semantic segmentation is a complex task that relies heavily on large amounts of annotated image data. However, annotating such data can be time-consuming and resource-intensive, especially in the medical domain. Active Learning (AL) is a popular approach that can help to reduce this burden by iteratively selecting images for annotation to improve the model performance. In the case of video data, it is important to consider the model uncertainty and the temporal nature of the sequences when selecting images for annotation. This work proposes a novel AL strategy for surgery video segmentation, \COALSamp{}, COrrelation-aWare Active Learning. Our approach involves projecting images into a latent space that has been fine-tuned using contrastive learning and then selecting a fixed number of representative images from local clusters of video frames. We demonstrate the effectiveness of this approach on two video datasets of surgical instruments and three real-world video datasets. The datasets and code will be made publicly available upon receiving necessary approvals.
This work presents an abstract framework for the design, implementation, and analysis of the multiscale spectral generalized finite element method (MS-GFEM), a particular numerical multiscale method originally proposed in [I. Babuska and R. Lipton, Multiscale Model.\;\,Simul., 9 (2011), pp.~373--406]. MS-GFEM is a partition of unity method employing optimal local approximation spaces constructed from local spectral problems. We establish a general local approximation theory demonstrating exponential convergence with respect to local degrees of freedom under certain assumptions, with explicit dependence on key problem parameters. Our framework applies to a broad class of multiscale PDEs with $L^{\infty}$-coefficients in both continuous and discrete, finite element settings, including highly indefinite problems (convection-dominated diffusion, as well as the high-frequency Helmholtz, Maxwell and elastic wave equations with impedance boundary conditions), and higher-order problems. Notably, we prove a local convergence rate of $O(e^{-cn^{1/d}})$ for MS-GFEM for all these problems, improving upon the $O(e^{-cn^{1/(d+1)}})$ rate shown by Babuska and Lipton. Moreover, based on the abstract local approximation theory for MS-GFEM, we establish a unified framework for showing low-rank approximations to multiscale PDEs. This framework applies to the aforementioned problems, proving that the associated Green's functions admit an $O(|\log\epsilon|^{d})$-term separable approximation on well-separated domains with error $\epsilon>0$. Our analysis improves and generalizes the result in [M. Bebendorf and W. Hackbusch, Numerische Mathematik, 95 (2003), pp.~1-28] where an $O(|\log\epsilon|^{d+1})$-term separable approximation was proved for Poisson-type problems.
Factor models have been widely used to summarize the variability of high-dimensional data through a set of factors with much lower dimensionality. Gaussian linear factor models have been particularly popular due to their interpretability and ease of computation. However, in practice, data often violate the multivariate Gaussian assumption. To characterize higher-order dependence and nonlinearity, models that include factors as predictors in flexible multivariate regression are popular, with GP-LVMs using Gaussian process (GP) priors for the regression function and VAEs using deep neural networks. Unfortunately, such approaches lack identifiability and interpretability and tend to produce brittle and non-reproducible results. To address these problems by simplifying the nonparametric factor model while maintaining flexibility, we propose the NIFTY framework, which parsimoniously transforms uniform latent variables using one-dimensional nonlinear mappings and then applies a linear generative model. The induced multivariate distribution falls into a flexible class while maintaining simple computation and interpretation. We prove that this model is identifiable and empirically study NIFTY using simulated data, observing good performance in density estimation and data visualization. We then apply NIFTY to bird song data in an environmental monitoring application.
Many datasets suffer from missing values due to various reasons,which not only increases the processing difficulty of related tasks but also reduces the accuracy of classification. To address this problem, the mainstream approach is to use missing value imputation to complete the dataset. Existing imputation methods estimate the missing parts based on the observed values in the original feature space, and they treat all features as equally important during data completion, while in fact different features have different importance. Therefore, we have designed an imputation method that considers feature importance. This algorithm iteratively performs matrix completion and feature importance learning, and specifically, matrix completion is based on a filling loss that incorporates feature importance. Our experimental analysis involves three types of datasets: synthetic datasets with different noisy features and missing values, real-world datasets with artificially generated missing values, and real-world datasets originally containing missing values. The results on these datasets consistently show that the proposed method outperforms the existing five imputation algorithms.To the best of our knowledge, this is the first work that considers feature importance in the imputation model.
In spatial blind source separation the observed multivariate random fields are assumed to be mixtures of latent spatially dependent random fields. The objective is to recover latent random fields by estimating the unmixing transformation. Currently, the algorithms for spatial blind source separation can only estimate linear unmixing transformations. Nonlinear blind source separation methods for spatial data are scarce. In this paper we extend an identifiable variational autoencoder that can estimate nonlinear unmixing transformations to spatially dependent data and demonstrate its performance for both stationary and nonstationary spatial data using simulations. In addition, we introduce scaled mean absolute Shapley additive explanations for interpreting the latent components through nonlinear mixing transformation. The spatial identifiable variational autoencoder is applied to a geochemical dataset to find the latent random fields, which are then interpreted by using the scaled mean absolute Shapley additive explanations.
This study addresses the significant challenge of developing efficient decoding algorithms for classifying steady-state visual evoked potentials (SSVEPs) in scenarios characterized by extreme scarcity of calibration data, where only one calibration is available for each stimulus target. To tackle this problem, we introduce a novel cross-subject dual-domain fusion network (CSDuDoFN) incorporating task-related and task-discriminant component analysis (TRCA and TDCA) for one-shot SSVEP classification. The CSDuDoFN framework is designed to comprehensively transfer information from source subjects, while TRCA and TDCA are employed to exploit the single available calibration of the target subject. Specifically, we develop multi-reference least-squares transformation (MLST) to map data from both source subjects and the target subject into the domain of sine-cosine templates, thereby mitigating inter-individual variability and benefiting transfer learning. Subsequently, the transformed data in the sine-cosine templates domain and the original domain data are separately utilized to train a convolutional neural network (CNN) model, with the adequate fusion of their feature maps occurring at distinct network layers. To further capitalize on the calibration of the target subject, source aliasing matrix estimation (SAME) data augmentation is incorporated into the training process of the ensemble TRCA (eTRCA) and TDCA models. Ultimately, the outputs of the CSDuDoFN, eTRCA, and TDCA are combined for SSVEP classification. The effectiveness of our proposed approach is comprehensively evaluated on three publicly available SSVEP datasets, achieving the best performance on two datasets and competitive performance on one. This underscores the potential for integrating brain-computer interface (BCI) into daily life.
We propose a novel and simple spectral method based on the semi-discrete Fourier transforms to discretize the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$. Numerical analysis and experiments are provided to study its performance. Our method has the same symbol $|\xi|^\alpha$ as the fractional Laplacian $(-\Delta)^\frac{\alpha}{2}$ at the discrete level, and thus it can be viewed as the exact discrete analogue of the fractional Laplacian. This {\it unique feature} distinguishes our method from other existing methods for the fractional Laplacian. Note that our method is different from the Fourier pseudospectral methods in the literature, which are usually limited to periodic boundary conditions (see Remark \ref{remark0}). Numerical analysis shows that our method can achieve a spectral accuracy. The stability and convergence of our method in solving the fractional Poisson equations were analyzed. Our scheme yields a multilevel Toeplitz stiffness matrix, and thus fast algorithms can be developed for efficient matrix-vector products. The computational complexity is ${\mathcal O}(2N\log(2N))$, and the memory storage is ${\mathcal O}(N)$ with $N$ the total number of points. Extensive numerical experiments verify our analytical results and demonstrate the effectiveness of our method in solving various problems.
Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191