Determinantal point processes (a.k.a. DPPs) have recently become popular tools for modeling the phenomenon of negative dependence, or repulsion, in data. However, our understanding of an analogue of a classical parametric statistical theory is rather limited for this class of models. In this work, we investigate a parametric family of Gaussian DPPs with a clearly interpretable effect of parametric modulation on the observed points. We show that parameter modulation impacts the observed points by introducing directionality in their repulsion structure, and the principal directions correspond to the directions of maximal (i.e. the most long ranged) dependency. This model readily yields a novel and viable alternative to Principal Component Analysis (PCA) as a dimension reduction tool that favors directions along which the data is most spread out. This methodological contribution is complemented by a statistical analysis of a spiked model similar to that employed for covariance matrices as a framework to study PCA. These theoretical investigations unveil intriguing questions for further examination in random matrix theory, stochastic geometry and related topics.
相關內容
The U.S. Bureau of Labor Statistics allows public access to much of the data acquired through its Occupational Requirements Survey (ORS). This data can be used to draw inferences about the requirements of various jobs and job classes within the United States workforce. However, the dataset contains a multitude of missing observations and estimates, which somewhat limits its utility. Here, we propose a method by which to impute these missing values that leverages many of the inherent features present in the survey data, such as known population limit and correlations between occupations and tasks. An iterative regression fit, implemented with a recent version of XGBoost and executed across a set of simulated values drawn from the distribution described by the known values and their standard deviations reported in the survey, is the approach used to arrive at a distribution of predicted values for each missing estimate. This allows us to calculate a mean prediction and bound said estimate with a 95% confidence interval. We discuss the use of our method and how the resulting imputations can be utilized to inform and pursue future areas of study stemming from the data collected in the ORS. Finally, we conclude with an outline of WIGEM, a generalized version of our weighted, iterative imputation algorithm that could be applied to other contexts.
Stochastic epidemic models provide an interpretable probabilistic description of the spread of a disease through a population. Yet, fitting these models when the epidemic process is only partially observed is a notoriously difficult task due to the intractability of the likelihood for many classical models. To remedy this issue, this article introduces a novel data-augmented MCMC algorithm for fast and exact Bayesian inference for the stochastic SIR model given discretely observed infection incidence counts. In a Metropolis-Hastings step, new event times of the latent data are jointly proposed from a surrogate process that closely resembles the SIR, and from which we can efficiently generate epidemics compatible with the observed data. The proposed DA-MCMC algorithm is fast and, since the latent data are generated from a faithful approximation of the target model, a large portion thereof can be updated per iteration without prohibitively lowering the acceptance rate. We find that the method explores the high-dimensional latent space efficiently and scales to outbreaks with hundreds of thousands of individuals, and we show that the Markov chain underlying the algorithm is uniformly ergodic. We validate its performance via thorough simulation experiments and a case study on the 2013-2015 Ebola outbreak in Western Africa.
Structural matrix-variate observations routinely arise in diverse fields such as multi-layer network analysis and brain image clustering. While data of this type have been extensively investigated with fruitful outcomes being delivered, the fundamental questions like its statistical optimality and computational limit are largely under-explored. In this paper, we propose a low-rank Gaussian mixture model (LrMM) assuming each matrix-valued observation has a planted low-rank structure. Minimax lower bounds for estimating the underlying low-rank matrix are established allowing a whole range of sample sizes and signal strength. Under a minimal condition on signal strength, referred to as the information-theoretical limit or statistical limit, we prove the minimax optimality of a maximum likelihood estimator which, in general, is computationally infeasible. If the signal is stronger than a certain threshold, called the computational limit, we design a computationally fast estimator based on spectral aggregation and demonstrate its minimax optimality. Moreover, when the signal strength is smaller than the computational limit, we provide evidences based on the low-degree likelihood ratio framework to claim that no polynomial-time algorithm can consistently recover the underlying low-rank matrix. Our results reveal multiple phase transitions in the minimax error rates and the statistical-to-computational gap. Numerical experiments confirm our theoretical findings. We further showcase the merit of our spectral aggregation method on the worldwide food trading dataset.
The Stochastic Volatility (SV) model and its variants are widely used in the financial sector while recurrent neural network (RNN) models are successfully used in many large-scale industrial applications of Deep Learning. Our article combines these two methods in a non-trivial way and proposes a model, which we call the Statistical Recurrent Stochastic Volatility (SR-SV) model, to capture the dynamics of stochastic volatility. The proposed model is able to capture complex volatility effects (e.g., non-linearity and long-memory auto-dependence) overlooked by the conventional SV models, is statistically interpretable and has an impressive out-of-sample forecast performance. These properties are carefully discussed and illustrated through extensive simulation studies and applications to five international stock index datasets: The German stock index DAX30, the Hong Kong stock index HSI50, the France market index CAC40, the US stock market index SP500 and the Canada market index TSX250. An user-friendly software package together with the examples reported in the paper are available at \url{//github.com/vbayeslab}.
Continuous determinantal point processes (DPPs) are a class of repulsive point processes on $\mathbb{R}^d$ with many statistical applications. Although an explicit expression of their density is known, it is too complicated to be used directly for maximum likelihood estimation. In the stationary case, an approximation using Fourier series has been suggested, but it is limited to rectangular observation windows and no theoretical results support it. In this contribution, we investigate a different way to approximate the likelihood by looking at its asymptotic behaviour when the observation window grows towards $\mathbb{R}^d$. This new approximation is not limited to rectangular windows, is faster to compute than the previous one, does not require any tuning parameter, and some theoretical justifications are provided. It moreover provides an explicit formula for estimating the asymptotic variance of the associated estimator. The performances are assessed in a simulation study on standard parametric models on $\mathbb{R}^d$ and compare favourably to common alternative estimation methods for continuous DPPs.
We give a review of recent ANOVA-like procedures for testing group differences based on data in a metric space and present a new such procedure. Our statistic is based on the classic Levene's test for detecting differences in dispersion. It uses only pairwise distances of data points and and can be computed quickly and precisely in situations where the computation of barycenters ("generalized means") in the data space is slow, only by approximation or even infeasible. We show the asymptotic normality of our test statistic and present simulation studies for spatial point pattern data, in which we compare the various procedures in a 1-way ANOVA setting. As an application, we perform a 2-way ANOVA on a data set of bubbles in a mineral flotation process.
In health-pollution cohort studies, accurate predictions of pollutant concentrations at new locations are needed, since the locations of fixed monitoring sites and study participants are often spatially misaligned. For multi-pollution data, principal component analysis (PCA) is often incorporated to obtain low-rank (LR) structure of the data prior to spatial prediction. Recently developed predictive PCA modifies the traditional algorithm to improve the overall predictive performance by leveraging both LR and spatial structures within the data. However, predictive PCA requires complete data or an initial imputation step. Nonparametric imputation techniques without accounting for spatial information may distort the underlying structure of the data, and thus further reduce the predictive performance. We propose a convex optimization problem inspired by the LR matrix completion framework and develop a proximal algorithm to solve it. Missing data are imputed and handled concurrently within the algorithm, which eliminates the necessity of a separate imputation step. We show that our algorithm has low computational burden and leads to reliable predictive performance as the severity of missing data increases.
A determinantal point process (DPP) on a collection of $M$ items is a model, parameterized by a symmetric kernel matrix, that assigns a probability to every subset of those items. Recent work shows that removing the kernel symmetry constraint, yielding nonsymmetric DPPs (NDPPs), can lead to significant predictive performance gains for machine learning applications. However, existing work leaves open the question of scalable NDPP sampling. There is only one known DPP sampling algorithm, based on Cholesky decomposition, that can directly apply to NDPPs as well. Unfortunately, its runtime is cubic in $M$, and thus does not scale to large item collections. In this work, we first note that this algorithm can be transformed into a linear-time one for kernels with low-rank structure. Furthermore, we develop a scalable sublinear-time rejection sampling algorithm by constructing a novel proposal distribution. Additionally, we show that imposing certain structural constraints on the NDPP kernel enables us to bound the rejection rate in a way that depends only on the kernel rank. In our experiments we compare the speed of all of these samplers for a variety of real-world tasks.
Humans and animals show remarkable flexibility in adjusting their behaviour when their goals, or rewards in the environment change. While such flexibility is a hallmark of intelligent behaviour, these multi-task scenarios remain an important challenge for machine learning algorithms and neurobiological models alike. Factored representations can enable flexible behaviour by abstracting away general aspects of a task from those prone to change, while nonparametric methods provide a principled way of using similarity to past experiences to guide current behaviour. Here we combine the successor representation (SR), that factors the value of actions into expected outcomes and corresponding rewards, with evaluating task similarity through nonparametric inference and clustering the space of rewards. The proposed algorithm improves SR's transfer capabilities by inverting a generative model over tasks, while also explaining important neurobiological signatures of place cell representation in the hippocampus. It dynamically samples from a flexible number of distinct SR maps while accumulating evidence about the current reward context, and outperforms competing algorithms in settings with both known and unsignalled rewards changes. It reproduces the "flickering" behaviour of hippocampal maps seen when rodents navigate to changing reward locations, and gives a quantitative account of trajectory-dependent hippocampal representations (so-called splitter cells) and their dynamics. We thus provide a novel algorithmic approach for multi-task learning, as well as a common normative framework that links together these different characteristics of the brain's spatial representation.
Precise user and item embedding learning is the key to building a successful recommender system. Traditionally, Collaborative Filtering(CF) provides a way to learn user and item embeddings from the user-item interaction history. However, the performance is limited due to the sparseness of user behavior data. With the emergence of online social networks, social recommender systems have been proposed to utilize each user's local neighbors' preferences to alleviate the data sparsity for better user embedding modeling. We argue that, for each user of a social platform, her potential embedding is influenced by her trusted users. As social influence recursively propagates and diffuses in the social network, each user's interests change in the recursive process. Nevertheless, the current social recommendation models simply developed static models by leveraging the local neighbors of each user without simulating the recursive diffusion in the global social network, leading to suboptimal recommendation performance. In this paper, we propose a deep influence propagation model to stimulate how users are influenced by the recursive social diffusion process for social recommendation. For each user, the diffusion process starts with an initial embedding that fuses the related features and a free user latent vector that captures the latent behavior preference. The key idea of our proposed model is that we design a layer-wise influence propagation structure to model how users' latent embeddings evolve as the social diffusion process continues. We further show that our proposed model is general and could be applied when the user~(item) attributes or the social network structure is not available. Finally, extensive experimental results on two real-world datasets clearly show the effectiveness of our proposed model, with more than 13% performance improvements over the best baselines.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191