Daniel Khashabi, Mojtaba Ziyadi, Feng Liang
In this work we propose a heteroscedastic generalization to RVM, a fast Bayesian framework for regression, based on some recent similar works. We use variational approximation and expectation propagation to tackle the problem. The work is still under progress and we are examining the results and comparing with the previous works.
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Statistical modeling and inference problems with sample sizes substantially smaller than the number of available covariates are challenging. Chakraborty et al. (2012) did a full hierarchical Bayesian analysis of nonlinear regression in such situations using relevance vector machines based on reproducing kernel Hilbert space (RKHS). But they did not provide any theoretical properties associated with their procedure. The present paper revisits their problem, introduces a new class of global-local priors different from theirs, and provides results on posterior consistency as well as posterior con…
The Support Vector Machine (SVM) of Vapnik (1998) has become widely established as one of the leading approaches to pattern recognition and machine learning. It expresses predictions in terms of a linear combination of kernel functions centred on a subset of the training data, known as support vectors. Despite its widespread success, the SVM suffers from some important limitations, one of the most significant being that it makes point predictions rather than generating predictive distributions. Recently Tipping (1999) has formulated the Relevance Vector Machine (RVM), a probabilistic model who…
Relevance vector machine (RVM) is a popular sparse Bayesian learning model typically used for prediction. Recently it has been shown that improper priors assumed on multiple penalty parameters in RVM may lead to an improper posterior. Currently in the literature, the sufficient conditions for posterior propriety of RVM do not allow improper priors over the multiple penalty parameters. In this article, we propose a single penalty relevance vector machine (SPRVM) model in which multiple penalty parameters are replaced by a single penalty and we consider a semi Bayesian approach for fitting the S…
The problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional empirical Bayes estimators. By taking a different approach, that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is str…
Relevance Vector Machine (RVM) is a supervised learning algorithm extended from Support Vector Machine (SVM) based on the Bayesian sparsity model. Compared with the regression problem, RVM classification is difficult to be conducted because there is no closed-form solution for the weight parameter posterior. Original RVM classification algorithm used Newton's method in optimization to obtain the mode of weight parameter posterior then approximated it by a Gaussian distribution in Laplace's method. It would work but just applied the frequency methods in a Bayesian framework. This paper proposes…
Using a Bayesian approach, we consider the problem of recovering sparse signals under additive sparse and dense noise. Typically, sparse noise models outliers, impulse bursts or data loss. To handle sparse noise, existing methods simultaneously estimate the sparse signal of interest and the sparse noise of no interest. For estimating the sparse signal, without the need of estimating the sparse noise, we construct a robust Relevance Vector Machine (RVM). In the RVM, sparse noise and ever present dense noise are treated through a combined noise model. The precision of combined noise is modeled b…
A main goal of regression is to derive statistical conclusions on the conditional distribution of the output variable Y given the input values x. Two of the most important characteristics of a single distribution are location and scale. Support vector machines (SVMs) are well established to estimate location functions like the conditional median or the conditional mean. We investigate the estimation of scale functions by SVMs when the conditional median is unknown, too. Estimation of scale functions is important e.g. to estimate the volatility in finance. We consider the median absolute deviat…
Vector autoregressive (VAR) models assume linearity between the endogenous variables and their lags. This assumption might be overly restrictive and could have a deleterious impact on forecasting accuracy. As a solution, we propose combining VAR with Bayesian additive regression tree (BART) models. The resulting Bayesian additive vector autoregressive tree (BAVART) model is capable of capturing arbitrary non-linear relations between the endogenous variables and the covariates without much input from the researcher. Since controlling for heteroscedasticity is key for producing precise density f…
Bayesian algorithmic perfumery: A Hierarchical Relevance Vector Machine for the Estimation of Personalized Fragrance Preferences based on Three Sensory Layers and Jungian Personality Archetypes
本研究提出一种结合层级相关向量机与荣格人格原型的贝叶斯算法，用于个性化香氛推荐，通过三层香气偏好与人格特质关联，实现自适应可解释推荐。
We consider the problem of multivariate density deconvolution where the distribution of a random vector needs to be estimated from replicates contaminated with conditionally heteroscedastic measurement errors. We propose a conceptually straightforward yet fundamentally novel and highly robust approach to multivariate density deconvolution by stochastically rotating the replicates toward the corresponding true latent values. We also address the additionally significantly challenging problem of accommodating conditionally heteroscedastic measurement errors in this newly introduced framework. We …
The expressive power of Bayesian kernel-based methods has led them to become an important tool across many different facets of artificial intelligence, and useful to a plethora of modern application domains, providing both power and interpretability via uncertainty analysis. This article introduces and discusses two methods which straddle the areas of probabilistic Bayesian schemes and kernel methods for regression: Gaussian Processes and Relevance Vector Machines. Our focus is on developing a common framework with which to view these methods, via intermediate methods a probabilistic version o…
A Novel Theoretical Analysis for Clustering Heteroscedastic Gaussian Data without Knowledge of the Number of Clusters
本文提出一种新的成本函数用于估计聚类中心，证明在测量向量数量和聚类中心距离足够大时，固定点即为聚类中心，并引入Wald核进行聚类，提出CENTRE-X算法在复杂度和性能上优于传统算法。
In the context of Gaussian process regression with functional inputs, it is common to treat the input as a vector. The parameter space becomes prohibitively complex as the number of functional points increases, effectively becoming a hindrance for automatic relevance determination in high-dimensional problems. Generalizing a framework for time-varying inputs, we introduce the asymmetric Laplace functional weight (ALF): a flexible, parametric function that drives predictive relevance over the index space. Automatic dynamic relevance determination (ADRD) is achieved with three unknowns per input…
We study the problem of online generalized linear regression in the stochastic setting, where the label is generated from a generalized linear model with possibly unbounded additive noise. We provide a sharp analysis of the classical follow-the-regularized-leader (FTRL) algorithm to cope with the label noise. More specifically, for $σ$-sub-Gaussian label noise, our analysis provides a regret upper bound of $O(σ^2 d \log T) + o(\log T)$, where $d$ is the dimension of the input vector, $T$ is the total number of rounds. We also prove a $Ω(σ^2d\log(T/d))$ lower bound for stochastic online linear …
Relevance vector machine (RVM) can be seen as a probabilistic version of support vector machines which is able to produce sparse solutions by linearly weighting a small number of basis functions instead using all of them. Regardless of a few merits of RVM such as giving probabilistic predictions and relax of parameter tuning, it has poor prediction for test instances that are far away from the relevance vectors. As a solution, we propose a new combination of RVM and k-nearest neighbor (k-NN) rule which resolves this issue with regionally dealing with every test instance. In our settings, we ob…
We propose in this contribution a method for l one regularization in prototype based relevance learning vector quantization (LVQ) for sparse relevance profiles. Sparse relevance profiles in hyperspectral data analysis fade down those spectral bands which are not necessary for classification. In particular, we consider the sparsity in the relevance profile enforced by LASSO optimization. The latter one is obtained by a gradient learning scheme using a differentiable parametrized approximation of the $l_{1}$-norm, which has an upper error bound. We extend this regularization idea also to the mat…
This paper aims to decrease the time complexity of multi-output relevance vector regression from O(VM^3) to O(V^3+M^3), where V is the number of output dimensions, M is the number of basis functions, and V<M. The experimental results demonstrate that the proposed method is more competitive than the existing method, with regard to computation time. MATLAB codes are available at http://www.mathworks.com/matlabcentral/fileexchange/49131.
In the variational relevance vector machine, the gamma distribution is representative as a hyperprior over the noise precision of automatic relevance determination prior. Instead of the gamma hyperprior, we propose to use the inverse gamma hyperprior with a shape parameter close to zero and a scale parameter not necessary close to zero. This hyperprior is associated with the concept of a weakly informative prior. The effect of this hyperprior is investigated through regression to non-homogeneous data. Because it is difficult to capture the structure of such data with a single kernel function, …
Accurate Bayesian inference for tail risk extrapolation in time series
本文提出一种基于广义帕累托分布的贝叶斯框架，用于时间序列尾部风险外推，提供渐近诚实的可信区域，并在模拟和实际数据中优于现有方法。
This study combines two different learning paradigms, k-nearest neighbor (k-NN) rule, as memory-based learning paradigm and relevance vector machines (RVM), as statistical learning paradigm. This combination is performed in kernel space and is called k-relevance vector (k-RV). The purpose is to improve the performance of k-NN rule. The proposed model significantly prunes irrelevant attributes. We also introduced a new parameter, responsible for early stopping of iterations in RVM. We show that the new parameter improves the classification accuracy of k-RV. Intensive experiments are conducted o…
Multi-label learning has attracted the attention of the machine learning community. The problem conversion method Binary Relevance converts a familiar single label into a multi-label algorithm. The binary relevance method is widely used because of its simple structure and efficient algorithm. But binary relevance does not consider the links between labels, making it cumbersome to handle some tasks. This paper proposes a multi-label learning algorithm that can also be used for single-label classification. It is based on standard support vector machines and changes the original single decision h…
The challenges for non-intrusive methods for Polynomial Chaos modeling lie in the computational efficiency and accuracy under a limited number of model simulations. These challenges can be addressed by enforcing sparsity in the series representation through retaining only the most important basis terms. In this work, we present a novel sparse Bayesian learning technique for obtaining sparse Polynomial Chaos expansions which is based on a Relevance Vector Machine model and is trained using Variational Inference. The methodology shows great potential in high-dimensional data-driven settings usin…
With the increased need for multi-center magnetic resonance imaging studies, problems arise related to differences in hardware and software between centers. Namely, current algorithms for brain volume quantification are unreliable for the longitudinal assessment of volume changes in this type of setting. Currently most methods attempt to decrease this issue by regressing the scanner- and/or center-effects from the original data. In this work, we explore a novel approach to harmonize brain volume measurements by using only image descriptors. First, we explore the relationships between volumes a…
The task of automatic language identification (LID) involving multiple dialects of the same language family in the presence of noise is a challenging problem. In these scenarios, the identity of the language/dialect may be reliably present only in parts of the temporal sequence of the speech signal. The conventional approaches to LID (and for speaker recognition) ignore the sequence information by extracting long-term statistical summary of the recording assuming an independence of the feature frames. In this paper, we propose a neural network framework utilizing short-sequence information in …
In many problem settings, parameter vectors are not merely sparse but dependent in such a way that non-zero coefficients tend to cluster together. We refer to this form of dependency as "region sparsity." Classical sparse regression methods, such as the lasso and automatic relevance determination (ARD), which model parameters as independent a priori, and therefore do not exploit such dependencies. Here we introduce a hierarchical model for smooth, region-sparse weight vectors and tensors in a linear regression setting. Our approach represents a hierarchical extension of the relevance determina…
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