This paper presents the recurrent estimation of distributions (RED) for modeling real-valued data in a semiparametric fashion. RED models make two novel uses of recurrent neural networks (RNNs) for density estimation of general real-valued data. First, RNNs are used to transform input covariates into a latent space to better capture conditional dependencies in inputs. After, an RNN is used to compute the conditional distributions of the latent covariates. The resulting model is efficient to train, compute, and sample from, whilst producing normalized pdfs. The effectiveness of RED is shown via several real-world data experiments. Our results show that RED models achieve a lower held-out negative log-likelihood than other neural network approaches across multiple dataset sizes and dimensionalities. Further context of the efficacy of RED is provided by considering anomaly detection tasks, where we also observe better performance over alternative models.
Nonparametric density estimation has been a well studied problem in statistics and machine learning @cite_0 . Unfortunately, nonparametric approaches like kernel density estimation suffer greatly from the curse of dimensionality and do not perform well when data does not have a small number of dimensions ( @math ). To alleviate this, several semiparametric approaches have been explored. Such approaches include forest density estimation @cite_15 , which assumes that the data has a forest (i.e. a collection of trees) structured graph. This assumption leads to a density which factorizes in a first order Markovian fashion through a tree traversal of the graph. Another common semiparametric approach is to use a nonparanormal type model @cite_21 . This approach uses a Gaussian copula with a rank-based transformation and a sparse precision matrix. While both approaches are well-understood theoretically, their strong assumptions often lead to inflexible models.
@cite_0 @cite_15 @cite_21
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This paper develops a nonparametric density estimator with parametric overtones. Suppose f(x, θ) is some family of densities, indexed by a vector of parameters θ. We define a local kernel-smoothed likelihood function which, for each x, can be used to estimate the best local parametric approximant to the true density. This leads to a new density estimator of the form f(x, o(x)), thus inserting the best local parameter estimate for each new value of x. When the bandwidth used is large, this amounts to ordinary full likelihood parametric density estimation, while for moderate and small bandwidths the method is essentially nonparametric, using only local properties of data and the model. Alternative ways more general than via the local likelihood are also described. The methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class. Properties of this new semiparametric estimator are investigated. Our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias. The new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed, while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct. Other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework. The methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression. Non-parametric density estimation is an important technique in probabilistic modeling and reasoning with uncertainty. We present a method for learning mixtures of polynomials (MoPs) approximations of one-dimensional and multidimensional probability densities from data. The method is based on basis spline interpolation, where a density is approximated as a linear combination of basis splines. We compute maximum likelihood estimators of the mixing coefficients of the linear combination. The Bayesian information criterion is used as the score function to select the order of the polynomials and the number of pieces of the MoP. The method is evaluated in two ways. First, we test the approximation fitting. We sample artificial datasets from known one-dimensional and multidimensional densities and learn MoP approximations from the datasets. The quality of the approximations is analyzed according to different criteria, and the new proposal is compared with MoPs learned with Lagrange interpolation and mixtures of truncated basis functions. Second, the proposed method is used as a non-parametric density estimation technique in Bayesian classifiers. Two of the most widely studied Bayesian classifiers, i.e., the naive Bayes and tree-augmented naive Bayes classifiers, are implemented and compared. Results on real datasets show that the non-parametric Bayesian classifiers using MoPs are comparable to the kernel density-based Bayesian classifiers. We provide a free R package implementing the proposed methods. Kernel density estimation is the most widely-used practical method for accurate nonparametric density estimation. However, long-standing worst-case theoretical results showing that its performance worsens exponentially with the dimension of the data have quashed its application to modern high-dimensional datasets for decades. In practice, it has been recognized that often such data have a much lower-dimensional intrinsic structure. We propose a small modification to kernel density estimation for estimating probability density functions on Riemannian submanifolds of Euclidean space. Using ideas from Riemannian geometry, we prove the consistency of this modified estimator and show that the convergence rate is determined by the intrinsic dimension of the submanifold. We conclude with empirical results demonstrating the behavior predicted by our theory. Recent methods for estimating sparse undirected graphs for real-valued data in high dimensional problems rely heavily on the assumption of normality. We show how to use a semiparametric Gaussian copula---or "nonparanormal"---for high dimensional inference. Just as additive models extend linear models by replacing linear functions with a set of one-dimensional smooth functions, the nonparanormal extends the normal by transforming the variables by smooth functions. We derive a method for estimating the nonparanormal, study the method's theoretical properties, and show that it works well in many examples. We study graph estimation and density estimation in high dimensions, using a family of density estimators based on forest structured undirected graphical models. For density estimation, we do not assume the true distribution corresponds to a forest; rather, we form kernel density estimates of the bivariate and univariate marginals, and apply Kruskal's algorithm to estimate the optimal forest on held out data. We prove an oracle inequality on the excess risk of the resulting estimator relative to the risk of the best forest. For graph estimation, we consider the problem of estimating forests with restricted tree sizes. We prove that finding a maximum weight spanning forest with restricted tree size is NP-hard, and develop an approximation algorithm for this problem. Viewing the tree size as a complexity parameter, we then select a forest using data splitting, and prove bounds on excess risk and structure selection consistency of the procedure. Experiments with simulated data and microarray data indicate that the methods are a practical alternative to Gaussian graphical models. Building a model of large-scale terrain that can adequately handle uncertainty and incompleteness in a statistically sound way is a challenging problem. This work proposes the use of Gaussian processes as models of large-scale terrain. The proposed model naturally provides a multiresolution representation of space, incorporates and handles uncertainties aptly, and copes with incompleteness of sensory information. Gaussian process regression techniques are applied to estimate and interpolate (to fill gaps in occluded areas) elevation information across the field. The estimates obtained are the best linear unbiased estimates for the data under consideration. A single nonstationary (neural network) Gaussian process is shown to be powerful enough to model large and complex terrain, effectively handling issues relating to discontinuous data. A local approximation method based on a “moving window” methodology and implemented using k-dimensional (KD)-trees is also proposed. This enables the approach to handle extremely large data sets, thereby completely addressing its scalability issues. Experiments are performed on large-scale data sets taken from real mining applications. These data sets include sparse mine planning data, which are representative of a global positioning system–based survey, as well as dense laser scanner data taken at different mine sites. Further, extensive statistical performance evaluation and benchmarking of the technique has been performed through cross-validation experiments. They conclude that for dense and-or flat data, the proposed approach will perform very competitively with grid-based approaches using standard interpolation techniques and triangulated irregular networks using triangle-based interpolation techniques; for sparse and-or complex data, however, it would significantly outperform them. © 2009 Wiley Periodicals, Inc. We introduce and investigate a new model of learning probability distributions from independent draws. Our model is inspired by the popular Probably Approximately Correct (PAC) model for learning boolean functions from labeled examples [24], in the sense that we emphasize efficient and approximate learning, and we study the learnability of restricted classes of target distributions. The dist ribut ion classes we examine are often defined by some simple computational mechanism for transforming a truly random string of input bits (which is not visible to the learning algorithm) into the stochastic observation (output) seen by the learning algorithm. In this paper, we concentrate on discrete distributions over O, I n. The problem of inferring an approximation to an unknown probability distribution on the basis of independent draws has a long and complex history in the pattern recognition and statistics literature. For instance, the problem of estimating the parameters of a Gaussian density in highdimensional space is one of the most studied statistical problems. Distribution learning problems have often been investigated in the context of unsupervised learning, in which a linear mixture of two or more distributions is generating the observations, and the final goal is not to model the distributions themselves, but to predict from which distribution each observation was drawn. Data clustering methods are a common tool here. There is also a large literature on nonpararnetric density estimation, in which no assumptions are made on the unknown target density. Nearest-neighbor approaches to the unsupervised learning problem often arise in the nonparametric setting. While we obviously cannot do justice to these areas here, the books of Duda and Hart [9] and Vapnik [25] provide excellent overviews and introductions to the pattern recognition work, as well as many pointers for further reading. See also Izenman’s recent survey article [16]. Roughly speaking, our work departs from the traditional statistical and pattern recognition approaches in two ways. First, we place explicit emphasis on the comput ationrd complexity of distribution learning. It seems fair to say that while previous research has provided an excellent understanding of the information-theoretic issues involved in dis- Visual features are commonly modeled with probability density functions in computer vision problems, but current methods such as a mixture of Gaussians and kernel density estimation suffer from either the lack of flexibility by fixing or limiting the number of Gaussian components in the mixture or large memory requirement by maintaining a nonparametric representation of the density. These problems are aggravated in real-time computer vision applications since density functions are required to be updated as new data becomes available. We present a novel kernel density approximation technique based on the mean-shift mode finding algorithm and describe an efficient method to sequentially propagate the density modes over time. Although the proposed density representation is memory efficient, which is typical for mixture densities, it inherits the flexibility of nonparametric methods by allowing the number of components to be variable. The accuracy and compactness of the sequential kernel density approximation technique is illustrated by both simulations and experiments. Sequential kernel density approximation is applied to online target appearance modeling for visual tracking, and its performance is demonstrated on a variety of videos. We consider the problem of estimating a low-rank signal matrix from noisy measurements under the assumption that the distribution of the data matrix belongs to an exponential family. In this setting, we derive generalized Stein's unbiased risk estimation (SURE) formulas that hold for any spectral estimators which shrink or threshold the singular values of the data matrix. This leads to new data-driven spectral estimators, whose optimality is discussed using tools from random matrix theory and through numerical experiments. Under the spiked population model and in the asymptotic setting where the dimensions of the data matrix are let going to infinity, some theoretical properties of our approach are compared to recent results on asymptotically optimal shrinking rules for Gaussian noise. It also leads to new procedures for singular values shrinkage in finite-dimensional matrix denoising for Gamma-distributed and Poisson-distributed measurements. Statistical inference and information processing of high-dimensional data often require an efficient and accurate estimation of their second-order statistics. With rapidly changing data, limited processing power and storage at the acquisition devices, it is desirable to extract the covariance structure from a single pass over the data and a small number of stored measurements. In this paper, we explore a quadratic (or rank-one) measurement model which imposes minimal memory requirements and low computational complexity during the sampling process, and is shown to be optimal in preserving various low-dimensional covariance structures. Specifically, four popular structural assumptions of covariance matrices, namely, low rank, Toeplitz low rank, sparsity, jointly rank-one and sparse structure, are investigated, while recovery is achieved via convex relaxation paradigms for the respective structure. The proposed quadratic sampling framework has a variety of potential applications, including streaming data processing, high-frequency wireless communication, phase space tomography and phase retrieval in optics, and noncoherent subspace detection. Our method admits universally accurate covariance estimation in the absence of noise, as soon as the number of measurements exceeds the information theoretic limits. We also demonstrate the robustness of this approach against noise and imperfect structural assumptions. Our analysis is established upon a novel notion called the mixed-norm restricted isometry property (RIP- @math ), as well as the conventional RIP- @math for near-isotropic and bounded measurements. In addition, our results improve upon the best-known phase retrieval (including both dense and sparse signals) guarantees using PhaseLift with a significantly simpler approach. Density estimation with Gaussian Mixture Models is a popular generative technique used also for clustering. We develop a framework to incorporate side information in the form of equivalence constraints into the model estimation procedure. Equivalence constraints are defined on pairs of data points, indicating whether the points arise from the same source (positive constraints) or from different sources (negative constraints). Such constraints can be gathered automatically in some learning problems, and are a natural form of supervision in others. For the estimation of model parameters we present a closed form EM procedure which handles positive constraints, and a Generalized EM procedure using a Markov net which handles negative constraints. Using publicly available data sets we demonstrate that such side information can lead to considerable improvement in clustering tasks, and that our algorithm is preferable to two other suggested methods using the same type of side information. Consider non-parametric estimation of a decreasing density function f under the random (right) censorship model. Alternatively, consider estimation of a monotone increasing (or decreasing) hazard rate i based on randomly right censored data. We show that the non-paramet- ric maximum likelihood estimator of the density f (introduced by Laslett, 1982) is asymptotically equivalent to the estimator obtained by differentiating the least concave majorant of the Kaplan-Meier estimator, the non-parametric maximum likelihood estimator of the distribution function F in the larger model without any monotonicity assumption. A similar result is shown to hold for the non-parametric maximum likelihood estimator of an increasing hazard rate A: the non-parametric maximum likelihood estimator of A (introduced in the uncensored case by Prakasa Rao, 1970) is asymptotically equivalent to the estimator obtained by differentiation of the greatest convex minorant of the Nelson-Aalen estimator, the non-parametric maximum likelihood estimator of the cumulative hazard function A in the larger model without any monotonicity assumption. In proving these asymptotic equivalences, we also establish the asymptotic distributions of the different estimators at a fixed point at which the monotonicity assumption is strictly satisfied. In this paper, we consider the cluster estimation problem under the stochastic block model. We show that the semidefinite programming (SDP) formulation for this problem achieves an error rate that decays exponentially in the signal-to-noise ratio. The error bound implies weak recovery in the sparse graph regime with bounded expected degrees as well as exact recovery in the dense regime. An immediate corollary of our results yields error bounds under the censored block model. Moreover, these error bounds are robust, continuing to hold under heterogeneous edge probabilities and a form of the so-called monotone attack. Significantly, this error rate is achieved by the SDP solution itself without any further pre- or post-processing and improves upon existing polynomially decaying error bounds proved using the Grothendieck’s inequality. Our analysis builds on two key ingredients: 1) showing that the graph has a well-behaved spectrum, even in the sparse regime, after discounting an exponentially small number of edges and 2) an order-statistics argument that governs the final error rate. Both arguments highlight the implicit regularization effect of the SDP formulation. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practical limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n^2). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p^2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such methods, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performance than existing algorithms, for any given problem instance, and not only for worst-case situations. We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may be treated as noisy linear observations, and thus the signal may be estimated using the generalized Lasso. This is appealing because of the abundance of efficient, specialized solvers for this program. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear observations with linear observations will be greatly reduced when using the signal structure in the reconstruction. We allow general signal structure, only assuming that the signal belongs to some set K in R^n. We consider the single-index model of non-linearity. Our theory allows the non-linearity to be discontinuous, not one-to-one and even unknown. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown covariance matrix. As special cases of our results, we recover near-optimal theory for noisy linear observations, and also give the first theoretical accuracy guarantee for 1-bit compressed sensing with unknown covariance matrix of the measurement vectors. Summary Background—Most machine learning approaches only provide a classification for binary responses. However, probabilities are required for risk estimation using individual patient characteristics. It has been shown recently that every statistical learning machine known to be consistent for a nonparametric regression problem is a probability machine that is provably consistent for this estimation problem. Objectives—The aim of this paper is to show how random forests and nearest neighbors can be used for consistent estimation of individual probabilities. Methods—Two random forest algorithms and two nearest neighbor algorithms are described in detail for estimation of individual probabilities. We discuss the consistency of random forests, nearest neighbors and other learning machines in detail. We conduct a simulation study to illustrate the validity of the methods. We exemplify the algorithms by analyzing two well-known data sets on the diagnosis of appendicitis and the diagnosis of diabetes in Pima Indians. Results—Simulations demonstrate the validity of the method. With the real data application, we show the accuracy and practicality of this approach. We provide sample code from R packages in which the probability estimation is already available. This means that all calculations can be performed using existing software. Conclusions—Random forest algorithms as well as nearest neighbor approaches are valid machine learning methods for estimating individual probabilities for binary responses. Freely available implementations are available in R and may be used for applications. Pictorial structure (PS) models are extensively used for part-based recognition of scenes, people, animals and multi-part objects. To achieve tractability, the structure and parameterization of the model is often restricted, for example, by assuming tree dependency structure and unimodal, data-independent pairwise interactions. These expressivity restrictions fail to capture important patterns in the data. On the other hand, local methods such as nearest-neighbor classification and kernel density estimation provide non-parametric flexibility but require large amounts of data to generalize well. We propose a simple semi-parametric approach that combines the tractability of pictorial structure inference with the flexibility of non-parametric methods by expressing a subset of model parameters as kernel regression estimates from a learned sparse set of exemplars. This yields query-specific, image-dependent pose priors. We develop an effective shape-based kernel for upper-body pose similarity and propose a leave-one-out loss function for learning a sparse subset of exemplars for kernel regression. We apply our techniques to two challenging datasets of human figure parsing and advance the state-of-the-art (from 80 to 86 on the Buffy dataset [8]), while using only 15 of the training data as exemplars. We introduce a new nonparametric clustering model which combines the recently proposed distance-dependent Chinese restaurant process (dd-CRP) and non-linear, spectral methods for dimensionality reduction. Our model retains the ability of nonparametric methods to learn the number of clusters from data. At the same time it addresses two key limitations of nonparametric Bayesian methods: modeling data that are not exchangeable and have many correlated features. Spectral methods use the similarity between documents to map them into a low-dimensional spectral space where we then compare several clustering methods. Our experiments on handwritten digits and text documents show that nonparametric methods such as the CRP or dd-CRP can perform as well as or better than k-means and also recover the true number of clusters. We improve the performance of the dd-CRP in spectral space by incorporating the original similarity matrix in its prior. This simple modification results in better performance than all other methods we compared to. We offer a new formulation and first experimental evaluation of a general Gibbs sampler for mixture modeling with distance-dependent CRPs. This paper proposes a novel method to apply the standard graph cut technique to segmenting multimodal tensor valued images. The Riemannian nature of the tensor space is explicitly taken into account by first mapping the data to a Euclidean space where non-parametric kernel density estimates of the regional distributions may be calculated from user initialized regions. These distributions are then used as regional priors in calculating graph edge weights. Hence this approach utilizes the true variation of the tensor data by respecting its Riemannian structure in calculating distances when forming probability distributions. Further, the non-parametric model generalizes to arbitrary tensor distribution unlike the Gaussian assumption made in previous works. Casting the segmentation problem in a graph cut framework yields a segmentation robust with respect to initialization on the data tested. We consider supervised learning problems within the positive-definite kernel framework, such as kernel ridge regression, kernel logistic regression or the support vector machine. With kernels leading to infinite-dimensional feature spaces, a common practi cal limiting difficulty is the necessity of computing the kernel matrix, which most frequently leads to algorithms with running time at least quadratic in the number of observations n, i.e., O(n 2 ). Low-rank approximations of the kernel matrix are often considered as they allow the reduction of running time complexities to O(p 2 n), where p is the rank of the approximation. The practicality of such methods thus depends on the required rank p. In this paper, we show that in the context of kernel ridge regression, for approximations based on a random subset of columns of the original kernel matrix, the rank p may be chosen to be linear in the degrees of freedom associated with the problem, a quantity which is classically used in the statistical analysis of such method s, and is often seen as the implicit number of parameters of non-parametric estimators. This result enables simple algorithms that have sub-quadratic running time complexity, but provably exhibit the same predictive performancethan existing algorithms, for any given problem instance, and not only for worst-case situations.