Subspace-valued functions arise in a wide range of problems, including parametric reduced order modeling (PROM). In PROM, each parameter point can be associated with a subspace, which is used for Petrov-Galerkin projections of large system matrices. Previous efforts to approximate such functions use interpolations on manifolds, which can be inaccurate and slow. To tackle this, we propose a novel Bayesian nonparametric model for subspace prediction: the Gaussian Process Subspace regression (GPS) model. This method is extrinsic and intrinsic at the same time: with multivariate Gaussian distributions on the Euclidean space, it induces a joint probability model on the Grassmann manifold, the set of fixed-dimensional subspaces. The GPS adopts a simple yet general correlation structure, and a principled approach for model selection. Its predictive distribution admits an analytical form, which allows for efficient subspace prediction over the parameter space. For PROM, the GPS provides a probabilistic prediction at a new parameter point that retains the accuracy of local reduced models, at a computational complexity that does not depend on system dimension, and thus is suitable for online computation. We give four numerical examples to compare our method to subspace interpolation, as well as two methods that interpolate local reduced models. Overall, GPS is the most data efficient, more computationally efficient than subspace interpolation, and gives smooth predictions with uncertainty quantification.
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We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.
Over the last years, robotic cloth manipulation has gained relevance within the research community. While significant advances have been made in robotic manipulation of rigid objects, the manipulation of non-rigid objects such as cloth garments is still a challenging problem. The uncertainty on how cloth behaves often requires the use of model-based approaches. However, cloth models have a very high dimensionality. Therefore, it is difficult to find a middle point between providing a manipulator with a dynamics model of cloth and working with a state space of tractable dimensionality. For this reason, most cloth manipulation approaches in literature perform static or quasi-static manipulation. In this paper, we propose a variation of Gaussian Process Dynamical Models (GPDMs) to model cloth dynamics in a low-dimensional manifold. GPDMs project a high-dimensional state space into a smaller dimension latent space which is capable of keeping the dynamic properties. Using such approach, we add control variables to the original formulation. In this way, it is possible to take into account the robot commands exerted on the cloth dynamics. We call this new version Controlled Gaussian Process Dynamical Model (CGPDM). Moreover, we propose an alternative parametric structure for the model, that is richer than the one employed in previous GPDM realizations. The modeling capacity of our proposal has been tested in both a simulated and a real scenario, where CGPDM proved to be capable of generalizing over a wide range of movements and correctly predicting the cloth motions obtained by previously unseen sequences of control actions.
In this paper, a functional partial quantile regression approach, a quantile regression analog of the functional partial least squares regression, is proposed to estimate the function-on-function linear quantile regression model. A partial quantile covariance function is first used to extract the functional partial quantile regression basis functions. The extracted basis functions are then used to obtain the functional partial quantile regression components and estimate the final model. In our proposal, the functional forms of the discretely observed random variables are first constructed via a finite-dimensional basis function expansion method. The functional partial quantile regression constructed using the functional random variables is approximated via the partial quantile regression constructed using the basis expansion coefficients. The proposed method uses an iterative procedure to extract the partial quantile regression components. A Bayesian information criterion is used to determine the optimum number of retained components. The proposed functional partial quantile regression model allows for more than one functional predictor in the model. However, the true form of the proposed model is unspecified, as the relevant predictors for the model are unknown in practice. Thus, a forward variable selection procedure is used to determine the significant predictors for the proposed model. Moreover, a case-sampling-based bootstrap procedure is used to construct pointwise prediction intervals for the functional response. The predictive performance of the proposed method is evaluated using several Monte Carlo experiments under different data generation processes and error distributions. Through an empirical data example, air quality data are analyzed to demonstrate the effectiveness of the proposed method.
We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression models in the literature as special cases. Based on the theory of reproducing kernel Hilbert spaces (RKHS), we propose a penalized least squares estimator that can accommodate functional variables observed on discrete sample points. Besides a conventional smoothness penalty, a group Lasso-type penalty is further imposed to induce sparsity in the high-dimensional vector predictors. We derive finite sample theoretical guarantees and show that the excess prediction risk of our estimator is minimax optimal. Furthermore, our analysis reveals an interesting phase transition phenomenon that the optimal excess risk is determined jointly by the smoothness and the sparsity of the functional regression coefficients. A novel efficient optimization algorithm based on iterative coordinate descent is devised to handle the smoothness and group penalties simultaneously. Simulation studies and real data applications illustrate the promising performance of the proposed approach compared to the state-of-the-art methods in the literature.
Point processes in time have a wide range of applications that include the claims arrival process in insurance or the analysis of queues in operations research. Due to advances in technology, such samples of point processes are increasingly encountered. A key object of interest is the local intensity function. It has a straightforward interpretation that allows to understand and explore point process data. We consider functional approaches for point processes, where one has a sample of repeated realizations of the point process. This situation is inherently connected with Cox processes, where the intensity functions of the replications are modeled as random functions. Here we study a situation where one records covariates for each replication of the process, such as the daily temperature for bike rentals. For modeling point processes as responses with vector covariates as predictors we propose a novel regression approach for the intensity function that is intrinsically nonparametric. While the intensity function of a point process that is only observed once on a fixed domain cannot be identified, we show how covariates and repeated observations of the process can be utilized to make consistent estimation possible, and we also derive asymptotic rates of convergence without invoking parametric assumptions.
We consider the classical problems of estimating the mean of an $n$-dimensional normally (with identity covariance matrix) or Poisson distributed vector under the squared loss. In a Bayesian setting the optimal estimator is given by the prior-dependent conditional mean. In a frequentist setting various shrinkage methods were developed over the last century. The framework of empirical Bayes, put forth by Robbins (1956), combines Bayesian and frequentist mindsets by postulating that the parameters are independent but with an unknown prior and aims to use a fully data-driven estimator to compete with the Bayesian oracle that knows the true prior. The central figure of merit is the regret, namely, the total excess risk over the Bayes risk in the worst case (over the priors). Although this paradigm was introduced more than 60 years ago, little is known about the asymptotic scaling of the optimal regret in the nonparametric setting. We show that for the Poisson model with compactly supported and subexponential priors, the optimal regret scales as $\Theta((\frac{\log n}{\log\log n})^2)$ and $\Theta(\log^3 n)$, respectively, both attained by the original estimator of Robbins. For the normal mean model, the regret is shown to be at least $\Omega((\frac{\log n}{\log\log n})^2)$ and $\Omega(\log^2 n)$ for compactly supported and subgaussian priors, respectively, the former of which resolves the conjecture of Singh (1979) on the impossibility of achieving bounded regret; before this work, the best regret lower bound was $\Omega(1)$. In addition to the empirical Bayes setting, these results are shown to hold in the compound setting where the parameters are deterministic. As a side application, the construction in this paper also leads to improved or new lower bounds for density estimation of Gaussian and Poisson mixtures.
Multitask Gaussian processes (MTGP) are the Gaussian process (GP) framework's solution for multioutput regression problems in which the $T$ elements of the regressors cannot be considered conditionally independent given the observations. Standard MTGP models assume that there exist both a multitask covariance matrix as a function of an intertask matrix, and a noise covariance matrix. These matrices need to be approximated by a low rank simplification of order $P$ in order to reduce the number of parameters to be learnt from $T^2$ to $TP$. Here we introduce a novel approach that simplifies the multitask learning by reducing it to a set of conditioned univariate GPs without the need for any low rank approximations, therefore completely eliminating the requirement to select an adequate value for hyperparameter $P$. At the same time, by extending this approach with both a hierarchical and an approximate model, the proposed extensions are capable of recovering the multitask covariance and noise matrices after learning only $2T$ parameters, avoiding the validation of any model hyperparameter and reducing the overall complexity of the model as well as the risk of overfitting. Experimental results over synthetic and real problems confirm the advantages of this inference approach in its ability to accurately recover the original noise and signal matrices, as well as the achieved performance improvement in comparison to other state of art MTGP approaches. We have also integrated the model with standard GP toolboxes, showing that it is computationally competitive with state of the art options.
In an instrumental variable model, the score statistic can be bounded for any alternative in parts of the parameter space. These regions involve a constraint on the first-stage regression coefficients and the reduced-form covariance matrix. Consequently, the Lagrange Multiplier test can have power close to size, despite being efficient under standard asymptotics. This information loss limits the power of conditional tests which use only the Anderson-Rubin and the score statistic. The conditional quasi-likelihood ratio test also suffers severe losses because it can be bounded for any alternative. A necessary condition for drastic power loss to occur is that the Hermitian of the reduced-form covariance matrix has eigenvalues of opposite signs. These cases are denoted impossibility designs (ID). We show this happens in practice, by applying our theory to the problem of inference on the intertemporal elasticity of substitution (IES). Of eleven countries studied by Yogo (2004} and Andrews (2016), nine are consistent with ID at the 95\% level.
We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.
We propose a new approach to inverse reinforcement learning (IRL) based on the deep Gaussian process (deep GP) model, which is capable of learning complicated reward structures with few demonstrations. Our model stacks multiple latent GP layers to learn abstract representations of the state feature space, which is linked to the demonstrations through the Maximum Entropy learning framework. Incorporating the IRL engine into the nonlinear latent structure renders existing deep GP inference approaches intractable. To tackle this, we develop a non-standard variational approximation framework which extends previous inference schemes. This allows for approximate Bayesian treatment of the feature space and guards against overfitting. Carrying out representation and inverse reinforcement learning simultaneously within our model outperforms state-of-the-art approaches, as we demonstrate with experiments on standard benchmarks ("object world","highway driving") and a new benchmark ("binary world").
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