arXiv subjects

Aleksandr Y. Aravkin

Publications and source records attributed to Aleksandr Y. Aravkin.

At least 55 records · Page 3Linked to original sources

arXiv2014

Sparse Quantile Huber Regression for Efficient and Robust Estimation

We consider new formulations and methods for sparse quantile regression in the high-dimensional setting. Quantile regression plays an important role in many applications, including outlier-robust exploratory analysis in gene selection. In addition, the sparsity consideration in quantile regression enables the exploration of the entire conditional distribution of the response variable given the predictors and therefore yields a more comprehensive view of the important predictors. We propose a generalized OMP algorithm for variable selection, taking the misfit loss to be either the traditional quantile loss or a smooth version we call quantile Huber, and compare the resulting greedy approaches with convex sparsity-regularized formulations. We apply a recently proposed interior point methodology to efficiently solve all convex formulations as well as convex subproblems in the generalized OMP setting, pro- vide theoretical guarantees of consistent estimation, and demonstrate the performance of our approach using empirical studies of simulated and genomic datasets.

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arXiv2014

Semistochastic Quadratic Bound Methods

Partition functions arise in a variety of settings, including conditional random fields, logistic regression, and latent gaussian models. In this paper, we consider semistochastic quadratic bound (SQB) methods for maximum likelihood inference based on partition function optimization. Batch methods based on the quadratic bound were recently proposed for this class of problems, and performed favorably in comparison to state-of-the-art techniques. Semistochastic methods fall in between batch algorithms, which use all the data, and stochastic gradient type methods, which use small random selections at each iteration. We build semistochastic quadratic bound-based methods, and prove both global convergence (to a stationary point) under very weak assumptions, and linear convergence rate under stronger assumptions on the objective. To make the proposed methods faster and more stable, we consider inexact subproblem minimization and batch-size selection schemes. The efficacy of SQB methods is demonstrated via comparison with several state-of-the-art techniques on commonly used datasets.

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arXiv2013

Outlier robust system identification: a Bayesian kernel-based approach

In this paper, we propose an outlier-robust regularized kernel-based method for linear system identification. The unknown impulse response is modeled as a zero-mean Gaussian process whose covariance (kernel) is given by the recently proposed stable spline kernel, which encodes information on regularity and exponential stability. To build robustness to outliers, we model the measurement noise as realizations of independent Laplacian random variables. The identification problem is cast in a Bayesian framework, and solved by a new Markov Chain Monte Carlo (MCMC) scheme. In particular, exploiting the representation of the Laplacian random variables as scale mixtures of Gaussians, we design a Gibbs sampler which quickly converges to the target distribution. Numerical simulations show a substantial improvement in the accuracy of the estimates over state-of-the-art kernel-based methods.

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arXiv2013

Improvements to deep convolutional neural networks for LVCSR

Deep Convolutional Neural Networks (CNNs) are more powerful than Deep Neural Networks (DNN), as they are able to better reduce spectral variation in the input signal. This has also been confirmed experimentally, with CNNs showing improvements in word error rate (WER) between 4-12% relative compared to DNNs across a variety of LVCSR tasks. In this paper, we describe different methods to further improve CNN performance. First, we conduct a deep analysis comparing limited weight sharing and full weight sharing with state-of-the-art features. Second, we apply various pooling strategies that have shown improvements in computer vision to an LVCSR speech task. Third, we introduce a method to effectively incorporate speaker adaptation, namely fMLLR, into log-mel features. Fourth, we introduce an effective strategy to use dropout during Hessian-free sequence training. We find that with these improvements, particularly with fMLLR and dropout, we are able to achieve an additional 2-3% relative improvement in WER on a 50-hour Broadcast News task over our previous best CNN baseline. On a larger 400-hour BN task, we find an additional 4-5% relative improvement over our previous best CNN baseline.

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arXiv2013

Accelerating Hessian-free optimization for deep neural networks by implicit preconditioning and sampling

Hessian-free training has become a popular parallel second or- der optimization technique for Deep Neural Network training. This study aims at speeding up Hessian-free training, both by means of decreasing the amount of data used for training, as well as through reduction of the number of Krylov subspace solver iterations used for implicit estimation of the Hessian. In this paper, we develop an L-BFGS based preconditioning scheme that avoids the need to access the Hessian explicitly. Since L-BFGS cannot be regarded as a fixed-point iteration, we further propose the employment of flexible Krylov subspace solvers that retain the desired theoretical convergence guarantees of their conventional counterparts. Second, we propose a new sampling algorithm, which geometrically increases the amount of data utilized for gradient and Krylov subspace iteration calculations. On a 50-hr English Broadcast News task, we find that these methodologies provide roughly a 1.5x speed-up, whereas, on a 300-hr Switchboard task, these techniques provide over a 2.3x speedup, with no loss in WER. These results suggest that even further speed-up is expected, as problems scale and complexity grows.

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arXiv2013

Kalman smoothing and block tridiagonal systems: new connections and numerical stability results

The Rauch-Tung-Striebel (RTS) and the Mayne-Fraser (MF) algorithms are two of the most popular smoothing schemes to reconstruct the state of a dynamic linear system from measurements collected on a fixed interval. Another (less popular) approach is the Mayne (M) algorithm introduced in his original paper under the name of Algorithm A. In this paper, we analyze these three smoothers from an optimization and algebraic perspective, revealing new insights on their numerical stability properties. In doing this, we re-interpret classic recursions as matrix decomposition methods for block tridiagonal matrices. First, we show that the classic RTS smoother is an implementation of the forward block tridiagonal (FBT) algorithm (also known as Thomas algorithm) for particular block tridiagonal systems. We study the numerical stability properties of this scheme, connecting the condition number of the full system to properties of the individual blocks encountered during standard recursion. Second, we study the M smoother, and prove it is equivalent to a backward block tridiagonal (BBT) algorithm with a stronger stability guarantee than RTS. Third, we illustrate how the MF smoother solves a block tridiagonal system, and prove that it has the same numerical stability properties of RTS (but not those of M). Finally, we present a new hybrid RTS/M (FBT/BBT) smoothing scheme, which is faster than MF, and has the same numerical stability guarantees of RTS and MF.

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arXiv2013

The connection between Bayesian estimation of a Gaussian random field and RKHS

Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation. Given the noisy measurements, the RKHS estimate represents the posterior mean (minimum variance estimate) of a Gaussian random field with covariance proportional to the kernel associated with the RKHS. In this paper, we provide a statistical interpretation when more general losses are used, such as absolute value, Vapnik or Huber. Specifically, for any finite set of sampling locations (including where the data were collected), the MAP estimate for the signal samples is given by the RKHS estimate evaluated at these locations.

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arXiv2013

Fast Dual Variational Inference for Non-Conjugate LGMs

Latent Gaussian models (LGMs) are widely used in statistics and machine learning. Bayesian inference in non-conjugate LGMs is difficult due to intractable integrals involving the Gaussian prior and non-conjugate likelihoods. Algorithms based on variational Gaussian (VG) approximations are widely employed since they strike a favorable balance between accuracy, generality, speed, and ease of use. However, the structure of the optimization problems associated with these approximations remains poorly understood, and standard solvers take too long to converge. We derive a novel dual variational inference approach that exploits the convexity property of the VG approximations. We obtain an algorithm that solves a convex optimization problem, reduces the number of variational parameters, and converges much faster than previous methods. Using real-world data, we demonstrate these advantages on a variety of LGMs, including Gaussian process classification, and latent Gaussian Markov random fields.

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arXiv2013

Variational properties of value functions

Regularization plays a key role in a variety of optimization formulations of inverse problems. A recurring theme in regularization approaches is the selection of regularization parameters, and their effect on the solution and on the optimal value of the optimization problem. The sensitivity of the value function to the regularization parameter can be linked directly to the Lagrange multipliers. This paper characterizes the variational properties of the value functions for a broad class of convex formulations, which are not all covered by standard Lagrange multiplier theory. An inverse function theorem is given that links the value functions of different regularization formulations (not necessarily convex). These results have implications for the selection of regularization parameters, and the development of specialized algorithms. Numerical examples illustrate the theoretical results.

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arXiv2013

Sparse/Robust Estimation and Kalman Smoothing with Nonsmooth Log-Concave Densities: Modeling, Computation, and Theory

We introduce a class of quadratic support (QS) functions, many of which play a crucial role in a variety of applications, including machine learning, robust statistical inference, sparsity promotion, and Kalman smoothing. Well known examples include the l2, Huber, l1 and Vapnik losses. We build on a dual representation for QS functions using convex analysis, revealing the structure necessary for a QS function to be interpreted as the negative log of a probability density, and providing the foundation for statistical interpretation and analysis of QS loss functions. For a subclass of QS functions called piecewise linear quadratic (PLQ) penalties, we also develop efficient numerical estimation schemes. These components form a flexible statistical modeling framework for a variety of learning applications, together with a toolbox of efficient numerical methods for inference. In particular, for PLQ densities, interior point (IP) methods can be used. IP methods solve nonsmooth optimization problems by working directly with smooth systems of equations characterizing their optimality. The efficiency of the IP approach depends on the structure of particular applications. We consider the class of dynamic inverse problems using Kalman smoothing, where the aim is to reconstruct the state of a dynamical system with known process and measurement models starting from noisy output samples. In the classical case, Gaussian errors are assumed in the process and measurement models. The extended framework allows arbitrary PLQ densities to be used, and the proposed IP approach solves the generalized Kalman smoothing problem while maintaining the linear complexity in the size of the time series, just as in the Gaussian case. This extends the computational efficiency of classic algorithms to a much broader nonsmooth setting, and includes many recently proposed robust and sparse smoothers as special cases.

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arXiv2013

Robust and Trend Following Student's t Kalman Smoothers

We present a Kalman smoothing framework based on modeling errors using the heavy tailed Student's t distribution, along with algorithms, convergence theory, open-source general implementation, and several important applications. The computational effort per iteration grows linearly with the length of the time series, and all smoothers allow nonlinear process and measurement models. Robust smoothers form an important subclass of smoothers within this framework. These smoothers work in situations where measurements are highly contaminated by noise or include data unexplained by the forward model. Highly robust smoothers are developed by modeling measurement errors using the Student's t distribution, and outperform the recently proposed L1-Laplace smoother in extreme situations with data containing 20% or more outliers. A second special application we consider in detail allows tracking sudden changes in the state. It is developed by modeling process noise using the Student's t distribution, and the resulting smoother can track sudden changes in the state. These features can be used separately or in tandem, and we present a general smoother algorithm and open source implementation, together with convergence analysis that covers a wide range of smoothers. A key ingredient of our approach is a technique to deal with the non-convexity of the Student's t loss function. Numerical results for linear and nonlinear models illustrate the performance of the new smoothers for robust and tracking applications, as well as for mixed problems that have both types of features.

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arXiv2013

Linear system identification using stable spline kernels and PLQ penalties

The classical approach to linear system identification is given by parametric Prediction Error Methods (PEM). In this context, model complexity is often unknown so that a model order selection step is needed to suitably trade-off bias and variance. Recently, a different approach to linear system identification has been introduced, where model order determination is avoided by using a regularized least squares framework. In particular, the penalty term on the impulse response is defined by so called stable spline kernels. They embed information on regularity and BIBO stability, and depend on a small number of parameters which can be estimated from data. In this paper, we provide new nonsmooth formulations of the stable spline estimator. In particular, we consider linear system identification problems in a very broad context, where regularization functionals and data misfits can come from a rich set of piecewise linear quadratic functions. Moreover, our anal- ysis includes polyhedral inequality constraints on the unknown impulse response. For any formulation in this class, we show that interior point methods can be used to solve the system identification problem, with complexity O(n3)+O(mn2) in each iteration, where n and m are the number of impulse response coefficients and measurements, respectively. The usefulness of the framework is illustrated via a numerical experiment where output measurements are contaminated by outliers.

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arXiv2013

Optimization viewpoint on Kalman smoothing, with applications to robust and sparse estimation

In this paper, we present the optimization formulation of the Kalman filtering and smoothing problems, and use this perspective to develop a variety of extensions and applications. We first formulate classic Kalman smoothing as a least squares problem, highlight special structure, and show that the classic filtering and smoothing algorithms are equivalent to a particular algorithm for solving this problem. Once this equivalence is established, we present extensions of Kalman smoothing to systems with nonlinear process and measurement models, systems with linear and nonlinear inequality constraints, systems with outliers in the measurements or sudden changes in the state, and systems where the sparsity of the state sequence must be accounted for. All extensions preserve the computational efficiency of the classic algorithms, and most of the extensions are illustrated with numerical examples, which are part of an open source Kalman smoothing Matlab/Octave package.

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arXiv2013

Convex vs nonconvex approaches for sparse estimation: GLasso, Multiple Kernel Learning and Hyperparameter GLasso

The popular Lasso approach for sparse estimation can be derived via marginalization of a joint density associated with a particular stochastic model. A different marginalization of the same probabilistic model leads to a different non-convex estimator where hyperparameters are optimized. Extending these arguments to problems where groups of variables have to be estimated, we study a computational scheme for sparse estimation that differs from the Group Lasso. Although the underlying optimization problem defining this estimator is non-convex, an initialization strategy based on a univariate Bayesian forward selection scheme is presented. This also allows us to define an effective non-convex estimator where only one scalar variable is involved in the optimization process. Theoretical arguments, independent of the correctness of the priors entering the sparse model, are included to clarify the advantages of this non-convex technique in comparison with other convex estimators. Numerical experiments are also used to compare the performance of these approaches.

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arXiv2012

Sparse seismic imaging using variable projection

We consider an important class of signal processing problems where the signal of interest is known to be sparse, and can be recovered from data given auxiliary information about how the data was generated. For example, a sparse Green's function may be recovered from seismic experimental data using sparsity optimization when the source signature is known. Unfortunately, in practice this information is often missing, and must be recovered from data along with the signal using deconvolution techniques. In this paper, we present a novel methodology to simultaneously solve for the sparse signal and auxiliary parameters using a recently proposed variable projection technique. Our main contribution is to combine variable projection with sparsity promoting optimization, obtaining an efficient algorithm for large-scale sparse deconvolution problems. We demonstrate the algorithm on a seismic imaging example.

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arXiv2012

Estimating Nuisance Parameters in Inverse Problems

Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. Structure present in these problems allows efficient optimization strategies - a well known example is variable projection, where nonlinear least squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood (ML) and maximum a posteriori likelihood (MAP) problems with nuisance parameters, such as variance or degrees of freedom. As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, degree of freedom (d.o.f.) parameter estimation in the context of robust inverse problems, automatic calibration, and optimal experimental design. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large- scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.

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arXiv2011

Robust and Trend-following Kalman Smoothers using Student's t

We propose two nonlinear Kalman smoothers that rely on Student's t distributions. The T-Robust smoother finds the maximum a posteriori likelihood (MAP) solution for Gaussian process noise and Student's t observation noise, and is extremely robust against outliers, outperforming the recently proposed l1-Laplace smoother in extreme situations (e.g. 50% or more outliers). The second estimator, which we call the T-Trend smoother, is able to follow sudden changes in the process model, and is derived as a MAP solver for a model with Student's t-process noise and Gaussian observation noise. We design specialized methods to solve both problems which exploit the special structure of the Student's t-distribution, and provide a convergence theory. Both smoothers can be implemented with only minor modifications to an existing L2 smoother implementation. Numerical results for linear and nonlinear models illustrating both robust and fast tracking applications are presented.

stat.CO↗

arXiv2011

A statistical and computational theory for robust and sparse Kalman smoothing

Kalman smoothers reconstruct the state of a dynamical system starting from noisy output samples. While the classical estimator relies on quadratic penalization of process deviations and measurement errors, extensions that exploit Piecewise Linear Quadratic (PLQ) penalties have been recently proposed in the literature. These new formulations include smoothers robust with respect to outliers in the data, and smoothers that keep better track of fast system dynamics, e.g. jumps in the state values. In addition to L2, well known examples of PLQ penalties include the L1, Huber and Vapnik losses. In this paper, we use a dual representation for PLQ penalties to build a statistical modeling framework and a computational theory for Kalman smoothing. We develop a statistical framework by establishing conditions required to interpret PLQ penalties as negative logs of true probability densities. Then, we present a computational framework, based on interior-point methods, that solves the Kalman smoothing problem with PLQ penalties and maintains the linear complexity in the size of the time series, just as in the L2 case. The framework presented extends the computational efficiency of the Mayne-Fraser and Rauch-Tung-Striebel algorithms to a much broader non-smooth setting, and includes many known robust and sparse smoothers as special cases.

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