arXiv subjects

Aleksandr Aravkin

Publications and source records attributed to Aleksandr Aravkin.

At least 37 records · Page 2Linked to original sources

arXiv2021

A nonconvex optimization approach to IMRT planning with dose-volume constraints

Fluence map optimization for intensity-modulated radiation therapy planning can be formulated as a large-scale inverse problem with competing objectives and constraints associated with the tumors and organs-at-risk. Unfortunately, clinically relevant dose-volume constraints are nonconvex, so standard algorithms for convex problems cannot be directly applied. While prior work focused on convex approximations for these constraints, we propose a novel relaxation approach to handle nonconvex dose-volume constraints. We develop efficient, provably convergent algorithms based on partial minimization, and show how to adapt them to handle maximum-dose constraints and infeasible problems. We demonstrate our approach using the CORT dataset, and show that it is easily adaptable to radiation treatment planning with dose-volume constraints for multiple tumors and organs-at-risk.

math.OC↗

arXiv2021

Promoting global stability in data-driven models of quadratic nonlinear dynamics

Modeling realistic fluid and plasma flows is computationally intensive, motivating the use of reduced-order models for a variety of scientific and engineering tasks. However, it is challenging to characterize, much less guarantee, the global stability (i.e., long-time boundedness) of these models. The seminal work of Schlegel and Noack (JFM, 2015) provided a theorem outlining necessary and sufficient conditions to ensure global stability in systems with energy-preserving, quadratic nonlinearities, with the goal of evaluating the stability of projection-based models. In this work, we incorporate this theorem into modern data-driven models obtained via machine learning. First, we propose that this theorem should be a standard diagnostic for the stability of projection-based and data-driven models, examining the conditions under which it holds. Second, we illustrate how to modify the objective function in machine learning algorithms to promote globally stable models, with implications for the modeling of fluid and plasma flows. Specifically, we introduce a modified "trapping SINDy" algorithm based on the sparse identification of nonlinear dynamics (SINDy) method. This method enables the identification of models that, by construction, only produce bounded trajectories. The effectiveness and accuracy of this approach are demonstrated on a broad set of examples of varying model complexity and physical origin, including the vortex shedding in the wake of a circular cylinder.

physics.flu-dyn↗

arXiv2021

Analysis of Truncated Orthogonal Iteration for Sparse Eigenvector Problems

A wide range of problems in computational science and engineering require estimation of sparse eigenvectors for high dimensional systems. Here, we propose two variants of the Truncated Orthogonal Iteration to compute multiple leading eigenvectors with sparsity constraints simultaneously. We establish numerical convergence results for the proposed algorithms using a perturbation framework, and extend our analysis to other existing alternatives for sparse eigenvector estimation. We then apply our algorithms to solve the sparse principle component analysis problem for a wide range of test datasets, from simple simulations to real-world datasets including MNIST, sea surface temperature and 20 newsgroups. In all these cases, we show that the new methods get state of the art results quickly and with minimal parameter tuning.

math.NA↗

arXiv2020

Non-smooth Variable Projection

Variable projection solves structured optimization problems by completely minimizing over a subset of the variables while iterating over the remaining variables. Over the last 30 years, the technique has been widely used, with empirical and theoretical results demonstrating both greater efficacy and greater stability compared to competing approaches. Classic examples have exploited closed-form projections and smoothness of the objective function. We extend the approach to problems that include non-smooth terms, and where the projection subproblems can only be solved inexactly by iterative methods. We propose an inexact adaptive algonrithm for solving such problems and analyze its computational complexity. Finally, we show how the theory can be used to design methods for selected problems occurring frequently in machine-learning and inverse problems.

math.OC↗

arXiv2020

Time-varying Autoregression with Low Rank Tensors

We present a windowed technique to learn parsimonious time-varying autoregressive models from multivariate timeseries. This unsupervised method uncovers interpretable spatiotemporal structure in data via non-smooth and non-convex optimization. In each time window, we assume the data follow a linear model parameterized by a system matrix, and we model this stack of potentially different system matrices as a low rank tensor. Because of its structure, the model is scalable to high-dimensional data and can easily incorporate priors such as smoothness over time. We find the components of the tensor using alternating minimization and prove that any stationary point of this algorithm is a local minimum. We demonstrate on a synthetic example that our method identifies the true rank of a switching linear system in the presence of noise. We illustrate our model's utility and superior scalability over extant methods when applied to several synthetic and real-world example: two types of time-varying linear systems, worm behavior, sea surface temperature, and monkey brain datasets.

stat.ML↗

arXiv2020

Stable and Robust LQR Design via Scenario Approach

Linear Quadratic Regulator (LQR) design is one of the most classical optimal control problems, whose well-known solution is an input sequence expressed as a state-feedback. In this work, finite-horizon and discrete-time LQR is solved under stability constraints and uncertain system dynamics. The resulting feedback controller balances cost value and closed-loop stability. Robustness of the solution is modeled using the scenario approach, without requiring any probabilistic description of the uncertainty in the system matrices. The new methods are tested and compared on the Leslie growth model, where we control population size while minimizing a suitable finite-horizon cost function.

math.OC↗

arXiv2019

Preliminary Results in Current Profile Estimation and Doppler-aided Navigation for Autonomous Underwater Gliders

This paper describes the development and experimental results of navigation algorithms for an autonomous underwater glider (AUG) that uses an on-board acoustic Doppler current profiler (ADCP). AUGs are buoyancy-driven autonomous underwater vehicles that use small hydrofoils to make forward progress while profiling vertically. During each dive, which can last up to 6 hours, the Seaglider AUG used in this experiment typically reaches the depth of 1000 m and travels 3-6 km horizontally through the water, relying solely on dead-reckoning. Horizontal through-the-water (TTW) progress of AUG is 20-30 cm/s, which is comparable to the speed of the stronger ocean currents. Underwater navigation of an AUG in the presence of unknown advection therefore presents a considerable challenge. We develop two related formulations for post-processing. Both use ADCP observations, through the water velocity estimates, and GPS fixes to estimate current profiles. However, while the first solves an explicit inverse problem for the current profiles only, the second solves a deconvolution problem that infers both states and current profiles using a state-space model. Both approaches agree on their estimates of the ocean current profile through which the AUG was flown using measurements of current relative to the AUG from the ADCP, and estimates of the AUGs TTW velocity from a hydrodynamic model. The result is a complete current profile along the AUGs trajectory, as well as over-the-ground (OTG) velocities for the AUG that can be used for more accurate subsea positioning. Results are demonstrated using 1 MHz ADCP data collected from a Seaglider AUG deployed for 49 days off the north coast of Alaska during August and September 2017. The results are compared to ground truth data from the top 40 meters of the water column, from a moored, upward-facing 600 kHz ADCP.

math.OC↗

arXiv2019

On the Global Minimizers of Real Robust Phase Retrieval with Sparse Noise

We study a class of real robust phase retrieval problems under a Gaussian assumption on the coding matrix when the received signal is sparsely corrupted by noise. The goal is to establish conditions on the sparsity under which the input vector can be exactly recovered. The recovery problem is formulated as the minimization of the $\ell_1$ norm of the residual. The main contribution is a robust phase retrieval counterpart to the seminal paper by Candes and Tao on compressed sensing ($\ell_1$ regression) [Decoding by linear programming. IEEE Transactions on Information Theory, 51(12):4203-4215, 2005]. Our analysis depends on a key new property on the coding matrix which we call the {Absolute Range Property} (ARP). This property is an analogue to the Null Space Property (NSP) in compressed sensing. When the residuals are computed using squared magnitudes, we show that ARP follows from a standard Restricted Isometry Property (RIP). However, when the residuals are computed using absolute magnitudes, a new and very different kind of RIP or growth property is required. We conclude by showing that the robust phase retrieval objectives are sharp with respect to their minimizers with high probability.

cs.IT↗

arXiv2019

Offline state estimation for hybrid systems via nonsmooth variable projection

A hybrid dynamical system switches between dynamic regimes at time- or state-triggered events. We propose an offline algorithm that simultaneously estimates discrete and continuous components of a hybrid system's state. We formulate state estimation as a continuous optimization problem by relaxing the discrete component and use a robust loss function to accommodate large changes in the continuous component during switching events. Subsequently, we develop a novel nonsmooth variable projection algorithm with Gauss-Newton updates to solve the state estimation problem and prove the algorithm's global convergence to stationary points. We demonstrate the effectiveness of our approach on simple piecewise-linear and -nonlinear mechanical systems undergoing intermittent impact.

math.OC↗

arXiv2019

Robust Singular Smoothers For Tracking Using Low-Fidelity Data

Tracking underwater autonomous platforms is often difficult because of noisy, biased, and discretized input data. Classic filters and smoothers based on standard assumptions of Gaussian white noise break down when presented with any of these challenges. Robust models (such as the Huber loss) and constraints (e.g. maximum velocity) are used to attenuate these issues. Here, we consider robust smoothing with singular covariance, which covers bias and correlated noise, as well as many specific model types, such as those used in navigation. In particular, we show how to combine singular covariance models with robust losses and state-space constraints in a unified framework that can handle very low-fidelity data. A noisy, biased, and discretized navigation dataset from a submerged, low-cost inertial measurement unit (IMU) package, with ultra short baseline (USBL) data for ground truth, provides an opportunity to stress-test the proposed framework with promising results. We show how robust modeling elements improve our ability to analyze the data, and present batch processing results for 10 minutes of data with three different frequencies of available USBL position fixes (gaps of 30 seconds, 1 minute, and 2 minutes). The results suggest that the framework can be extended to real-time tracking using robust windowed estimation.

math.OC↗

arXiv2019

M-estimation with the Trimmed l1 Penalty

We study high-dimensional estimators with the trimmed $\ell_1$ penalty, which leaves the $h$ largest parameter entries penalty-free. While optimization techniques for this nonconvex penalty have been studied, the statistical properties have not yet been analyzed. We present the first statistical analyses for $M$-estimation and characterize support recovery, $\ell_\infty$ and $\ell_2$ error of the trimmed $\ell_1$ estimates as a function of the trimming parameter $h$. Our results show different regimes based on how $h$ compares to the true support size. Our second contribution is a new algorithm for the trimmed regularization problem, which has the same theoretical convergence rate as the difference of convex (DC) algorithms, but in practice is faster and finds lower objective values. Empirical evaluation of $\ell_1$ trimming for sparse linear regression and graphical model estimation indicate that trimmed $\ell_1$ can outperform vanilla $\ell_1$ and non-convex alternatives. Our last contribution is to show that the trimmed penalty is beneficial beyond $M$-estimation, and yields promising results for two deep learning tasks: input structures recovery and network sparsification.

math.ST↗

arXiv2018

Basis Pursuit Denoise with Nonsmooth Constraints

Level-set optimization formulations with data-driven constraints minimize a regularization functional subject to matching observations to a given error level. These formulations are widely used, particularly for matrix completion and sparsity promotion in data interpolation and denoising. The misfit level is typically measured in the l2 norm, or other smooth metrics. In this paper, we present a new flexible algorithmic framework that targets nonsmooth level-set constraints, including L1, Linf, and even L0 norms. These constraints give greater flexibility for modeling deviations in observation and denoising, and have significant impact on the solution. Measuring error in the L1 and L0 norms makes the result more robust to large outliers, while matching many observations exactly. We demonstrate the approach for basis pursuit denoise (BPDN) problems as well as for extensions of BPDN to matrix factorization, with applications to interpolation and denoising of 5D seismic data. The new methods are particularly promising for seismic applications, where the amplitude in the data varies significantly, and measurement noise in low-amplitude regions can wreak havoc for standard Gaussian error models.

math.OC↗

arXiv2018

A Relaxed Optimization Approach for Cardinality-Constrained Portfolio Optimization

A cardinality-constrained portfolio caps the number of stocks to be traded across and within groups or sectors. These limitations arise from real-world scenarios faced by fund managers, who are constrained by transaction costs and client preferences as they seek to maximize return and limit risk. We develop a new approach to solve cardinality-constrained portfolio optimization problems, extending both Markowitz and conditional value at risk (CVaR) optimization models with cardinality constraints. We derive a continuous relaxation method for the NP-hard objective, which allows for very efficient algorithms with standard convergence guarantees for nonconvex problems. For smaller cases, where brute force search is feasible to compute the globally optimal cardinality- constrained portfolio, the new approach finds the best portfolio for the cardinality-constrained Markowitz model and a very good local minimum for the cardinality-constrained CVaR model. For higher dimensions, where brute-force search is prohibitively expensive, we find feasible portfolios that are nearly as efficient as their non-cardinality constrained counterparts.

math.OC↗

arXiv2018

Relax-and-split method for nonsmooth nonconvex problems

We develop and analyze a new `relax-and-split' (RS) approach for compositions of separable nonconvex nonsmooth functions with linear maps. RS uses a relaxation technique together with partial minimization, and brings classic techniques including direct factorization, matrix decompositions, and fast iterative methods to bear on nonsmooth nonconvex problems. We also extend the approach to trimmed nonconvex-composite formulations; the resulting Trimmed RS (TRS) can fit models while detecting outliers in the data. We then test RS and TRS on a diverse set of applications: (1) phase retrieval, (2) stochastic shortest path problems, (3) semi-supervised classification, and (4) new clustering approaches. RS/TRS can be applied to models with very weak functional assumptions, are easy to implement, competitive with existing methods, and enable a new level of modeling formulations to be put forward to address emerging challenges in the mathematical sciences.

math.OC↗

arXiv2018

Relaxation algorithms for matrix completion, with applications to seismic travel-time data interpolation

Travel time tomography is used to infer the underlying three-dimensional wavespeed structure of the Earth by fitting seismic travel time data collected at surface stations. Data interpolation and denoising techniques are important pre-processing steps that use prior knowledge about the data, including parsimony in the frequency and wavelet domains, low-rank structure of matricizations, and local smoothness. We show how local smoothness structure can be combined with low rank constraints using level-set optimization formulations, and develop a new relaxation algorithm that can efficiently solve these joint problems. In the seismology setting, we use the approach to interpolate missing stations and de-noise observed stations. The new approach is competitive with alternative algorithms, and offers new functionality to interpolate observed data using both smoothness and low rank structure in the presence of data fitting constraints.

math.OC↗

arXiv2018

Simultaneous shot inversion for nonuniform geometries using fast data interpolation

Stochastic optimization is key to efficient inversion in PDE-constrained optimization. Using 'simultaneous shots', or random superposition of source terms, works very well in simple acquisition geometries where all sources see all receivers, but this rarely occurs in practice. We develop an approach that interpolates data to an ideal acquisition geometry while solving the inverse problem using simultaneous shots. The approach is formulated as a joint inverse problem, combining ideas from low-rank interpolation with full-waveform inversion. Results using synthetic experiments illustrate the flexibility and efficiency of the approach.

math.OC↗

arXiv2017

Robust and scalable methods for the dynamic mode decomposition

The dynamic mode decomposition (DMD) is a broadly applicable dimensionality reduction algorithm that approximates a matrix containing time-series data by the outer product of a matrix of exponentials, representing Fourier-like time dynamics, and a matrix of coefficients, representing spatial structures. This interpretable spatio-temporal decomposition is commonly computed using linear algebraic techniques in its simplest formulation or a nonlinear optimization procedure within the variable projection framework. For data with sparse outliers or data which are not well-represented by exponentials in time, the standard Frobenius norm fit of the data creates significant biases in the recovered time dynamics. As a result, practitioners are left to clean such defects from the data manually or to use a black-box cleaning approach like robust PCA. As an alternative, we propose a framework and a set of algorithms for incorporating robust features into the nonlinear optimization used to compute the DMD itself. The algorithms presented are flexible, allowing for regu- larizers and constraints on the optimization, and scalable, using a stochastic approach to decrease the computational cost for data in high dimensional space. Both synthetic and real data examples are provided.

math.OC↗

arXiv2017

Adapting Regularized Low Rank Models for Parallel Architectures

We introduce a reformulation of regularized low-rank recovery models to take advantage of GPU, multiple CPU, and hybridized architectures. Low-rank recovery often involves nuclear-norm minimization through iterative thresholding of singular values. These models are slow to fit and difficult to parallelize because of their dependence on computing a singular value decomposition at each iteration. Regularized low-rank recovery models also incorporate non-smooth terms to separate structured components (e.g. sparse outliers) from the low-rank component, making these problems more difficult. Using Burer-Monteiro splitting and marginalization, we develop a smooth, non-convex formulation of regularized low-rank recovery models that can be fit with first-order solvers. We develop a computable certificate of convergence for this non-convex program, and use it to establish bounds on the suboptimality of any point. Using robust principal component analysis (RPCA) as an example, we include numerical experiments showing that this approach is an order-of-magnitude faster than existing RPCA solvers on the GPU. We also show that this acceleration allows new applications for RPCA, including real-time background subtraction and MR image analysis.

math.OC↗