Randomized Dynamic Mode Decomposition

This paper presents a randomized algorithm for computing the near-optimal low-rank dynamic mode decomposition (DMD). Randomized algorithms are emerging techniques to compute low-rank matrix approximations at a fraction of the cost of deterministic algorithms, easing the computational challenges arising in the area of `big data'. The idea is to derive a small matrix from the high-dimensional data, which is then used to efficiently compute the dynamic modes and eigenvalues. The algorithm is presented in a modular probabilistic framework, and the approximation quality can be controlled via oversampling and power iterations. The effectiveness of the resulting randomized DMD algorithm is demonstrated on several benchmark examples of increasing complexity, providing an accurate and efficient approach to extract spatiotemporal coherent structures from big data in a framework that scales with the intrinsic rank of the data, rather than the ambient measurement dimension. For this work we assume that the dynamics of the problem under consideration is evolving on a low-dimensional subspace that is well characterized by a fast decaying singular value spectrum

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.Comment: 8 pages, 2 figure

Analyzing sparse dictionaries for online learning with kernels

Many signal processing and machine learning methods share essentially the same linear-in-the-parameter model, with as many parameters as available samples as in kernel-based machines. Sparse approximation is essential in many disciplines, with new challenges emerging in online learning with kernels. To this end, several sparsity measures have been proposed in the literature to quantify sparse dictionaries and constructing relevant ones, the most prolific ones being the distance, the approximation, the coherence and the Babel measures. In this paper, we analyze sparse dictionaries based on these measures. By conducting an eigenvalue analysis, we show that these sparsity measures share many properties, including the linear independence condition and inducing a well-posed optimization problem. Furthermore, we prove that there exists a quasi-isometry between the parameter (i.e., dual) space and the dictionary's induced feature space.Comment: 10 page

Best Subset Selection via a Modern Optimization Lens

In the last twenty-five years (1990-2014), algorithmic advances in integer optimization combined with hardware improvements have resulted in an astonishing 200 billion factor speedup in solving Mixed Integer Optimization (MIO) problems. We present a MIO approach for solving the classical best subset selection problem of choosing $k$ out of $p$ features in linear regression given $n$ observations. We develop a discrete extension of modern first order continuous optimization methods to find high quality feasible solutions that we use as warm starts to a MIO solver that finds provably optimal solutions. The resulting algorithm (a) provides a solution with a guarantee on its suboptimality even if we terminate the algorithm early, (b) can accommodate side constraints on the coefficients of the linear regression and (c) extends to finding best subset solutions for the least absolute deviation loss function. Using a wide variety of synthetic and real datasets, we demonstrate that our approach solves problems with $n$ in the 1000s and $p$ in the 100s in minutes to provable optimality, and finds near optimal solutions for $n$ in the 100s and $p$ in the 1000s in minutes. We also establish via numerical experiments that the MIO approach performs better than {\texttt {Lasso}} and other popularly used sparse learning procedures, in terms of achieving sparse solutions with good predictive power.Comment: This is a revised version (May, 2015) of the first submission in June 201

SISSO: a compressed-sensing method for identifying the best low-dimensional descriptor in an immensity of offered candidates

The lack of reliable methods for identifying descriptors - the sets of parameters capturing the underlying mechanisms of a materials property - is one of the key factors hindering efficient materials development. Here, we propose a systematic approach for discovering descriptors for materials properties, within the framework of compressed-sensing based dimensionality reduction. SISSO (sure independence screening and sparsifying operator) tackles immense and correlated features spaces, and converges to the optimal solution from a combination of features relevant to the materials' property of interest. In addition, SISSO gives stable results also with small training sets. The methodology is benchmarked with the quantitative prediction of the ground-state enthalpies of octet binary materials (using ab initio data) and applied to the showcase example of predicting the metal/insulator classification of binaries (with experimental data). Accurate, predictive models are found in both cases. For the metal-insulator classification model, the predictive capability are tested beyond the training data: It rediscovers the available pressure-induced insulator->metal transitions and it allows for the prediction of yet unknown transition candidates, ripe for experimental validation. As a step forward with respect to previous model-identification methods, SISSO can become an effective tool for automatic materials development.Comment: 11 pages, 5 figures, in press in Phys. Rev. Material