arXiv subjects
Publications and source records attributed to Jakub Marecek.
In tracking of time-varying low-rank models of time-varying matrices, we present a method robust to both uniformly-distributed measurement noise and arbitrarily-distributed ``sparse'' noise. In theory, we bound the tracking error. In practice, our use of randomised coordinate descent is scalable and allows for encouraging results on changedetection net, a benchmark.
Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, actual applications have been restricted by the fact the training data are obtained using traditional PDE solvers. Thereby, the uses of deep-learning techniques were limited to domains, where the PDE solver was applicable. We demonstrate a deep-learning framework for air-pollution monitoring and forecasting that provides the ability to train across different model domains, as well as a reduction in the run-time by two orders of magnitude. It presents a first-of-a-kind implementation that combines deep-learning and domain-decomposition techniques to allow model deployments extend beyond the domain(s) on which the it has been trained.
Many systems exhibit a mixture of continuous and discrete dynamics. We consider a family of mixed-integer non-convex non-linear optimisation problems obtained in discretisations of optimal control of such systems. For this family, a branch-and-bound algorithm solves the discretised problem to global optimality. As an example, we consider water systems, where variations in flow and variations in water levels are continuous, while decisions related to fixed-speed pumps and whether gates that may be opened and closed are discrete. We show that the related optimal-control problems come from the family we introduce -- and implement deterministic solvers with global convergence guarantees.
Many real-world monitoring and surveillance applications require non-trivial anomaly detection to be run in the streaming model. We consider an incremental-learning approach, wherein a deep-autoencoding (DAE) model of what is normal is trained and used to detect anomalies at the same time. In the detection of anomalies, we utilise a novel thresholding mechanism, based on value at risk (VaR). We compare the resulting convolutional neural network (CNN) against a number of subspace methods, and present results on changedetection net.
Within the study of uncertain dynamical systems, iterated random functions are a key tool. There, one samples a family of functions according to a stationary distribution. Here, we introduce an extension, where one sample functions according to a time-varying distribution over the family of functions. For such iterated piecewise-stationary random functions on Polish spaces, we prove a number of results, including a bound on the tracking error.
We study the projection onto the set of feasible inputs and the set of feasible solutions of a polynomial optimisation problem (POP). Our motivation is increasing the robustness of solvers for POP: Without a priori guarantees of feasibility of a particular instance, one should like to perform the projection onto the set of feasible inputs prior to running a solver. Without a certificate of optimality, one should like to project the output of the solver onto the set of feasible solutions subsequently. We study the computational complexity, formulations, and convexifications of the projections. Our results are illustrated on IEEE test cases of Alternating Current Optimal Power Flow (ACOPF) problem.
Time-varying non-convex continuous-valued non-linear constrained optimization is a fundamental problem. We study conditions wherein a momentum-like regularising term allow for the tracking of local optima by considering an ordinary differential equation (ODE). We then derive an efficient algorithm based on a predictor-corrector method, to track the ODE solution.
Low-rank methods for semidefinite programming (SDP) have gained a lot of interest recently, especially in machine learning applications. Their analysis often involves determinant-based or Schatten-norm penalties, which are hard to implement in practice due to high computational efforts. In this paper, we propose Entropy Penalized Semi-definite programming (EP-SDP) which provides a unified framework for a wide class of penalty functions used in practice to promote a low-rank solution. We show that EP-SDP problems admit efficient numerical algorithm having (almost) linear time complexity of the gradient iteration which makes it useful for many machine learning and optimization problems. We illustrate the practical efficiency of our approach on several combinatorial optimization and machine learning problems.
There has been much recent interest in hierarchies of progressively stronger convexifications of polynomial optimisation problems (POP). These often converge to the global optimum of the POP, asymptotically, but prove challenging to solve beyond the first level in the hierarchy for modest instances. We present a finer-grained variant of the Lasserre hierarchy, together with first-order methods for solving the convexifications, which allow for efficient warm-starting with solutions from lower levels in the hierarchy.
Across smart-grid and smart-city application domains, there are many problems where an ensemble of agents is to be controlled such that both the aggregate behaviour and individual-level perception of the system's performance are acceptable. In many applications, traditional PI control is used to regulate aggregate ensemble performance. Our principal contribution in this note is to demonstrate that PI control may not be always suitable for this purpose, and in some situations may lead to a loss of ergodicity for closed-loop systems. Building on this observation, a theoretical framework is proposed to both analyse and design control systems for the regulation of large scale ensembles of agents with a probabilistic intent. Examples are given to illustrate our results.
We compare three formulations of stationary equations of the Kuramoto model as systems of polynomial equations. In the comparison, we present bounds on the numbers of real equilibria based on the work of Bernstein, Kushnirenko, and Khovanskii, and performance of methods for the optimisation over the set of equilibria based on the work of Lasserre, both of which could be of independent interest.
In scheduling and timetabling applications, the mutual-exclusion constraint stipulates that certain pairs of tasks that cannot be executed at the same time. This corresponds to the vertex colouring problem in graph theory, for which there are well-known semidefinite programming (SDP) relaxations. In practice, however, the mutual-exclusion constraint is typically combined with many other constraints, whose SDP representability has not been studied. We present SDP relaxations for a variety of mutual-exclusion scheduling and timetabling problems, starting from a bound on the number of tasks executed within each period, which corresponds to graph colouring bounded in the number of uses of each colour. In theory, this provides the strongest known bounds for these problems that are computable to any precision in time polynomial in the dimensions. In practice, we report encouraging computational results on random graphs, Knesser graphs, ``forbidden intersection'' graphs, the Toronto benchmark, and the International Timetabling Competition.
Kalman filter is a key tool for time-series forecasting and analysis. We show that the dependence of a prediction of Kalman filter on the past is decaying exponentially, whenever the process noise is non-degenerate. Therefore, Kalman filter may be approximated by regression on a few recent observations. Surprisingly, we also show that having some process noise is essential for the exponential decay. With no process noise, it may happen that the forecast depends on all of the past uniformly, which makes forecasting more difficult. Based on this insight, we devise an on-line algorithm for improper learning of a linear dynamical system (LDS), which considers only a few most recent observations. We use our decay results to provide the first regret bounds w.r.t. to Kalman filters within learning an LDS. That is, we compare the results of our algorithm to the best, in hindsight, Kalman filter for a given signal. Also, the algorithm is practical: its per-update run-time is linear in the regression depth.
We consider a setting, where the output of a linear dynamical system (LDS) is, with an unknown but fixed probability, replaced by noise. There, we present a robust method for the prediction of the outputs of the LDS and identification of the samples of noise, and prove guarantees on its statistical performance. One application lies in anomaly detection: the samples of noise, unlikely to have been generated by the dynamics, can be flagged to operators of the system for further study.
The extraction and understanding of temporal events and their relations are major challenges in natural language processing. Processing text on a sentence-by-sentence or expression-by-expression basis often fails, in part due to the challenge of capturing the global consistency of the text. We present an ensemble method, which reconciles the outputs of multiple classifiers of temporal expressions across the text using integer programming. Computational experiments show that the ensemble improves upon the best individual results from two recent challenges, SemEval-2013 TempEval-3 (Temporal Annotation) and SemEval-2016 Task 12 (Clinical TempEval).
The unit commitment with transmission constraints in the alternating-current (AC) model is a challenging mixed-integer non-linear optimisation problem. We present an approach based on decomposition of a Mixed-Integer Semidefinite Programming (MISDP) problem into a mixed-integer quadratic (MIQP) master problem and a semidefinite programming (SDP) sub-problem. Between the master problem and the sub-problem, we pass novel classes of cuts. We analyse finite convergence to the optimum of the MISDP and report promising computational results on a test case from the Canary Islands, Spain.
We consider the applicability of the data from operators of cellular systems to modelling demand for transportation. While individual-level data may contain precise paths of movement, stringent privacy rules prohibit their use without consent. Presence data aggregate the individual-level data to information on the numbers of transactions at each base transceiver station (BTS) per each time period. Our work is aimed at demonstrating value of such aggregate data for mobility management while maintaining privacy of users. In particular, given mobile subscriber activity aggregated to short time intervals for a zone, a convex optimisation problem estimates most likely transitions between zones. We demonstrate the method on presence data from Warsaw, Poland, and compare with official demand estimates obtained with classical econometric methods.
We consider the problem of estimating means of two Gaussians in a 2-Gaussian mixture, which is not balanced and is corrupted by noise of an arbitrary distribution. We present a robust algorithm to estimate the parameters, together with upper bounds on the numbers of samples required for the estimate to be correct, where the bounds are parametrised by the dimension, ratio of the mixing coefficients, a measure of the separation of the two Gaussians, related to Mahalanobis distance, and a condition number of the covariance matrix. In theory, this is the first sample-complexity result for imbalanced mixtures corrupted by adversarial noise. In practice, our algorithm outperforms the vanilla Expectation-Maximisation (EM) algorithm in terms of estimation error.