arXiv subjects

Mario Sznaier

Publications and source records attributed to Mario Sznaier.

At least 37 records · Page 2Linked to original sources

arXiv2024

Data-Driven Superstabilizing Control under Quadratically-Bounded Errors-in-Variables Noise

The Error-in-Variables model of system identification/control involves nontrivial input and measurement corruption of observed data, resulting in generically nonconvex optimization problems. This paper performs full-state-feedback stabilizing control of all discrete-time linear systems that are consistent with observed data for which the input and measurement noise obey quadratic bounds. Instances of such quadratic bounds include elementwise norm bounds (at each time sample), energy bounds (across the entire signal), and chance constraints arising from (sub)gaussian noise. Superstabilizing controllers are generated through the solution of a sum-of-squares hierarchy of semidefinite programs. A theorem of alternatives is employed to eliminate the input and measurement noise process, thus improving tractability.

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arXiv2024

Solving Masked Jigsaw Puzzles with Diffusion Vision Transformers

Solving image and video jigsaw puzzles poses the challenging task of rearranging image fragments or video frames from unordered sequences to restore meaningful images and video sequences. Existing approaches often hinge on discriminative models tasked with predicting either the absolute positions of puzzle elements or the permutation actions applied to the original data. Unfortunately, these methods face limitations in effectively solving puzzles with a large number of elements. In this paper, we propose JPDVT, an innovative approach that harnesses diffusion transformers to address this challenge. Specifically, we generate positional information for image patches or video frames, conditioned on their underlying visual content. This information is then employed to accurately assemble the puzzle pieces in their correct positions, even in scenarios involving missing pieces. Our method achieves state-of-the-art performance on several datasets.

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arXiv2024

Analysis and Control of Input-Affine Dynamical Systems using Infinite-Dimensional Robust Counterparts

Input-affine dynamical systems often arise in control and modeling scenarios, such as the data-driven case when state-derivative observations are recorded under bounded noise. Common tasks in system analysis and control include optimal control, peak estimation, reachable set estimation, and maximum control invariant set estimation. Existing work poses these types of problems as infinite-dimensional linear programs in auxiliary functions with sum-of-squares tightenings. The bottleneck in most of these programs is the Lie derivative nonnegativity constraint posed over the time-state-control set. Decomposition techniques to improve tractability by eliminating the control variables include vertex decompositions (switching), or facial decompositions in the case where the polytopic set is a scaled box. This work extends the box-facial decomposition technique to allow for a robust-counterpart decomposition of semidefinite representable sets (e.g. polytopes, ellipsoids, and projections of spectahedra). These robust counterparts are proven to be equivalent to the original Lie constraint under mild compactness and regularity constraints. Efficacy is demonstrated under peak/distance/reachable set data-driven analysis problems and Region of Attraction maximizing control.

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arXiv2024

Peak Value-at-Risk Estimation of Stochastic Processes using Occupation Measures

This paper formulates algorithms to upper-bound the maximum Value-at-Risk (VaR) of a state function along trajectories of stochastic processes. The VaR is upper bounded by two methods: minimax tail-bounds (Cantelli/Vysochanskij-Petunin) and Expected Shortfall/Conditional Value-at-Risk (ES). Tail-bounds lead to a infinite-dimensional Second Order Cone Program (SOCP) in occupation measures, while the ES approach creates a Linear Program (LP) in occupation measures. Under compactness and regularity conditions, there is no relaxation gap between the infinite-dimensional convex programs and their nonconvex optimal-stopping stochastic problems. Upper-bounds on the SOCP and LP are obtained by a sequence of semidefinite programs through the moment-Sum-of-Squares hierarchy. The VaR-upper-bounds are demonstrated on example continuous-time and discrete-time polynomial stochastic processes.

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arXiv2023

On the Hardness of Learning to Stabilize Linear Systems

Inspired by the work of Tsiamis et al. \cite{tsiamis2022learning}, in this paper we study the statistical hardness of learning to stabilize linear time-invariant systems. Hardness is measured by the number of samples required to achieve a learning task with a given probability. The work in \cite{tsiamis2022learning} shows that there exist system classes that are hard to learn to stabilize with the core reason being the hardness of identification. Here we present a class of systems that can be easy to identify, thanks to a non-degenerate noise process that excites all modes, but the sample complexity of stabilization still increases exponentially with the system dimension. We tie this result to the hardness of co-stabilizability for this class of systems using ideas from robust control.

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arXiv2023

Inferring Relational Potentials in Interacting Systems

Systems consisting of interacting agents are prevalent in the world, ranging from dynamical systems in physics to complex biological networks. To build systems which can interact robustly in the real world, it is thus important to be able to infer the precise interactions governing such systems. Existing approaches typically discover such interactions by explicitly modeling the feed-forward dynamics of the trajectories. In this work, we propose Neural Interaction Inference with Potentials (NIIP) as an alternative approach to discover such interactions that enables greater flexibility in trajectory modeling: it discovers a set of relational potentials, represented as energy functions, which when minimized reconstruct the original trajectory. NIIP assigns low energy to the subset of trajectories which respect the relational constraints observed. We illustrate that with these representations NIIP displays unique capabilities in test-time. First, it allows trajectory manipulation, such as interchanging interaction types across separately trained models, as well as trajectory forecasting. Additionally, it allows adding external hand-crafted potentials at test-time. Finally, NIIP enables the detection of out-of-distribution samples and anomalies without explicit training. Website: https://energy-based-model.github.io/interaction-potentials.

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arXiv2023

Data-Driven Superstabilization of Linear Systems under Quantization

This paper focuses on the stabilization and regulation of linear systems affected by quantization in state-transition data and actuated input. The observed data are composed of tuples of current state, input, and the next state's interval ranges based on sensor quantization. Using an established characterization of input-logarithmically-quantized stabilization based on robustness to sector-bounded uncertainty, we formulate a nonconservative infinite-dimensional linear program that enforces superstabilization of all possible consistent systems under assumed priors. We solve this problem by posing a pair of exponentially-scaling linear programs, and demonstrate the success of our method on example quantized systems.

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arXiv2023

Superstabilizing Control of Discrete-Time ARX Models under Error in Variables

This paper applies a polynomial optimization based framework towards the superstabilizing control of an Autoregressive with Exogenous Input (ARX) model given noisy data observations. The recorded input and output values are corrupted with L-infinity bounded noise where the bounds are known. This is an instance of Error in Variables (EIV) in which true internal state of the ARX system remains unknown. The consistency set of ARX models compatible with noisy data has a bilinearity between unknown plant parameters and unknown noise terms. The requirement for a dynamic compensator to superstabilize all consistent plants is expressed using polynomial nonnegativity constraints, and solved using sum-of-squares (SOS) methods in a converging hierarchy of semidefinite programs in increasing size. The computational complexity of this method may be reduced by applying a Theorem of Alternatives to eliminate the noise terms. Effectiveness of this method is demonstrated on control of example ARX models.

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arXiv2023

Cross-view Action Recognition via Contrastive View-invariant Representation

Cross view action recognition (CVAR) seeks to recognize a human action when observed from a previously unseen viewpoint. This is a challenging problem since the appearance of an action changes significantly with the viewpoint. Applications of CVAR include surveillance and monitoring of assisted living facilities where is not practical or feasible to collect large amounts of training data when adding a new camera. We present a simple yet efficient CVAR framework to learn invariant features from either RGB videos, 3D skeleton data, or both. The proposed approach outperforms the current state-of-the-art achieving similar levels of performance across input modalities: 99.4% (RGB) and 99.9% (3D skeletons), 99.4% (RGB) and 99.9% (3D Skeletons), 97.3% (RGB), and 99.2% (3D skeletons), and 84.4%(RGB) for the N-UCLA, NTU-RGB+D 60, NTU-RGB+D 120, and UWA3DII datasets, respectively.

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arXiv2023

Peak Estimation of Time Delay Systems using Occupation Measures

This work proposes a method to compute the maximum value obtained by a state function along trajectories of a Delay Differential Equation (DDE). An example of this task is finding the maximum number of infected people in an epidemic model with a nonzero incubation period. The variables of this peak estimation problem include the stopping time and the original history (restricted to a class of admissible histories). The original nonconvex DDE peak estimation problem is approximated by an infinite-dimensional Linear Program (LP) in occupation measures, inspired by existing measure-based methods in peak estimation and optimal control. This LP is approximated from above by a sequence of Semidefinite Programs (SDPs) through the moment-Sum of Squares (SOS) hierarchy. Effectiveness of this scheme in providing peak estimates for DDEs is demonstrated with provided examples

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arXiv2023

Data-Driven Stabilizing and Robust Control of Discrete-Time Linear Systems with Error in Variables

This work presents a sum-of-squares (SOS) based framework to perform data-driven stabilization and robust control tasks on discrete-time linear systems where the full-state observations are corrupted by L-infinity bounded input, measurement, and process noise (error in variable setting). Certificates of state-feedback superstability, quadratic stability or positive stability of all plants in a consistency set are provided by solving a feasibility program formed by polynomial nonnegativity constraints. Under mild compactness and data-collection assumptions, SOS tightenings in rising degree will converge to recover the true superstabilizing or positive stabilizing controller, with some conservatism introduced for quadratic stabilizability. The performance of this SOS method is improved through the application of a theorem of alternatives while retaining tightness, in which the unknown noise variables are eliminated from the consistency set description. This SOS feasibility method is extended to provide worst-case-optimal robust controllers under H2 control costs. The consistency set description may be broadened to include cases where the data and process are affected by a combination of L-infinity bounded measurement, process, and input noise.

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arXiv2023

Data-Driven Control of Positive Linear Systems using Linear Programming

This paper presents a linear-programming based algorithm to perform data-driven stabilizing control of linear positive systems. A set of state-input-transition observations is collected up to magnitude-bounded noise. A state feedback controller and dual linear copositive Lyapunov function are created such that the set of all data-consistent plants is contained within the set of all stabilized systems. This containment is certified through the use of the Extended Farkas Lemma and solved via Linear Programming. Sign patterns and sparsity structure for the controller may be imposed using linear constraints. The complexity of this algorithm scales in a polynomial manner with the number of states and inputs. Effectiveness is demonstrated on example systems.

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arXiv2023

Peak Estimation of Hybrid Systems with Convex Optimization

Peak estimation of hybrid systems aims to upper bound extreme values of a state function along trajectories, where this state function could be different in each subsystem. This finite-dimensional but nonconvex problem may be lifted into an infinite-dimensional linear program (LP) in occupation measures with an equal objective under mild finiteness/compactness and smoothness assumptions. This LP may in turn be approximated by a convergent sequence of upper bounds attained from solutions of Linear Matrix Inequalities (LMIs) using the Moment-Sum-of-Squares hierarchy. The peak estimation problem is extended to problems with uncertainty and safety settings, such as measuring the distance of closest approach between points along hybrid system trajectories and unsafe sets.

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arXiv2023

Robust Data-Driven Safe Control using Density Functions

This paper presents a tractable framework for data-driven synthesis of robustly safe control laws. Given noisy experimental data and some priors about the structure of the system, the goal is to synthesize a state feedback law such that the trajectories of the closed loop system are guaranteed to avoid an unsafe set even in the presence of unknown but bounded disturbances (process noise). The main result of the paper shows that for polynomial dynamics, this problem can be reduced to a tractable convex optimization by combining elements from polynomial optimization and the theorem of alternatives. This optimization provides both a rational control law and a density function safety certificate. These results are illustrated with numerical examples.

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arXiv2023

Bounding the Distance to Unsafe Sets with Convex Optimization

This work proposes an algorithm to bound the minimum distance between points on trajectories of a dynamical system and points on an unsafe set. Prior work on certifying safety of trajectories includes barrier and density methods, which do not provide a margin of proximity to the unsafe set in terms of distance. The distance estimation problem is relaxed to a Monge-Kantorovich type optimal transport problem based on existing occupation-measure methods of peak estimation. Specialized programs may be developed for polyhedral norm distances (e.g. L1 and Linfinity) and for scenarios where a shape is traveling along trajectories (e.g. rigid body motion). The distance estimation problem will be correlatively sparse when the distance objective is separable.

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arXiv2022

Data-Driven Gain Scheduling Control of Linear Parameter-Varying Systems using Quadratic Matrix Inequalities

This paper synthesizes a gain-scheduled controller to stabilize all possible Linear Parameter-Varying (LPV) plants that are consistent with measured input/state data records. Inspired by prior work in data informativity and LTI stabilization, a set of Quadratic Matrix Inequalities is developed to represent the noise set, the class of consistent LPV plants, and the class of stabilizable plants. The bilinearity between unknown plants and `for all' parameters is avoided by vertex enumeration of the parameter set. Effectiveness and computational tractability of this method is demonstrated on example systems.

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arXiv2022

Euclidean Distance Bounds for LMI Analytic Centers using a Novel Bound on the Lambert Function

Linear matrix inequalities (LMIs) are ubiquitous in modern control theory, as well as in a variety of other fields in science and engineering. Their analytic centers, i.e. the maximum determinant elements of the feasible set spanned by these LMIs, are the solution of many well-known problems in statistics, communications, geometry, and control and can be approximated to arbitrary precision by semidefinite programs (SDPs). The quality of these approximations is measured with respect to the difference in log-determinant of both the exact and the approximate solution to these SDPs, a quantity that follows directly from the duality theory of semidefinite programming. However, in many applications the relevant parameters are functions of the entries of the LMI argument $X$. In these cases it is of interest to develop metrics that quantify the quality of approximate solutions based on the error on these parameters, something that the log-determinant error fails to capture due to the non-linear interaction of all the matrix entries. In this work we develop upper bounds on the Frobenius norm error between suboptimal solutions $X_f$ and the exact optimizer $X_*$ of maximum determinant problems, a metric that provides a direct translation to the entry-wise error of $X$ and thus to the relevant parameters of the application. We show that these bounds can be expressed through the use of the Lambert function $W(x)$, i.e. the solution of the equation $W(x) e^{W(x)}= x$, and derive novel bounds for one of its branches to generate efficient closed-form bounds on the Euclidean distance to the LMI analytic center. Finally, we test the quality of these bounds numerically in the context of interior point methods termination criteria.

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arXiv2022

A Globally Convergent LP and SOCP-based algorithm for Semidefinite Programming

Semidefinite programs (SDP) are one of the most versatile frameworks in numerical optimization, serving as generalizations of many conic programs and as relaxations of NP-hard combinatorial problems. Their main drawback is their computational and memory complexity, which sets a practical limit to the size of problems solvable by off-the-shelf SDP solvers. To circumvent this fact, many algorithms have been proposed to exploit the structure of particular problems and increase the scalability of SDPs for those problem instances. Progress has been less steep, however, for general-case SDPs. In this paper, motivated by earlier results by Ahmadi and Hall, we show that a general SDP can be solved to $ε$-optimality, in polynomial time, by performing a sequence of less computationally demanding Linear or Second Order Cone programs. In addition, we provide a bound on the number of iterations required to achieve $ε$-optimality. These results are illustrated using random SDPs and well-known problems from the SDPLib dataset.

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