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This paper considers the problem of error in variables identification for switched affine models. Since it is well known that this problem is generically NP hard, several relaxations have been proposed in the literature. However, while these approaches work well for low dimensional systems with few subsystems, they scale poorly with both the number of subsystems and their memory. To address this difficulty, we propose a computationally efficient alternative, based on embedding the data in the manifold of positive semidefinite matrices, and using a manifold metric there to perform the identification. Our main result shows that, under dwell-time assumptions, the proposed algorithm is convergent, in the sense that it is guaranteed to identify the system for suitably low noise. In scenarios with larger noise levels, we provide experimental results showing that the proposed method outperforms existing ones. The paper concludes by illustrating these results with academic examples and a non-trivial application: action video segmentation. 

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 dis- cover 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. 

One of the long-term objectives of Machine Learning is to endow machines with the capacity of structuring and interpreting the world as we do. This is particularly challenging in scenes involving time series, such as video sequences, since seemingly different data can correspond to the same underlying dynamics. Recent approaches seek to decompose video sequences into their composing objects, attributes and dynamics in a self-supervised fashion, thus simplifying the task of learning suitable features that can be used to analyze each component. While existing methods can successfully disentangle dynamics from other components, there have been relatively few efforts in learning parsimonious representations of these underlying dynamics. In this paper, motivated by recent advances in non-linear identification, we propose a method to decompose a video into moving objects, their attributes and the dynamic modes of their trajectories. We model video dynamics as the output of a Koopman operator to be learned from the available data. In this context, the dynamic information contained in the scene is encapsulated in the eigenvalues and eigenvectors of the Koopman operator, providing an interpretable and parsimonious representation. We show that such decomposition can be used for instance to perform video analytics, predict future frames or generate synthetic video. We test our framework in a variety of datasets that encompass different dynamic scenarios, while illustrating the novel features that emerge from our dynamic modes decomposition: Video dynamics interpretation and user manipulation at test-time. We successfully forecast challenging object trajectories from pixels, achieving competitive performance while drawing useful insights. 

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. 

This paper addresses the problem of identification of error in variables switched linear models from experimental input/output data. This problem is known to be generically NP hard and thus computationally expensive to solve. To address this difficulty, several relaxations have been proposed in the past few years. While solvable in polynomial time these (convex) relaxations tend to scale poorly with the number of points and number/order of the subsystems, effectively limiting their applicability to scenarios with relatively small number of data points. To address this difficulty, in this paper we propose an efficient method that only requires performing (number of subsystems) singular value decompositions of matrices whose size is independent of the number of points. The underlying idea is to obtain a sum-of-squares polynomial approximation of the support of each subsystem one-at-a-time, and use these polynomials to segment the data into sets, each generated by a single subsystem. As shown in the paper, exploiting ideas from Christoffel's functions allows for finding these polynomial approximations simply by performing SVDs. The parameters of each subsystem can then be identified from the segmented data using existing error-in-variables (EIV) techniques. 

In this paper we propose a data-driven fault detection framework for semi-supervised scenarios where labeled training data from the system under consideration (the “target”) is imbalanced (e.g. only relatively few labels are available from one of the classes), but data from a related system (the “source”) is readily available. An example of this situation is when a generic simulator is available, but needs to be tuned on a case-by-case basis to match the parameters of the actual system. The goal of this paper is to work with the statistical distribution of the data without necessitating system identification. Our main result shows that if the source and target domain are related by a linear transformation (a common assumption in domain adaptation), the problem of designing a classifier that minimizes a miss-classification loss over the joint source and target domains reduces to a convex optimization subject to a single (non-convex) equality constraint. This second-order equality constraint can be recast as a rank-1 optimization problem, where the rank constraint can be efficiently handled through a reweighted nuclear norm surrogate. These results are illustrated with a practical application: fault detection in additive manufacturing (industrial 3D printing). The proposed method is able to exploit simulation data (source domain) to substantially outperform classifiers tuned using only data from a single domain. 

This paper addresses the problem of subspace clustering in the presence of outliers. Typically, this scenario is handled through a regularized optimization, whose computational complexity scales polynomially with the size of the data. Further, the regularization terms need to be manually tuned to achieve optimal performance. To circumvent these difficulties, in this paper we propose an outlier removal algorithm based on evaluating a suitable sum-ofsquares polynomial, computed directly from the data. This algorithm only requires performing two singular value decompositions of fixed size, and provides certificates on the probability of misclassifying outliers as inliers.