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1. Nonlinear Data-Driven Control via State-Dependent Representations

   https://doi.org/10.1109/CDC45484.2021.9682919  

   Dai, T.; Sznaier, M. (December 2021, 60th IEEE Conf. Decision and Control)

   Recently, there has been renewed interest in data-driven control, that is, the design of controllers directly from observed data. In the case of linear time-invariant (LTI) systems, several approaches have been proposed that lead to tractable optimization problems. On the other hand, the case of nonlinear dynamics is considerably less developed, with existing approaches limited to at most rational dynamics and requiring the solution to a computationally expensive Sum of Squares (SoS) optimization. Since SoS problems typically scale combinatorially with the size of the problem, these approaches are limited to relatively low order systems. In this paper, we propose an alternative, based on the use of state-dependent representations. This idea allows for synthesizing data-driven controllers by solving at each time step an on-line optimization problem whose complexity is comparable to the LTI case. Further, the proposed approach is not limited to rational dynamics. The main result of the paper shows that the feasibility of this on-line optimization problem guarantees that the proposed controller renders the origin a globally asymptotically stable equilibrium point of the closed-loop system. These results are illustrated with some simple examples. The paper concludes by briefly discussing the prospects for adding performance criteria. 

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2. Hilbert’s 17th problem in free skew fields

   https://doi.org/10.1017/fms.2021.54  

   Volčič, Jurij (January 2020, Forum of Mathematics, Sigma)

   Abstract This paper solves the rational noncommutative analogue of Hilbert’s 17th problem: if a noncommutative rational function is positive semidefinite on all tuples of Hermitian matrices in its domain, then it is a sum of Hermitian squares of noncommutative rational functions. This result is a generalisation and culmination of earlier positivity certificates for noncommutative polynomials or rational functions without Hermitian singularities. More generally, a rational Positivstellensatz for free spectrahedra is given: a noncommutative rational function is positive semidefinite or undefined at every matricial solution of a linear matrix inequality $$L\succeq 0$$ if and only if it belongs to the rational quadratic module generated by L . The essential intermediate step toward this Positivstellensatz for functions with singularities is an extension theorem for invertible evaluations of linear matrix pencils. 

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3. Learning Stochastic Dynamical Systems via Bridge Sampling

   https://doi.org/10.1007/978-3-030-39098-3_14  

   Bhat, Harish S; Rawat, Shagun (January 2020, Advanced Analytics and Learning on Temporal Data. AALTD 2019. Lecture Notes in Computer Science)

   We develop algorithms to automate discovery of stochastic dynamical system models from noisy, vector-valued time series. By discovery, we mean learning both a nonlinear drift vector field and a diagonal diffusion matrix for an Itô stochastic differential equation in Rd . We parameterize the vector field using tensor products of Hermite polynomials, enabling the model to capture highly nonlinear and/or coupled dynamics. We solve the resulting estimation problem using expectation maximization (EM). This involves two steps. We augment the data via diffusion bridge sampling, with the goal of producing time series observed at a higher frequency than the original data. With this augmented data, the resulting expected log likelihood maximization problem reduces to a least squares problem. We provide an open-source implementation of this algorithm. Through experiments on systems with dimensions one through eight, we show that this EM approach enables accurate estimation for multiple time series with possibly irregular observation times. We study how the EM method performs as a function of the amount of data augmentation, as well as the volume and noisiness of the data. 

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4. Sparse High-Dimensional Matrix-Valued Graphical Model Learning from Dependent Data

   https://doi.org/10.1109/SSP53291.2023.10208070  

   Tugnait, Jitendra K. (July 2023, 2023 IEEE Statistical Signal Processing Workshop (SSP))

   We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional, stationary matrix-variate Gaussian time series. All past work on matrix graphical models assume that i.i.d. observations of matrix-variate are available. Here we allow dependent observations. We consider a sparse-group lasso based frequency-domain formulation of the problem with a Kronecker-decomposable power spectral density (PSD), and solve it via an alternating direction method of multipliers (ADMM) approach. The problem is bi-convex which is solved via flip-flop optimization. We provide sufficient conditions for local convergence in the Frobenius norm of the inverse PSD estimators to the true value. This results also yields a rate of convergence. We illustrate our approach using numerical examples. 

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5. Identifiability of Label Noise Transition Matrix

   Liu, Yang; Cheng, Hao; Zhang, Kun (July 2023, International Conference on Machine Learning (ICML) 2023)

   The noise transition matrix plays a central role in the problem of learning with noisy labels. Among many other reasons, a large number of existing solutions rely on access to it. Identifying and estimating the transition matrix without ground truth labels is a critical and challenging task. When label noise transition depends on each instance, the problem of identifying the instance-dependent noise transition matrix becomes substantially more challenging. Despite recent works proposing solutions for learning from instance-dependent noisy labels, the field lacks a unified understanding of when such a problem remains identifiable. The goal of this paper is to characterize the identifiability of the label noise transition matrix. Building on Kruskal's identifiability results, we are able to show the necessity of multiple noisy labels in identifying the noise transition matrix for the generic case at the instance level. We further instantiate the results to explain the successes of the state-of-the-art solutions and how additional assumptions alleviated the requirement of multiple noisy labels. Our result also reveals that disentangled features are helpful in the above identification task and we provide empirical evidence. 

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