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  1. We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

     
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    Free, publicly-accessible full text available March 1, 2025
  2. We present a novel framework for learning cost-efficient latent representations in problems with highdimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms we account for key nonlinear interactions existing in the data thereby reducing the problem’s intrinsic dimensionality. Two methods are introduced for learning the representation of such low-dimensional, polynomial manifolds for embedding the data. The manifold parametrization coefficients can be obtained by regression via either a proper orthogonal decomposition or an alternating minimization based approach. Our numerical results focus on the one-dimensional Korteweg-de Vries equation where accounting for nonlinear correlations in the data was found to lower the representation error by up to two orders of magnitude compared to linear dimension reduction techniques. 
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    Free, publicly-accessible full text available December 13, 2024
  3. An extensively studied phenomenon of the past few years in training deep networks is the implicit bias of gradient descent towards parsimonious solutions. In this work, we further investigate this phenomenon by narrowing our focus to deep matrix factorization, where we reveal surprising low-dimensional structures in the learning dynamics when the target matrix is low-rank. Specifically, we show that the evolution of gradient descent starting from arbitrary orthogonal initialization only affects a minimal portion of singular vector spaces across all weight matrices. In other words, the learning process happens only within a small invariant subspace of each weight matrix, despite the fact that all parameters are updated throughout training. From this, we provide rigorous justification for low-rank training in a specific, yet practical setting. In particular, we demonstrate that we can construct compressed factorizations that are equivalent to full-width, deep factorizations throughout training for solving low-rank matrix completion problems efficiently. 
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    Free, publicly-accessible full text available November 6, 2024
  4. Principal component analysis (PCA) is a key tool in the field of data dimensionality reduction that is useful for various data science problems. However, many applications involve heterogeneous data that varies in quality due to noise characteristics associated with different sources of the data. Methods that deal with this mixed dataset are known as heteroscedastic methods. Current methods like HePPCAT make Gaussian assumptions of the basis coefficients that may not hold in practice. Other methods such as Weighted PCA (WPCA) assume the noise variances are known, which may be difficult to know in practice. This paper develops a PCA method that can estimate the sample-wise noise variances and use this information in the model to improve the estimate of the subspace basis associated with the low-rank structure of the data. This is done without distributional assumptions of the low-rank component and without assuming the noise variances are known. Simulations show the effectiveness of accounting for such heteroscedasticity in the data, the benefits of using such a method with all of the data versus retaining only good data, and comparisons are made against other PCA methods established in the literature like PCA, Robust PCA (RPCA), and HePPCAT. Code available at https://github.com/javiersc1/ALPCAH. 
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    Free, publicly-accessible full text available July 10, 2024
  5. Learning how to effectively control unknown dynamical systems from data is crucial for intelligent autonomous systems. This task becomes a significant challenge when the underlying dynamics are changing with time. Motivated by this challenge, this paper considers the problem of controlling an unknown Markov jump linear system (MJS) to optimize a quadratic objective in a data-driven way. By taking a model-based perspective, we consider identification-based adaptive control for MJS. We first provide a system identification algorithm for MJS to learn the dynamics in each mode as well as the Markov transition matrix, underlying the evolution of the mode switches, from a single trajectory of the system states, inputs, and modes. Through mixing-time arguments, sample complexity of this algorithm is shown to be O(1/T−−√). We then propose an adaptive control scheme that performs system identification together with certainty equivalent control to adapt the controllers in an episodic fashion. Combining our sample complexity results with recent perturbation results for certainty equivalent control, we prove that when the episode lengths are appropriately chosen, the proposed adaptive control scheme achieves O(T−−√) regret. Our proof strategy introduces innovations to handle Markovian jumps and a weaker notion of stability common in MJSs. Our analysis provides insights into system theoretic quantities that affect learning accuracy and control performance. Numerical simulations are presented to further reinforce these insights. 
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  6. Real-world control applications often involve complex dynamics subject to abrupt changes or variations. Markov jump linear systems (MJS) provide a rich framework for modeling such dynamics. Despite an extensive history, theoretical understanding of parameter sensitivities of MJS control is somewhat lacking. Motivated by this, we investigate robustness aspects of certainty equivalent model-based optimal control for MJS with a quadratic cost function. Given the uncertainty in the system matrices and in the Markov transition matrix is bounded by ϵ and η respectively, robustness results are established for (i) the solution to coupled Riccati equations and (ii) the optimal cost, by providing explicit perturbation bounds that decay as O(ε+η) and O((ε+η)2) respectively. 
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  7. We propose a new fast streaming algorithm for the tensor completion problem of imputing missing entries of a lowtubal-rank tensor using the tensor singular value decomposition (t-SVD) algebraic framework. We show the t-SVD is a specialization of the well-studied block-term decomposition for third-order tensors, and we present an algorithm under this model that can track changing free submodules from incomplete streaming 2-D data. The proposed algorithm uses principles from incremental gradient descent on the Grassmann manifold of subspaces to solve the tensor completion problem with linear complexity and constant memory in the number of time samples. We provide a local expected linear convergence result for our algorithm. Our empirical results are competitive in accuracy but much faster in compute time than state-of-the-art tensor completion algorithms on real applications to recover temporal chemo-sensing and MRI data under limited sampling. 
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  8. The K-subspaces (KSS) method is a generalization of the K-means method for subspace clustering. In this work, we present local convergence analysis and a recovery guarantee for KSS, assuming data are generated by the semi-random union of subspaces model, where N points are randomly sampled from K ≥ 2 overlapping subspaces. We show that if the initial assignment of the KSS method lies within a neighborhood of a true clustering, it converges at a superlinear rate and finds the correct clustering within (log logN) iterations with high probability. Moreover, we propose a thresholding inner-product based spectral method for initialization and prove that it produces a point in this neighborhood. We also present numerical results of the studied method to support our theoretical developments. 
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  9. AutoRegressive eXogenous (ARX) models form one of the most important model classes in control theory, econometrics, and statistics, but they are yet to be understood in terms of their finite sample identification analysis. The technical challenges come from the strong statistical dependency not only between data samples at different time instances but also between elements within each individual sample. In this work, for ARX models with potentially unknown orders, we study how ordinary least squares (OLS) estimator performs in terms of identifying model parameters from data collected from either a single length-T trajectory or N i.i.d. trajectories. Our main results show that as long as the orders of the model are chosen optimistically, i.e., we are learning an over-parameterized model compared to the ground truth ARX, the OLS will converge with the optimal rate O(1/√T) (or O(1/√N)) to the true (low-order) ARX parameters. This occurs without the aid of any regularization, thus is referred to as self-regularization. Our results imply that the oracle knowledge of the true orders and usage of regularizers are not necessary in learning ARX models — over-parameterization is all you need 
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