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1. Effect of data length on time delay and embedding dimension for calculating the Lyapunov exponent in walking

   https://doi.org/10.1098/rsif.2020.0311  

   Hussain, Victoria Smith; Spano, Mark L.; Lockhart, Thurmon E. (July 2020, Journal of The Royal Society Interface)

   The Lyapunov exponent (LyE) is a trending measure for characterizing gait stability. Previous studies have shown that data length has an effect on the resultant LyE, but the origin of why it changes is unknown. This study investigates if data length affects the choice of time delay and embedding dimension when reconstructing the phase space, which is a requirement for calculating the LyE. The effect of three different preprocessing methods on reconstructing the gait attractor was also investigated. Lumbar accelerometer data were collected from 10 healthy subjects walking on a treadmill at their preferred walking speed for 30 min. Our results show that time delay was not sensitive to the amount of data used during calculation. However, the embedding dimension had a minimum data requirement of 200 or 300 gait cycles, depending on the preprocessing method used, to determine the steady-state value of the embedding dimension. This study also found that preprocessing the data using a fixed number of strides or a fixed number of data points had significantly different values for time delay compared to a time series that used a fixed number of normalized gait cycles, which have a fixed number of data points per stride. Thus, comparing LyE values should match the method of calculation using either a fixed number of strides or a fixed number of data points. 

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2. Sample Complexity Bound for Evaluating the Robust Observer’s Performance under Coprime Factors Uncertainty

   Sabau, Serban; Zhang, Yifei; Ukil, Sourav Kumar (June 2023, Proceedings of Machine Learning Research)

   Lawrence, N (Ed.)

   This paper addresses the end-to-end sample complexity bound for learning in closed loop the state estimator-based robust H2 controller for an unknown (possibly unstable) Linear Time Invariant (LTI) system, when given a fixed state-feedback gain. We build on the results from Ding et al. (1994) to bridge the gap between the parameterization of all state-estimators and the celebrated Youla parameterization. Refitting the expression of the relevant closed loop allows for the optimal linear observer problem given a fixed state feedback gain to be recast as a convex problem in the Youla parameter. The robust synthesis procedure is performed by considering bounded additive model uncertainty on the coprime factors of the plant, such that a min-max optimization problem is formulated for the robust H2 controller via an observer approach. The closed-loop identification scheme follows Zhang et al. (2021), where the nominal model of the true plant is identified by constructing a Hankel-like matrix from a single time-series of noisy, finite length input-output data by using the ordinary least squares algorithm from Sarkar et al. (2020). Finally, a H∞ bound on the estimated model error is provided, as the robust synthesis procedure requires bounded additive uncertainty on the coprime factors of the model. Reference Zhang et al. (2022b) is the extended version of this paper. 

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3. Unsupervised Learning of PCFGs with Normalizing Flow

   Jin, Lifeng; Doshi-Velez, Finale; Miller, Timothy; Schwartz, Lane; Schuler, William (January 2019, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics)

   Unsupervised PCFG inducers hypothesize sets of compact context-free rules as explanations for sentences. PCFG induction not only provides tools for low-resource languages, but also plays an important role in modeling language acquisition (Bannard et al., 2009; Abend et al. 2017). However, current PCFG induction models, using word tokens as input, are unable to incorporate semantics and morphology into induction, and may encounter issues of sparse vocabulary when facing morphologically rich languages. This paper describes a neural PCFG inducer which employs context embeddings (Peters et al., 2018) in a normalizing flow model (Dinh et al., 2015) to extend PCFG induction to use semantic and morphological information. Linguistically motivated sparsity and categorical distance constraints are imposed on the inducer as regularization. Experiments show that the PCFG induction model with normalizing flow produces grammars with state-of-the-art accuracy on a variety of different languages. Ablation further shows a positive effect of normalizing flow, context embeddings and proposed regularizers. 

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4. Information Theoretically Optimal Sample Complexity of Learning Dynamical Directed Acyclic Graphs

   Veedu, Mishfad; Deka, Deepjyoti; Salapaka, Murti (May 2024, Proceedings of Machine Learning Research)

   Dasgupta, Sanjoy; Mandt, Stephen; Li, Yingzhen i (Ed.)

   In this article, the optimal sample complexity of learning the underlying interactions or dependencies of a Linear Dynamical System (LDS) over a Directed Acyclic Graph (DAG) is studied. We call such a DAG underlying an LDS as dynamical DAG (DDAG). In particular, we consider a DDAG where the nodal dynamics are driven by unobserved exogenous noise sources that are wide-sense stationary (WSS) in time but are mutually uncorrelated, and have the same power spectral density (PSD). Inspired by the static DAG setting, a metric and an algorithm based on the PSD matrix of the observed time series are proposed to reconstruct the DDAG. It is shown that the optimal sample complexity (or length of state trajectory) needed to learn the DDAG is n = Θ(q log(p/q)), where p is the number of nodes and q is the maximum number of parents per node. To prove the sample complexity upper bound, a concentration bound for the PSD estimation is derived, under two different sampling strategies. A matching min-max lower bound using generalized Fano’s inequality also is provided, thus showing the order optimality of the proposed algorithm. The codes used in the paper are available at https://github.com/Mishfad/Learning-Dynamical-DAGs 

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5. A Note on Exploratory Item Factor Analysis by Singular Value Decomposition

   https://doi.org/10.1007/s11336-020-09704-7  

   Zhang, Haoran; Chen, Yunxiao; Li, Xiaoou (June 2020, Psychometrika)

   null (Ed.)

   Abstract We revisit a singular value decomposition (SVD) algorithm given in Chen et al. (Psychometrika 84:124–146, 2019b) for exploratory item factor analysis (IFA). This algorithm estimates a multidimensional IFA model by SVD and was used to obtain a starting point for joint maximum likelihood estimation in Chen et al. (2019b). Thanks to the analytic and computational properties of SVD, this algorithm guarantees a unique solution and has computational advantage over other exploratory IFA methods. Its computational advantage becomes significant when the numbers of respondents, items, and factors are all large. This algorithm can be viewed as a generalization of principal component analysis to binary data. In this note, we provide the statistical underpinning of the algorithm. In particular, we show its statistical consistency under the same double asymptotic setting as in Chen et al. (2019b). We also demonstrate how this algorithm provides a scree plot for investigating the number of factors and provide its asymptotic theory. Further extensions of the algorithm are discussed. Finally, simulation studies suggest that the algorithm has good finite sample performance. 

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