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1. State aggregation learning from Markov transition data

   Duan, Yaqi ; Ke, Zheng Tracy ; Wang, Mengdi ( October 2019 , Advances in neural information processing systems)

   null (Ed.)

   State aggregation is a popular model reduction method rooted in optimal control. It reduces the complexity of engineering systems by mapping the system’s states into a small number of meta-states. The choice of aggregation map often depends on the data analysts’ knowledge and is largely ad hoc. In this paper, we propose a tractable algorithm that estimates the probabilistic aggregation map from the system’s trajectory. We adopt a soft-aggregation model, where each meta-state has a signature raw state, called an anchor state. This model includes several common state aggregation models as special cases. Our proposed method is a simple two- step algorithm: The first step is spectral decomposition of empirical transition matrix, and the second step conducts a linear transformation of singular vectors to find their approximate convex hull. It outputs the aggregation distributions and disaggregation distributions for each meta-state in explicit forms, which are not obtainable by classical spectral methods. On the theoretical side, we prove sharp error bounds for estimating the aggregation and disaggregation distributions and for identifying anchor states. The analysis relies on a new entry-wise deviation bound for singular vectors of the empirical transition matrix of a Markov process, which is of independent interest and cannot be deduced from existing literature. The application of our method to Manhattan traffic data successfully generates a data-driven state aggregation map with nice interpretations. 

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2. Alignment of density maps in Wasserstein distance

   https://doi.org/10.1017/S2633903X24000059  

   Singer, Amit ; Yang, Ruiyi ( January 2024 , Biological Imaging)

   In this article, we propose an algorithm for aligning three-dimensional objects when represented as density maps, motivated by applications in cryogenic electron microscopy. The algorithm is based on minimizing the 1-Wasserstein distance between the density maps after a rigid transformation. The induced loss function enjoys a more benign landscape than its Euclidean counterpart and Bayesian optimization is employed for computation. Numerical experiments show improved accuracy and efficiency over existing algorithms on the alignment of real protein molecules. In the context of aligning heterogeneous pairs, we illustrate a potential need for new distance functions.

    

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3. Better Together: Online Probabilistic Clique Change Detection in 3D Landmark-Based Maps

   Bateman, Sam ; Harlow, Kyle ; Heckman, Christoffer ( October 2020 , Proceedings of the IEEERSJ International Conference on Intelligent Robots and Systems)

   Many modern simultaneous localization and mapping (SLAM) techniques rely on sparse landmark-based maps due to their real-time performance. However, these techniques frequently assert that these landmarks are fixed in position over time, known as the static-world assumption. This is rarely, if ever, the case in most real-world environments. Even worse, over long deployments, robots are bound to observe traditionally static landmarks change, for example when an autonomous vehicle encounters a construction zone. This work addresses this challenge, accounting for changes in complex three-dimensional environments with the creation of a probabilistic filter that operates on the features that give rise to landmarks. To accomplish this, landmarks are clustered into cliques and a filter is developed to estimate their persistence jointly among observations of the landmarks in a clique. This filter uses estimated spatial-temporal priors of geometric objects, allowing for dynamic and semi-static objects to be removed from a formally static map. The proposed algorithm is validated in a 3D simulated environment. 

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4. Functional Maps Representation On Product Manifolds

   https://doi.org/10.1111/cgf.13598  

   Rodolà, E. ; Lähner, Z. ; Bronstein, A. M. ; Bronstein, M. M. ; Solomon, J. ( January 2019 , Computer Graphics Forum)

   We consider the tasks of representing, analysing and manipulating maps between shapes. We model maps as densities over the product manifold of the input shapes; these densities can be treated as scalar functions and therefore are manipulable using the language of signal processing on manifolds. Being a manifold itself, the product space endows the set of maps with a geometry of its own, which we exploit to define map operations in the spectral domain; we also derive relationships with other existing representations (soft maps and functional maps). To apply these ideas in practice, we discretize product manifolds and their Laplace–Beltrami operators, and we introduce localized spectral analysis of the product manifold as a novel tool for map processing. Our framework applies to maps defined between and across 2D and 3D shapes without requiring special adjustment, and it can be implemented efficiently with simple operations on sparse matrices.

    

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5. How can classical multidimensional scaling go wrong?

   Sonthalia, Rishi ; Van Buskirk, Greg ; Raichel, Benjamin ; Gilbert, Anna ( January 2021 , Advances in neural information processing systems)

   Given a matrix D describing the pairwise dissimilarities of a data set, a common task is to embed the data points into Euclidean space. The classical multidimensional scaling (cMDS) algorithm is a widespread method to do this. However, theoretical analysis of the robustness of the algorithm and an in-depth analysis of its performance on non-Euclidean metrics is lacking. In this paper, we derive a formula, based on the eigenvalues of a matrix obtained from D, for the Frobenius norm of the difference between D and the metric Dcmds returned by cMDS. This error analysis leads us to the conclusion that when the derived matrix has a significant number of negative eigenvalues, then ∥D−Dcmds∥F, after initially decreasing, willeventually increase as we increase the dimension. Hence, counterintuitively, the quality of the embedding degrades as we increase the dimension. We empirically verify that the Frobenius norm increases as we increase the dimension for a variety of non-Euclidean metrics. We also show on several benchmark datasets that this degradation in the embedding results in the classification accuracy of both simple (e.g., 1-nearest neighbor) and complex (e.g., multi-layer neural nets) classifiers decreasing as we increase the embedding dimension.Finally, our analysis leads us to a new efficiently computable algorithm that returns a matrix Dl that is at least as close to the original distances as Dt (the Euclidean metric closest in ℓ2 distance). While Dl is not metric, when given as input to cMDS instead of D, it empirically results in solutions whose distance to D does not increase when we increase the dimension and the classification accuracy degrades less than the cMDS solution. 

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