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1. High dimensional, robust, unsupervised record linkage

   https://doi.org/10.21307/stattrans-2020-034  

   Bera, Sabyasachi ; Chatterjee, Snigdhansu ( August 2020 , Statistics in Transition New Series)

   Okrasa, Włodzimierz ; Lahiri, Partha (Ed.)

   We develop a technique for record linkage on high dimensional data, where the two datasets may not have any common variable, and there may be no training set available. Our methodology is based on sparse, high dimensional principal components. Since large and high dimensional datasets are often prone to outliers and aberrant observations, we propose a technique for estimating robust, high dimensional principal components. We present theoretical results validating the robust, high dimensional principal component estimation steps, and justifying their use for record linkage. Some numeric results and remarks are also presented.

    

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2. SGOOP-d: Estimating kinetic distances and reaction coordinate dimensionality for rare event systems from biased/unbiased simulations

   Tsai, Sun-Ting ; Smith, Zachary ; Tiwary, Pratyush ( April 2021 , ArXivorg)

   null (Ed.)

   Understanding kinetics including reaction pathways and associated transition rates is an important yet difficult problem in numerous chemical and biological systems especially in situations with multiple competing pathways. When these high-dimensional systems are projected on low-dimensional coordinates often needed for enhanced sampling or for interpretation of simulations and experiments, one often ends up losing the kinetic connectivity of the underlying high-dimensional landscape. Thus in the low-dimensional projection metastable states might appear closer or further than they actually are. To deal with this issue, in this work we develop a formalism that learns a multi-dimensional yet minimally complex reaction coordinate (RC) for generic high-dimensional systems, such that when projected along this RC, all possible kinetically relevant pathways can be demarcated and the true high-dimensional connectivity is maintained. One of the defining attributes of our method lies in that it can work on long unbiased simulations as well as biased simulations often needed for rare event systems. We demonstrate the utility of the method by studying a range of model systems including conformational transitions in a small peptide Ace-Ala3-Nme, where we show how SGOOPderived two-dimensional and three-dimensional reaction coordinate can capture the kinetics for 23 and all 28 out of the 28 dominant state-to-state-transitions respectively. 

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3. Leafwise quasigeodesic foliations in dimension three and the funnel property

   Chanda, Anindya ; Fenley, Sergio ( November 2023 , Ergodic theory dynamical systems)

   We construct one dimensional foliations which are subfoliations of two dimensional foliations in 3-manifolds. The subfoliation is by quasigeodesics in each two dimensional leaf, but it is not funnel: not all quasigeodesics share a common ideal point in most leaves. 

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4. Neural Set Function Extensions: Learning with Discrete Functions in High Dimensions

   Nikolaos Karalias, Joshua Robinson ( November 2022 , Neural Information Processing Systems (NeurIPS))

   Integrating functions on discrete domains into neural networks is key to develop- ing their capability to reason about discrete objects. But, discrete domains are (I) not naturally amenable to gradient-based optimization, and (II) incompatible with deep learning architectures that rely on representations in high-dimensional vector spaces. In this work, we address both difficulties for set functions, which capture many important discrete problems. First, we develop a framework for extending set functions onto low-dimensional continuous domains, where many extensions are naturally defined. Our framework subsumes many well-known extensions as special cases. Second, to avoid undesirable low-dimensional neural network bottlenecks, we convert low-dimensional extensions into representations in high-dimensional spaces, taking inspiration from the success of semidefinite programs for combinatorial optimization. Empirically, we observe benefits of our extensions for unsupervised neural combinatorial optimization, in particular with high-dimensional representations. 

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5. On Deep Generative Models for Approximation and Estimation of Distributions on Manifolds

   Biraj Dahal, Alexander Havrilla ( December 2022 , Advances in Neural Information Processing Systems)

   Deep generative models have experienced great empirical successes in distribution learning. Many existing experiments have demonstrated that deep generative networks can efficiently generate high-dimensional complex data from a low-dimensional easy-to-sample distribution. However, this phenomenon can not be justified by existing theories. The widely held manifold hypothesis speculates that real-world data sets, such as natural images and signals, exhibit low-dimensional geometric structures. In this paper, we take such low-dimensional data structures into consideration by assuming that data distributions are supported on a low-dimensional manifold. We prove approximation and estimation theories of deep generative networks for estimating distributions on a low-dimensional manifold under the Wasserstein-1 loss. We show that the Wasserstein-1 loss converges to zero at a fast rate depending on the intrinsic dimension instead of the ambient data dimension. Our theory leverages the low-dimensional geometric structures in data sets and justifies the practical power of deep generative models. We require no smoothness assumptions on the data distribution which is desirable in practice. 

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