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1. Multivariate fidelities

   https://doi.org/10.1088/1751-8121/adc645  

   Nuradha, Theshani; Mishra, Hemant K; Leditzky, Felix; Wilde, Mark M (April 2025, Journal of Physics A: Mathematical and Theoretical)

   The bivariate classical fidelity is a widely used measure of the similarity of two probability distributions. There exist a few extensions of the notion of the bivariate classical fidelity to more than two probability distributions; herein we call these extensions multivariate classical fidelities, with some examples being the Matusita multivariate fidelity and the average pairwise fidelity. Hitherto, quantum generalizations of multivariate classical fidelities have not been systematically explored, even though there are several well known generalizations of the bivariate classical fidelity to quantum states, such as the Uhlmann and Holevo fidelities. The main contribution of our paper is to introduce a number of multivariate quantum fidelities and show that they satisfy several desirable properties that are natural extensions of those of the Uhlmann and Holevo fidelities. We propose several variants that reduce to the average pairwise fidelity for commuting states, including the average pairwisez-fidelities, the multivariate semi-definite programming (SDP) fidelity, and a multivariate fidelity inspired by an existing secrecy measure. The second one is obtained by extending the SDP formulation of the Uhlmann fidelity to more than two states. All of these variants satisfy the following properties: (i) reduction to multivariate classical fidelities for commuting states, (ii) the data-processing inequality, (iii) invariance under permutations of the states, (iv) its values are in the interval $[0,1]$; they are faithful, that is, their values are equal to one if and only if all the states are equal, and they satisfy orthogonality, that is their values are equal to zero if and only if the states are mutually orthogonal to each other, (v) direct-sum property, (vi) joint concavity, and (vii) uniform continuity bounds under certain conditions. Furthermore, we establish inequalities relating these different variants, indeed clarifying that all these definitions coincide with the average pairwise fidelity for commuting states. We also introduce another multivariate fidelity called multivariate log-Euclidean fidelity, which is a quantum generalization of the Matusita multivariate fidelity. We also show that it satisfies most of the desirable properties listed above, it is a function of a multivariate log-Euclidean divergence, and it has an operational interpretation in terms of quantum hypothesis testing with an arbitrarily varying null hypothesis. Lastly, we propose multivariate generalizations of Matsumoto’s geometric fidelity and establish several properties of them. 

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2. Multivariate Goppa codes

   https://doi.org/10.1109/TIT.2022.3201692  

   Lopez, Hiram H.; Matthews, Gretchen L. (January 2022, IEEE Transactions on Information Theory)
3. A multivariate to multivariate approach for voxel‐wise genome‐wide association analysis

   https://doi.org/10.1002/sim.10101  

   Wu, Qiong; Zhang, Yuan; Huang, Xiaoqi; Ma, Tianzhou; Hong, L Elliot; Kochunov, Peter; Chen, Shuo (August 2024, Statistics in Medicine)

   The joint analysis of imaging‐genetics data facilitates the systematic investigation of genetic effects on brain structures and functions with spatial specificity. We focus on voxel‐wise genome‐wide association analysis, which may involve trillions of single nucleotide polymorphism (SNP)‐voxel pairs. We attempt to identify underlying organized association patterns of SNP‐voxel pairs and understand the polygenic and pleiotropic networks on brain imaging traits. We propose abi‐cliquegraph structure (ie, a set of SNPs highly correlated with a cluster of voxels) for the systematic association pattern. Next, we develop computational strategies to detect latent SNP‐voxelbi‐cliquesand an inference model for statistical testing. We further provide theoretical results to guarantee the accuracy of our computational algorithms and statistical inference. We validate our method by extensive simulation studies, and then apply it to the whole genome genetic and voxel‐level white matter integrity data collected from 1052 participants of the human connectome project. The results demonstrate multiple genetic loci influencing white matter integrity measures on splenium and genu of the corpus callosum. 

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4. Temporal Multimodal Multivariate Learning

   https://doi.org/10.1145/3534678.3539159  

   Park, Hyoshin; Darko, Justice; Deshpande, Niharika; Pandey, Venktesh; Su, Hui; Ono, Masahiro; Barkley, Dedrick; Folsom, Larkin; Posselt, Derek; Chien, Steve (August 2022, ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD2022))

   We introduce temporal multimodal multivariate learning, a new family of decision making models that can indirectly learn and transfer online information from simultaneous observations of a probability distribution with more than one peak or more than one outcome variable from one time stage to another. We approximate the posterior by sequentially removing additional uncertainties across different variables and time, based on data-physics driven correlation, to address a broader class of challenging time-dependent decision-making problems under uncertainty. Extensive experiments on real-world datasets ( i.e., urban traffic data and hurricane ensemble forecasting data) demonstrate the superior performance of the proposed targeted decision-making over the state-of-the-art baseline prediction methods across various settings. 

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5. Longitudinal multivariate normative comparisons

   https://doi.org/10.1002/sim.8850  

   Wang, Zheng; Cheng, Yu; Seaberg, Eric C.; Rubin, Leah H.; Levine, Andrew J.; Becker, James T. (March 2021, Statistics in Medicine)

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