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In multiple dimensions, the conservation of momentum applies in each direction independently. Hence, to solve collisions in multiple dimensions, we should write down the momentum conservation in each direction separately. To help understand collisions in multiple dimensions, consider an example.
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Cross-Modal Multivariate Pattern Analysis
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Unsupervised Metric Fusion Over Multiview Data by Graph Random Walk-Based Cross-View Diffusion.

Yang Wang, Wenjie Zhang, Lin Wu

    IEEE Transactions on Neural Networks and Learning Systems
    |December 17, 2015
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    Summary

    This study introduces a novel cross-view fusion algorithm for learning similarity metrics in multiview data. The method uses graph random walks to fuse multiple similarity measures, improving distance learning for computer vision tasks.

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    Area of Science:

    • Computer Vision
    • Machine Learning
    • Data Science

    Background:

    • Learning optimal similarity metrics is vital for computer vision tasks.
    • Multiview data offers complementary information through diverse feature representations.
    • Existing methods often struggle with effectively fusing information from multiple views.

    Purpose of the Study:

    • To propose a cross-view fusion algorithm for learning a robust similarity metric for multiview data.
    • To enhance distance learning by exploiting graph structures and random walks.
    • To develop a scalable and adaptive approach for multiview similarity learning.

    Main Methods:

    • Constructing multiple graphs, each representing an individual view.
    • Employing graph random walk for similarity propagation and metric improvement.
    • Implementing an anchor graph representation for scalability and sparsity.
    • Dynamically learning view-specific coefficients for adaptive fusion.
    • Utilizing a heuristic approach to control iteration count and prevent over-smoothing.

    Main Results:

    • The proposed algorithm effectively fuses multiple similarity measures into an optimal distance metric.
    • The method demonstrates scalability to large datasets through sparse representations.
    • Adaptive learning of view coefficients balances the influence of different views.
    • Experimental validation on real-world datasets confirms the approach's effectiveness and efficiency.

    Conclusions:

    • The developed cross-view fusion algorithm provides an effective solution for multiview similarity learning.
    • The graph random walk-based approach enhances distance metric learning by leveraging data structure.
    • The method offers a scalable, adaptive, and robust solution for computer vision applications dealing with multiview data.