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Abstract In this article, we review the mathematical foundations of convolutional neural nets (CNNs) with the goals of: (i) highlighting connections with techniques from statistics, signal processing, linear algebra, differential equations, and optimization, (ii) demystifying underlying computations, and (iii) identifying new types of applications. CNNs are powerful machine learning models that highlight features from grid data to make predictions (regression and classification). The grid data object can be represented as vectors (in 1D), matrices (in 2D), or tensors (in 3D or higher dimensions) and can incorporate multiple channels (thus providing high flexibility in the input data representation). CNNs highlight features from the grid data by performing convolution operations with different types of operators. The operators highlight different types of features (e.g., patterns, gradients, geometrical features) and are learned by using optimization techniques. In other words, CNNs seek to identify optimal operators that best map the input data to the output data. A common misconception is that CNNs are only capable of processing image or video data but their application scope is much wider; specifically, datasets encountered in diverse applications can be expressed as grid data. Here, we show how to apply CNNs to new types of applications such as optimal control, flow cytometry, multivariate process monitoring, and molecular simulations.
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Wedge reversion antisymmetry and 41 types of physical quantities in arbitrary dimensions
It is shown that there are 41 types of multivectors representing physical quantities in non-relativistic physics in arbitrary dimensions within the formalism of Clifford algebra. The classification is based on the action of three symmetry operations on a general multivector: spatial inversion, 1 , time-reversal, 1′, and a third that is introduced here, namely wedge reversion, 1 † . It is shown that the traits of `axiality' and `chirality' are not good bases for extending the classification of multivectors into arbitrary dimensions, and that introducing 1 † would allow for such a classification. Since physical properties are typically expressed as tensors, and tensors can be expressed as multivectors, this classification also indirectly classifies tensors. Examples of these multivector types from non-relativistic physics are presented.
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- Award ID(s):
- 1807768
- PAR ID:
- 10179570
- Date Published:
- Journal Name:
- Acta Crystallographica Section A Foundations and Advances
- Volume:
- 76
- Issue:
- 3
- ISSN:
- 2053-2733
- Page Range / eLocation ID:
- 318 to 327
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1107/S205327332000217X
