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kpknudson.com | ||
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jdh.hamkins.org
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| | | | | Please feel free to post comments or questions here about the ideas on this site. | |
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gowers.wordpress.com
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| | | | | It's been a while since I have written a post in the "somewhat philosophical" category, which is where I put questions like "How can one statement be stronger than an another, equivalent, statement?" This post is about a question that I've intended for a long time to sort out in my mind but have found... | |
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www.quantamagazine.org
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| | | | | Number theorist Andrew Granville on what mathematics really is - and why objectivity is never quite within reach. | |
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www.depthfirstlearning.com
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| | | [AI summary] The provided text is a detailed exploration of the mathematical and statistical foundations of neural networks, focusing on the Jacobian matrix, its spectral properties, and the implications for dynamical isometry. The key steps and results are as follows: 1. **Jacobian and Spectral Analysis**: The Jacobian matrix $ extbf{J} $ of a neural network is decomposed into $ extbf{J} = extbf{W} extbf{D} $, where $ extbf{W} $ is the weight matrix and $ extbf{D} $ is a diagonal matrix of derivatives. The spectral properties of $ extbf{J} extbf{J}^T $ are analyzed using the $ S $-transform, which captures the behavior of the eigenvalues of the Jacobian matrix. 2. **$ S $-Transform Derivation**: The $ S $-transform of $ extbf{J} extbf{J}^T $ is... | ||
