數學是關於數量、結構、變化等主題的探索。

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第一章 判別式神經網絡 Discriminative Neural Networks

自2012年以來,深度神經網絡已經徹底改變了機器學習。盡管這項技術相對較老,但近年來在文字、聲音、圖像和視頻識別方麵取得了令人矚目的進步。考慮到這些方法的利害關係,在數學和算法之間的銜接問題就出現了。在本文中,我將解釋這些網絡的結構以及它們的監督學習的關鍵概念。

1.1 算法和數學學習 Algorithmics and mathematics of learning 1.2 判別式神經網絡 Discriminative neural networks 1.3 神經網絡監督學習 Supervised learning of a neural network 1.4 神經網絡效率 The efficiency of neural networks

第二章 生成式神經網絡 Generative Neural Networks

在前一篇文章中,我們了解了如何以監督的方式訓練神經網絡。這使得有效地解決分類問題成為可能,例如圖像識別。也許更令人驚訝的是,這些神經網絡也以一種無人監督的方式被用來自動生成“虛擬”文本或圖像,這通常被稱為“深度偽造”。在第二篇文章中,我將把生成神經網絡的學習和最優運輸理論聯係起來。這個問題在18世紀由加斯帕德·蒙格提出,然後在20世紀中葉由列昂尼德·坎托羅維奇重新闡述。現在,它已經成為解決數據科學中重要問題的首選工具。

2.1 Generative neural networks 2.2 Unsupervised learning of generative networks 2.3 Monge’s optimal transport 2.4 The optimal transport of Kantorovitch 2.5 Adversarial networks

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This study proposes a low-complexity interpretable classification system. The proposed system contains three main modules including feature extraction, feature reduction, and classification. All of them are linear. Thanks to the linear property, the extracted and reduced features can be inversed to original data, like a linear transform such as Fourier transform, so that one can quantify and visualize the contribution of individual features towards the original data. Also, the reduced features and reversibility naturally endure the proposed system ability of data compression. This system can significantly compress data with a small percent deviation between the compressed and the original data. At the same time, when the compressed data is used for classification, it still achieves high testing accuracy. Furthermore, we observe that the extracted features of the proposed system can be approximated to uncorrelated Gaussian random variables. Hence, classical theory in estimation and detection can be applied for classification. This motivates us to propose using a MAP (maximum a posteriori) based classification method. As a result, the extracted features and the corresponding performance have statistical meaning and mathematically interpretable. Simulation results show that the proposed classification system not only enjoys significant reduced training and testing time but also high testing accuracy compared to the conventional schemes.

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