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Title:  基於深度殘差神經網路的流形嵌入與霍奇排名的連續性之探討 Manifold embedding with deep ResNet and a study of the continuity of HodgeRank 
Authors:  林澤佑 Lin, TseYu 
Contributors:  蔡炎龍 劉宣谷 Tsai, YenLung Liu, HsuanKu 林澤佑 Lin, TseYu 
Keywords:  深層神經網路 深度殘差網路 移動邊界問題 流形重建 霍奇排名 組合霍奇理論 拉普拉斯矩陣 同儕互評 Deep neural network Deep residual network Moving boundary problem Manifold reconstruction HodgeRank Combinatorial Hodge theory Graph Laplacian Peer assessment 
Date:  2020 
Issue Date:  20210303 
Abstract:  仿造人腦的功能性，深層神經網路被建立用於萃取高階訊息。從數學的觀點來看，神經網路可以視為在適當定義遇上近似任意函數的近似器。 為了展示深層神經網路的威力，我們在本論文的第一部分考慮神經網路的兩種不同形式的應用。第一種應用是源於衍生性金融商品的定價模型，而另一種應用則是將基於仿射空間的流形重建演算法改寫為一殘差神經網路的學習過程，而這樣的改寫提供了深層神經網路在幾何演算法上的潛在應用。 本論文的第二個部分，我們聚焦在 HodgeRank，一個基於組合霍奇理論的逐對排名演算法。我們首先會回顧組合貨奇理論的背景知識，接著，我們考慮 HodgeRank 在線上同儕互評上之應用。最後，將 HodgeRank 視為 MoorePenrose 廣義逆算子與矩陣向量乘法的合成函數，我們可以探討 HodgeRank 的連續性。最後，我們從圖的角度證明了關於 HodgeRank 的一個連續性定理。 Deep neural networks are modeled to extract higherlevel information in a way that is like the function of the human brain. From a mathematical perspective, neural networks are function approximators, which can approximate any function on a suitable domain. In the first part of this dissertation, we consider two different tasks to demonstrate the power of deep neural networks. One task is derived from a option pricing model of financial derivatives while another task is to rewrite an affine subspaces based manifold reconstruction algorithm to a learning process of a deep residual network. Such reformulation offers a possibility for potential application of deep neural networks to various geometrical related algorithms. In the second part, we focus on the HodgeRank, a pairwise ranking method based on the combinatorial Hodge theory. We first quick review the background of combinatorial Hodge theory, then a real world application of HodgeRank to online peer assessment is provided. Finally, by considering HodgeRank as a composition of MoorePenrose generalized inverse and matrixvector product, we can study the continuity of HodgeRank. In terms of graph, a theorem of continuity of the HodgeRank is provided in the end. 
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