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Manifold embedding with deep ResNet and a study of the continuity of HodgeRank
Deep neural network
Deep residual network
Moving boundary problem
Combinatorial Hodge theory
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 Moore-Penrose 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 higher-level 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 Moore-Penrose generalized inverse and matrix-vector 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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