本論文探討在極度觀測稀疏情況下，時間感知推薦系統中的低秩張量補全問題。傳統的矩陣分解方法通常是非凸的，且容易陷入局部極小值。為了克服此問題，我們引入了來自高光譜影像處理中的「約翰橢球（John Ellipsoid）」框架 cite{lin2023hyperspectral} cite{lin2018maximum}，將非凸分解轉換為凸的幾何最佳化問題。
為了建立此幾何方法，我們回顧了凸幾何中必要的數學知識。由於人類的行為限制具有單形（simplex）的幾何特性，透過利用這點，我們的方法能決定性地識別資料凸包（convex hull）的幾何接觸點，進而解析地還原極端的潛在時間模式。藉由數值實驗，我們的幾何外推方法展現了良好的還原能力，並避開了傳統迭代演算法的病態問題。

This thesis addresses the low-rank tensor completion problem in time-aware recommender systems under extreme observational sparsity. In conventional matrix factorization, it is often non-convex and prone to local minima. To overcome this, we adopt the so-called John Ellipsoid framework cite{lin2023hyperspectral} cite{lin2018maximum} from hyperspectral imaging to transform the non-convex factorization into a strictly convex geometric optimization problem.
To establish this geometric approach, we review the necessary mathematical knowledge from convex geometry. By exploiting the probability-based simplex geometry inherent in human behavioral constraints, our method deterministically identifies the geometric contact points of the data convex hull to analytically recover the extreme latent temporal patterns. Numerical experiments demonstrate that our geometric extrapolation approach guarantees recovery and successfully circumvents the ill-conditioning issues of traditional iterative algorithms.

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