克羅內克近似廣泛應用於信號處理、機器學習、量子計算和數據壓縮。在本文中，我們探討了無約束和有約束的克羅內克近似問題，並特別關注於有約束情況下的正交結構。我們應用梯度下降法和投影梯度法來構建優化的梯度流。在理論上，我們證明了構建的梯度流收斂到局部最優解。在數值上，我們提供了實驗結果來說明我們方法的有效性和效率。

Kronecker approximations are widely used in signal processing, machine learning, quantum computing, and data compression. In this article, we explore both unconstrained and constrained Kronecker approximation problems, focusing on orthogonal structures for the constrained case. We apply gradient descent and projected gradient methods to build the gradient flow for optimization. Theoretically, we demonstrate that the constructed flow converges to a local optimum. Numerically, we provide experimental results to illustrate the effectiveness and efficiency of our method.

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