The Schatten quasi-norm is an approximation of the rank, which is tighter than the nuclear norm. However, most Schatten quasi-norm minimization (SQNM) algorithms suffer from high computational cost to compute the singular value decomposition (SVD) of large matrices at each iteration. In this paper, we prove that for any p, $p_1$, $p_2>0$ satisfying $1/p=1/p_1+1/p_2$, the Schatten p-(quasi-)norm of any matrix is equivalent to minimizing the product of the Schatten $p_1$-(quasi-)norm and Schatten $p_2$-(quasi-)norm of its two much smaller factor matrices. Then, we present and prove the equivalence between the product and its weighted sum formulations for two cases: $p_1=p_2$ and $p_1\ne p_2$. In particular, when $p>1/2$, there is an equivalence between the Schatten p-quasi-norm of any matrix and the Schatten $2p$-norms of its two factor matrices. We further extend the theoretical results of two factor matrices to the cases of three and more factor matrices, from which we can see that for any $0<p<1$, the Schatten p-quasi-norm of any matrix is the minimization of the mean of the Schatten $(\lfloor 1/p\rfloor +1)p$-norms of $\lfloor 1/p\rfloor +1$ factor matrices, where $\lfloor 1/p\rfloor$ denotes the largest integer not exceeding $1/p$.

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Schatten准规范的统一可扩展等效公式

Schatten准规范是等级的近似，比核规范更严格。但是，大多数Schatten准规范最小化（SQNM）算法在每次迭代中都要计算大型矩阵的奇异值分解（SVD）的计算成本很高。在本文中，我们证明对于任何p，$p_{\mathrm{1个}}$， $p_2>0$ 满意的 $\mathrm{1个}/p=\mathrm{1个}/p_{\mathrm{1个}}+\mathrm{1个}/p_2$，任何矩阵的Schatten p-（拟-）范数都等于最小化Schatten的乘积$p_{\mathrm{1个}}$-（准）范数和Schatten $p_2$-两个较小因子矩阵的-（准）范数。然后，我们提出并证明两种情况下乘积与其加权和公式之间的等价关系：$p_{\mathrm{1个}}=p_2$ 和 $p_{\mathrm{1个}}\ne p_2$。特别是当$p>\mathrm{1个}/2$，任何矩阵的Schatten p-拟范数与Schatten之间存在等价关系$2p$-其两个因子矩阵的范数。我们进一步将两个因子矩阵的理论结果扩展到三个或更多因子矩阵的情况，从中我们可以看到$0<p<\mathrm{1个}$，任何矩阵的Schatten p-拟范数就是Schatten均值的最小值$（\lfloor \mathrm{1个}/p\rfloor +\mathrm{1个}）p$-规范 $\lfloor \mathrm{1个}/p\rfloor +\mathrm{1个}$ 因子矩阵，其中 $\lfloor \mathrm{1个}/p\rfloor$ 表示不超过的最大整数 $\mathrm{1个}/p$。