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→近似动力学可逆性
==近似动力学可逆性==
==近似动力学可逆性==
由于绝大部分马尔科夫链的状态转移矩阵（TPM）都不是置换矩阵，所以我们需要一个指标来刻画一个TPM靠近可逆的置换矩阵程度的指标，这就是近似动力学可逆性。
假设马尔科夫链的TPM为P，奇异值为<math>
假设马尔科夫链的TPM为P，奇异值为<math>
(\sigma_{1}\ge\sigma_{2}\ge...\ge\sigma_{N}\ge0)
(\sigma_{1}\ge\sigma_{2}\ge...\ge\sigma_{N}\ge0)
</math>时，<math>
</math>时，<math>
\Gamma_{\alpha}
\Gamma_{\alpha}
</math>是P的沙滕范数（Schatten norm）；当<math>
</math>是P的[[沙滕范数]]（Schatten norm）；当<math>
0<\alpha<1
0<\alpha<1
</math>时，<math>
</math>时，<math>
\Gamma_{\alpha}
\Gamma_{\alpha}
</math>是P的准范数（quasinorm）<ref>Schatten norm from Wikipedia. https://en.wikipedia.org/wiki/Schatten norm</ref><ref>Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM review 52(3), 471–501 (2010)</ref><ref>Chi, Y., Lu, Y.M., Chen, Y.: Nonconvex optimization meets low-rank matrix factorization: An overview. IEEE Transactions on Signal Processing 67(20), 52395269 (2019)</ref><ref name="Cui">Cui, S., Wang, S., Zhuo, J., Li, L., Huang, Q., Tian, Q.: Towards discriminability and diversity: Batch nuclear-norm maximization under label insufficient situations. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3941–3950 (2020)</ref>。使用这个定义来刻画近似动力学可逆性是合理的，因为完全动力学可逆性可以通过最大化<math>
</math>是P的准范数（quasinorm）<ref>Schatten norm from Wikipedia. https://en.wikipedia.org/wiki/Schatten norm</ref><ref>Recht, B., Fazel, M., Parrilo, P.A.: Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM review 52(3), 471–501 (2010)</ref><ref>Chi, Y., Lu, Y.M., Chen, Y.: Nonconvex optimization meets low-rank matrix factorization: An overview. IEEE Transactions on Signal Processing 67(20), 52395269 (2019)</ref><ref name="Cui">Cui, S., Wang, S., Zhuo, J., Li, L., Huang, Q., Tian, Q.: Towards discriminability and diversity: Batch nuclear-norm maximization under label insufficient situations. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3941–3950 (2020)</ref>。使用这个定义来刻画近似动力学可逆性是合理的，因为完全动力学可逆性可以通过最大化<math>\Gamma_{\alpha}</math>来得到。
\Gamma_{\alpha}
</math>来得到。