A Strict Gap Between Relaxed and Partition-Constrained Spectral Compression in a Six-State Lumpable Markov Chain
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2026
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| author | Kiriukhin, Oleg |
| author_facet | Kiriukhin, Oleg |
| contents | This paper studies a finite reversible lumpable Markov chain for which relaxed spectral compression yields a larger determinant than partition-constrained compression. For a symmetric six-state lumpable chain and the positive operator $T=P^2$, I compare the relaxed benchmark \begin{equation*} \mathfrak D^{\mathrm{rel}}_3(T):=\sup_{U^*U=I_3}\det(U^*TU) \end{equation*} and the partition-constrained benchmark \begin{equation*} \sup_{\mathcal A\,\mathrm{3\text{-}partition}}\det Q_{\mathcal A}(T), \qquad Q_{\mathcal A}(T)=H_{\mathcal A}^*TH_{\mathcal A}. \end{equation*} Here the partition-constrained benchmark is the compression induced by normalized indicator vectors of genuine partitions of the state space. I derive closed formulas for the two analytically central partition families, prove strict upper bounds for both in a local-mode-dominated regime, and combine these bounds with an exhaustive enumeration of all $90$ partitions into three nonempty cells in an explicit six-state model. For this model, one obtains a strict global gap: \begin{equation*} \sup_{\mathcal A}\det Q_{\mathcal A}(T)<\mathfrak D^{\mathrm{rel}}_3(T). \end{equation*} Thus, in this model, indicator-based partition frames are strictly weaker than relaxed orthonormal frames even after global partition-constrained optimization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_10820 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Strict Gap Between Relaxed and Partition-Constrained Spectral Compression in a Six-State Lumpable Markov Chain Kiriukhin, Oleg Probability Econometrics Combinatorics Statistics Theory This paper studies a finite reversible lumpable Markov chain for which relaxed spectral compression yields a larger determinant than partition-constrained compression. For a symmetric six-state lumpable chain and the positive operator $T=P^2$, I compare the relaxed benchmark \begin{equation*} \mathfrak D^{\mathrm{rel}}_3(T):=\sup_{U^*U=I_3}\det(U^*TU) \end{equation*} and the partition-constrained benchmark \begin{equation*} \sup_{\mathcal A\,\mathrm{3\text{-}partition}}\det Q_{\mathcal A}(T), \qquad Q_{\mathcal A}(T)=H_{\mathcal A}^*TH_{\mathcal A}. \end{equation*} Here the partition-constrained benchmark is the compression induced by normalized indicator vectors of genuine partitions of the state space. I derive closed formulas for the two analytically central partition families, prove strict upper bounds for both in a local-mode-dominated regime, and combine these bounds with an exhaustive enumeration of all $90$ partitions into three nonempty cells in an explicit six-state model. For this model, one obtains a strict global gap: \begin{equation*} \sup_{\mathcal A}\det Q_{\mathcal A}(T)<\mathfrak D^{\mathrm{rel}}_3(T). \end{equation*} Thus, in this model, indicator-based partition frames are strictly weaker than relaxed orthonormal frames even after global partition-constrained optimization. |
| title | A Strict Gap Between Relaxed and Partition-Constrained Spectral Compression in a Six-State Lumpable Markov Chain |
| topic | Probability Econometrics Combinatorics Statistics Theory |
| url | https://arxiv.org/abs/2604.10820 |