A Strict Gap Between Relaxed and Partition-Constrained Spectral Compression in a Six-State Lumpable Markov Chain

Abstract: 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.

• The paper shows that relaxed spectral compression achieves higher determinant values compared to partition-constrained methods in a six-state model.
• It develops closed-form determinant formulas for two structured partition families using a symmetric lumpable Markov chain framework.
• Exhaustive partition enumeration confirms a global spectral gap, highlighting potential limitations of traditional partition-based aggregation.

Determinant Gap in Spectral Compression: Partition-Constrained vs. Relaxed Frames in Lumpable Markov Chains

Problem Setting and Motivation

The paper establishes a strict separation between two state compression paradigms for finite, reversible, lumpable Markov chains: (1) spectral compression optimized over arbitrary orthonormal frames, and (2) compression via normalized indicator vectors arising from genuine partitions (partition-constrained compression). Focusing on a symmetric six-state model with natural block partitioning, the analysis concerns the positive operator $T = P^2$, where $P$ is the transition matrix, and benchmarks the spectral content captured by the determinant maximization over both approaches.

A principal metric is the determinant of the compressed operator, as this equals the product of the three largest eigenvalues in the relaxed case and quantifies the jointly captured spectral content in rank-3 reductions.

Model Specification and Lumpability Structure

The six-state symmetric Markov chain is explicitly constructed as a $3 \times 3$ block matrix, where each block corresponds to a pair of states. The partition into blocks $\{B_1, B_2, B_3\}$ (each of cardinality 2) satisfies lumpability conditions: transitions from a state in a block into other blocks depend only on the block indices, making aggregation via this block partition Markovian.

The spectral decomposition exploited by the paper reveals an invariant "macro" subspace spanned by block indicator vectors and a "local" subspace corresponding to within-block modes. The spectrum of $P$ consists of macro eigenvalues $1, \kappa_2, \kappa_3$ and local-mode eigenvalues $\beta_1, \beta_2, \beta_3$, whence $T = P^2$ has eigenvalues $1, \kappa_2^2, \kappa_3^2, t_1, t_2, t_3$ ($t_r = \beta_r^2$).

Compression Benchmarks and Partition Enumeration

The relaxed spectral compression benchmark,

$P$0

is maximized by projecting onto the eigenvectors associated with the top three eigenvalues of $P$1 (2604.10820). In contrast, partition-constrained compression utilizes normalized indicator vectors of partitions $P$2, yielding compression matrices $P$3.

Crucially, in the six-state model, there are exactly 90 unordered partitions into three nonempty cells (Stirling number $P$4), constituting the feasible set for partition-constrained optimization.

Closed Determinant Formulas and Analytical Gap

Two analytically central structured partition families are isolated:

• The (1,1,4) family: Forms by splitting one block into two singletons and merging the other blocks into a four-cell. The determinant formula for such partitions is

$P$5

where $P$6 is the local-mode eigenvalue.

• The (1,2,3) family: Keeps one block intact, splits another, and merges a singleton into the third block. The determinant formula is

$P$7

with $P$8 (macro block squared).

Using spectral bounds, the paper proves that, under a "local-mode-dominated" ordering ($P$9, $3 \times 3$0), the determinant for any structured family partition is strictly below the relaxed spectral benchmark.

Global Gap Certification in Explicit Model

For explicit parameter choices, the spectrum is computed:

• Macro eigenvalues: $3 \times 3$1, $3 \times 3$2, $3 \times 3$3
• Local-mode magnitudes: $3 \times 3$4, $3 \times 3$5, $3 \times 3$6

Thus, $3 \times 3$7 and $3 \times 3$8. Exhaustive enumeration across all 90 partitions yields:

• Relaxed benchmark: $3 \times 3$9
• Largest partition determinant: $\{B_1, B_2, B_3\}$0
• Natural block partition determinant: $\{B_1, B_2, B_3\}$1

This certifies a strict global gap: $\{B_1, B_2, B_3\}$2

Implications and Future Directions

The results demonstrate that in this explicit six-state lumpable chain, partition-constrained spectral compression—standard in state aggregation for Markov processes—is intrinsically suboptimal relative to relaxed spectral compression. This has implications for algorithmic state aggregation in modeling and numerical analysis: constraints to indicator-based partitions may forfeit spectral content that could be captured by more flexible orthonormal compressions.

Theoretical implications include encouragement for further characterization of parameter regimes and partition structures where such gaps universally persist. Practically, this motivates exploration of relaxations in state aggregation protocols, potentially informing model reduction in large-scale stochastic systems or learning-based approaches to Markov state space compression.

Potential future developments include parameter-robust gap theorems, analytical extensions beyond finite certificate arguments, and integration of relaxed spectral compression into scalable algorithms for Markov process abstraction.

Conclusion

This study provides a finite, explicit certificate for a strict determinant gap between partition-constrained and relaxed spectral state compression in a six-state reversible, lumpable Markov chain. Closed formulas and analytic bounds establish suboptimality of partition-indicator frames, confirmed by exhaustive optimization. The work suggests the need for broader analytical guarantees for spectral gaps and signals practical caution against exclusive reliance on partition-based state aggregations in complex stochastic systems.