Identifiability and Estimation in Continuous Lyapunov Models

Abstract: Cross-sectional observations from a dynamical system can be modeled via steady-state distributions of Markov processes. The major challenge is then to determine whether the process parameters can be identified and estimated from the steady-state distributions. We study this problem for continuous Lyapunov models that arise as steady-state distributions of the solution to a multivariate stochastic differential equation, whose linear drift matrix is parametrized by a directed graph. We derive equations for the cumulant tensors of any order for this distribution, which generalize the well-known covariance Lyapunov equation. Under a non-Gaussianity assumption we prove generic identifiability of the drift matrix for any connected graph using the equations for the higher-order cumulants. Based on the identifiability result, we propose a new semiparametric estimator of the drift matrix, and we derive its asymptotic distribution. A simulation study demonstrates the asymptotic validity of the estimator but shows that it is only accurate for relatively large sample sizes, illustrating the hardness of the unconstrained estimation problem.

• continuous Lyapunov models are algebraic objects that describe the relationship between the drift matrix M, covariance, and cumulants of a multivar TC(L sde) via Lyapunov equations with non diagonal cumulants of a Lévy process.
• This paper shows that rescaling time invalidates precise identifiability: drift matrix parameters up to joint positive scaling are identifiable from 2nd and r-th order cumulants which is the best possible identifiability under non-Gaussianity, assuming diagonal cumulant matrices.
• Semiparametric estimator dependence matrix suffices to exhibit positive parametised cases to show non diagonal block generically has rank d²-1 hence implying low rank linear dependency across the block differential.

This paper studies identifiability and estimation of the drift matrix in continuous Lyapunov models: steady-state distributions of solutions to the multivariate stochastic differential equation $dX_t = MX_t\,dt + dZ_t$, where $M$ is a stable matrix whose sparsity pattern is encoded by a directed graph $G$ and $Z$ is a Lévy process. The resulting distributions are $M$-selfdecomposable (2603.17142). Prior work established that when $Z$ is Brownian motion (so observations are Gaussian), drift parameters are identifiable only under strong assumptions, with known counterexamples to generic identifiability from the covariance Lyapunov equation alone. This paper gives the first positive identifiability result for the drift under non-Gaussianity, using higher-order cumulants.

Higher-order Lyapunov equations

The central algebraic object is a generalization of the classical covariance Lyapunov equation $M\Sigma + \Sigma M^T + \mathcal{C}_2 = 0$. Using the Tucker mode product, the authors show that if the Lévy process has finite $k$-th-order moment, then the $k$-th-order cumulant tensor $\mathcal{K}$ of the steady-state distribution solves

$M$0

where $M$1 (2603.17142). For stable $M$2, this equation has a unique solution given by an integral against $M$3 across all modes. The paper also derives trek rules of independent interest that express cumulant entries as sums over graph-theoretic treks, generalizing the trek rule for the covariance equation; these yield vanishing criteria for cumulant entries based on absence of treks in mixed graphs with directed and blunt edges.

Generic identifiability up to scaling

Identifiability cannot hold exactly: rescaling time shows $M$4 for all $M$5, so identification up to a common positive scaling is the best possible result without knowing cumulants of $M$6. Connectedness is also necessary — for a graph with $M$7 connected components, the relevant rank drops by exactly $M$8, so component-wise scalings are unidentifiable.

The main theorem states that for any connected graph with all self-loops present, and assuming $M$9 and $G$0 ($G$1) are diagonal with non-zero diagonal entries (the non-Gaussianity condition), there exists a proper algebraic subset $G$2 of the parameter space outside of which $G$3 is identifiable up to joint scaling from the pair $G$4. The exception set is thus lower-dimensional, closed, and has open dense complement.

The proof strategy reorganizes the vectorized second- and $G$5-th-order Lyapunov equations into a homogeneous linear system whose kernel contains $G$6; identifiability reduces to showing that the off-diagonal block $G$7 generically has rank $G$8. Since cumulant entries are rational functions of the parameters, it suffices to exhibit one parameter choice where the rank holds. The proof constructs such a point on a polytree subgraph with self-loops, diagonal identity cumulants, and $G$9, unit weights on non-loop edges, and uses a specialized trek rule plus a combinatorial sum identity to show the lowest-degree coefficient of the determinant polynomial in $Z$0 is non-zero. Notably, the proof also implies the exception set is non-empty even for connected graphs, since disconnected subgraphs embed into the model; whether $Z$1 contains anything beyond parameters disconnecting the graph remains open. The self-loop assumption is likewise not fully resolved — computational examples show it can be relaxed on some cyclic graphs, but no general characterization exists.

A complementary result shows that if some $Z$2-th-order cumulant of the Lévy process is known and diagonal, $Z$3 is generically identifiable for any graph with self-loops, dropping the connectedness requirement.

Estimation

The identification equation suggests a semiparametric estimator: given any matrix $Z$4 built from population cumulants with $Z$5, the least singular value estimator takes $Z$6 as the right singular vector of the plug-in estimator $Z$7 corresponding to its smallest singular value, sign-fixed by requiring negative trace. Stability of $Z$8 is not imposed and is not guaranteed.

The asymptotic theory is complete: consistency holds whenever the required moments exist, and under finite $Z$9-th-order moments the estimator is asymptotically normal with an explicit asymptotic covariance matrix,

$M$0

where $M$1 is the asymptotic covariance of empirical cumulants and $M$2 the Moore–Penrose inverse. A corollary rewrites this compactly via matrices $M$3, avoiding explicit construction of large Kronecker products, which facilitates practical variance estimation. The derivation rests on the Jacobian of the smallest-eigenvalue map at a simple zero eigenvalue.

Finite-sample behavior

Simulations use compound Poisson Lévy processes with beta jumps and a two-parameter family of stable drift matrices indexed by a correlation parameter $M$4 and asymmetry parameter $M$5, with dimensions $M$6 and sample sizes up to $M$7. The $M$8-scaled bias vanishes and the scaled root mean squared error converges to the theoretical asymptotic standard deviation, validating the theory. However, the finite-sample bias is substantial, particularly for high correlation ($M$9), and accurate estimation requires fairly large samples; the problem is harder still for $Z$0 or symmetric $Z$1 ($Z$2). Adding fourth-order cumulants did not uniformly help — it increased error for $Z$3 and had mixed effects otherwise — so the trade-off between including more cumulant orders and statistical efficiency is unresolved.

Limitations and open questions

Several assumptions bear directly on applicability. The identifiability theorem requires diagonal $Z$4 and $Z$5 with non-zero diagonals; extending to non-diagonal cumulants (blunt edges) would enable latent-variable handling via marginalization, but marginalized drifts need not be stable and marginalized covariances need not be positive definite. Whether the self-loop assumption can be dropped in general, and whether the exceptional set consists solely of parameters that disconnect the graph, are both open. Finally, the paper does not address measurement noise, which realistic applications to single-cell data would require, and the demonstrated need for large sample sizes mirrors known difficulties of higher-order-cumulant methods in non-Gaussian causal discovery.

Conclusion

The paper establishes generic identifiability of the drift matrix in continuous Lyapunov models from cross-sectional data under non-Gaussianity, resolving a previously negative identification landscape with a positive result valid for all connected graphs. It supplies the full asymptotic distribution of a natural singular-value-based estimator together with an implementable variance formula. The gap between the asymptotic guarantees and the demanding finite-sample requirements quantified in the simulation study defines the principal practical limitation of the approach.