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Observability conditions for neural state-space models with eigenvalues and their roots of unity

T0 review · 4 major / 6 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper claims that observability of diagonalizable neural state-space models reduces to distinct eigenvalues plus one nonzero row of CV, with Mamba as the cleanest case.

desk verdict A useful idea — enforce Mamba observability via distinct eigenvalues and nonzero C V rows — but the headline theorem is false because it ignores the discretization, and the Fourier high-probability results are admitted sketches. read the letter →

arxiv 2504.15758 v2 pith:SIJVQLE4 submitted 2025-04-22 cs.LG cs.SYeess.SYmath.DSmath.OC

Andrew Gracyk This is my paper · ORCID
classification cs.LGcs.SYeess.SYmath.DSmath.OC MSC 93B0768T0715A1842A38
keywords observabilityneuralstate-spacemodelsMambacontroltheoryeigenvaluesrootsofunityFouriertransformHautuslemma
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that observability—the ability to recover the initial hidden state of a neural state-space model from its outputs—can be enforced by small, cheap loss terms instead of by checking the full observability matrix. The cleanest result is for the Mamba architecture: if $A$ is diagonalizable with distinct eigenvalues and, after diagonalizing, one row of the output matrix $C$ has no zero entries, then the pair $(C,A)$ is observable; this converts observability into two simple quantities, the eigenvalue gaps and the entries of $CV$. For Fourier-based state-space models the paper gives two conditions under which observability holds with high probability: distinct eigenvalues plus frequency-dependent kernel spaces that are pairwise non-nested, enforced by losses on a diagonal matrix $\Psi$. It also shows that a permutation matrix with distinct roots of unity as eigenvalues makes the system observable, and that small perturbations preserve that property. If these conditions are correct, observability can be certified at roughly $O(n^2)$-scale computation rather than at the $O(n^3m^2+n^3)$ cost of forming the observability matrix.

What carries the argument

The load-bearing object is the observability matrix $\mathcal{O}=[C;CA;\ldots;CA^{n-1}]$ and the difficulty that $C$ is wide ($n>m$), so $\mathcal{O}$ cannot be full column rank without the powers of $A$ shuffling $C$'s columns. The paper's main engine for bypassing $\mathcal{O}$ is diagonalization followed by a Kronecker product: when $A=V\operatorname{diag}(\lambda_i)V^{-1}$, the Mamba discretized transition is $\overline{A}=V\operatorname{diag}(e^{\Delta\lambda_i})V^{-1}$, and the observability blocks reduce to $\Phi_i\otimes[\widetilde{C}]_i$, with $\Phi$ the Vandermonde matrix of the exponentials $e^{\Delta\lambda_i}$. Distinct eigenvalues make $\Phi$'s columns independent, and Lemma 6 transfers that independence to the Kronecker products with nonzero columns of $\widetilde{C}=CV$; hence the first row of $\widetilde{C}$ alone can certify full rank. The Fourier results use a related diagonal matrix $\Psi_j$ derived from the closed-form geometric series of the input kernel, whose entries depend nonlinearly on frequency $j$ and eigenvalue $\lambda_i$, so that different Fourier bins see different row spaces of $CV\Psi_jV^{-1}$. The permutation results use the Birkhoff–von Neumann theorem: a doubly stochastic matrix with distinct roots of unity as eigenvalues must be a permutation matrix.

What would settle it

Take a small concrete system, say $n=4$, $m=2$, choose a diagonalizable $A$ with distinct eigenvalues and a full-rank $C$ so that the Fourier losses in Theorem 4 are exactly satisfied, and compute the rows $CV\Psi_jV^{-1}$ for $j=1,\ldots,L-1$. If every pair of kernels is non-nested but the joint row space has dimension less than $n$, Theorem 4's high-probability claim fails in that instance; the paper's own Figure 12 indicates the union ranks $n$ for one random sample set, so the search is for a counterexample to that pattern, including with $m$ much smaller than $n$. For Theorem 5, the same check is deterministic: verify numerically that with distinct eigenvalues and a nonzero first row of $CV$, the matrix $\Gamma=(e^{\Delta\lambda_i k}\widetilde{C}_{1i})_{k,i}$ has full column rank.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 5. Let $A=V\operatorname{diag}(\lambda_1,\ldots,\lambda_n)V^{-1}$ be diagonalizable with $V$ nonsingular, let $C\in\mathbb{R}^{m\times n}$, and set $\widetilde{C}=CV$. If the loss $L=\mathrm{relu}(c-\min_{k_1\ne k_2}|\lambda_{k_1}-\lambda_{k_2}|)+\sum_j\mathrm{relu}(c-|\widetilde{C}_{1j}|)$ is exactly satisfied, then $(C,A)$ is observable. The reason is that exponentiating $A$ turns the observability matrix into blocks $\Phi_i\otimes[\widetilde{C}]_i$, where $\Phi$ is an $n\times n$ Vandermonde matrix with entries $e^{\Delta\lambda_i k}$; distinct eigenvalues make the columns of $\Phi$ independent, and a nonzero entry in the first row of $\widetilde{C}$ makes each Kronecker product contribute a new independent direction, so the full observability matrix has column rank $n$. The same argument works with any row of $\widetilde{C}$, and the paper also states the Fourier-transform analogues (Theorems 3 and 4) in which pairwise non-nested kernels $CV\Psi_jV^{-1}$ are claimed to force full rank by a pigeonhole argument, with observability holding "with high probability."

Load-bearing premise

The proof of the Fourier-based claims (Theorems 3 and 4) rests on the unproven assertion that if the kernels $\ker(CV\Psi_{j_1}V^{-1})$ are pairwise non-nested across frequency bins $j$, then the union of the corresponding row spaces spans $\mathbb{R}^n$; the paper itself calls this pigeonhole argument "not airtight." If that combinatorial fill-out assumption fails, the Fourier losses do not imply observability.

Editorial extensions

If this is right

  • Mamba systems can be made observable by training against the two-term loss on eigenvalue gaps and first-row entries of $CV$; the cost is roughly $O(n^2+mn)$, versus $O(n^4)$ for the Hautus-based loss and far more for the full observability matrix.
  • Since full column rank is an open condition, eigenvalue matching to roots of unity does not need to be exact: small perturbation matrices $\Xi$ preserve observability once $\|\Xi\|_F$ is below a threshold (Theorem 2).
  • For Fourier/convolutional state-space models, adding losses that force distinct eigenvalues, distinct $\Psi_j$ diagonals, and angular dispersion makes observability likely, with the probability improving as $m$ approaches $n$.
  • The coupled $A$-$B$ parameterization cuts the number of learned parameters and makes powers $A^k$ evaluate by diagonal scaling; the accompanying update satisfies a Robbins-Monro condition under orthogonality, whereas the direct update does not satisfy a contraction for large Lipschitz constants.
  • One output channel is enough: the first row of $CV$ with no zeros suffices for observability, so observability can be certified by a single scalar output sequence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Inference: the first-row condition in Theorem 5 could be randomized across training: if the rows of $CV$ are random, any single row has zero entries with very small probability under a continuous distribution, so for many random initializations the local certificate already holds; a testable extension is to make the loss choose the row of $CV$ that is most uniformly nonzero.
  • Inference: because the Fourier "high probability" is unquantified, a natural next step is to bound the probability that $L-1$ random diagonal scalings of a fixed wide matrix fill the row space, using union bounds over row-space intersections; this would turn the pigeonhole heuristic into a theorem for large $n$ and $m/n$ near 1.
  • Inference: the permutation/root-of-unity condition suggests a quantization-friendly design: if $A$ is exactly a permutation with distinct roots of unity, the precision needed is only enough to keep the perturbed matrix inside the open set of full-rank observability matrices, so observability could survive fixed-point arithmetic.
  • Inference: the parameter-sharing algorithm's divergence between scaled and unscaled updates suggests a testable recipe for other state-space architectures: scale parameter increments by $\|B_k-B_{k-1}\|^{1/q}$ to smooth gradients, and watch for contraction failure (large Lipschitz) in the unscaled variant.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes methods to enforce observability in neural state-space models and the Mamba architecture by adding loss terms derived from control-theoretic conditions: full-rank observability matrices, permutation/root-of-unity structure, Fourier-domain kernel distinctness, a Vandermonde-based Hautus-type criterion for Mamba, and a coupled A/B parameterization with claimed Robbins-Monro convergence. The main results are conditional statements of the form 'if a certain loss is exactly satisfied, then the pair is observable (with high probability).' The paper also includes empirical illustrations and a limitations section. The central Mamba result and the Fourier 'with high probability' results are not established rigorously as stated, and one of the main theorems is false.

Significance. If the results were correct, the paper would provide practical, computationally efficient loss terms for enforcing observability in modern sequence models, which is a legitimate and useful goal. The paper does explicitly formulate candidate losses and tests them in small experiments, and the permutation-based sufficient conditions (Theorem 1 and related lemmas) are plausible. However, the central Mamba observability theorem is false as stated, and the Fourier-based high-probability claims rest on informal, admittedly non-airtight pigeonhole reasoning rather than proofs. As a result, the paper's main contributions cannot currently be relied upon, and the significance is limited to identifying potential directions rather than establishing them.

major comments (4)
  1. [Theorem 5, Appendix C] Theorem 5 is false as stated. The proof constructs a Vandermonde matrix from the exponentials e^{Delta lambda_j}, but the loss in Eq. (53) enforces only that the lambda_j are distinct. Distinct lambda_j do not imply distinct e^{Delta lambda_j}. A concrete counterexample is n=2, Delta=1, A=[[0,pi],[-pi,0]], and C=[1,1]. The eigenvalues are +i pi and -i pi, so min_{k1,k2}|lambda_k1 - lambda_k2| = 2 pi, and with the natural eigenvector matrix V the columns of C V = [1+i, 1-i] both have modulus sqrt(2). For any positive constant below sqrt(2), the loss (53) is exactly zero. Yet exp(A) = -I, so the observability matrix [C; C exp(Delta A)] = [[1,1],[-1,-1]] has rank 1, and the pair (C, exp(Delta A)) is not observable. The theorem needs an additional condition such as distinctness of e^{Delta lambda_j} or a sampling constraint like Delta |Im(lambda_i - lambda_j)| < 2 pi for all i,j. The related assertion in Section 3.1 that observability of (C,A) is equivalent to observability of (C,e^{Delta A}) is likewise false without such a condition.
  2. [Section 4.2 and Appendix B.2] Theorems 3 and 4 claim observability 'with high probability,' but no probabilistic model or formal proof is supplied. The key step is the claim that non-nested kernels ker(C V Psi_{j1} V^{-1}) across j imply that the concatenated row spaces have dimension n. In Appendix B.2 this 'pigeonhole-type argument' is stated as a Claim with a sketch, and the text itself says the argument 'is not airtight.' No distribution over C, V, or Lambda is defined, and no probability estimate is derived. These high-probability statements are therefore not established, and the two Fourier-based results are conditional on an unproved combinatorial assumption.
  3. [Corollary to Theorem 3, Appendix B.1] The Corollary states that the full-rank condition on (I - e^{A L Delta t}) can be relaxed by requiring det(F[Ce^{A k Delta t} B]_j) > 0 for all j. The paper's own counterexample in the same appendix contradicts this: with C=(I_m 0), A=I_n, B=(I_m 0)^T, the matrix F[Ce^{A t} B]_j is proportional to the identity for every j, so all determinants are positive, yet the system is not observable because A=I and the columns of C never permute. Thus the Corollary's claimed relaxation is false as stated.
  4. [Theorem 6, Section 4.6] Theorem 6 assumes in Eq. (61) that the expected parameter differences decay as E[||(U_{k+1},Q_{k+1},S_{k+1}) - (U_k,Q_k,S_k)||] ~ Theta(1/k^q). This is essentially the convergence behavior the theorem is supposed to establish or derive, but no argument is provided for why gradient descent on f produces this rate. The proof then verifies the Robbins-Monro summability conditions under that assumption. The result is thus conditional on an unproved rate, and the paragraph describing the assumption as 'reasonable in certain nonconvex settings' does not make it a theorem. The algorithmic-convergence contribution needs either a proof of Eq. (61) or an explicit reformulation as a conditional statement with the rate as a hypothesis.
minor comments (6)
  1. [Equation (10)] In the convolutional recurrence, the second line appears to have a typo: y_k = sum_{i=0}^k C A^i B y_{k-i} should presumably be C A^i B x_{k-i}.
  2. [Equation (27)] The notation L != ... in the Hautus-loss subsection is confusing; the intended meaning appears to be that this loss is incomplete, but the symbol '!=' is not standard and should be replaced with words or a different annotation.
  3. [Theorems 3 and 4] The hypotheses state that A is full column rank while also writing A in R^{n x n}; for a square matrix full column rank is equivalent to invertibility, so this phrasing is redundant and should be clarified.
  4. [Definitions of theta_k] The term 'angular part of the complex exponentials' used in the Fourier losses is not precisely defined; the manuscript should specify whether it is the argument in (-pi,pi], a wrapped phase, or something else.
  5. [References] The reference list includes items such as 'OpenAI. Chatgpt.' and 'Undermind. Undermind ai-powered search for scientific literature.' These are not appropriate scholarly references and should be removed or replaced.
  6. [Figures and reproducibility] Several figures are described as supporting empirical claims, but no code or detailed experimental protocol is provided, and some captions refer to appendix equations by number without clear cross-reference; this makes the empirical sections difficult to reproduce or verify.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the main claims are conditional sufficient conditions; the proof gaps and the discretization issue in Theorem 5 are correctness problems, not self-referential reductions.

full rationale

The paper's central claims are conditional statements of the form 'if a specified loss is exactly zero, then the pair (C,A) is observable.' The losses are defined from classical rank conditions, the Hautus test, or Vandermonde/Kronecker structure; they are not fitted to observability outcomes and then relabeled as predictions. There is no load-bearing self-citation: the references to Mamba and S4 are external architecture citations, and no prior work by the author is invoked to justify a theorem. Theorem 5's loss (Eq. 53) enforces distinct eigenvalues and nonzero entries of CV; in the diagonalized coordinates this is essentially the classical Hautus eigenvector condition, and the paper supplies a Vandermonde-matrix argument rather than importing the result by fiat. Theorem 5's additional claim about the discretized pair (C, A') with A' = exp(Delta A) is not circular, but it is mathematically unsound as stated: distinct eigenvalues lambda_j do not imply distinct exponentials e^{Delta lambda_j}, since they can collide when Delta(lambda_i - lambda_j) = 2*pi*i*k. Thus the Vandermonde block over e^{Delta lambda_j} can collapse; a concrete counterexample satisfies the loss and fails observability. This is a false-theorem/correctness gap, not a reduction of the conclusion to the hypothesis. Theorems 3 and 4 rely on an admitted informal pigeonhole argument ('this argument is not airtight') and on the paper's own experiments to justify the high-probability kernel condition. That is a missing proof and an evidentiary weakness, but not circularity, because the claimed high-probability observability is not equivalent by construction to the kernel non-nesting hypothesis. Theorem 6 is a conditional convergence result that assumes a decay rate and then verifies the Robbins-Monro summability conditions; the conclusion follows from the stated hypothesis rather than from fitting or self-reference. Overall, the paper has no significant circularity under the required standard of exhibiting a specific equation-level reduction of a claimed result to its inputs.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims depend chiefly on classical control-theory facts plus a set of structural assumptions on A, C, and V. The most fragile additions are the informal high-probability row-space filling assertion and the unproven decay-rate assumption in the Robbins-Monro theorem. The paper introduces no new physical entities.

free parameters (4)
  • positive constants in relu losses = unspecified; 0.05 in Figure 2
    Each proposed loss uses relu(c - ...) with an unspecified c. The central claims require the loss to be exactly zero, and the value of c changes which matrices satisfy the condition.
  • step size Delta = chosen by user; 0.1 in Figure 3
    Delta enters the discretization and the Fourier kernel definitions. The paper treats it as a tunable hyperparameter and does not derive it from the data.
  • function f in the B construction = not specified
    The shared-parameter construction defines B = f(diag(lambda))^{1/2}(U tensor I)S, but f is never defined. This leaves a significant part of the parameterization unspecified.
  • integer p with n = p m = not specified
    The Kronecker construction requires n to be divisible by p. The choice of p is a free structural assumption that changes the parameter count.
assumptions (5)
  • standard math Observability is equivalent to the observability matrix having full column rank; Hautus lemma; Birkhoff-von Neumann theorem; Bauer-Fike theorem; Hoffman-Wielandt inequality; Cayley transform equivalence for (C,A) and (C, exp(Delta A)).
    These classical results are invoked throughout Sections 2 and 4 and in the appendix proofs.
  • domain assumption A is diagonalizable with distinct eigenvalues, C is full row rank, A is full column rank, V is surjective, and Q,U are orthogonal where required.
    Theorems 3, 4, and 5 require these structural restrictions on the model matrices, stated in the theorem hypotheses.
  • domain assumption The 'permutation-invariant property' of C stated in Theorem 1 is well-defined and holds for the C matrices of interest.
    This condition is used in Theorem 1 and Lemma 5, but its formal statement is ambiguous and no constructive characterization is given.
  • ad hoc to paper The high-probability row-space filling assertion: enough kernel changes across j imply the concatenated row spaces have dimension n.
    Theorems 3 and 4 rely on this 'pigeonhole-type argument,' which the paper itself calls 'not airtight.' It is not proven.
  • ad hoc to paper In Theorem 6, the optimizer parameter differences decay as E[||(U_{k+1},Q_{k+1},S_{k+1}) - (U_k,Q_k,S_k)||] ~ Theta(1/k^q) with 1/2 <= q <= 1.
    This decay assumption is not derived; it is essentially the convergence behavior the algorithm is supposed to produce, making the theorem conditional on its own conclusion.

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Cite this review

Pith. "Pith review of Observability conditions for neural state-space models with eigenvalues and their roots of unity." pith.science (2026). https://pith.science/paper/SIJVQLE4

@misc{pith2026250415758,
  author       = {Pith},
  title        = {Pith review of: Observability conditions for neural state-space models with eigenvalues and their roots of unity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SIJVQLE4}},
  note         = {Machine review of arXiv:2504.15758}
}
read the original abstract

We operate through the lens of ordinary differential equations and control theory to study the concept of observability in the context of neural state-space models and the Mamba architecture. We develop strategies to enforce observability, which are tailored to a learning context, specifically where the hidden states are learnable at initial time, in conjunction to over its continuum, and high-dimensional. We also highlight our methods emphasize eigenvalues, roots of unity, or both. Our methods effectuate computational efficiency when enforcing observability, sometimes at great scale. We formulate observability conditions in machine learning based on classical control theory and discuss their computational complexity. Our nontrivial results are fivefold. We discuss observability through the use of permutations in neural applications with learnable matrices without high precision. We present two results built upon the Fourier transform that effect observability with high probability up to the randomness in the learning. These results are worked with the interplay of representations in Fourier space and their eigenstructure, nonlinear mappings, and the observability matrix. We present a result for Mamba that is similar to a Hautus-type condition, but instead employs an argument using a Vandermonde matrix instead of eigenvectors. Our final result is a shared-parameter construction of the Mamba system, which is computationally efficient in high exponentiation. We develop a training algorithm with this coupling, showing it satisfies a Robbins-Monro condition under certain orthogonality, while a more classical training procedure fails to satisfy a contraction with high Lipschitz constant.

Figures

Figures reproduced from arXiv: 2504.15758 by the authors.

Figure 1
Figure 1. We illustrate observability: ambient output states may be used to learn the initial [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. This figures illustrates the Fourier kernel loss as in equation [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. This figure highlights a few important concepts. First, this figure illustrates the number [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: We illustrate that Theorem 4 is empirically valid. (a) illustrates observability and the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: In (a), regarding Theorem 4, as we can see, satisfying the distinct eigenvalues condition [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: This figure helps demonstrate that the Hautus lemma eigenvector test indeed yields an [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: We illustrate the phenomenon that Ψj is more likely to change the rowspace than simply taking powers of Λ. We randomly generate 50 matrices C, V and compare the rowspace across all n 2 possibilities when multiplied together using both Ψj and Λ taken to a power. We have…
Figure 8
Figure 8. Figure 8: We illustrate the same phenomenon as Figure [PITH_FULL_IMAGE:figures/full_fig_p034_8.png]
Figure 9
Figure 9. Figure 9: This figure goes hand-in-hand with Figure [PITH_FULL_IMAGE:figures/full_fig_p035_9.png]
Figure 10
Figure 10. Figure 10: We compare the eigenvalues in a prototypical example of our method (a) versus [PITH_FULL_IMAGE:figures/full_fig_p035_10.png]
Figure 11
Figure 11. Figure 11: We train a Mamba state-space model on simple data of mappings between sinusoidal [PITH_FULL_IMAGE:figures/full_fig_p036_11.png]
Figure 12
Figure 12. Figure 12: We illustrate the rank of the rows concatenated, or the union of their space, with our [PITH_FULL_IMAGE:figures/full_fig_p036_12.png]
Figure 13
Figure 13. Figure 13: We illustrate results upon our proposed algorithm versus a more traditional algorithm. [PITH_FULL_IMAGE:figures/full_fig_p041_13.png]
Figure 14
Figure 14. Figure 14: We illustrate training loss with our proposed algorithm versus a classical objective. [PITH_FULL_IMAGE:figures/full_fig_p042_14.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.