{"id":"8d2bb16d-6457-4677-8c95-ed38e3fb4ffe","arxiv_id":"2504.15758","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"A set of sufficient conditions and training losses for enforcing observability in neural state-space models, with one clean Mamba condition and several unproven high-probability Fourier results.","lead":"This paper proposes training losses and parameter constraints that aim to make the hidden states of Mamba-style neural sequence models recoverable from their outputs, a property called observability. It derives conditions based on eigenvalues, permutation matrices, and the Fourier transform, with one clean sufficient condition for Mamba and several high-risk probabilistic claims.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5 is false as stated: enforcing distinct eigenvalues of A does not enforce distinct eigenvalues of the discretized A' = exp(Δ A), so the Vandermonde columns can collapse and observability of the Mamba system is not guaranteed.","rationale":"I agree with the reader's overall rejection: the paper's advertised high-probability observability results are not established, and the Theorem 6 argument assumes the convergence rate it is supposed to prove. However, the most decisive problem I find is different from the reader's stated weakest assumption. The reader treats Theorem 5 as a clean sufficient condition and locates the main weakness in the unproved pigeonhole argument behind Theorems 3 and 4. That pigeonhole issue is real and serious. But Theorem 5 itself contains a concrete, checkable false claim: the proof's Vandermonde matrix is built from e^{Δλ_j}, while the theorem only enforces distinct λ_j. Because the exponential map is periodic along the imaginary axis, distinct λ_j can produce repeated e^{Δλ_j}, making the Vandermonde columns linearly dependent and the discretized observability matrix rank-deficient. The 2×2 real-matrix example above satisfies all hypotheses and the loss is exactly zero, yet (C, A') is not observable. This is an internal inconsistency, not a matter of probability or an admittedly non-airtight sketch. It matters for the Mamba application because the discretized pair (C, A') is what is actually used in the architecture. The fix is straightforward, but it changes the theorem's content: observability of the discretized system requires distinct eigenvalues of A' or an explicit sampling-rate condition. I therefore agree with rejection, but the most load-bearing concern is the false statement in Theorem 5, not only the unformalized pigeonhole claim.","tokens_in":33619,"tokens_out":9292,"duration_ms":92446,"concrete_test":"Run the 2×2 counterexample: set Δ = 1, A = [[0, π], [−π, 0]], C = [1, 1], and V = [[1, 1], [i, −i]]. Compute the loss in (53) with positive constant 0.1; it is exactly 0. Then compute A' = exp(A) and the rank of the observability matrix [C; C A']; it is 1, not 2. If this reproduces, Theorem 5 is refuted as stated. Then test the natural repair by adding relu(c − min_{i≠j} |e^{Δλ_i} − e^{Δλ_j}|) to the loss and verify that this additional term is strictly positive for the counterexample and restores full rank of the observability matrix.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing concern is that Theorem 5's sufficient condition does not imply observability of the discretized Mamba pair (C, A'), because the proof in Appendix C builds a Vandermonde matrix from e^{Δλ_j} while the enforced loss only makes the λ_j distinct. Distinct λ_j do not imply distinct e^{Δλ_j}: if Δ(λ_i − λ_j) = 2π i k, the exponentials coincide. A concrete counterexample satisfies every hypothesis of Theorem 5: take n = 2, m = 1, Δ = 1, A = [[0, π], [−π, 0]], C = [1, 1], and V the eigenvector matrix of A. The eigenvalues are ±iπ, so min_{i≠j} |λ_i − λ_j| = 2π, and C V = [1+i, 1−i] has both entries of modulus √2. With positive constant 0.1, the loss (53) is exactly 0. But A' = exp(A) = −I, so the observability matrix [C; C A'] = [[1,1], [−1,−1]] has rank 1: the pair (C, A') is not observable. Thus the stated conclusion that the Mamba discretization is observable fails. The fix is not merely a proof gap: the loss must enforce distinct eigenvalues of A', or a sampling constraint such as Δ |Im(λ_i − λ_j)| < 2π for all i ≠ j. This is central because Mamba inference and training use the discretized A'.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":34008,"tokens_out":5517,"duration_ms":56347,"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":[{"comment":"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.","section":"Theorem 5, Appendix C"},{"comment":"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.","section":"Section 4.2 and Appendix B.2"},{"comment":"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.","section":"Corollary to Theorem 3, Appendix B.1"},{"comment":"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.","section":"Theorem 6, Section 4.6"}],"minor_comments":[{"comment":"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}.","section":"Equation (10)"},{"comment":"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.","section":"Equation (27)"},{"comment":"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.","section":"Theorems 3 and 4"},{"comment":"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.","section":"Definitions of theta_k"},{"comment":"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.","section":"References"},{"comment":"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.","section":"Figures and reproducibility"}],"recommendation":"reject","confidential_remarks":"The manuscript repeatedly relies on informal 'high probability' and 'sketch' arguments in its core theorems, including an explicit admission that a central argument is 'not airtight.' More importantly, Theorem 5, which is the paper's main Mamba observability result, is false as stated. While some of the permutation-based sufficient conditions may be salvageable, the central claims need substantial new mathematical work rather than local revision. I would also recommend that the editor check the reference list and the overall presentation quality before considering any future resubmission."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Andrew, quick take on arXiv:2504.15758. The paper has a genuinely useful idea — enforce observability of a Mamba state-space model by penalizing distinct eigenvalues and nonzero entries of C V — but the main Mamba theorem is false as stated, and the Fourier high-probability results are explicitly not airtight. I'd send it back rather than forward.\n\nWhat's actually new: the one-row sufficient condition (Theorem 5) is a clean way to avoid building the full observability matrix, and the Ψ_j row-space diversity losses plus the shared-parameter construction (Theorem 6) are legitimate extensions. The permutation/root-of-unity lemmas are correct as far as I can tell, though narrow. The paper is honest about many of its own gaps, which I respect.\n\nThe soft spots are serious. First, Theorem 5 confuses distinct eigenvalues of A with distinct eigenvalues of A' = exp(ΔA). The proof's Vandermonde matrix is built from e^{Δλ_j}, but the loss only enforces distinct λ_j. The stress-test counterexample is right: n=2, Δ=1, A = [[0,π],[-π,0]], C=[1,1] satisfies the loss exactly but A' = -I and (C,A') is unobservable. That's not a proof gap; it's a false theorem, and it's the paper's headline result. Second, Theorems 3 and 4 rely on a pigeonhole argument that the authors themselves call 'not airtight' (Section 4.2 and Appendix B.2). The high-probability statements are never formalized, so those results are conjectures, not theorems. Third, the Corollary to Theorem 3 is contradicted by the paper's own counterexample (C=(I_m 0), A=I_n) where det F[Ce^{AkΔt}B]_j > 0 for all j but the system is unobservable. Fourth, Theorem 6 assumes the Θ(1/k^q) decay it is supposed to establish, so the 'Robbins-Monro' conclusion is conditional on the very rate at issue.\n\nThere's a salvageable paper in here: fix Theorem 5 by enforcing distinct e^{Δλ_j} or a sampling bound, either prove or drop the high-probability claims, and clean up the corollary. As it stands, the central advertised results don't hold up, and the empirical figures don't compensate. I would not cite this in its current form, and I wouldn't spend a referee cycle on it as-is. If the author repairs the Mamba condition, it becomes a reasonable niche contribution for people working on SSM training stability.","headline":"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.","tokens_in":34467,"tokens_out":4112,"would_cite":false,"duration_ms":36011,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B07","68T07","15A18","42A38"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["observability","neural state-space models","Mamba","control theory","eigenvalues","roots of unity","Fourier transform","Hautus lemma"],"falsifier":"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.","tokens_in":33367,"feed_emoji":"🎛️","tokens_out":8535,"duration_ms":74641,"temperature":0.7,"pith_summary":"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.","feed_headline":"Distinct eigenvalues plus one nonzero row make Mamba observable","feed_subtitle":"A Vandermonde argument turns observability into a cheap loss on eigenvalues and one row of CV.","key_machinery":"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.","core_discovery":"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.\"","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the Mamba architecture and its discretization $A'=\\exp(\\Delta A)$, $B'=(\\Delta A)^{-1}(\\exp(\\Delta A)-I)\\Delta B$, which Theorems 5-7 target.","marker":"[19]"},{"why":"Supplies the Fourier/convolutional state-space setup and the kernels $K$ analyzed in Theorems 3 and 4.","marker":"[20]"},{"why":"Gives the classical duality definition of observability via controllability and the observability matrix used throughout the paper.","marker":"[15]"},{"why":"Bauer-Fike theorem, used in Lemma 3 to show that eigenvalues close to roots of unity imply the matrix is close to a permutation matrix.","marker":"[4]"},{"why":"Birkhoff-von Neumann theorem, used in Lemma 2 to force a doubly stochastic matrix with distinct root-of-unity eigenvalues to be a permutation matrix.","marker":"[12]"},{"why":"Provides the Robbins-Monro/stochastic approximation condition that Theorem 6 verifies for the coupled training algorithm.","marker":"[43]"},{"why":"Hoffman-Wielandt inequality, used in Lemma 4 to show that observability matrices sufficiently close to a full-rank one remain full rank.","marker":"[11]"},{"why":"Supplies the formula for derivatives of eigenvectors, used in Lemma 10 to lower-bound the Lipschitz constant of the eigendecomposition in Theorem 7.","marker":"[39]"},{"why":"Supplies the formula for derivatives of eigenvalues, used alongside [39] in Lemma 10.","marker":"[1]"}],"fun_headline_variants":["Cheap observability via eigenvalue separation and one nonzero row","Vandermonde argument: distinct eigenvalues plus one nonzero row force observability","Eigenvalue gaps plus one nonzero row of CV: cheap observability condition","Roots of unity and Vandermonde: cheap observability in Mamba","One nonzero row in CV plus distinct eigenvalues = observable Mamba"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Cheap observability via eigenvalue separation and one nonzero row","Vandermonde argument: distinct eigenvalues plus one nonzero row force observability","Eigenvalue gaps plus one nonzero row of CV: cheap observability condition","Roots of unity and Vandermonde: cheap observability in Mamba","One nonzero row in CV plus distinct eigenvalues = observable Mamba"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001237,"raw_usage":{"total_tokens":5162,"prompt_tokens":1113,"completion_tokens":4049,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":729,"completion_tokens_details":{"reasoning_tokens":3955}},"tokens_in":729,"tokens_out":4049,"duration_ms":24372,"temperature":1.0,"reasoning_tokens":3955,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:19:30.969082+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the classical duality definition of observability via controllability and the observability matrix used throughout the paper."},{"cited_title":"URL https://en.wikipedia.org/wiki/Bauer%E2%80%93Fike_ theorem","cited_arxiv_id":null,"evidence_quote":"Bauer-Fike theorem, used in Lemma 3 to show that eigenvalues close to roots of unity imply the matrix is close to a permutation matrix."},{"cited_title":"Doubly stochastic matrix","cited_arxiv_id":null,"evidence_quote":"Birkhoff-von Neumann theorem, used in Lemma 2 to force a doubly stochastic matrix with distinct root-of-unity eigenvalues to be a permutation matrix."},{"cited_title":"The hoffman-wielandt inequality","cited_arxiv_id":null,"evidence_quote":"Hoffman-Wielandt inequality, used in Lemma 4 to show that observability matrices sufficiently close to a full-rank one remain full rank."},{"cited_title":"Derivative of eigenvectors of a matrix with respect to its components","cited_arxiv_id":null,"evidence_quote":"Supplies the formula for derivatives of eigenvectors, used in Lemma 10 to lower-bound the Lipschitz constant of the eigendecomposition in Theorem 7."},{"cited_title":"StackExchange","cited_arxiv_id":null,"evidence_quote":"Supplies the formula for derivatives of eigenvalues, used alongside [39] in Lemma 10."}],"review_version":1}