{"id":"e900c91d-e7a9-437c-978e-6543bbd3651b","arxiv_id":"2411.17729","paper_version":3,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives a finite-product approximation of an LTI transfer function and gives a cascade time-domain algorithm, but the advertised unconditional stability and 2L cost bound are inconsistent with the derivation.","lead":"An algorithm is proposed for applying state space model transfer functions using a cascade of matrix multiplications, based on a classical product identity. The paper claims it works even when the state matrix has eigenvalues outside the unit circle, but that claim is not supported by the paper's own derivation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 1 requires the largest singular value γ of A to be strictly below 1, yet the abstract and Section 6 drop this hypothesis and advertise accuracy even when eigenvalues exceed 1; for |λ|>1 the cascade tail grows like λ^{2^{N+1}}, so the central claim of unconditional approximation is unsupported.","rationale":"The paper's genuine contribution is the finite-product factorization of ∑_{k=0}^{2^{N+1}-1}(z^{-1}A)^k, which is a valid causal convolution algorithm and is numerically stable in the sense that no feedback recursion is used. For matrices with largest singular value below 1, Lemma 1 supplies the advertised tail bound and the method is sound. The load-bearing flaw is the unannounced removal of that hypothesis when the abstract and Section 6 claim that approximate state matrices may have eigenvalues exceeding 1. This is not merely a gap in exposition: the error analysis in Lemma 1 breaks, and the approximation objective itself is unattainable for |λ|>1 because the target function has nonvanishing anti-causal Fourier coefficients. The scalar counterexample is decisive and requires no code or external benchmark. The complexity discrepancy is also real and follows directly from the update rules, though it is secondary to the approximation failure. The preprint provides no numerical experiments or error tables that could rescue the eigenvalue claim, and the Mathematica example in Section 4 uses only a stable HiPPO matrix. I therefore find no reason to change the Reader's reject verdict; the central advertised properties remain unsupported unless the hypotheses are restricted to γ<1 and the complexity claim is corrected.","tokens_in":6112,"tokens_out":10761,"duration_ms":109725,"concrete_test":"Run the scalar counterexample. Set A=[1.01], B=C=[1], D=[0], so H(z)=1/(1−1.01 z^{-1}) and H(1)=−100. For N=0,1,2,..., compute H_N(1)=∑_{k=0}^{2^{N+1}-1}1.01^k and plot the relative error |H_N(1)−H(1)|/|H(1)|. The error grows like 1.01^{2^{N+1}}, demonstrating that Lemma 1's hypothesis γ<1 cannot be relaxed and no user-selected accuracy is achieved. As a secondary check, instrument the pseudocode for L=2^K and count matrix-vector multiplications by A^{2^{n−1}}; the total is ∑_{n=1}^K (L−2^{n−1}) ≈ L log_2 L, not 2L.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive soft spot is the jump from Lemma 1 in Section 2 to the claims in the abstract and Section 6. Lemma 1 bounds the tail error of the polynomial product by γ^{2^{N+1}}/(1−γ), which exists only if the largest singular value γ of A is strictly less than 1. The paper then claims that within the desired accuracy some eigenvalues of an approximate state matrix may exceed 1, and that the cascade remains an unconditionally stable approximation of H(z). For a scalar eigenvalue λ with |λ|>1, the telescoping identity (I−X)∏_{n=0}^N(I+X^{2^n}) = I−X^{2^{N+1}} gives H_N(1) = ∑_{k=0}^{2^{N+1}-1} λ^k = (λ^{2^{N+1}}−1)/(λ−1), whereas H(1)=1/(1−λ). For λ=1.01, H(1)=−100 while H_N(1) grows like 100·1.01^{2^{N+1}}; no N can reach any fixed accuracy. Equivalently, for |λ|>1 the rational function H(z) has an anti-causal Laurent expansion on the unit circle, so a causal FIR polynomial cannot converge uniformly to it. Thus the advertised benefit of allowing eigenvalues above 1 fails exactly where Lemma 1's hypothesis is violated. A secondary inconsistency is the arithmetic complexity: the update rules (3.2)–(3.4) apply A^{2^{n−1}} to L−2^{n−1} columns at stage n, giving O(L log L) total matrix-vector products, not the claimed 2L; Section 6's description of applying A^2 to every second entry does not match the pseudocode.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a cascade algorithm for applying the transfer function H(z) = C(I_m - z^{-1}A)^{-1}B + D of a linear time-invariant system as a convolution in the time domain. The method replaces the inverse (I_m - z^{-1}A)^{-1} by the finite product \\prod_{n=0}^N (I + z^{-2^n} A^{2^n}), whose time-domain realization updates a sequence by repeatedly applying the powers A, A^2, A^4, ... . The abstract and Section 6 claim that the resulting algorithm is unconditionally stable, that it requires no more than 2L matrix-vector multiplications to compute L states, and that it remains accurate even when an approximate state matrix has some eigenvalues with absolute value greater than 1. Section 4 reports a numerical example with a discretized HiPPO matrix whose eigenvalues are all below 1.","tokens_in":6445,"tokens_out":6405,"duration_ms":64884,"significance":"The underlying algebraic identity is elegant and correct in the stable case: for a matrix A with spectral radius below 1, the telescoping product gives a uniform polynomial approximation of (I - z^{-1}A)^{-1} on the unit circle, and Lemma 1 quantifies this approximation in terms of the largest singular value \\gamma < 1. If the advertised complexity and stability properties were true, the method would be a useful alternative to direct recurrence for long-range state-space models. However, the central advertised property -- accuracy with eigenvalues exceeding 1 -- is false, and the complexity count in the abstract and Section 6 is inconsistent with the algorithm's own update rules. The paper contains no code, no reproducible experiments, and no numerical test of the claimed eigenvalue-above-1 regime, so the positive aspects are limited to the derivation of Lemma 1 and the cascade identity itself.","major_comments":[{"comment":"The central claim of unconditional accuracy with eigenvalues above 1 is unsupported and, in fact, false. Lemma 1 assumes that the largest singular value \\gamma of A is strictly less than 1, and its tail bound \\gamma^{2^{N+1}}/(1-\\gamma) exists only in that regime. The abstract and Section 6, however, state that within the desired accuracy some eigenvalues of an approximate state matrix may have absolute value exceeding 1 and that the cascade remains an unconditionally stable approximation of H(z). For a scalar eigenvalue \\lambda > 1 the telescoping identity gives H_N(1) = \\sum_{k=0}^{2^{N+1}-1} \\lambda^k = (\\lambda^{2^{N+1}}-1)/(\\lambda-1), whereas H(1) = 1/(1-\\lambda). With \\lambda = 1.01, H(1) = -100 while H_N(1) grows like 100 \\cdot 1.01^{2^{N+1}}, so no choice of N yields any fixed accuracy. Thus the claimed benefit of allowing structured approximations with eigenvalues above 1 fails exactly where Lemma 1's hypothesis is violated, and this is a load-bearing error for the paper's main message.","section":"Section 2, Lemma 1; abstract; Section 6"},{"comment":"The claimed arithmetic complexity is inconsistent with the stated algorithm. At stage n the update rule applies A^{2^{n-1}} to L - 2^{n-1} columns, so the total number of matrix-vector multiplications is \\sum_{n} (L - 2^{n-1}) over the relevant range, which is O(L log L), not O(L) and not the abstract's 'no more than 2L'. For example, for L = 8 the numbers are 7 + 6 + 4 = 17 matrix-vector multiplications, already exceeding 16. Section 6's description that 'A is applied to every entry, A^2 to every second entry, A^4 to every fourth entry, etc.' does not match Eqs. (3.3)-(3.4), where A^2 is applied to L-2 columns and A^4 to L-4 columns. The complexity claims in the abstract and Section 3 consequently need to be corrected, and the claimed 'factor not exceeding two' advantage over the standard recursion is not supported.","section":"Section 3, Eqs. (3.2)-(3.4); abstract; Section 6"}],"minor_comments":[{"comment":"Equation (3.1) writes the direct convolution as y_\\ell = \\sum_{k=0}^\\ell C A^{\\ell-k} B u_k + D u_k; the last term should be D u_\\ell, not D u_k.","section":"Eq. (3.1)"},{"comment":"The sentence that applying the matrix polynomial 'as a cascade in time domain requires only N+1 matrix-vector multiplications' is at best ambiguous and, taken literally, is inconsistent with the algorithm in Section 3, which requires O(L log L) such multiplications for L outputs.","section":"Section 2, after Eq. (2.2)"},{"comment":"The numerical example only treats a stable discretized HiPPO matrix whose eigenvalues are all below 1; it provides no evidence for the advertised scenario of an approximate state matrix with eigenvalues greater than 1 in modulus.","section":"Section 4"},{"comment":"The structured approximation proposals (partitioned low-rank, wavelet non-standard form, multiwavelets) are described only as possibilities; no concrete algorithm, error analysis, or numerical demonstration is provided for any of them.","section":"Section 5"}],"recommendation":"reject","confidential_remarks":"The paper is not circular and the author's prior FIR-filter works are cited as background rather than as the target result. The reason for rejection is that the central correctness claim -- accuracy when eigenvalues exceed 1 -- is mathematically false, and the complexity claim is contradicted by the manuscript's own update formulas. These are not local presentational issues. A revision that restricted the theorem to \\gamma < 1 and corrected the complexity count would remove the advertised advantages that motivate the paper."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Greg's paper is a short note that rewrites his earlier FIR-cascade approximation of IIR filters in state-space language. The core construction is correct for the stable case: when the largest singular value γ of A is below 1, Lemma 1's error bound holds and the cascade update rules in Section 3 are a clean way to apply the truncated polynomial. That part is not new—it is the same product factorization from his own [6,7]—but it is clearly presented.\n\nThe abstract, however, makes a stronger claim: the algorithm is unconditionally stable and 'it is not necessary to assure that an approximate state matrix has all eigenvalues with absolute values strictly less than 1.' That claim is false. Lemma 1 requires γ<1. For a scalar eigenvalue λ>1, the truncated product grows like λ^{2^{N+1}}, so no N gives a fixed accuracy; the tail term cannot be small. The stress-test counterexample is right. The paper does not offer a separate argument that covers that case.\n\nThere is also a complexity error. The update rules apply A^{2^{n−1}} to L−2^{n−1} columns at stage n. Summing over n gives O(L log L) matrix-vector products, not the promised 2L. Section 6's own description—A to every entry, A^2 to every second, A^4 to every fourth—confirms the log factor.\n\nThe paper has no experiments or code; the only example is a brief Mathematica check. The references are appropriate and include the author's prior work. So the stable-case material is fine, but the advertised advantage for unstable approximations is the main point, and it fails. A revised version limited to the stable case might be publishable as a short note, but the current claims cannot stand.\n\nMy recommendation: this does not deserve referee time as is. Desk reject or return for major revision with the unsupported claims removed.","headline":"A correct variant of the author's old FIR-cascade trick for stable matrices, with an advertised extension to |λ|>1 that is false and a complexity claim off by a log factor.","tokens_in":7002,"tokens_out":3615,"would_cite":false,"duration_ms":48442,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65F60","93C05"],"pacs":[],"model":"deepseek-v4-flash","headline":"An unconditionally stable cascade algorithm applies state-space transfer functions in time at about twice the cost of the standard recursion.","keywords":["state space models","linear time-invariant systems","transfer function","cascade algorithm","matrix polynomial","structured matrices","HiPPO matrix","IIR/FIR approximation"],"falsifier":"For scalar systems, set a = 1.01 (eigenvalue above 1), B = C = 1, D = 0, and compare the cascade output H_N with the true stable target H(z) = 1/(1 - $z^{{-1}}$ a_0) for a_0 = 0.99. The tail (a/z)^{$2^{{N+1}}$} grows with N, so the error ‖H - H_N‖ on |z| = 1 will not decrease as N increases; observing that would refute the unconditional-stability claim.","tokens_in":5832,"feed_emoji":"🧮","tokens_out":10265,"duration_ms":93624,"temperature":0.7,"pith_summary":"This paper is about applying the transfer function of a linear time-invariant state space model as a convolution in time, without running the recursion that becomes unstable when the state matrix has an eigenvalue close to or above 1 in modulus. The proposed algorithm truncates the infinite product (I - $z^{{-1}}$A)^{-1} = ∏_{n≥0}(I + ($z^{{-1}}$A)^{2^n}) to N+1 factors, forming a matrix polynomial of degree $2^{{N+1}}$-1. In the time domain this polynomial acts as a cascade that reuses intermediate matrix-vector products, so L output samples cost at most about 2L matrix-vector multiplications, compared with L for the standard recursion. The paper's advertised payoff is unconditional stability: structured approximations of the state matrix that would break the eigenvalue condition no longer have to be rejected, so cheaper O(m) or O(m log m) matrix-vector multiplication schemes become available. A numerical example with a HiPPO matrix of size 100 shows that N=14 achieves degree 32767 with 15 matrix-vector multiplications.","feed_headline":"State-space convolution stays stable past the eigenvalue cliff","feed_subtitle":"A cascade of matrix-polynomial factors keeps long-range models stable for roughly twice the arithmetic.","key_machinery":"The load-bearing object is the infinite product identity (I - $z^{{-1}}$A)^{-1} = ∏_{n=0}^{∞} (I + ($z^{{-1}}$A)^{2^n}), valid on the unit circle when the spectral radius of A is below 1. The algorithm truncates this product after N+1 factors, giving a matrix polynomial of degree $2^{{N+1}}$-1, and applies it via a cascade: at level n, the matrix $A^{{2^{n-1}}$} acts only on entries spaced $2^{{n-1}}$ apart, reusing earlier intermediate vectors. That re-use is what caps the total work at about 2L matrix-vector products for L time steps, independent of the polynomial degree.","core_discovery":"The paper's central claim is that the rational transfer function H(z) = C(I - $z^{{-1}}$A)^{-1}B + D of a discrete-time LTI system can be replaced, to any user-selected accuracy, by the matrix polynomial H_N(z) = C ∏_{n=0}^{N}(I + ($z^{{-1}}$A)^{2^n})B + D, and that this polynomial can be applied as a cascade in the time domain using only about 2L matrix-vector multiplications for L output samples. Because the cascade only ever multiplies by powers of A and never iterates a recursion with a potentially unstable state matrix, the paper asserts the algorithm is unconditionally stable: even if a structured approximation of A used inside the product has eigenvalues with magnitude above 1, the convolution remains bounded. This removes the constraint that approximate state matrices must be confined to the unit disk, and thereby widens the range of structured matrix representations that can accelerate the matrix-vector product.","pith_inferences":["The unconditional-stability claim as stated in the abstract goes beyond what Lemma 1 proves, since the error bound requires the largest singular value γ to stay below 1; for an approximate eigenvalue above 1 the tail (z^{-1}A)^{2^{N+1}} grows, so a separate error analysis would be needed to support the claim.","For normal state matrices the cascade identity reduces to the scalar geometric identity on each eigenvalue, so stability depends only on the spectral radius; for highly non-normal matrices the singular-value condition is the binding constraint, and the cascade may be preferable to diagonalization when eigenvectors are ill-conditioned.","The cascade's reuse pattern suggests natural extensions to quantized or randomized matrix-vector products, since stability no longer couples to the spectrum of the approximate state matrix.","A direct numerical test on scalar systems with a = 1.01 would separate the paper's stability claim from its accuracy claim; the paper's example stays within γ<1 and so does not exercise the advertised beyond-unit-circle regime."],"forward_implications":["The eigenvalue restriction that confines standard SSM recursions to the open unit disk can be lifted in the cascade formulation, so structured approximations such as diagonal-plus-low-rank, partitioned low-rank, or wavelet non-standard forms can be used with less fear of instability.","The cost of the cascade is at most a factor of two above the direct recursion, and the algorithm is parallel-friendly since each level's updates act on disjoint index sets.","The approximation accuracy is controlled by N: the truncation error decays like γ^{2^{N+1}}/(1-γ), so any user-selected relative error can be met by choosing N of order log_2 log(1/ε).","For long sequences, the cascade's L + L/2 + L/4 + ... pattern of matrix-vector products amounts to about 2L operations, independent of the degree of the approximating polynomial."],"supporting_citations":[{"why":"Supplies the factored FIR approximation identity and error analysis that Lemma 1 is built on.","marker":"[6]"},{"why":"Extends the factored FIR design to high-accuracy IIR/FIR filters; the cascade stability properties are inherited from this method.","marker":"[7]"},{"why":"Defines the structured state space (S4) setting and the HiPPO matrix, motivating the need to apply transfer functions with nearly unit eigenvalues.","marker":"[1]"},{"why":"Introduces the HiPPO matrix used in the paper's numerical example.","marker":"[9]"},{"why":"Describes approximate diagonalization of SSMs, the structured approximation context where eigenvalues can drift outside the unit circle.","marker":"[4]"},{"why":"Provides the partitioned low-rank representation used as an example of O(m log m) matrix-vector multiplication.","marker":"[10]"},{"why":"Provides the non-standard form wavelet representation used for O(m) matrix-vector multiplication.","marker":"[11]"}],"fun_headline_variants":["Matrix polynomial cascade tames unstable state matrices","Eigenvalue-proof convolution for state space models","Stable SSM convolution via cascade of powers","Unconditionally stable SSM convolution in 2L matvecs","Cascade polynomial keeps long-range models stable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument rests on the assumption that the state matrix has all singular values below 1, which makes the cascade product converge; the paper's promise that eigenvalues above 1 are acceptable is not backed by that convergence proof.","fun_headline_variants_meta":{"raw":{"variants":["Matrix polynomial cascade tames unstable state matrices","Eigenvalue-proof convolution for state space models","Stable SSM convolution via cascade of powers","Unconditionally stable SSM convolution in 2L matvecs","Cascade polynomial keeps long-range models stable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1438,"prompt_tokens":1025,"completion_tokens":413,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":641,"completion_tokens_details":{"reasoning_tokens":339}},"tokens_in":641,"tokens_out":413,"duration_ms":4469,"temperature":1.0,"reasoning_tokens":339,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:58:55.445788+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For scalar systems, set a = 1.01 (eigenvalue above 1), B = C = 1, D = 0, and compare the cascade output H_N with the true stable target H(z) = 1/(1 - $z^{{-1}}$ a_0) for a_0 = 0.99. The tail (a/z)^{$2^{{N+1}}$} grows with N, so the error ‖H - H_N‖ on |z| = 1 will not decrease as N increases; observing that would refute the unconditional-stability claim.","supporting_citations":[{"cited_title":"On factored FIR approximation of IIR filters ,","cited_arxiv_id":null,"evidence_quote":"Supplies the factored FIR approximation identity and error analysis that Lemma 1 is built on."},{"cited_title":"On the design of hi ghly accurate and efficient IIR and FIR filters,","cited_arxiv_id":null,"evidence_quote":"Extends the factored FIR design to high-accuracy IIR/FIR filters; the cascade stability properties are inherited from this method."},{"cited_title":"Hippo: Recurr ent memory with optimal polynomial projections,","cited_arxiv_id":null,"evidence_quote":"Introduces the HiPPO matrix used in the paper's numerical example."},{"cited_title":"Wave propagation using bas es for bandlimited functions,","cited_arxiv_id":null,"evidence_quote":"Provides the partitioned low-rank representation used as an example of O(m log m) matrix-vector multiplication."},{"cited_title":"Fast wavelet tr ansforms and numerical algorithms, I,","cited_arxiv_id":null,"evidence_quote":"Provides the non-standard form wavelet representation used for O(m) matrix-vector multiplication."}],"review_version":1}