{"id":"9cdadba7-1d22-400b-a5b9-c14941638de1","arxiv_id":"2506.06256","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Quadratic extended and unscented Kalman filters are derived by augmenting the measurement with its Kronecker square in the update.","lead":"This paper derives a way to upgrade standard Kalman filters so the state update can curve with the measurement, using the measurement's square in addition to the measurement itself. The authors report accuracy gains in a space-relative-navigation simulation and argue the upgrade applies to many existing filters.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The vector-measurement update is undefined as written: Eq. (22) has singular P_YY for m>1 because δy⊗δy contains duplicate entries, yet Eqs. (20)-(23) require P^{-1}; the CW example uses m=2, leaving the central numerical claim unsupported.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: singular P_YY for vector measurements. This is an internal inconsistency in the paper's mathematical formulation, not merely a disagreement with existing consensus. The augmented covariance is singular by construction for m>1 because the Kronecker square duplicates cross terms; the paper never defines a pseudo-inverse or a reduced basis. Since the numerical application uses a two-dimensional measurement vector, Eqs. (20)-(23) and (34)-(36) cannot be evaluated as written for the paper's main demonstration. The concern is concrete and falsifiable by a rank computation. A pseudo-inverse or a duplicate-free quadratic basis may repair the algorithm, and the qualitative claim might survive, but that repair is not in the manuscript. Therefore the reader's REJECT verdict remains appropriate; no adjustment is needed. I also considered whether the QEKF's degeneracy to the EKF under Gaussian priors is more load-bearing, but that is explicitly acknowledged in Section VII.A and does not undermine the QUKF comparison. The singular-inverse issue is the decisive gap.","tokens_in":10311,"tokens_out":5246,"duration_ms":55362,"concrete_test":"Compute the numerical rank of P_YY from Eq. (22) for the CW measurement setup (m=2, nondegenerate prior and noise) using the paper's definitions. If rank(P_YY) < 6, then P_YY is singular and K = PxY P_YY^{-1} in Eq. (34) is undefined. A natural follow-up is to re-run the CW example after replacing δy[2] with a duplicate-free quadratic basis (δy1^2, δy2^2, δy1δy2) and a pseudo-inverse, checking whether the reported QEKF/QUKF improvements persist.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the quadratic update (Eq. 23) yields accurate vector-measurement updates. This requires the augmented covariance P_YY in Eq. (22) to be invertible. For any measurement dimension m>1, δy[2]=δy⊗δy contains identical pairs (δy_iδy_j and δy_jδy_i), so the random vector [δy; δy[2]] has exact linear dependencies. Consequently, Py[2]y[2] and the full P_YY are singular: for each pair i≠j, a coefficient vector with +1 in the ij slot and -1 in the ji slot annihilates δy[2]. Subtracting v(Pyy)v(Pyy)^T in the lower block does not remove this degeneracy. Since P_YY is singular, P^{-1} in Eqs. (20), (23), and (34) does not exist; no pseudo-inverse or regularization is specified. This is not a minor technicality. The Clohessy-Wiltshire experiment uses a two-dimensional measurement vector, so the main numerical demonstration cannot be reproduced from the equations as written. The paper itself notes that earlier polynomial-update work avoided vector measurements because of matrix singularity, but it does not resolve the analogous singularity in its own formulation. The scalar example is well-posed but cannot validate the vector-measurement claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes quadratic extensions of the extended and unscented Kalman filters. The authors derive a quadratic MMSE estimator from the orthogonality principle, with the update g(y) = E[x] + B(y-E[y]) + C((y-E[y])⊗(y-E[y]) - vec(P_yy)). They compute the required moments using linearization (QEKF) and the unscented transform (QUKF), and evaluate the filters on a scalar toy problem and on a Clohessy-Wiltshire relative-navigation problem with two angle measurements. The paper claims that these quadratic updates outperform their linear counterparts and that the formulation can be generalized to other linear estimators.","tokens_in":10577,"tokens_out":10488,"duration_ms":91029,"significance":"The idea of upgrading linear Kalman updates with quadratic terms is of practical interest, and the derivation is self-contained and mostly clear. The scalar experiment shows a meaningful RMSE improvement for the QUKF, and the method, if valid for vector measurements, could be applied to other filters such as CKF and EnKF. However, the vector-measurement case is not well defined as written, and the main numerical application is affected. The paper's claims are therefore only partially supported.","major_comments":[{"comment":"For measurement dimension m>1, the augmented covariance matrix P_YY in Eq. (22) is singular by construction. The vector δy[2]=δy⊗δy contains identical products (e.g., δy_1 δy_2 and δy_2 δy_1), so there exist nonzero vectors e such that the linear combination e^T[δy; δy[2]] is identically zero. Therefore e^T P_YY e = 0, and P_YY^{-1} in Eqs. (20), (23), and (34) does not exist. The subtraction of v(P_yy)v(P_yy)^T in the lower-right block does not remove this degeneracy. Since the Clohessy-Wiltshire experiment in Section VII-B uses a two-dimensional measurement vector (Eqs. (77)-(78)), the central numerical claim cannot be reproduced from the equations as written. The paper notes that earlier polynomial updates were restricted to scalar measurements because of matrix singularity (Section I), but it does not propose a pseudo-inverse, regularization, or a reduced set of unique quadratic monomials.","section":"Section II, Eqs. (20)-(23) and Section VII-B"},{"comment":"The scalar example does not support the claimed benefits for QEKF: because the prior is Gaussian and the Isserlis assumption (Eqs. (45)-(46)) is used, S_xxx=0 and hence P_xy[2]=0 (Eq. (39)), so the QEKF update in Eq. (35) reduces to the EKF update. In Fig. 2 the QEKF and EKF bars are identical. Thus the only valid demonstration of a quadratic improvement is the QUKF in the scalar example and the (undefined) vector example. The conclusion in Section VIII that 'the QEKF and QUKF propose significant improvements' is therefore not supported for QEKF by the presented evidence.","section":"Section VII-A and Section VIII"},{"comment":"If a pseudo-inverse of P_YY were used to repair the vector case, the covariance update P_xx^+ = P_xx^- - K P_YY K^T would require additional justification. For a rank-deficient regressor, the minimum-norm solution of the normal equations does not in general give the posterior covariance of the resulting estimator; the error covariance of the quadratic estimator differs from this expression unless the regressor covariance is nonsingular. The paper does not provide such a justification.","section":"Section IV, Eq. (36)"}],"minor_comments":[{"comment":"The right-hand side as typeset appears to be [P_xy P_xy[2]]^T, but the product [B* C*] P_YY is an n_x × (m+m^2) matrix; the right-hand side should be [P_xy P_xy[2]] (without transpose). Please correct the notation.","section":"Eq. (19)"},{"comment":"The phrase 'the Quadratic Unscented Kalman Filter (UKF)' should read '(QUKF)'.","section":"Section II, final paragraph"},{"comment":"The symbol δY is used both for the measurement deviation Y_k^{(i)} - y_k and for the augmented deviation that includes the Kronecker square; using a separate symbol (e.g., δY_aug) would avoid ambiguity.","section":"Eq. (60)"},{"comment":"The word 'respectfully' should be 'respectively'.","section":"Figure 5 caption"},{"comment":"Please state explicitly whether the two measurement noises η1 and η2 are mutually independent and independent over time, as this is used implicitly in the covariance expressions (Eqs. (66)-(68)).","section":"Section VII-B"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' own previous work (Refs. [19], [20], [22]) for the underlying polynomial-update concept. This is not inappropriate, but the novelty relative to those papers could be stated more clearly. In addition, the singular-vector-measurement issue is the sort of problem that would block publication in most journals; the authors should be encouraged to either restrict the claims to scalar measurements or develop a valid vector formulation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe core problem with this paper is not the algebra but the fact that the central algorithm is undefined for vector measurements. For any m>1, δy⊗δy contains duplicate entries, so the augmented covariance P_YY in Eq. (22) is singular. The paper uses ordinary inverses in Eqs. (20), (23), and (34) and never mentions a pseudo-inverse or regularization. The Clohessy–Wiltshire example uses a two-dimensional measurement, so the main numerical demonstration cannot be reproduced from the equations as written. The authors even note in the introduction that previous polynomial updates were tested only with scalar measurements to avoid matrix singularity; they didn't resolve that issue here.\n\nWhat is worth taking seriously: the QMMSE estimator itself is known from the authors' prior work, and the orthogonality-principle derivation is clean for scalar measurements. The new contribution is the evaluation of the required high-order moments via linearization (QEKF) and sigma points (QUKF). The scalar example is instructive, and the QUKF shows a real ~25% RMSE improvement over the UKF because it captures skewness that a linear estimator misses.\n\nThe soft spots are significant. The QEKF collapses to the EKF under Gaussian assumptions, leaving the non-Gaussian vector case as the only place where it could matter—and that is exactly the case with the singular P_YY. The paper also omits Monte Carlo run counts, UKF tuning parameters, and code, and the moment expression in Eq. (44) is stated without derivation. These are fixable, but together they undercut the abstract's claim of general benefits.\n\nI agree with the reader's REJECT verdict, but this is not a desk-reject paper. The flaw is identifiable and repairable: use a generalized inverse and specify the estimator uniquely, restrict to scalar measurements, or add a regularized formulation. Add the missing experimental details and show the moment derivations. A serious referee could guide that revision. I recommend sending it to peer review.\n\nNet: interesting idea, honest derivation for the scalar case, but the vector-measurement claim is unsupported as written.","headline":"The vector-measurement update is undefined as written because P_YY is singular for m>1, so the CW results are unsupported, but the scalar case and the QUKF idea show enough merit to warrant a major revision.","tokens_in":11117,"tokens_out":6332,"would_cite":false,"duration_ms":58209,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that replacing the linear measurement update of the Extended and Unscented Kalman Filters with a quadratic estimator that also uses the square of the measurement residual produces more accurate state estimates, with the…","keywords":["Extended Kalman Filter","Unscented Kalman Filter","quadratic update","polynomial estimator","minimum mean square error","Clohessy–Wiltshire equations","non-Gaussian noise","Kronecker product"],"falsifier":"Compute $P_{YY}$ from Eq. (22) for a two-dimensional measurement with a Gaussian prior and any nonzero measurement noise; the second block column contains identical rows for the products $\\delta y_1 \\delta y_2$ and $\\delta y_2 \\delta y_1$, so the determinant of $P_{YY}$ is identically zero. A Monte Carlo run of the Clohessy–Wiltshire example using the formulas exactly as printed would therefore fail at the inversion step, exposing that the singular case is never handled. Conversely, a run inserting a least-squares pseudo-inverse would show whether the quadratic updates still beat the linear ones once the singularity is bypassed.","tokens_in":10013,"feed_emoji":"🛰️","tokens_out":8155,"duration_ms":66292,"temperature":0.7,"pith_summary":"Common nonlinear filters—the Extended and Unscented Kalman Filters—still produce an estimate that is a linear function of the measurement, which is only a first-order stand-in for the true minimum mean square error estimator. The paper proposes a quadratic estimator that adds the Kronecker square $\\delta y^{[2]} = \\delta y \\otimes \\delta y$ of the measurement residual to the update, and derives the optimal coefficients from the orthogonality principle. It then builds two working filters, the QEKF and QUKF, that keep the Kalman recursion but use augmented covariances that incorporate skewness and kurtosis of the prior and noise. In a scalar example the QUKF achieves about 25% lower root-mean-square error than the UKF, and in a satellite relative-navigation problem both quadratic filters outperform their linear counterparts.","feed_headline":"Adding the measurement square cuts Kalman update error by ~25%","feed_subtitle":"Extending the Extended and Unscented Kalman Filters to quadratic updates beats their linear versions on nonlinear measurements.","key_machinery":"The load-bearing object is the quadratic minimum-mean-square-error estimator and its augmented covariance block $P_{YY}$ (Eq. 22) with cross-covariance $P_{xY}$ (Eq. 21). Here $\\delta y^{[2]} = \\delta y \\otimes \\delta y$ is the Kronecker square of the measurement residual, $\\mathrm{v}(P_{yy})$ is the stack operator applied to the measurement covariance, and $K = P_{xY} P_{YY}^{-1}$ acts like a Kalman gain on the augmented residual $[\\delta y;\\, \\delta y^{[2]} - \\mathrm{v}(P_{yy})]$. The argument works because the quadratic term carries third- and fourth-order moment information—skewness and kurtosis—that the linear update discards, and because both the linearization route (QEKF) and the $\\sigma$-point route (QUKF) can be made to supply those moments.","core_discovery":"The central claim is that the linear measurement update shared by the EKF and UKF is the reason these filters leave accuracy on the table, and that a modest modification—replacing the linear estimator $g_L(y)$ with the parabolic $g_Q(y) = \\mathbb{E}[x] + B\\, \\delta y + C\\, \\delta y^{[2]}$—recovers part of the nonlinear conditional mean. The optimal constants $B^*$ and $C^*$ follow from the orthogonality principle: they are the augmented Kalman gain $K = P_{xY} P_{YY}^{-1}$, where the augmented matrices contain cross-covariances with $\\delta y^{[2]}$ and the covariance of $\\delta y^{[2]}$ itself, offset by $\\mathrm{v}(P_{yy})$. The QEKF evaluates these moments by linearizing the measurement model and applying Isserlis' formula for Gaussian kurtosis, while the QUKF evaluates them by propagating $\\sigma$ points and their Kronecker squares. The update equations retain the familiar Kalman form, with the residual vector augmented by $\\delta y^{[2]} - \\mathrm{v}(P_{yy})$.","pith_inferences":["The paper leaves unsigned the fact that $\\delta y \\otimes \\delta y$ duplicates entries whenever the measurement dimension $m$ exceeds 1 (e.g., $\\delta y_1 \\delta y_2$ and $\\delta y_2 \\delta y_1$ are identical), making the augmented covariance $P_{YY}$ singular by construction and the gain $P_{xY} P_{YY}^{-1}$ undefined as written; a pseudo-inverse, a de-duplication to $m(m+1)/2$ terms, or a small ","The claimed ~25% improvement is measured only in the scalar example; the Clohessy–Wiltshire experiment shows lower error qualitatively but does not report a comparable error reduction, so the size of the benefit in realistic problems remains open.","The paper's scalar comparison also shows QEKF reducing to EKF under a Gaussian prior; an interesting test would be to run QEKF with a genuinely skewed prior to see whether a linearization-based quadratic update can match the sigma-point-based one.","A consistency check in the rank-deficient case would be whether the update $P_{xx}^+ = P_{xx}^- - K P_{YY} K^T$ remains symmetric positive semidefinite when $P_{YY}$ is pseudo-inverted."],"forward_implications":["For nonlinear measurement models or non-Gaussian measurement noise, the quadratic update should systematically beat the corresponding linear filter, since a parabola can bend toward the conditional mean where a straight line cannot.","Because the update formulas are written in terms of moments only, they transfer directly to other linear estimators such as the Cubature, Central Difference, and Ensemble Kalman Filters, upgrading them to quadratic versions without changing their prediction steps.","The added computational cost is small relative to particle or Gaussian-sum filters, since only an augmented covariance of dimension $m + m^2$ needs to be built and inverted at each step.","Generalizing the same derivation to cubic and higher polynomial residuals should continue improving accuracy, trending toward the true MMSE estimator as the polynomial order grows."],"supporting_citations":[{"why":"Defines the linear filtering recursion that the proposed quadratic update extends.","marker":"[1]"},{"why":"Provides the unscented transformation and sigma-point machinery the QUKF uses for its moments.","marker":"[4]"},{"why":"First quadratic update by augmenting the measurement with its square, the idea this paper re-derives with a different mechanism.","marker":"[16]"},{"why":"Extends the quadratic update to arbitrary polynomial orders, establishing the polynomial-filtering line this paper continues.","marker":"[17]"},{"why":"Polynomial extended Kalman filter whose scalar-only treatment flags the vector-measurement singularity the paper inherits.","marker":"[18]"},{"why":"Derives polynomial MMSE estimators with differential algebra, providing the theoretical basis for the quadratic gain structure.","marker":"[19]"},{"why":"Shows how to evaluate polynomial updates without carrying higher-order moments, the practical computation the QEKF and QUKF adopt.","marker":"[20]"},{"why":"Supplies the formula for Gaussian fourth-order moments used in the QEKF's Isserlis-based kurtosis.","marker":"[21]"},{"why":"Extends polynomial updates to conditional covariance estimation, informing the covariance update used here.","marker":"[22]"}],"fun_headline_variants":["Quadratic update reduces Kalman filter error on nonlinear measurements","Measurement square improves Extended and Unscented Kalman filters","QEKF and QUKF: quadratic estimators beat linear for state updates","Extending Kalman filters to quadratic updates lowers estimation error"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The update rule requires the augmented covariance matrix $P_{YY}$ to be invertible, but for any measurement of dimension greater than one the measurement-square vector contains duplicate entries, so $P_{YY}$ is singular and the inverse in the gain formula does not exist as written.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic update reduces Kalman filter error on nonlinear measurements","Measurement square improves Extended and Unscented Kalman filters","QEKF and QUKF: quadratic estimators beat linear for state updates","Extending Kalman filters to quadratic updates lowers estimation error"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000281,"raw_usage":{"total_tokens":1638,"prompt_tokens":889,"completion_tokens":749,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":505,"completion_tokens_details":{"reasoning_tokens":679}},"tokens_in":505,"tokens_out":749,"duration_ms":7130,"temperature":1.0,"reasoning_tokens":679,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:58:29.871112+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $P_{YY}$ from Eq. (22) for a two-dimensional measurement with a Gaussian prior and any nonzero measurement noise; the second block column contains identical rows for the products $\\delta y_1 \\delta y_2$ and $\\delta y_2 \\delta y_1$, so the determinant of $P_{YY}$ is identically zero. A Monte Carlo run of the Clohessy–Wiltshire example using the formulas exactly as printed would therefore fail at the inversion step, exposing that the singular case is never handled. Conversely, a run inserting a least-squares pseudo-inverse would show whether the quadratic updates still beat the linear ones once the singularity is bypassed.","supporting_citations":[{"cited_title":"Optimal quadratic fil- tering of linear discrete-time non-gaussian systems,","cited_arxiv_id":null,"evidence_quote":"First quadratic update by augmenting the measurement with its square, the idea this paper re-derives with a different mechanism."},{"cited_title":"Polynomial filtering of discrete-time stochastic linear systems with multiplicative state noise,","cited_arxiv_id":null,"evidence_quote":"Extends the quadratic update to arbitrary polynomial orders, establishing the polynomial-filtering line this paper continues."},{"cited_title":"Polynomial extended kalman filter,","cited_arxiv_id":null,"evidence_quote":"Polynomial extended Kalman filter whose scalar-only treatment flags the vector-measurement singularity the paper inherits."},{"cited_title":"Recursive polynomial minimum mean- square error estimation with applications to orbit determination,","cited_arxiv_id":null,"evidence_quote":"Derives polynomial MMSE estimators with differential algebra, providing the theoretical basis for the quadratic gain structure."},{"cited_title":"Nonlinear filtering with a polynomial series of gaussian random variables,","cited_arxiv_id":null,"evidence_quote":"Shows how to evaluate polynomial updates without carrying higher-order moments, the practical computation the QEKF and QUKF adopt."},{"cited_title":"Estimation of the conditional state and covariance with taylor polynomials,","cited_arxiv_id":null,"evidence_quote":"Extends polynomial updates to conditional covariance estimation, informing the covariance update used here."}],"review_version":1}