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REVIEW 3 major objections 5 minor 22 references

Quadratic Extended and Unscented Kalman Filter Updates

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.06256 v1 pith:JPMPL4ZO submitted 2025-06-06 cs.IT eess.SPmath.IT

Simone Servadio , Chiran Cherian This is my paper
classification cs.ITeess.SPmath.IT
keywords ExtendedKalmanFilterUnscentedquadraticupdatepolynomialestimatorminimummeansquareerrorClohessy–Wiltshireequationsnon-GaussiannoiseKroneckerproduct
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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 $\delta y^{[2]} - \mathrm{v}(P_{yy})$.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 5 minor

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.

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 (3)
  1. [Section II, Eqs. (20)-(23) and Section VII-B] 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.
  2. [Section VII-A and Section VIII] 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.
  3. [Section IV, Eq. (36)] 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.
minor comments (5)
  1. [Eq. (19)] 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.
  2. [Section II, final paragraph] The phrase 'the Quadratic Unscented Kalman Filter (UKF)' should read '(QUKF)'.
  3. [Eq. (60)] 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.
  4. [Figure 5 caption] The word 'respectfully' should be 'respectively'.
  5. [Section VII-B] 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)).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the QMMSE update is derived self-contained from the orthogonality principle; self-citations are contextual, not load-bearing. A separate well-posedness flaw affects the vector-measurement update.

full rationale

The central derivation is self-contained. The paper starts from the standard orthogonality condition E[(x - g_Q*(y)) g_Q(y)^T] = 0 (Eq. 5), substitutes the three test estimators (Eqs. 6-9), obtains the normal equations (Eqs. 10-11), assembles them into the augmented linear system (Eq. 19), solves for K = P_xY P_YY^{-1} (Eq. 20), and substitutes back to obtain the quadratic estimator g_Q*(y) = E[x] + P_xY P_YY^{-1} [δy; δy[2] - v(P_yy)] (Eq. 23). No parameter is fitted to the reported test data and then renamed a prediction; the gains B* and C* are computed from prior moments, via linearization for the QEKF and sigma points for the QUKF. The reported accuracy improvements, including the roughly 25% scalar RMSE improvement, are therefore an honest consequence of enlarging the estimator family rather than a circular fit. The self-citations ([19], [20], [22]) provide context and elementary tensor identities (e.g., S_xxx = P_x[2]x and K_xxxx = P_x[2]x[2] for null-mean distributions), but the derivation does not reduce to those citations. Separately, the paper itself acknowledges that prior polynomial updates were 'tested only with scalar measurements, avoiding the issue of matrix singularity'; the present formulation has the same by-construction singularity for any vector measurement, because δy[2] = δy⊗δy contains duplicated entries for m > 1, making P_YY in Eq. (22) singular and P_YY^{-1} in Eqs. (20), (23), and (34) nonexistent as written. The Clohessy-Wiltshire example uses a two-dimensional measurement vector, so that numerical claim is not reproducible from the stated equations. This is a correctness and well-posedness concern, not a circularity concern, and it is reflected here only by not raising the circularity score above 2. The score of 2 reflects the cluster of non-load-bearing self-citations, not any actual circular derivation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central derivation relies on the standard orthogonality principle, on assumed knowledge and independence of noise statistics, on an unspecified approximation for state prior high-order moments, and on an implicit invertibility assumption for P_YY that fails for vector measurements. No new physical entities are introduced. The QUKF's scaling parameters are user-chosen and unreported.

free parameters (1)
  • UKF scaling parameters α, β, κ = not reported
    The QUKF update depends on the sigma-point spread (Eqs. 47-52); the numerical Clohessy-Wiltshire test does not state the chosen values, so the reported accuracy is conditional on unstated tuning.
assumptions (4)
  • standard math Orthogonality principle characterizes the optimal quadratic estimator (Eqs. 5-23).
    Standard Hilbert-space projection result for MMSE estimation, invoked in Section II.
  • domain assumption Measurement noise moments (covariance, skewness, kurtosis) are exactly known and noise is independent of state.
    Used in Section IV ('Full knowledge of the moments of the noise is assumed') and in the moment factorizations of Eqs. 42-44 and 66-68.
  • ad hoc to paper State prior skewness and kurtosis are available; for a Gaussian prior, S_xxx=0 and K_xxxx follows Isserlis' formula (Eqs. 45-46).
    The EKF/UKF propagate only mean and covariance, so high-order moments are not updated; Section IV says they 'can be approximated' without specifying a recursion. This makes the QEKF non-recursive for non-Gaussian priors.
  • ad hoc to paper The augmented covariance P_YY is nonsingular and its inverse in Eqs. 20 and 34 exists.
    Required to define the gain; false for vector measurements because δy[2] contains duplicate entries. No pseudo-inverse is mentioned (Eq. 22, Sections IV and V).

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Cite this review

Pith. "Pith review of Quadratic Extended and Unscented Kalman Filter Updates." pith.science (2026). https://pith.science/paper/JPMPL4ZO

@misc{pith2026250606256,
  author       = {Pith},
  title        = {Pith review of: Quadratic Extended and Unscented Kalman Filter Updates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JPMPL4ZO}},
  note         = {Machine review of arXiv:2506.06256}
}
read the original abstract

Common filters are usually based on the linear approximation of the optimal minimum mean square error estimator. The Extended and Unscented Kalman Filters handle nonlinearity through linearization and unscented transformation, respectively, but remain linear estimators, meaning that the state estimate is a linear function of the measurement. This paper proposes a quadratic approximation of the optimal estimator, creating the Quadratic Extended and Quadratic Unscented Kalman Filter. These retain the structure of their linear counterpart, but include information from the measurement square to obtain a more accurate estimate. Numerical results show the benefits in accuracy of the new technique, which can be generalized to upgrade other linear estimators to their quadratic versions.

Figures

Figures reproduced from arXiv: 2506.06256 by the authors.

Figure 1
Figure 1. Representation of the state-measurement joint distribution and of the [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. RMSE among selected estimators B. Clohessy–Wiltshire Relative Motion The Clohessy–Wiltshire (CW) equations, also known as the Hill’s equations, describe the relative motion of a chaser spacecraft with respect to a target spacecraft in a circular orbit. Assuming the target is in a circular orbit and a local￾vertical-local-horizontal (LVLH) coordinate frame centered on the target, the linearized equations of motion fo… view at source ↗
Figure 3
Figure 3. Target’s relative motion with respect to the chief satellite. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Monte Carlo Consistency Analysis for the Convergence of the QEKF. [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Error Standard Deviation Comparison between the linear estimators (EKF, UKF) and the Quadratic Estimators (QEKF, QUKF) [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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