Invariant Smoother (IS)

• Invariant Smoother (IS) is a class of smoothing methods for trajectory estimation on Lie groups that exploits invariant error coordinates and group-compatible retractions.
• It formulates the MAP solution via left-invariant perturbations, enabling state-independent linearizations and consistent covariance propagation.
• Applications in inertial localization and visual–inertial SLAM demonstrate faster convergence, improved accuracy, and enhanced observability consistency compared to traditional methods.

Searching arXiv for papers on invariant smoothing, Lie-group smoothing, and related invariant RTS/fixed-lag methods. Invariant Smoother (IS) denotes a class of smoothing methods for estimation problems on matrix Lie groups in which the trajectory is optimized using invariant error coordinates and group-compatible retractions. In the recent literature, the term covers batch Gauss–Newton smoothing with left-invariant perturbations (Chauchat et al., 2022), bias-aware inertial localization on the Two Frames Group (TFG) (Chauchat et al., 2023), right-invariant fixed-lag smoothing for visual–inertial SLAM (Huai et al., 2021), and Rauch–Tung–Striebel-type forward–backward smoothing on matrix Lie groups (Laan et al., 2024). Across these variants, the defining objective is not merely to optimize on a manifold, but to exploit Lie-group structure so that process or measurement linearizations become state-estimate independent, or at least less state-dependent, under appropriate group-affine and invariant-measurement assumptions. Earlier invariant-estimation work on $SE(2)$ with perfect odometry and noisy position measurements already exhibited the central geometric phenomenon—exact preservation of the kinematic manifold induced by deterministic motion—even though that work is a recursive IEKF analysis rather than a batch MAP smoother (Chauchat et al., 2018).

1. Definition and terminological boundaries

Invariant smoothing belongs to the estimation-and-control meaning of “smoother”: it is concerned with trajectory estimation from all observations, or from a sliding window of observations, when the state naturally evolves on a Lie group and when the error representation is chosen to respect the underlying symmetry. In this sense, IS is best understood as the smoothing analogue of invariant filtering.

The term should not be conflated with other uses of “smoother.” In numerical linear algebra, a “parameter-dependent smoother” is a multigrid component for parameter-dependent linear systems, with damped Richardson, damped Jacobi, and low-rank exponential-sum approximations as the relevant constructions; that literature is not about invariant estimation on Lie groups (Grasedyck et al., 2020). Nor is IS the same as the linear continuous-time fixed-interval smoother theory in which the “true type” of a smoother is analyzed through a variational formulation and Euler–Lagrange equations; that work studies polynomial steady-state tracking and shows that an optimal smoother behaves, in that sense, like a filter of double the type (0802.0130).

Within invariant estimation itself, an important precursor is the $SE(2)$ navigation problem with deterministic odometry and noisy position measurements. There the left-invariant error

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$

and body-frame innovation preserve the exact manifold

$R(\hat\theta_t)^T\hat x_t=b_t,$

where the standard EKF does not (Chauchat et al., 2018). This result is historically important because it isolates the structural benefit that later invariant smoothers seek to preserve in batch and fixed-lag settings.

2. Lie-group MAP formulation

A representative IS formulation places the trajectory on a matrix Lie group $G$, with states $(\chi_i)_{i=0}^n$, and writes

$$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$

The trajectory estimate is then the MAP solution

$$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$

Invariant smoothing linearizes this problem through a left perturbation

$\chi_i=\hat{\chi}_i\exp(\xi_i),$

so that the update is multiplicative on the group rather than additive in Euclidean coordinates (Chauchat et al., 2023).

The resulting linearized objective has the sparse least-squares form

$$\tilde{\mathcal C} = \|\mathbf p_0+\xi_0\|_{\widetilde{\mathbf P}_0}^2 +\sum_i \|\hat u_i-\mathbf F_i\xi_i+\xi_{i+1}\|_{\mathbf Q_i}^2 +\sum_k \|\hat n_k+\mathbf H_k \Xi\|_{\mathbf N_k}^2,$$

with

$SE(2)$0

and $SE(2)$1 the concatenation of all increments. The transformed prior covariance is

$SE(2)$2

where $SE(2)$3 is the Jacobian associated with the Lie-group retraction (Chauchat et al., 2023).

The defining structural assumption is group-affine dynamics. In the discrete-time formulation,

$SE(2)$4

with $SE(2)$5 and $SE(2)$6 an automorphism. This yields the log-linear property

$SE(2)$7

Because perturbations propagate linearly in the Lie algebra, the process residual

$SE(2)$8

linearizes as

$SE(2)$9

and the state update after solving the linear subproblem is

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$0

This is the essential batch IS mechanism (Chauchat et al., 2022).

3. Structural properties of invariant error coordinates

The central claim of IS is structural rather than merely numerical. In the low-process-noise limit, the process model ceases to be a soft penalty and becomes an equality constraint. In the deterministic setting

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$1

the zero-noise theorem for invariant smoothing states that, if initialized accordingly, every iteration satisfies

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$2

and, when the prior support is a Lie-subalgebra-supported subspace $$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$3, all iterations remain in the reachable subspace generated by that support (Chauchat et al., 2022).

The algebraic reason is that the invariant Gauss–Newton increment obeys

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$4

hence

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$5

The update therefore preserves the exact deterministic relation instead of merely making its linearized residual small. This is the distinctive zero-noise property of IS.

A second structural advantage concerns Jacobian dependence. In the invariant RTS framework, if the continuous-time process model

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$6

is group affine in the sense that

$$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$7

and if the measurement model is matched to the chosen left- or right-invariant error, then the linearized error propagation is state independent (Laan et al., 2024). In practice, this means that the process and measurement Jacobians can become state-estimate independent under the ideal assumptions, or less state-dependent when the assumptions are only approximately satisfied.

A third structural theme is observability consistency in fixed-lag smoothing. For right-invariant fixed-lag smoothing on $$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$8, the unobservable directions correspond to $$\begin{pmatrix} \hat\theta_t-\theta_t\ R(\theta_t)^T(\hat x_t-x_t) \end{pmatrix}$$9 DOF global translation and $R(\hat\theta_t)^T\hat x_t=b_t,$0 DOF global rotation about gravity. Under the right-invariant error, the nullspace block for each navigation state is

$R(\hat\theta_t)^T\hat x_t=b_t,$1

which does not depend on the linearization point. By contrast, in a conventional formulation the corresponding block

$R(\hat\theta_t)^T\hat x_t=b_t,$2

depends on the current estimate, and marginalization then injects spurious information into the gauge directions. The right-invariant formulation eliminates this mechanism and thereby resolves the consistency issue without FEJ-style Jacobian freezing (Huai et al., 2021).

4. Principal algorithmic realizations

The batch Gauss–Newton form of IS is the most direct realization. It iterates over a current trajectory estimate, introduces tangent increments through

$R(\hat\theta_t)^T\hat x_t=b_t,$3

solves the sparse linearized problem in the stacked variables $R(\hat\theta_t)^T\hat x_t=b_t,$4, and retracts back to the group with

$R(\hat\theta_t)^T\hat x_t=b_t,$5

Its novelty lies in the invariant retraction and in the process Jacobian $R(\hat\theta_t)^T\hat x_t=b_t,$6 induced by group-affine dynamics, not in an exotic solver architecture.

The invariant Rauch–Tung–Striebel smoother (IRTS) is the forward–backward counterpart. It performs a forward invariant EKF pass followed by a backward RTS pass on the Lie algebra. For left-invariant filtering,

$R(\hat\theta_t)^T\hat x_t=b_t,$7

while the backward left-invariant smoothing innovation is

$R(\hat\theta_t)^T\hat x_t=b_t,$8

with state update

$R(\hat\theta_t)^T\hat x_t=b_t,$9

The right-invariant version uses the corresponding right-invariant innovations and left multiplication on the update. Covariance recursion retains the classical RTS form, while the state correction becomes Lie-group consistent (Laan et al., 2024).

The right-invariant fixed-lag smoother (RI-FLS) embeds the navigation state in $G$0 and keeps the standard fixed-lag nonlinear least-squares and marginalization pipeline. The innovation is geometric rather than additive: the IMU residual is

$G$1

with navigation error defined through

$G$2

The solver itself remains an ordinary fixed-lag factor-graph solver; the invariant content resides in the residual and Jacobian definitions (Huai et al., 2021).

5. Inertial localization and bias-inclusive group design

Inertial navigation is the principal application domain in the present literature because the state naturally combines rotational and translational quantities and because high-grade IMUs create precisely the low-process-noise regime in which conventional smoothers become delicate. For unbiased inertial navigation, the state

$G$3

is embedded in

$G$4

and the dynamics admit the group-affine decomposition

$G$5

$G$6

This is why $G$7 is described as the appropriate embedding for low-noise inertial alignment (Chauchat et al., 2022).

Biases complicate the picture because they are body-frame variables. The TFG-based extension therefore replaces the older $G$8 treatment with a group law in which the accelerometer and gyroscope biases transform as body-frame quantities. For $G$9, the TFG composition is

$(\chi_i)_{i=0}^n$0

where $(\chi_i)_{i=0}^n$1, and the accelerometer-bias part obeys the analogous TFG law (Chauchat et al., 2023).

The corresponding exponential map is

$(\chi_i)_{i=0}^n$2

with logarithm

$(\chi_i)_{i=0}^n$3

Under this retraction, an orientation increment rotates the biases, so orientation and bias are no longer artificially decoupled.

The full discrete inertial model is

$(\chi_i)_{i=0}^n$4

The paper proves that the IMU equations in $(\chi_i)_{i=0}^n$5 with accelerometer bias are group affine in the TFG sense whenever $(\chi_i)_{i=0}^n$6, but not in general for the full $(\chi_i)_{i=0}^n$7 model with gyro bias, because the state appears inside

$(\chi_i)_{i=0}^n$8

Thus the TFG-based formulation is not a claim of exact group-affinity in all cases; it is a more principled group design for the bias-inclusive state (Chauchat et al., 2023).

6. Empirical behavior, comparisons, and limitations

The experimental record of IS is strongest in regimes with low process noise, poor initialization, or gauge-related consistency issues. In simulated in-motion alignment with IMU at $(\chi_i)_{i=0}^n$9 Hz, GPS at $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$0 Hz, initial yaw error $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$1, and one Gauss–Newton iteration per update, the low-noise IS study reports that, for window size $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$2, Forster et al. becomes inconsistent, IS and GTSAM converge to similar final solutions, but IS converges faster and is more consistent, while GTSAM exceeds the $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$3 bound between $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$4 and $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$5 seconds; for window size $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$6, Forster et al. converges to the same estimates as IS and GTSAM (Chauchat et al., 2022).

For TFG-based inertial localization on KITTI raw data with IMU at $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$7 Hz, position measurements at $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$8 Hz, and $$\chi_0 \sim \mathcal{N}_L(\bar{\chi}, \mathbf P_0), \qquad \chi_{i+1}=f_i(\chi_i)\exp(w_i), \qquad y_k = h_k(\chi_{I_k}) + n_k.$$9 Monte Carlo runs, the main reported metric is the ratio of consistent trajectories. With window size $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$0, the reported ratios for sequences $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$1 are: TFG $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$2; imperfect $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$3-based IS $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$4; and GTSAM $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$5. For larger windows the differences shrink, but the TFG method remains competitive and often best or tied (Chauchat et al., 2023).

For right-invariant fixed-lag smoothing in visual–inertial SLAM, the clearest evidence is covariance consistency. In the torus simulation, the reference NEES values are $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$6 for position, $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$7 for orientation, and $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$8 for pose. The reported averages over the last $$(\chi_i)_i^* = \arg\max_{(\chi_i)_i} \mathbb P\!\left((\chi_i)_i \mid y_0,\dots,y_n\right).$$9 seconds are: Incremental FLS $\chi_i=\hat{\chi}_i\exp(\xi_i),$0; Batch FLS $\chi_i=\hat{\chi}_i\exp(\xi_i),$1; iSAM2 $\chi_i=\hat{\chi}_i\exp(\xi_i),$2; and RI-FLS $\chi_i=\hat{\chi}_i\exp(\xi_i),$3. On EuRoC, the same paper reports comparable real-data accuracy rather than uniformly superior odometry, which is consistent with its central claim that the main gain is consistency, not necessarily lower ATE on every sequence (Huai et al., 2021).

For IRTS, the reported pattern is robustness under poor initialization. On the Starry Night dataset, IRTS and MRTS are comparable when initialization error is low, but IRTS substantially outperforms MRTS when initialization error is high. The same study further reports that one iteration of either smoother strongly outperforms one iteration of either Gauss–Newton method, and that even after $\chi_i=\hat{\chi}_i\exp(\xi_i),$4 Gauss–Newton iterations variability remains large (Laan et al., 2024).

The limitations are equally structural. Exact state-estimate independence requires a group-affine process model together with matching left- or right-invariant measurements; when those technical conditions fail, the gain is reduced estimate dependence rather than perfect invariance (Laan et al., 2024). The low-noise batch IS analysis assumes that the singular linearized system can already be solved correctly and explicitly delegates the ill-conditioned inversion problem to prior work (Chauchat et al., 2022). The TFG formulation does not make the full $\chi_i=\hat{\chi}_i\exp(\xi_i),$5 biased IMU model group affine in general (Chauchat et al., 2023). The RI-FLS consistency proof uses the approximation

$\chi_i=\hat{\chi}_i\exp(\xi_i),$6

and the same paper reports that exact Jacobians slightly worsen NEES (Huai et al., 2021).

Taken together, these works define IS as a geometrically structured approach to smoothing in which invariant error coordinates are used to preserve equality constraints in the zero-noise limit, maintain gauge-consistent nullspaces under marginalization, or reduce the estimate dependence of forward–backward linearizations. The unifying principle is that the Lie-group embedding and the choice of invariant error are not auxiliary implementation details; they are the source of the method’s distinctive theoretical and empirical behavior.