Recursive Total Least Squares

• RTLS is an online estimation method for errors-in-variables problems that compensates for noise in both regressors and observations by using recursive updates.
• It employs two main algebraic formulations—the minor-component view and the null-space view—to extract parameter estimates and improve tracking accuracy in applications like bearing-only target motion analysis.
• RTLS offers computational efficiency over batch TLS by leveraging recursive updates (including DCD approximations) that reduce arithmetic operations while ensuring convergence and stability.

Recursive total least squares (RTLS) is a class of online estimation methods for errors-in-variables (EIV) problems in which both the regressor and the observation are noisy, so ordinary least squares and ordinary recursive least squares are generally biased. In the formulations represented by adaptive FIR system identification and bearing-only target motion analysis (TMA), RTLS replaces repeated batch total least-squares solves with recursive updates driven by exponentially weighted data, a forgetting factor $0<\lambda<1$, and an augmented representation whose null-space or minor component encodes the parameter estimate (Arablouei et al., 2014, Li et al., 15 Aug 2025).

1. Conceptual basis in errors-in-variables estimation

The defining premise of RTLS is that the regression matrix is itself contaminated. In the FIR identification setting, the noiseless model is $y_n=\mathbf{x}_n^T\mathbf{h}$, but the observed variables are $\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$ and $\tilde y_n=y_n+v_n$; in the bearing-only TMA setting, the pseudo-linear observation is constructed from a noisy bearing $\tilde\theta_k$ and a noisy observer position $\tilde{\mathbf p}_{o,k}$, so both the scalar pseudo-measurement $\tilde y_k$ and the regressor row $\tilde{\mathbf h}_k^T$ are perturbed (Arablouei et al., 2014, Li et al., 15 Aug 2025).

This distinction is the reason RTLS differs fundamentally from RLS, PLKF, and ordinary least-squares formulations. Standard LS/RLS assumes that the regressor is exact and only the output is noisy; in EIV problems, that assumption is false, and the resulting estimate is biased. In the TMA formulation, this bias is tied directly to the fact that $\tilde y_k$ is noisy, $\tilde{\mathbf h}_k$ depends on noisy bearing measurements, and observer position noise contaminates the pseudo-linear relation. The 2025 bearing-only study explicitly positions TLS as more faithful to the pseudo-linear geometry because it models perturbations in both $y_n=\mathbf{x}_n^T\mathbf{h}$0 and $y_n=\mathbf{x}_n^T\mathbf{h}$1 rather than only output residuals (Li et al., 15 Aug 2025).

The canonical TLS objective is therefore written in perturbed-data form:

$y_n=\mathbf{x}_n^T\mathbf{h}$2

In the weighted and recursive variants studied on arXiv, this objective is implemented either through an augmented covariance/eigenvector formulation or through a generalized weighted TLS problem over an augmented data matrix (Arablouei et al., 2014, Li et al., 15 Aug 2025).

2. Algebraic formulations of RTLS

One common RTLS formulation is the minor-component view. In the 2014 DCD-RTLS paper, exponentially weighted statistics are defined by

$y_n=\mathbf{x}_n^T\mathbf{h}$3

These quantities form the augmented weighted covariance matrix

$y_n=\mathbf{x}_n^T\mathbf{h}$4

and the weighted TLS estimate is recovered from the eigenvector $y_n=\mathbf{x}_n^T\mathbf{h}$5 associated with the smallest-magnitude eigenvalue of $y_n=\mathbf{x}_n^T\mathbf{h}$6 via

$y_n=\mathbf{x}_n^T\mathbf{h}$7

In that formulation, RTLS is recursively estimating the minor component of an augmented covariance matrix (Arablouei et al., 2014).

A second formulation, used in bearing-only TMA, is the null-space view of generalized TLS. The target follows constant-velocity kinematics,

$y_n=\mathbf{x}_n^T\mathbf{h}$8

with time-invariant parameter vector

$y_n=\mathbf{x}_n^T\mathbf{h}$9

Bearing geometry implies an ideal pseudo-linear relation

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$0

where

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$1

and $\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$2. After injecting noisy bearing and noisy observer position, the stacked system becomes an EIV model,

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$3

The associated generalized TLS problem is

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$4

with $\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$5. In this view, the recursive algorithm is an online approximation to the batch TLS null-space relation (Li et al., 15 Aug 2025).

These two views are closely aligned. The 2025 paper states that batch TLS is tied to the smallest singular vector or eigenvector of a weighted augmented data matrix, while the 2014 paper makes that eigenvector relation explicit through inverse iteration on $\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$6 (Li et al., 15 Aug 2025, Arablouei et al., 2014).

3. Recursive update mechanisms

In the 2014 inverse-power formulation, the desired eigenvector corresponds to the smallest eigenvalue, so the update is based on one inverse iteration,

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$7

From this, the paper derives the RTLS coefficient recursion

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$8

and then applies Sherman–Morrison to avoid direct inversion. The resulting implementation depends on two auxiliary solves,

$\tilde{\mathbf{x}}_n=\mathbf{x}_n+\mathbf{u}_n$9

followed by

$\tilde y_n=y_n+v_n$0

The DCD-RTLS variant replaces the exact linear solves with dichotomous coordinate-descent iterations that use only additions and bit shifts inside the solver; the design parameters are $\tilde y_n=y_n+v_n$1, $\tilde y_n=y_n+v_n$2, and $\tilde y_n=y_n+v_n$3, and the paper reports that even with $\tilde y_n=y_n+v_n$4, $\tilde y_n=y_n+v_n$5, $\tilde y_n=y_n+v_n$6, the DCD approximation is sufficiently accurate in the tested scenarios (Arablouei et al., 2014).

In the 2025 TMA formulation, the recursion is written directly in augmented-row form. For

$\tilde y_n=y_n+v_n$7

the update is

$\tilde y_n=y_n+v_n$8

$\tilde y_n=y_n+v_n$9

$\tilde\theta_k$0

$\tilde\theta_k$1

The initialization is

$\tilde\theta_k$2

and the simulation setting uses $\tilde\theta_k$3 (Li et al., 15 Aug 2025).

The two recursive mechanisms reflect different numerical emphases. The DCD-RTLS paper derives its update from inverse power iteration and concentrates on reduced-complexity internal linear algebra, whereas the TMA paper updates a covariance-like matrix and an augmented null vector through an RLS-style matrix inversion lemma recursion. The 2025 paper explicitly notes that this is not a recursive Kalman state update in the usual Bayesian sense; it is a recursive approximation to the TLS null-space vector (Li et al., 15 Aug 2025).

4. Bearing-only target motion analysis as an RTLS application

In the bearing-only TMA problem, RTLS is applied to passive tracking with a moving observer. The observer obeys

$\tilde\theta_k$4

and measures only bearing,

$\tilde\theta_k$5

with noisy self-localization

$\tilde\theta_k$6

The core assumptions are constant target velocity over the estimation horizon, zero-mean Gaussian bearing noise, zero-mean Gaussian observer self-localization error, a first-order small-angle approximation for $\tilde\theta_k$7 and $\tilde\theta_k$8, and neglect of the cross-covariance between $\tilde\theta_k$9 and $\tilde{\mathbf p}_{o,k}$0 in the recursive weighting matrix (Li et al., 15 Aug 2025).

The pseudo-linearization step converts nonlinear bearing geometry into a linear relation in the constant-velocity trajectory parameters. Using the first-order approximations

$\tilde{\mathbf p}_{o,k}$1

the paper obtains perturbation models for both the pseudo-measurement and regressor. The scalar output perturbation covariance is

$\tilde{\mathbf p}_{o,k}$2

and the regressor perturbation covariance is

$\tilde{\mathbf p}_{o,k}$3

In practice, $\tilde{\mathbf p}_{o,k}$4 is replaced by $\tilde{\mathbf p}_{o,k}$5 (Li et al., 15 Aug 2025).

A distinctive feature of this application is that the recursion estimates a fixed parameter vector $\tilde{\mathbf p}_{o,k}$6 rather than a time-varying state directly. The current target state is reconstructed online by

$\tilde{\mathbf p}_{o,k}$7

This makes RTLS an online tracker through recursive estimation of static trajectory parameters under a constant-velocity model (Li et al., 15 Aug 2025).

The same paper couples RTLS to a circumnavigation controller intended to improve observability. The controller uses

$\tilde{\mathbf p}_{o,k}$8

with a radial term regulating standoff distance $\tilde{\mathbf p}_{o,k}$9 and a tangential term of strength $\tilde y_k$0. The paper emphasizes that observability improves when the observer has motion components perpendicular to the bearing direction, ideally orbiting the target, so the controller is designed to generate richer bearing geometry and thereby improve RTLS convergence and accuracy (Li et al., 15 Aug 2025).

5. Computational properties and analytical results

The computational motivation for RTLS is to retain the EIV fidelity of TLS without repeated batch decompositions. In the TMA formulation, the main savings come from avoiding a growing-history SVD and updating only a fixed-size $\tilde y_k$1 matrix through a rank-one RLS-style recursion. The paper states that RTLS is computationally more efficient than batch TLS, and the reconstruction notes that the per-step cost is effectively quadratic in the parameter dimension, $\tilde y_k$2; this quadratic-cost statement is explicitly identified there as an inference from the update formulas rather than a formal complexity theorem (Li et al., 15 Aug 2025).

The 2014 DCD-RTLS paper provides a much more explicit complexity analysis. For shift-structured FIR input, the method attains $\tilde y_k$3 complexity, and the per-iteration arithmetic counts are reported as $\tilde y_k$4 multiplications and $\tilde y_k$5 additions. For non-shift-structured input, the reported counts are $\tilde y_k$6 multiplications and $\tilde y_k$7 additions. These counts are lower than those of AIP, xRTLS, and kRTLS in the same comparisons, and the paper also reports fewer hardware gates, especially when $\tilde y_k$8 and $\tilde y_k$9 (Arablouei et al., 2014).

The principal analytical results in (Arablouei et al., 2014) concern convergence and stability. Under assumptions A1–A5, the exact RTLS recursion is shown to be convergent in the mean and asymptotically unbiased:

$\tilde{\mathbf h}_k^T$0

The paper also interprets RTLS as an RLS term plus a bias-compensation term,

$\tilde{\mathbf h}_k^T$1

in the large-$\tilde{\mathbf h}_k^T$2 approximation. Mean-square stability is linked to a sufficient lower bound on the forgetting factor,

$\tilde{\mathbf h}_k^T$3

and the steady-state MSD is given by

$\tilde{\mathbf h}_k^T$4

The paper notes that for $\tilde{\mathbf h}_k^T$5, the steady-state MSD is zero, so the algorithm is consistent in that case (Arablouei et al., 2014).

By contrast, the 2025 TMA paper does not provide a new theorem proving consistency, asymptotic unbiasedness in that specific bearing-only model, mean-square stability, or closed-loop convergence of the estimator-controller pair. Its guarantees are therefore empirical and qualitative rather than theorem-level (Li et al., 15 Aug 2025).

6. Empirical behavior, relations to other methods, and limitations

In the bearing-only study, RTLS is compared mainly against PLKF. Under $\tilde{\mathbf h}_k^T$6 and $\tilde{\mathbf h}_k^T$7, both methods can achieve circumnavigation, but RTLS reaches the orbital tracking regime faster; over 1000 Monte Carlo runs, RTLS has lower mean state error and shows less bias and tighter error distribution. With $\tilde{\mathbf h}_k^T$8 fixed and $\tilde{\mathbf h}_k^T$9 varied from $\tilde y_k$0 to $\tilde y_k$1, both methods worsen as bearing noise increases, but RTLS error grows more slowly and RTLS consistently has lower MSE. With $\tilde y_k$2 fixed and $\tilde y_k$3 varied from $\tilde y_k$4 to $\tilde y_k$5, both degrade with observer position noise, but RTLS remains more accurate and the degradation is more gradual. In a real-world experiment with a Crazyflie UAV observer, a RoboMaster ground robot target, target speed $\tilde y_k$6 constant, and controller parameters $\tilde y_k$7, $\tilde y_k$8, $\tilde y_k$9, RTLS converges faster and achieves lower position estimation error than PLKF (Li et al., 15 Aug 2025).

In the adaptive FIR setting, DCD-RTLS is compared with the exact RTLS recursion, kRTLS, xRTLS, and AIP. The reported learning curves of exact RTLS and DCD-RTLS are essentially indistinguishable, validating the approximation introduced by the DCD solves in the tested scenarios. DCD-RTLS converges faster than kRTLS, xRTLS, and AIP, while all considered RTLS variants have similar steady-state MSD because they are recursively estimating the same TLS minor component. The theoretical steady-state MSD is reported to match experiment well over varying input-noise variance $\tilde{\mathbf h}_k$0, forgetting factor $\tilde{\mathbf h}_k$1, and output-noise settings (Arablouei et al., 2014).

Several limitations recur across these formulations. In the TMA setting, the method assumes a constant-velocity target model, uses first-order trigonometric linearization, explicitly neglects the cross-covariance between $\tilde{\mathbf h}_k$2 and $\tilde{\mathbf h}_k$3, depends strongly on observability and trajectory geometry, and may encounter conditioning problems in $\tilde{\mathbf h}_k$4 or in normalization by $\tilde{\mathbf h}_k$5; no regularization strategy is specified (Li et al., 15 Aug 2025). In the DCD-RTLS setting, the analysis assumes the DCD internal solves are sufficiently accurate, performance can degrade when the smallest two eigenvalues of the augmented covariance are close, and practical behavior depends on the tuning of $\tilde{\mathbf h}_k$6 and on assumptions such as independence and temporal whiteness (Arablouei et al., 2014).

Two misconceptions are directly addressed by these papers. First, RTLS is not merely RLS with a different loss; its defining role is to compensate for regressor noise in EIV problems, and the 2014 analysis makes this explicit through the bias-compensation interpretation (Arablouei et al., 2014). Second, RTLS need not estimate a dynamic state directly: in the bearing-only formulation it estimates static trajectory parameters and reconstructs the current state afterward, while the forgetting factor only makes the method “possibly suitable for tracking time-varying targets,” which the paper presents as a practical heuristic rather than a formal result (Li et al., 15 Aug 2025).