Truncated Least Squares Formulation

• Truncated Least Squares (TLS) formulation is a robust estimation method that truncates large residuals to reduce the impact of outliers in regression and registration problems.
• It employs a convex relaxation via weighted least squares within a Branch and Bound framework, enabling rapid global optimization even under high outlier conditions.
• This method is pivotal for applications in point cloud registration, SLAM, and autonomous systems, providing certifiable optimality and computational efficiency.

Truncated Least Squares (TLS) formulation is a robust estimation approach designed to mitigate the influence of outliers in regression or registration problems by restricting the contribution of large residuals to a specified truncation threshold. This nonconvex model has become central in geometric computer vision, signal processing, and applied linear algebra, where high outlier rates are common and global optimality is critical for reliability in practical systems.

1. Core Formulation and Objective

The truncated least squares objective modifies the classical least squares by truncating each residual at a fixed threshold, thereby reducing the negative impact of gross outliers:

$$(R^*, t^*) = \underset{(R, t)\in\mathcal{R} \times \mathbb{R}^3}{\arg\min} \sum_{i=1}^N \min(\|R p_i - q_i + t\|^2, \epsilon^2)$$

where $\epsilon^2$ is the truncation threshold, $p_i$ and $q_i$ are corresponding points, and the rotation search may be restricted, e.g., to a fixed axis: $$\mathcal{R} = \{ \exp(\theta [n^*]_\times) : \theta \in [-\pi, \pi] \}$$. This structure is instrumental in robust point cloud registration, where non-truncated least squares is easily overwhelmed by adversarial outliers, but TLS can offer robustness rates up to 95% (Ivanov et al., 21 Aug 2025).

2. Convex Relaxation via Weighted Least Squares

Direct global optimization of the TLS objective is challenging due to the combinatorial nonconvexity induced by the truncation. The principal technical advance in (Ivanov et al., 21 Aug 2025) is the introduction of a linear-time convex relaxation, sometimes described as a "Weighted Least Squares" (WLS) relaxation (Editor's term) inside a Branch and Bound (BnB) framework.

For any search domain (such as a ball in the rotation or translation space), one computes for each residual $r_i$ an interval $[r^{\min}_i, r^{\max}_i]$. The relaxation then rewrites the truncated term as:

$$w_i = \frac{\epsilon^2 - r^{\min}_i}{r^{\max}_i - r^{\min}_i} \text{ if } r^{\max}_i > \epsilon^2, \quad w_i = 1 \text{ otherwise}$$

$$o_i = \begin{cases} 1 & r^{\min}_i > \epsilon^2 \ 0 & \text{otherwise} \end{cases}$$

$$\text{Relaxed Objective:} \quad \sum_{i=1}^N (1-o_i)\left[w_i r_i + (1-w_i) r^{\min}_i\right] + \sum_{i=1}^N o_i \epsilon^2$$

This convex surrogate maintains the property that no term will exceed $\epsilon^2$ within its domain, and the complete relaxation can be evaluated and minimized in linear time with respect to the number of correspondences. Unlike prior semidefinite programming (SDP) relaxations, this approach eschews costly lifted variable representations, making it amenable to real-time applications (Ivanov et al., 21 Aug 2025).

3. Search Space Contraction Techniques

To further improve computational efficiency in BnB, the contractor technique is employed, which eliminates subsets of the rotation (or translation) space that cannot possibly contain the global minimizer. For each datum, one defines a set of feasible transformations:

$$\mathcal{V}_i = \{ (R, t) \in \mathcal{R} \times \mathbb{R}^3 : \|R p_i - q_i + t\|^2 \le \epsilon^2 \}$$

The union $\mathcal{F} = \bigcup_i \mathcal{V}_i$ is the feasible region for BnB. In rotation-only registration, contraction reduces the search domain by collecting and uniting feasible rotation intervals, drastically decreasing the number of BnB nodes, without excluding the true solution. This logic derives from the principle that if every residual for a given subdomain exceeds $\epsilon^2$, that region can be pruned from the search (Ivanov et al., 21 Aug 2025).

4. Global Optimality and Empirical Validation

The combination of convex (WLS) relaxation and domain contraction enables highly efficient pruning, allowing provably globally optimal solutions in rotation-only TLS problems with up to $N=100$ points in less than half a second. Comparative analysis shows that this approach is two orders of magnitude faster than state-of-the-art SDP solvers like STRIDE, which may require several hundred seconds for similar instances. The paper further provides formal proof of global optimality and robust empirical evidence, even for adversarial problem instances where local minima closely approach the global minimum (Ivanov et al., 21 Aug 2025).

5. Applications and Efficiency Implications

The TLS framework and its linear-time relaxation are foundational to robust point cloud registration problems in robotics, autonomous vehicles, SLAM systems, and other domains where alignment must be robust to gross model errors and high outlier rates. The restriction to rotation about a fixed axis is particularly relevant for applications where absolute orientation (e.g., gravity) is known from an IMU. The method's scalability and efficiency imply that near-real-time, certifiably optimal registration is now achievable even at extreme outlier rates, provided the constraints (fixed axis, known correspondences) are met.

6. Tradeoffs and Current Limitations

While the approach achieves near real-time global optimality for rotation-only TLS registration, generalization to full 6DoF remains an open challenge. The core limitation is that the current contractor and convex relaxation techniques do not generalize directly to unconstrained spatial (rotation and translation) domains. Nonetheless, the framework sets a benchmark for efficient, robust registration in constrained settings.

7. Summary of Key Equations and Concepts

Aspect	Formula / Description	Significance
TLS Objective	$$(R^*, t^*)=\arg\min_{R,t}\sum\min(\|R p_i - q_i + t\|^2,\epsilon^2)$$	Robustness to outliers
WLS Relaxation	$\sum_{i=1}^N (1-o_i)[w_i r_i + (1-w_i) r^\min_i] + \sum_{i=1}^N o_i \epsilon^2$	Linear-time convex lower bound
Contractor Set	$\mathcal{F}=\bigcup_i \mathcal{V}_i$, with $\mathcal{V}_i$ as feasible transformations	Prunes search space without loss of solution

The truncated least squares formulation, when equipped with efficient convex relaxation and domain contraction, enables certifiably optimal registration under extreme outlier conditions with minimal computational overhead. This paradigm represents the state of the art in robust geometric estimation under fixed axis constraints (Ivanov et al., 21 Aug 2025).