Curvature-Driven Task Localization

• Curvature-Driven Task Localization is a set of methods that leverage intrinsic curvature information to accurately identify, localize, and transfer task-relevant features.
• Techniques integrate energy minimization, Fisher information proxies, and diffusion PDEs to achieve sub-pixel precision and shape-invariant performance.
• The approach enhances multi-task learning by reducing interference in model merging and enabling robust, real-time localization in complex, deformable environments.

Curvature-driven task localization refers to the set of methodologies that explicitly exploit curvature information—defined either on spatial structures (e.g., curves, surfaces, manifolds) or in the geometry of loss landscapes—to identify, localize, and transfer task-relevant features or capabilities within computational models or physical systems. This paradigm appears in diverse domains, including computer vision, robotics, multi-task learning, and parameter-space model merging. Curvature-driven formulations enable robust, sub-pixel localization, low interference in multi-task settings, shape-invariant manipulation, and geometric regularization by leveraging intrinsic geometric priors.

1. Mathematical Foundations of Curvature in Task Localization

Curvature-driven task localization methods formalize curvature as a regularizing or diagnostic signal on a spatial or functional domain.

In thin structure estimation (Marin et al., 2015), the energy is defined over discrete points $p_i$ with indicator variables $x_i$ and tangent line parameters $\ell_i$, incorporating curvature priors:

• Quadratic curvature:

$$E(L, X) = \sum_{(i, j) \in N} \kappa^2(\ell_i, \ell_j) x_i x_j + \sum_i \left[ \frac{1}{\sigma^2} \|\ell_i - p_i\|_+^2 + \lambda_i \right] x_i$$

• Absolute curvature: replace $\kappa^2$ by $|\kappa|$.

Here, $\kappa(\ell_i, \ell_j)$ is computed as the sum of distances between neighboring points and their tangents, normalized by Euclidean separation, thus directly encoding local curvature.

In model merging (Mahdavinia et al., 14 Sep 2025), curvature is abstracted as the diagonal (or low-rank) proxy for the Hessian or Fisher information matrix of the loss with respect to parameters. The Fast Fisher Grafting (FFG) method employs the Adam second-moment statistics to estimate parameter importance and sparsify task-specific updates via:

$s_{\tau, i} = (\Delta w_{\tau, i})^2 v_{\tau, i}$

where $v_{\tau, i}$ is the running estimate of squared gradients (serving as Fisher/Hessian proxy).

Diffused Orientation Fields (DOF) for robotic object-centric manipulation (Bilaloglu et al., 23 Nov 2025) cast curvature via the Laplace–Beltrami operator on a surface $\mathcal{M}$, solving:

$$\Delta_{\mathcal{M}} u(x) = 0, \quad x \in \mathcal{M}$$

with Dirichlet boundary conditions at keypoints, generating smoothly varying reference frames that naturally adapt to local surface curvature.

Curvature serves as both a regularizer (enforcing geometric simplicity) and a mechanism for geometric invariance or robustness (aligning model or robot actions to intrinsic structure).

2. Algorithms and Optimization Approaches

Curvature-driven task localization leverages specialized optimization and inference schemes.

The variational inference framework in (Marin et al., 2015) optimizes a lower bound $\mathcal{L}(q, \ell)$ over mean-field factors $q_i(x_i)$ and continuous variables $\ell_i$, alongside trust region methods (Levenberg–Marquardt) for minimizing highly nonconvex curvature energies. Block-coordinate updates alternate between tangent optimization and marginal probability estimation.

In model merging (Mahdavinia et al., 14 Sep 2025), FFG applies saliency-based pruning followed by curvature-aware aggregation:

• Compute task updates $\Delta w_{\tau, i}$.
• Score saliency $s_{\tau, i}$.
• Mask top-$k$ updates per task.
• Aggregate via:

$$w_{\text{merged}} = w_0 + \left( \sum_\tau P_\tau \right)^{-1} \sum_\tau P_\tau \Delta w'_\tau$$

with $$P_\tau = \operatorname{diag}(\sqrt{v_\tau + \epsilon})$$.

Diffused Orientation Fields (Bilaloglu et al., 23 Nov 2025) utilize discrete Laplacian construction on point clouds, sparse factorization for efficiency, and gradient extraction to build local reference frames. The Walk-on-Spheres algorithm extends orientation frames off-surface via Monte Carlo simulation, followed by quaternion averaging (Markley's method).

In multi-task graph networks (New et al., 2022), curvature of each task’s loss surface is probed through randomized estimation of the Hessian trace and spectrum using Hutchinson’s method and matrix-free Lanczos tridiagonalization.

3. Curvature-driven Localization in Computer Vision

Thin structure detection (Marin et al., 2015) employs curvature priors in energy minimization to localize edges, centerlines, surfaces, and vessels at sub-pixel accuracy. The discrete optimization incorporates both detection likelihoods and geometric regularization, supporting quadratic and absolute curvature models. Sub-pixel estimation is achieved by continuous tangent fitting and orthogonal projection.

Empirical gains include competitive F-score (F≈0.83) on edge benchmarks, sub-voxel vascular centerlines, and improved smoothness over local or greedy trackers. The trust region solver and parallelization support practical runtimes in 2D and 3D domains.

4. Curvature and Task Separation in Model Merging

The Fast Fisher Grafting procedure (Mahdavinia et al., 14 Sep 2025) exemplifies curvature-driven localization in neural model ensemble strategies. Curvature proxies derived from optimizer statistics enable:

• Identification and preservation of high-importance (high-curvature) parameter updates per capability.
• Pruning of conflicting or low-importance edits, minimizing interference in merged models.
• Emergence of extremely low-rank, overlapping masks in early attention and embedding layers, with denser structure in higher layers.
• Empirical increases in merged model performance and negative transfer reduction, as measured on benchmarks such as HumanEval and MATH.

Notably, stable-rank measurement (<1.3) validates rank-1 factor compression, showing the exploitation of shared curvature geometry across tasks.

5. Multi-task Learning and Curvature Landscape Heterogeneity

In multi-task graph networks (New et al., 2022), curvature-inspection reveals substantial heterogeneity in Hessian spectra across property prediction tasks. Fluctuations in the trace and spectral density of the Hessian correlate with differential loss surface geometry, where "stiff" (high-curvature) and "flat" (low-curvature) tasks coexist in shared-parameter optimizations.

This heterogeneity is posited to underlie poor multi-task performance, as a common optimizer cannot simultaneously manage conflicting local geometries. The paper introduces diagnostic tools for curvature measurement via matrix-free linear algebra, but does not implement curvature-adaptive training schemes, leaving such interventions as future work.

6. Object-centric Task Localization via Diffused Orientation Fields

In robotics, DOF (Bilaloglu et al., 23 Nov 2025) provides a principled approach to task localization on curved objects, overcoming the lack of a global frame. By solving a diffusion PDE conditioned on sparse keypoints, it supplies a field of smoothly varying SO(3) frames:

• Each manipulation task is parameterized in local DOF coordinates, rendering the task invariant to the object's shape up to sparse correspondence.
• Curvature, encoded via Laplace–Beltrami, acts as an intrinsic regularizer and filter: high-frequency, high-curvature features are exponentially suppressed with longer diffusion times.
• Task transfer across geometrically or topologically perturbed objects (e.g., random deformation, missing regions) exhibits up to a 6× reduction in velocity trajectory variance and robust performance under challenging perturbations.

The algorithm is efficient (factorization and query times admit real-time deployment) and theoretically grounded (heat-kernel error bounds, eigen-mode expansion).

7. Curvature-informed Landmark Localization in Robotics

Robotic ultrasound navigation for scoliosis (Victorova et al., 2021) demonstrates curvature-driven task localization in a closed-loop robotic context:

• The robot localizes spinous processes in real-time ultrasound via deep CNN inference, filtering detections to yield a smooth estimated curvature (lateral trajectory $x(y)$).
• Control commands (lateral velocity, probe pitch, force-based coupling) are directly modulated by this curvature estimate, keeping anatomical landmarks centered in the field of view.
• Coronal reconstructions and scoliosis angle measurements are derived from the curvature-following trajectory.
• Quantitative trials validate that robotic curvature-following outperforms manual scanning in localization error and trajectory smoothness, with consistent anatomical angle measurements.

This workflow integrates perception, curvature estimation, and action for robust task localization in a deformable, anatomical setting.

8. Significance and Future Directions

Curvature-driven task localization offers a unifying framework for exploiting geometric structure in data and models. It provides:

• Sub-pixel and sub-voxel localization in vision and biomedical applications.
• Modular, low-interference capability merging in large neural architectures.
• Geometric invariance and robustness in object-centric task transfer.
• Explanatory power for task separation and optimization dynamics in multi-task systems.

Outstanding challenges encompass the design and validation of curvature-adaptive training or merging schemes in multi-task learning (New et al., 2022), further theoretical integration of curvature-weighted diffusion in manipulation (Bilaloglu et al., 23 Nov 2025), and scalability to large-scale, high-dimensional systems. The empirical and theoretical foundation established in these cited works demonstrates critical progress toward geometric task localization in both computational and embodied AI systems.