Metamorphosis Representation Projection (MRP)

• MRP is a framework that integrates metamorphosis and projection techniques to disentangle geometric deformation from intrinsic template variation in complex systems.
• It employs coupled Euler–Poincaré equations and variational methods to optimize tasks in imaging, registration, shape analysis, and control.
• Additionally, MRP extends to machine unlearning by algebraically removing harmful features while preserving essential model functionalities.

Metamorphosis Representation Projection (MRP) encompasses a collection of geometric, analytical, and algorithmic methods that combine the conceptual framework of metamorphosis—originally developed for imaging sciences and geometric data analysis—with projection- or lifting-based operations that disentangle or manipulate structural and intrinsic components of complex representations. The core principle is to represent data or system states as points within a space that decomposes geometric deformation and intrinsic template variation, yielding powerful tools for matching, analysis, and even irreversible manipulation (such as machine unlearning) in a variety of domains including imaging, control theory, and machine learning.

1. Theoretical Foundations

MRP is deeply rooted in the metamorphosis framework, which, as formalized in the Euler–Poincaré theory, generalizes classical pattern matching approaches by upgrading the variational principle on deformation Lie groups to allow nontrivial “template–variation.” In the canonical setting, metamorphosis represents data by pairs $(g, \eta)$ where $g$ is a diffeomorphism (encoding geometric transformation) and $\eta$ is a dynamic template variable capturing intrinsic changes not expressible by $g$ alone. The dynamics are governed by variational principles with right–invariant Lagrangians

$\ell(u,\nu,n),$

with $u$ an Eulerian velocity in the Lie algebra, $\nu$ the template change rate, and $n$ the evolving image or structure. Euler–Poincaré reduction yields coupled evolution equations: $$\frac{d}{dt}\frac{\delta\ell}{\delta u} + \operatorname{ad}^*_u\frac{\delta\ell}{\delta u} + \frac{\delta\ell}{\delta n}\diamond n = 0,$$

$$\frac{d}{dt}\frac{\delta\ell}{\delta \nu} + u \star \frac{\delta\ell}{\delta \nu} - \frac{\delta\ell}{\delta n} = 0,$$

$\dot n = \nu + u n,$

where the operators $\diamond$ and $\star$ encode dual actions from the Lie group, and the form of $\ell$ governs the cost of deformation versus template change (0806.0870).

MRP exploits this structure by seeking projections, geodesic evolutions, or coordinate system changes that distinctly parameterize “deformation” and “intrinsic change,” often with constraints or regularization schemes that induce favorable analytic or computational properties. Notably, the metamorphosis model reveals deep analogies to order–parameter models in complex fluid dynamics, with the template variable acting as an internal state that is advected and coupled to overall momentum.

2. Applications in Imaging Science

MRP has had significant impact in imaging science, particularly in registration, shape analysis, and inverse problems. Classic metamorphosis theory yields coupled evolutionary PDEs for point-set data (landmark metamorphosis), images (interpreting both domain and intensity change), and densities (where representation requires measure-theoretic rigor). For example, image metamorphosis with Lagrangian

$$\ell(u,\nu,n) = \|u\|^2_\mathfrak{g} + \frac{1}{\sigma^2}\|\nu\|^2_{L^2}$$

produces evolution equations

$$L_h u_t = -\frac{1}{\sigma^2} \nu_t \nabla n_t, \quad \partial_t \nu_t + \sigma^{-2} \operatorname{div}(\nu_t u_t) = 0, \quad \partial_t n_t + (\nabla n_t)^T u_t = \sigma^2 z_t.$$

This dual evolution captures both deformation and intensity change—something not possible with pure diffeomorphic approaches (0806.0870).

Advanced variants generalize the metamorphosis model to manifold-valued images and statistical shape spaces, employing concepts such as Hadamard manifolds and joint convexity of Riemannian distance to guarantee the existence and convergence of optimal matching paths, and providing rigorous Mosco–convergence results for time-discrete schemes on such spaces (Effland et al., 2019).

In solving inverse imaging problems, e.g., tomography, metamorphosis-based MRP offers frameworks that combine geometric registration and intensity correction. In this setting, the method is analyzed as a regularization scheme—proofs establish existence, stability, and convergence of solutions, with explicit numerical demonstrations in 2D tomography, including cases of topological inconsistency and intensity mismatch between template and target (Barbara et al., 2018).

3. MRP in Attitude Representation and Control

The term MRP also refers to Modified Rodrigues Parameters, a minimal three-parameter global attitude representation for rigid body dynamics, used extensively in spacecraft and robotics. The MRP space results from stereographic projection of the quaternion sphere, yielding a double cover of $\mathrm{SO}(3)$ and resolving nonuniqueness and singularity issues through “shadow set” switching: $\sigma^s(t) = -\frac{\sigma(t)}{\|\sigma(t)\|^2}.$ This ensures singularity-free and shortest-path rotation representation, with the covering space given by the Alexandroff compactification of $\mathbb{R}^3$ (Martins et al., 2023).

MRP–based control strategies often embed hybrid path–lifting algorithms to ensure unambiguous extraction and continuous tracking of MRPs from the underlying rotation group. This involves designing hybrid state machines with hysteresis and discrete mode switches, thereby eliminating chattering and preventing the unwinding phenomenon identically:

• With delayed feedback, chattering at the switching boundary can be avoided by introducing a hysteretic boundary layer (e.g., $1 \leq \|\sigma(t)\| \leq 1+\varepsilon$) so that noisy measurements do not induce rapid switching (Samiei et al., 2015).
• Hybrid dynamic path-lifting mechanisms robustly ensure that the MRP stays within the principal domain, and theoretical analyses rigorously establish that stability (asymptotic and exponential) of the controller in the covering space translates to stability in $\mathrm{SO}(3)$, with precise exponential decay bounds (Martins et al., 2023).

These aspects are pivotal in practical applications including aerospace attitude control, UAV stabilization, and robotics.

4. MRP as a Machine Unlearning Paradigm

Recent developments extend MRP into the domain of machine unlearning for LLMs, where “projection” is executed algebraically at the level of learned representations. In this context, the Metamorphosis Representation Projection approach realizes irreversible removal of harmful information from the network’s internal states by inserting orthogonal projectors into selected network layers, typically after MLP blocks (Wu et al., 21 Aug 2025): $P = I - Q^T Q$ where $Q$ is a low-rank orthogonal matrix spanning the subspace aligned with “harmful” or undesired features. The transformation $h' = P h$ projects the hidden state $h$ onto the orthogonal complement of the subspace spanned by $Q$.

This method departs from traditional unlearning (which usually relies on parameter fine-tuning or suppression through negative losses) by ensuring that (i) the unwanted information is truly irrecoverable (algebraic irreversibility), and (ii) the projection preserves the rest of the model’s knowledge with minimal degradation. Initialization schemes and iterative updates maintain orthogonality and guarantee that subsequent re-training on similar data (relearning attacks) do not restore the erased components.

Experiments demonstrate:

• High stability and retention of useful skills even after multiple sequential unlearning operations (“continual unlearning” with stable Score4 = 0.905 on LLaMA2-7B).
• Robustness against relearning attacks, maintaining low test accuracy on unlearned content ($<0.4$ after 5 epochs of attack).
• Efficiency, with $\sim$0.1M trainable parameters and substantial runtime speedup compared to low-rank adaptation baselines.
• Consistently superior performance over state-of-the-art unlearning techniques including Gradient Ascent, EUL, NPO, and O3 (Wu et al., 21 Aug 2025).

MRP therefore represents a compelling approach for compliance-driven, privacy-centric, or safety-critical LLM deployments.

5. Variational and Computational Techniques

MRP often leverages variational principles not only in geometric registration and fluid-analogous pattern matching but also in numerical implementations:

• Infinite-dimensional optimization: The registration problem is posed using a kinetic energy term for velocity fields and a penalty for non-transportable intensity change. The control variable (e.g., velocity $u$) induces a “forward operator” via a PDE governing image evolution, with discretization achieved by Galerkin methods (e.g., piecewise polynomial finite elements in space-time domains) (Bock et al., 2020).
• Analytical solution of geodesic equations: By solving along characteristics of the optimal flow, solutions exploit analytic ansatz that interpolate fixed endpoints while obeying PDE constraints (advection, continuity).
• Gradient-based optimization: Efficient solution of high-dimensional projection or matching problems is enabled via adjoint-based automatic differentiation and gradient or quasi-Newton optimization, with numerical validation confirming convergence and robustness of the schemes (Bock et al., 2020).

In manifold-valued settings, the joint convexity of Riemannian distance (as in Hadamard manifolds) ensures existence of minimizing geodesic paths and supports convergence of time-discrete to time-continuous models in practical alternating descent schemes (Effland et al., 2019).

6. Broader Implications and Outlook

MRP provides a principled methodology to separate, manipulate, or project the geometric and intrinsic components of complex structured data. Its impact includes:

• Advanced registration, morphing, and statistical analysis in computational anatomy, medical imaging, and pattern recognition, with precise existence and optimality theorems even in singular or measure-theoretic regimes (0806.0870, Barbara et al., 2018).
• Robust, globally stable, and compact attitude representation and control in engineering domains, ensuring minimal representation, chattering avoidance, and elimination of unwinding effects (Samiei et al., 2015, Martins et al., 2023).
• Theoretical and practical advances in machine unlearning, with strong safety, privacy, and regulatory compliance guarantees in large-scale AI models (Wu et al., 21 Aug 2025).

Future directions suggested by the literature include extending MRP frameworks to broader classes of manifold-valued data, more general machine learning architectures, and further exploration of projection-based manipulation techniques for safety, fairness, and interpretability in artificial intelligence.