Nonlinear Conformal Regularization

• Nonlinear Conformal Regularization is a family of methods that use conformal geometry to control nonlinear deformations by enforcing isotropic scaling across different applications.
• It is applied in autoencoder decoders to ensure local isotropy, enabling latent curvature estimation and improving geometric fidelity in manifold learning.
• The approach extends to image registration, geometric flow, and field theory, providing rigorous constraints such as homeomorphism preservation and conformal invariance.

Searching arXiv for papers relevant to nonlinear conformal regularization. Searching arXiv for "nonlinear conformal regularization". Using arXiv search to retrieve papers on conformal regularization and related nonlinear geometric regularizers. Nonlinear conformal regularization denotes a family of constructions in which conformal structure is used to constrain, select, or stabilize nonlinear models. In the literature considered here, the term covers several distinct but related ideas: geometric regularization of autoencoder decoders so that they act conformally up to a scalar factor and expose a conformal factor and scalar curvature on the learned manifold; conformal-invariant hyperelastic regularisation for deformable image registration that enforces invertibility, orientation preservation, and topology preservation; regularized $n$-conformal heat flow in which an $n$-harmonic map flow is coupled to conformal evolution of the domain metric; a nonlinear automorphism of the 2D conformal algebra inducing continuous $\sqrt{T\bar T}$ deformations; and conformal-degree-preserving nonlinear Dirac equations in which admissible self-interactions are fixed by scaling and dimensionless coupling requirements (Couéraud et al., 28 Aug 2025, Zou et al., 2023, Park, 15 Feb 2025, Tempo et al., 2022, Alhaidari, 2012).

1. Conceptual scope and defining principles

A common geometric definition appears in the decoder setting. For a smooth map $f:(M,g)\to (N,h)$, conformality means

$f^\star h = c\,g$

for some smooth, strictly positive function $c:M\to\mathbb{R}$. In coordinates this is

$$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$

or equivalently

$J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$

This condition allows local stretching or shrinking, but only isotropically, by the same factor in every tangent direction. In the autoencoder literature, conformal regularization is explicitly positioned as less restrictive than isometry and more flexible for real data, where some deformation is expected (Couéraud et al., 28 Aug 2025).

Other uses of the term emphasize invariance rather than direct metric matching. In deformable registration, the relevant objects are conformal-invariant distortion measures built from $\nabla\Phi$, $\mathrm{Cof}\nabla\Phi$, and $n$0, embedded in a nonlinear-elastic energy (Zou et al., 2023). In the heat-flow setting, “conformal-direction metric evolution” means that the metric changes only by a scalar multiple of a fixed background metric, $n$1, so tensorial evolution is reduced to one scalar PDE for $n$2 (Park, 15 Feb 2025). In 2D conformal field theory, a nonlinear map mixing $n$3 and $n$4 preserves the conformal algebra and induces a continuous $n$5 deformation (Tempo et al., 2022). In nonlinear Dirac theory, conformal degree acts as a selection rule: requiring the kinetic and nonlinear terms to have the same conformal degree and the coupling to be dimensionless fixes the admissible power of the self-interaction (Alhaidari, 2012).

A plausible unifying description is that nonlinear conformal regularization uses conformal structure as a control variable. Depending on the application, that control variable may be a scalar conformal factor, a conformal-invariant distortion energy, a conformal metric mode, an automorphism of conformal generators, or a conformal-degree constraint.

2. Decoder regularization, conformal factor, and learned curvature

In dimensionality reduction with autoencoders, the decoder $n$6 is treated as a parametrization of the learned manifold $n$7. Standard reconstruction loss,

$n$8

does not guarantee good geometry: nearby latent codes can decode to outputs with strongly distorted distances, and neighboring data points can be mapped to codes that are far apart. Nonlinear conformal regularization addresses this by asking the decoder to behave like a conformal map, so that local deformation is isotropic rather than arbitrary (Couéraud et al., 28 Aug 2025).

The key quantity is

$n$9

whose eigenvalues $\sqrt{T\bar T}$0 describe stretching along principal latent directions. The regularizer is

$\sqrt{T\bar T}$1

where $\sqrt{T\bar T}$2 is smooth, positive, convex, and minimized at $\sqrt{T\bar T}$3, and $\sqrt{T\bar T}$4 is symmetric, homogeneous of degree $\sqrt{T\bar T}$5, with $\sqrt{T\bar T}$6. With the arithmetic mean and $\sqrt{T\bar T}$7, the regularizer becomes

$\sqrt{T\bar T}$8

and the conformal factor is

$\sqrt{T\bar T}$9

The minimum value $f:(M,g)\to (N,h)$0 is attained exactly when all eigenvalues are equal, $f:(M,g)\to (N,h)$1, so the decoder is conformal at $f:(M,g)\to (N,h)$2 (Couéraud et al., 28 Aug 2025).

This conformal factor has direct geometric meaning. When the decoder is conformal,

$f:(M,g)\to (N,h)$3

so $f:(M,g)\to (N,h)$4 measures local expansion or contraction of latent distances. The paper interprets it as a ratio of areas of small balls under the decoder. In the $f:(M,g)\to (N,h)$5 case, writing $f:(M,g)\to (N,h)$6, the scalar curvature of the learned manifold satisfies

$f:(M,g)\to (N,h)$7

After training, the curvature is estimated by computing latent codes, evaluating $f:(M,g)\to (N,h)$8, building a $f:(M,g)\to (N,h)$9-nearest-neighbor graph in latent space, forming the graph Laplacian $f^\star h = c\,g$0, and approximating

$f^\star h = c\,g$1

Implementation uses Hutchinson’s trace estimator,

$f^\star h = c\,g$2

with Rademacher samples emphasized as an unbiased low-variance option. The reported PyTorch tools are torch.func.jvp for Jacobian-vector products, torch.func.vjp for vector-Jacobian products, and torch.func.vmap and torch.func.jacfwd when the full Jacobian is needed. On the Swiss roll, the conformal regularizer improves $f^\star h = c\,g$3 and $f^\star h = c\,g$4 relative to the vanilla autoencoder and global isometry, and recovers the expected zero scalar curvature $f^\star h = c\,g$5 up to boundary effects. On CelebA, with latent dimension $f^\star h = c\,g$6, the nonlinear conformal regularizer and local isometry regularizer again give better-conditioned decoders than the global isometry regularizer, while reconstruction is limited by severe compression and the MSE stabilizes around $f^\star h = c\,g$7 (Couéraud et al., 28 Aug 2025).

Two limitations are explicit. First, curvature estimation is presented primarily for the $f^\star h = c\,g$8D latent case. Second, Jacobian-based quantities, Monte Carlo trace estimation, and graph Laplacians introduce computational cost and noise, and exact reproducibility can be difficult because of GPU nondeterminism in PyTorch/CUDA (Couéraud et al., 28 Aug 2025).

3. Conformal-invariant hyperelastic regularisation in deformable image registration

In deformable image registration, nonlinear conformal regularization appears as a conformal-invariant hyperelastic regulariser derived from nonlinear elasticity rather than from generic smoothness penalties. The registration problem is posed as

$f^\star h = c\,g$9

where $c:M\to\mathbb{R}$0 and $c:M\to\mathbb{R}$1 are source and target images, $c:M\to\mathbb{R}$2 is the deformation, and $c:M\to\mathbb{R}$3 is a convex bounded open set of class $c:M\to\mathbb{R}$4 satisfying the cone property required to apply Ball’s global invertibility results (Zou et al., 2023).

The regulariser is the stored-energy functional

$c:M\to\mathbb{R}$5

with

$c:M\to\mathbb{R}$6

where $c:M\to\mathbb{R}$7. The first two terms are described as conformal-invariant distortion measures. They penalise deviation from isotropic local scaling while remaining unchanged under conformal changes of variables (Zou et al., 2023).

The geometric consequences are central. Orientation-reversing mappings are forbidden because $c:M\to\mathbb{R}$8 whenever $c:M\to\mathbb{R}$9. The reciprocal Jacobian penalty $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$0 discourages singularities, while $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$1 controls extreme compression and expansion. Using Ball’s global invertibility results, the paper shows that finite-energy minimisers satisfy the conditions needed for homeomorphism, with

$$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$2

The paper also proves an existence theorem: problem $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$3 admits at least one minimiser in $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$4 (Zou et al., 2023).

The learning architecture represents the deformation by a coordinate MLP. Spatial coordinates $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$5 are mapped to $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$6, with sinusoidal activations of SIREN type and $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$7 in the reported experiments. The similarity term is negative normalized cross-correlation, implemented as local-window NCC. Optimization uses PyTorch, Adam, learning rate $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$8, 6000 epochs for COPD and 3000 for 4DCT, with 15,000 masked lung points per epoch for COPD and 10,000 for 4DCT. Reported runtime is about 1.7 minutes per 3D pair on COPD and 1.1 minutes on 4DCT (Zou et al., 2023).

Empirically, the method is evaluated on DIRLab COPD and DIRLab 4DCT with TRE based on 300 anatomical landmarks. Average TRE on COPD is $$h_{f(x)}\big(\mathrm{d}f_x(u),\mathrm{d}f_x(v)\big)=c(x)\,g_x(u,v),$$9 mm for the proposed method, compared with $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$0 mm for INR and larger errors for FE, PDD, VoxelMorph, and LapIRN. On 4DCT, average TRE is $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$1 mm, compared with $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$2 mm for INR and larger errors for the remaining baselines. Jacobian determinant visualisations show all-positive Jacobian determinants for the proposed method, while no regularisation and standard hyperelastic regularisation exhibit negative Jacobians or large expansions. The claim of topology-preserving, clinically meaningful transformations is therefore both theoretical and empirical, but it is tied to the model assumptions and deformation class $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$3, not to arbitrary deep registration architectures (Zou et al., 2023).

4. Regularized $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$4-conformal heat flow and conformal metric feedback

A different use of nonlinear conformal regularization appears in geometric analysis. The regularized $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$5-conformal heat flow couples a regularized $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$6-harmonic map flow to metric evolution in the conformal direction. Starting from the standard $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$7-harmonic map energy

$J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$8

the paper replaces the $J_f(x)^\mathsf{T}H\big(f(x)\big)J_f(x)=c(x)G(x).$9-energy density by a regularized density $\nabla\Phi$0 and defines the regularized $\nabla\Phi$1-energy $\nabla\Phi$2. The coupled system is

$\nabla\Phi$3

or, relative to the fixed background metric $\nabla\Phi$4,

$\nabla\Phi$5

with

$\nabla\Phi$6

The metric changes only through the scalar field $\nabla\Phi$7, so the feedback is purely conformal (Park, 15 Feb 2025).

The main theorem states: assume $\nabla\Phi$8 and $\nabla\Phi$9. For any $\mathrm{Cof}\nabla\Phi$0, there exists a smooth solution $\mathrm{Cof}\nabla\Phi$1 of $\mathrm{Cof}\nabla\Phi$2 on $\mathrm{Cof}\nabla\Phi$3 with initial condition $\mathrm{Cof}\nabla\Phi$4, $\mathrm{Cof}\nabla\Phi$5. In particular, the regularized $\mathrm{Cof}\nabla\Phi$6-conformal heat flow does not develop finite-time singularities, unlike the usual $\mathrm{Cof}\nabla\Phi$7-harmonic map flow (Park, 15 Feb 2025).

The regularizing mechanism combines energy dissipation with damping from the metric equation. The basic identity is

$\mathrm{Cof}\nabla\Phi$8

so the regularized energy is monotone decreasing. The volume density has the explicit form

$\mathrm{Cof}\nabla\Phi$9

which yields

$n$00

The paper further identifies a decisive sign condition,

$n$01

obtained by choosing $n$02 large enough relative to target curvature bounds. This allows the conformal metric feedback to dominate curvature-error terms (Park, 15 Feb 2025).

The analysis then develops local energy inequalities, differential inequalities for $n$03, bounds on higher powers of $n$04, elliptic estimates for $n$05, Sobolev bootstrapping, and a final Moser iteration. Under local smallness assumptions on $n$06, $n$07, and a bound on $n$08, the paper obtains a local $n$09 bound on $n$10, which rules out concentration. Global smoothness is then established by contradiction: if finite-time singularity occurred, it would force concentration of local energy, but the continuity estimate

$n$11

precludes the required energy drop. The result generalizes the “conformal heat flow prevents bubbling” phenomenon known for $n$12 to arbitrary $n$13 under the stated assumptions, with the explicit dimension restriction $n$14 used in short-time existence and higher-order estimates (Park, 15 Feb 2025).

5. Algebraic regularization and conformal-degree selection in field theory

In 2D conformal field theory, nonlinear conformal regularization takes the form of a nonlinear automorphism of the conformal algebra. Starting from two commuting Witt algebras generated by $n$15 and $n$16, the paper shows that the algebra is preserved by

$n$17

with real parameter $n$18. In the energy-momentum basis,

$n$19

the map becomes

$n$20

or equivalently

$n$21

The deformation preserves momentum density and induces the flow

$n$22

The same work gives a geometric reinterpretation: the deformed theory can be seen as the original CFT on a field-dependent curved metric with lapse and shift determined by variational derivatives of the deformed Hamiltonian, and the deformed conformal symmetries arise from diffeomorphisms satisfying modified conformal Killing equations (Tempo et al., 2022).

A distinct but structurally related selection principle appears in nonlinear Dirac theory. The spinor action

$n$23

is required to be scale invariant, so $n$24. For a spin-$n$25 field, the linear term is $n$26, the Dirac operator has degree $n$27, and the spinor field has conformal degree

$n$28

If the nonlinear self-interaction is required to have the same conformal degree as the kinetic term and the self-coupling is required to be dimensionless in relativistic units $n$29, then the allowed power is fixed by

$n$30

The general nonlinear Dirac equation is

$n$31

and in the massive case

$n$32

In $n$33 dimensions, $n$34 gives $n$35, so the interaction is quartic in the Lagrangian and cubic in the equation of motion. The allowed quartic structures include

$n$36

and the framework contains the massive Thirring and massive Gross–Neveu models as pure vector and pure scalar limits, respectively (Alhaidari, 2012).

Taken together, these field-theoretic constructions suggest a broader interpretation of nonlinear conformal regularization as a structural constraint on admissible nonlinearity. In one case, the constraint preserves the conformal algebra under a nonlinear map; in the other, it preserves conformal degree and dimensionless coupling in a nonlinear fermionic action.

6. Adjacent frameworks, boundaries, and recurring themes

An adjacent line of work concerns controlled learning of pointwise nonlinearities under explicit regularity and stability control. There, the trainable scalar nonlinearity $n$37 is optimized in $n$38 with a second-order total variation penalty $n$39 and slope-box constraints $n$40 a.e. The global optimum is achieved by adaptive nonuniform linear splines, and the same framework can enforce 1-Lipschitz stability, firm non-expansiveness, monotonicity, and invertibility. Its conceptual contribution includes a connection to nonlinear conformal regularization in the sense that the learned scalar nonlinearity can be identified either as the derivative of a scalar potential or as the proximal operator of a potential, which is useful for plug-and-play schemes, unrolled proximal gradient, and invertible flows (Unser et al., 2024).

A useful boundary case is provided by nonlinear sparsity regularization for ill-posed inverse problems. The penalty

$n$41

is nonconvex, promotes sparsity, and is studied in a Tikhonov framework for nonlinear operator equations $n$42. The paper establishes coercivity, weak lower semicontinuity, and Radon–Riesz for $n$43, sparse minimizers for $n$44, convergence rates of $n$45 and $n$46 under different nonlinear assumptions on $n$47, and an iterative half variation algorithm. This is a nonlinear regularization method, but it is not presented as conformal; its organizing principles are sparsity, coercivity, source conditions, and tangential cone-type conditions rather than conformal geometry or conformal invariance (Li et al., 22 Aug 2025).

Across the conformal literature proper, several recurring themes are explicit. Conformal regularization is repeatedly described as weaker than isometry but still meaningful, because it allows deformation while constraining anisotropy (Couéraud et al., 28 Aug 2025). It often yields additional geometric observables, most notably a conformal factor and, in the $n$48D latent case, scalar curvature (Couéraud et al., 28 Aug 2025). In some settings it provides hard guarantees rather than heuristic smoothing, such as homeomorphism in registration under Ball-type assumptions (Zou et al., 2023) or global smoothness for regularized $n$49-conformal heat flow when $n$50, $n$51, and $n$52 is large enough (Park, 15 Feb 2025). In relativistic field theory it acts less as a penalty term than as a compatibility condition that rules out arbitrary nonlinearities, fixing either the deformation flow or the interaction power through conformal structure (Tempo et al., 2022, Alhaidari, 2012).

These differences matter for interpretation. “Nonlinear conformal regularization” is not a single algorithm or a single variational penalty. It is a broader research theme in which nonlinear models are constrained by conformal geometry, conformal invariance, conformal algebra, or conformal degree, and the specific mathematical form depends strongly on whether the target problem is manifold learning, registration, geometric flow, or quantum field theory.