Hadamard Regularization: Methods & Applications

• Hadamard regularization is a collection of distinct techniques that regularize problems by exploiting structures like nonpositive curvature, singularity subtraction, or algebraic transforms.
• It includes methods such as Bregman divergence-based schemes on Hadamard manifolds, variational approaches in manifold-valued imaging, and finite-part prescriptions in quantum field theory.
• Additionally, in machine learning, Hadamard regularization leverages Walsh–Hadamard transforms and overparametrizations to induce sparsity and improve model performance.

Hadamard regularization denotes a family of technically distinct regularization procedures rather than a single method. In contemporary arXiv usage, the expression appears in at least five settings: proximal and equilibrium algorithms on Hadamard manifolds or Hadamard spaces, variational regularization for manifold-valued imaging, covariant point-splitting and regularized Hadamard expansions in quantum field theory on curved spacetime, Hadamard finite-part regularization of hypersingular kernels in open quantum systems, and machine-learning regularizers built from Walsh–Hadamard transforms or Hadamard overparametrizations. What unifies these usages is not a common formula but a common dependence on one of three structures: nonpositively curved geometry, Hadamard short-distance expansions, or Hadamard algebraic transforms (Sharma et al., 20 Jan 2026).

1. Terminological scope and principal meanings

The term is polysemous. In optimization on nonpositively curved spaces, it refers to regularization schemes that exploit the convexity structure of Hadamard manifolds or complete CAT(0) spaces. In quantum field theory, it refers to the subtraction of universal Hadamard singularities or to the regularized Hadamard expansion of two-point distributions. In open-system Schwinger–Keldysh theory, it refers to Hadamard’s finite part for hypersingular time-domain kernels. In machine learning, it can refer either to Walsh–Hadamard spectral penalties or to sparsity-inducing Hadamard overparametrization (Kumam et al., 2018).

Domain	Regularized object	Characteristic construction
Hadamard manifolds and spaces	equilibrium bifunctions or convex function sums	Bregman divergence, quasilinearization, or squared-distance proximal terms
Manifold-valued imaging	ROF-like variational energies	geodesic-distance TV terms and Douglas–Rachford splitting
Curved-spacetime QFT	two-point functions and stress tensors	Hadamard parametrix, point-splitting, regularized Hadamard expansion
Open quantum systems	hypersingular memory kernels	Hadamard finite part $\operatorname{Pf}$
Machine learning	functional spectra or sparse objectives	Walsh–Hadamard $L_1$ penalties and Hadamard-product reparametrizations

A common misconception is that all of these usages derive from the same mathematical operation. The literature does not support that reading. For example, “Hadamard regularization” in manifold-valued imaging is explicitly distinguished from Hadamard finite part integrals, whereas in open quantum systems “Hadamard regularization” means the partie finie of distributions such as $\operatorname{Pf}(1/t^2)$, and in Walsh–Hadamard regularization the reference is to the Hadamard transform rather than to CAT(0) geometry or curved-spacetime parametrices (Bergmann et al., 2015).

2. Proximal and equilibrium regularization on Hadamard manifolds and spaces

In the geometric-optimization literature, a Hadamard manifold is a complete, simply connected Riemannian manifold with nonpositive sectional curvature, and a Hadamard space is a complete CAT(0) metric space. These settings provide unique minimizing geodesics, convexity along geodesics, and nonexpansive metric geometry. Such properties support direct generalizations of Euclidean proximal methods, resolvents, and equilibrium algorithms (Banert, 2013).

For monotone equilibrium problems on Hadamard manifolds, the 2026 Bregman proximal construction defines the regularized bifunction

$$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$

where $D_\phi$ is the manifold Bregman divergence

$$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$

The associated iteration is

$$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$

and $x_{k+1}$ is the unique solution of the regularized equilibrium problem

$$F(x_{k+1},y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x_{k+1})\big)\ge 0 \quad \forall y\in C.$$

This construction is described as “Hadamard regularization” because the regularization term is built from the manifold Bregman divergence and depends on exponential and logarithm maps together with the Riemannian gradient (Sharma et al., 20 Jan 2026).

The central obstruction is geometric. On a generic Hadamard manifold with nonzero curvature, the Bregman regularization term is not geodesically convex in general. The paper addresses this by imposing geodesic convexity of the strict sublevel sets

$K_x:=\{\,y\in C:\tilde F(x,y)<0\,\},$

rather than convexity of $L_1$0 itself. Under the standing assumptions on $L_1$1, the Bregman function $L_1$2, the coercivity condition

$L_1$3

and bounded positive step sizes, the iterates satisfy a Fejér-type monotonicity

$L_1$4

together with

$L_1$5

and the whole sequence converges to a solution of the equilibrium problem (Sharma et al., 20 Jan 2026).

A related CAT(0)-space formulation regularizes equilibrium problems by the Berg–Nikolaev quasilinearization. For fixed $L_1$6,

$L_1$7

and the resolvent is

$L_1$8

Under monotonicity, convexity, lower semicontinuity, and domain assumptions, $L_1$9 is everywhere defined, single-valued, and nonexpansive, and the proximal iteration

$\operatorname{Pf}(1/t^2)$0

is Fejér monotone and $\operatorname{Pf}(1/t^2)$1-convergent to an equilibrium (Kumam et al., 2018).

A third usage in Hadamard spaces is the backward-backward regularization of the sum of two convex functions by the squared metric distance,

$\operatorname{Pf}(1/t^2)$2

The exact iteration

$\operatorname{Pf}(1/t^2)$3

generalizes the Hilbert-space backward-backward splitting. In Hadamard spaces the proximal maps are single-valued, firmly nonexpansive in the CAT(0) sense, and 1-Lipschitz, while the iterates satisfy square-summability of increments and $\operatorname{Pf}(1/t^2)$4-convergence; strong convergence follows if one term is uniformly convex (Banert, 2013).

In zero sectional curvature, the manifold Bregman divergence reduces to the classical Euclidean one, geodesics become straight lines, and the convexity of the regularization-induced sets $\operatorname{Pf}(1/t^2)$5 is automatic under the natural convexity condition on $\operatorname{Pf}(1/t^2)$6. This suggests that the curved setting is not a superficial generalization of Euclidean proximal regularization but a genuinely different regime in which curvature directly controls regularization well-posedness (Sharma et al., 20 Jan 2026).

3. Variational regularization for manifold-valued imaging

In image restoration on symmetric Hadamard manifolds, “Hadamard regularization” refers to variational regularization by manifold-valued total variation. The discrete ROF-like energy is

$\operatorname{Pf}(1/t^2)$7

Here $\operatorname{Pf}(1/t^2)$8 is a symmetric Hadamard manifold, the fidelity is quadratic in the geodesic distance, and the regularizer is an anisotropic first-order TV-like term defined by neighboring geodesic distances (Bergmann et al., 2015).

The algorithmic treatment uses a product-space Douglas–Rachford splitting. After decomposing the TV term into four local parts and introducing the diagonal set

$\operatorname{Pf}(1/t^2)$9

the optimization problem becomes

$$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$0

The manifold proximal mapping is defined by

$$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$1

and closed forms are available for the relevant distance-like terms. For example, if $$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$2, then

$$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$3

For pairwise terms $$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$4 or $$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$5, the proximal map acts by paired geodesic averaging (Bergmann et al., 2015).

A key technical issue is nonexpansiveness of reflections. In Euclidean space, reflections of convex lower semicontinuous functions are nonexpansive; on Hadamard manifolds this fails in general. The paper proves nonexpansiveness for reflections of several distance-like functions and for indicator reflections on constant-curvature manifolds, which yields convergence of the Douglas–Rachford algorithm in those cases. On variable-curvature SPD manifolds, the general theory does not apply because reflections of the indicator of the diagonal set can be expansive, but numerical convergence is still observed (Bergmann et al., 2015).

This usage of the term is geometrically different from equilibrium regularization. The object being regularized is not a bifunction or resolvent but an image-valued variational energy, and the principal regularizer is a sum of manifold distances. The commonality lies in the CAT(0)/Hadamard convexity of $$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$6 and $$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$7, which makes proximal splitting and convex analysis possible on nonpositively curved targets (Bergmann et al., 2015).

4. Hadamard point-splitting and regularized expansions in curved-spacetime quantum field theory

In quantum field theory on curved spacetime, Hadamard regularization is a local and covariant renormalization procedure based on the universal short-distance singular structure of two-point functions. For a Hadamard state, the local parametrix in four dimensions has the form

$$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$8

where $$\tilde{F}(x,y)=F(x,y)+\lambda\big(D_\phi(y,\bar{x})-D_\phi(y,x)-D_\phi(x,\bar{x})\big),$$9 is Synge’s world function, $D_\phi$0 and $D_\phi$1 are smooth geometric coefficients, and $D_\phi$2 is smooth and state-dependent. Point-splitting renormalization subtracts the singular $D_\phi$3 and $D_\phi$4 terms before taking the coincidence limit (Negro et al., 2024).

The regularized Hadamard expansion refines this by introducing

$D_\phi$5

with $D_\phi$6, together with regulator functions $D_\phi$7 and $D_\phi$8 in the expansion

$D_\phi$9

The regulator functions are determined by transport equations along null geodesics, and the resulting regularized Hadamard structure is preserved under Cauchy evolution to order $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$0. In this formulation, the ultraviolet regulator is encoded geometrically by the replacement $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$1, where $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$2 is an affine parameter difference along null geodesics (Finster et al., 2017).

At the level of renormalized stress tensors, Hadamard point-splitting is rigorously connected to other local schemes. For smooth static spacetimes, local $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$3-function regularization and Hadamard point-splitting are equivalent. DeWitt–Schwinger subtraction is likewise equivalent to Hadamard point-splitting in smooth Riemannian and Lorentzian spacetimes when the DeWitt–Schwinger prescription is reformulated as a local subtraction of the Hadamard singular structure plus a conservation-restoring local term. The symmetry of the Hadamard and Seeley–DeWitt coefficients is an essential ingredient in these equivalence results (Hack et al., 2012).

A recent application concerns tensor perturbations and the graviton stress tensor. The gauge-fixed graviton and ghost propagators are written in Hadamard form, their state-independent singularities are isolated covariantly, and local curvature counterterms are used to absorb the divergences. The paper emphasizes that averaging prescriptions such as the Isaacson or Misner–Thorne–Wheeler forms rely on prior scale separation and are therefore unsuitable as starting points for renormalization; if averaging is used, it should be invoked only after the Hadamard subtraction and renormalization procedure (Negro et al., 2024).

This branch of the literature uses “Hadamard regularization” in a sense entirely different from CAT(0) optimization. The relevant Hadamard structure is the singularity expansion of Green functions, not nonpositive curvature. The shared label is historical rather than structural.

5. Hadamard finite-part regularization in Schwinger–Keldysh open-system dynamics

In the Schwinger–Keldysh treatment of open quantum systems coupled to unstructured environments, “Hadamard regularization” means Hadamard’s finite part, $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$4, applied to hypersingular time-domain kernels. The basic setting is a damped quantum harmonic oscillator linearly coupled to a bosonic bath, with the bath integrated out to produce nonlocal retarded and Keldysh self-energies in the Kadanoff–Baym equations (Dolgner, 13 Mar 2026).

For algebraic poles stronger than $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$5, principal value is insufficient. Hadamard’s finite-part prescription defines, for example,

$$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$6

by subtracting the Taylor asymptotics of $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$7 near $$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$8, removing the divergent pieces explicitly, and taking the limit. Distributionally,

$$D_\phi(x,y)=\phi(x)-\phi(y)-\langle \operatorname{grad}\phi(y),\exp_y^{-1}(x)\rangle.$$9

This is the mechanism used to give meaning to the short-time singular kernels generated by wide-band baths (Dolgner, 13 Mar 2026).

In the Ohmic wide-band case, the relevant distributions are

$$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$0

Accordingly, the self-energies are decomposed as

$$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$1

The local distributional terms produce friction and frequency renormalization,

$$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$2

while the regular remainder retains the genuinely non-Markovian memory (Dolgner, 13 Mar 2026).

The computational consequence is a separation-of-scales formulation: the singular short-time pieces are absorbed into local renormalized parameters, and the remaining memory is advanced on a slow system timescale. The naive Kadanoff–Baym solver scales as $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$3 in the number of time steps, whereas the regularized slow-timescale algorithm yields $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$4 overall and can approach $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$5 with fast convolution quadrature. This makes it possible to retain low-temperature non-Markovianity and renormalization effects without fully resolving the fast bath scale (Dolgner, 13 Mar 2026).

A further source of confusion is terminological. In Schwinger–Keldysh theory, the “Hadamard correlator” $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$6 denotes the symmetrized two-point function, but the paper explicitly states that this object is unrelated to Hadamard’s finite-part regularization. Here the word “Hadamard” refers only to the partie finie of hypersingular kernels (Dolgner, 13 Mar 2026).

6. Walsh–Hadamard and Hadamard-product regularization in machine learning

In machine learning, the expression appears in two distinct but purely algebraic senses. The first is Walsh–Hadamard spectral regularization for Boolean-input neural networks. The second is sparsity-inducing Hadamard overparametrization.

For Boolean inputs, the Walsh–Hadamard framework expands a function over parity characters, with spectral degree given by Hamming weight. The regularizer proposed in the HashWH method penalizes the $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$7 norm of hashed Walsh–Hadamard bucket sums: $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$8 This is a scalable approximation to full-spectrum $$F_{k,t_k}(x,y):=F(x,y)+t_k\big(D_\phi(y,x_k)-D_\phi(y,x)-D_\phi(x,x_k)\big),$$9 regularization, designed to mitigate the low-degree spectral bias of neural networks on Boolean domains. The method does not assign explicit degree-wise weights; instead it promotes global sparsity in the Walsh spectrum, suppresses spurious low-degree coefficients, and can preserve higher-degree components when their amplitudes are large (Gorji et al., 2023).

The second usage starts from an explicitly regularized sparse objective

$x_{k+1}$0

and replaces it by the smooth surrogate

$x_{k+1}$1

where $x_{k+1}$2 is a smooth Hadamard-type map, such as the Hadamard Product Parametrization

$x_{k+1}$3

The induced sparse penalty is recovered exactly through a smooth variational form; for instance,

$x_{k+1}$4

The main equivalence theorem states that the base and surrogate problems have equal infima and matching local minima, so the smooth optimization transfer introduces no spurious local minima under the stated assumptions (Kolb et al., 2023).

These machine-learning constructions are not related to Hadamard manifolds, Hadamard point-splitting, or Hadamard finite parts. In one case the reference is to the Walsh–Hadamard transform matrix, and in the other to Hadamard products and powers. The shared terminology is therefore algebraic rather than geometric or microlocal. A plausible implication is that “Hadamard regularization” has become a contextual label whose exact meaning must be read from the surrounding formalism, not inferred from the phrase alone (Gorji et al., 2023).