Operator Sinkhorn Algorithm

• Operator Sinkhorn algorithms are iterative methods that generalize classical Sinkhorn scaling to operator-valued matrices, function spaces, and noncommutative settings.
• They alternate updates—such as positive definite matrix adjustments or continuous potential corrections—to enforce marginal, spectral, or projection constraints while enhancing numerical stability.
• Advanced variants integrate overrelaxation and implicit differentiation techniques, enabling robust applications in optimal transport, quantum symmetry, and Gaussian models.

Searching arXiv for papers on operator Sinkhorn algorithms and related analyses. The operator Sinkhorn algorithm denotes a family of alternating normalization procedures that extend classical Sinkhorn matrix scaling from nonnegative matrices to operator-valued, function-space, and noncommutative settings. In one prominent formulation, the operator Sinkhorn iteration (OSI) solves operator scaling problems for completely positive maps by alternating updates on positive definite matrices and, equivalently, by absorbing scalings directly into a tuple of matrices. In adjacent transport literatures, closely related “operator Sinkhorn” constructions act on continuous potentials, Gaussian parameters, or proximal operators rather than finite vectors, but preserve the same structural motif: alternating enforcement of marginal, spectral, or projection constraints through nonlinear operator updates (Soma et al., 2024, Berman, 2017, Akyildiz et al., 12 Mar 2025).

1. Canonical formulation: operator scaling for completely positive maps

In the operator-scaling setting, one is given matrices $A_1,\dots,A_k \in \mathbb{R}^{m \times n}$ and seeks invertible $L \in GL_m(\mathbb{R})$ and $R \in GL_n(\mathbb{R})$ such that the scaled matrices $\bar A_i = L A_i R^\top$ satisfy

$$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$

This can be rephrased through the completely positive map

$\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$

and its adjoint

$\Phi^*(X)=\sum_{i=1}^k A_i^\top X A_i.$

The fixed-point formulation seeks $X \succ 0$ and $Y \succ 0$ satisfying

$$X = \frac{1}{m}\left(\sum_{i=1}^k A_i Y A_i^\top\right)^{-1}, \qquad Y = \frac{1}{n}\left(\sum_{i=1}^k A_i^\top X A_i\right)^{-1}.$$

The operator Sinkhorn iteration alternates these two updates (Eisenmann et al., 13 Mar 2026).

A mathematically equivalent formulation continually updates and absorbs the scaling matrices into the $L \in GL_m(\mathbb{R})$0. This “on-the-fly” form is important because it maintains better conditioning of the computational objects; in the later acceleration literature, this preconditioning effect becomes the decisive numerical distinction between stable and unstable implementations (Eisenmann et al., 13 Mar 2026).

A useful way to situate the operator Sinkhorn algorithm is to distinguish several operator-level incarnations that appear in the literature:

Setting	State space	Characteristic update
Operator scaling	$L \in GL_m(\mathbb{R})$1 or scaled $L \in GL_m(\mathbb{R})$2	$L \in GL_m(\mathbb{R})$3, $L \in GL_m(\mathbb{R})$4
Continuous entropic transport	Potentials $L \in GL_m(\mathbb{R})$5 on function spaces	$L \in GL_m(\mathbb{R})$6 via log-integral operators
Generalized partial transport	Functions $L \in GL_m(\mathbb{R})$7 through $L \in GL_m(\mathbb{R})$8, $L \in GL_m(\mathbb{R})$9	KL prox-divide with clip or min
Gaussian Schrödinger bridges	Means/covariances	Riccati-type recursion
Quantum permutation setting	Matrices of projections	Alternating row/column operator normalization

A common misconception is that “operator Sinkhorn” refers to a single algorithmic template with a single state space. The cited literature instead uses the term for several closely related generalizations of Sinkhorn scaling: positive definite cone iterations for completely positive maps, nonlinear integral operators in continuous entropic transport, and noncommutative projection-normalization procedures (Soma et al., 2024, Berman, 2017, Nechita et al., 2019).

2. Overrelaxation, local rates, and numerical stability

The basic OSI is described as robust but only linearly convergent, and this motivated accelerated variants based on successive overrelaxation (SOR). For the positive definite cone formulation, Euclidean overrelaxation takes the form

$R \in GL_n(\mathbb{R})$0

while geodesic overrelaxation uses

$R \in GL_n(\mathbb{R})$1

and analogously for $R \in GL_n(\mathbb{R})$2. Cholesky-based overrelaxation interpolates in transformed coordinates rather than directly in the cone (Soma et al., 2024).

The local convergence analysis linearizes these schemes near a fixed point and shows that the different SOR variants are equivalent to first order. If $R \in GL_n(\mathbb{R})$3 denotes the spectral radius for the original OSI linearization, the SOR spectral radius is

$R \in GL_n(\mathbb{R})$4

with asymptotically optimal parameter

$R \in GL_n(\mathbb{R})$5

This is derived through linearization and Young’s SOR theorem (Soma et al., 2024).

For the geodesic version, a global convergence result is obtained in Hilbert metric for a specific range of relaxation parameters, provided the underlying maps are contractive. At the same time, the 2026 stability analysis identifies the main practical limitation of direct SOR implementations: if scaling is applied only at the end, the matrices entering Cholesky or geodesic computations can become ill-conditioned, leading to numerical instability and stagnation (Eisenmann et al., 13 Mar 2026).

The stable remedy is to merge SOR with classic OSI’s on-the-fly scaling. In exact arithmetic, the FPI-style SOR schemes and the OSI-style SOR schemes are mathematically equivalent, but they are numerically different. The stable variants immediately absorb new scaling factors into the $R \in GL_n(\mathbb{R})$6, so the next update is computed from more balanced matrices. Numerical experiments reported for ill-conditioned operator-scaling instances show that FPI-style SOR stagnates at moderate error levels such as $R \in GL_n(\mathbb{R})$7, whereas OSI-style SOR can reach errors such as $R \in GL_n(\mathbb{R})$8 (Eisenmann et al., 13 Mar 2026).

3. Continuous and infinite-dimensional operator formulations

A second major meaning of operator Sinkhorn arises in entropic optimal transport on continuous domains. In this setting, Sinkhorn updates are written as operators on functions rather than row and column scalings of a finite matrix. One formulation introduces

$R \in GL_n(\mathbb{R})$9

with iterates

$\bar A_i = L A_i R^\top$0

The discrete Sinkhorn algorithm is then interpreted as a time discretization of a continuous gradient flow in the log domain, and in a large-scale limit it converges to a parabolic Monge–Ampère equation (Berman, 2017).

For optimal transport on the torus, the continuous-time scaling limit yields

$\bar A_i = L A_i R^\top$1

The paper establishes uniform convergence of Sinkhorn iterates to the solution of this PDE and gives explicit error bounds and arithmetic complexity statements, including nearly linear per-iteration schemes on the torus and specialized schemes on the sphere (Berman, 2017).

More abstractly, Sinkhorn can be studied as a semigroup of nonlinear integral operators on weighted Banach spaces. A Lyapunov-drift and local-minorization framework yields contraction coefficients for general $\bar A_i = L A_i R^\top$2-divergences and proves exponential contraction of alternating Sinkhorn operators toward Schrödinger bridges. This operator-theoretic framework is explicitly designed for infinite-dimensional and unbounded settings, including cases where classical projective or finite-state analyses do not apply (Akyildiz et al., 12 Mar 2025).

The continuous-time limit can be analyzed even more directly. In the Sinkhorn flow, the entropy production rate is given by the exact identity

$\bar A_i = L A_i R^\top$3

The flow induces a reversible Markov dynamics on the target marginal, admits a Dirichlet form and a Poincaré inequality, and has a strictly positive spectral gap whenever $\bar A_i = L A_i R^\top$4. Exponential entropy decay is characterized by a logarithmic Sobolev inequality (Srinivasan et al., 14 Oct 2025).

4. Gaussian operator Sinkhorn and Schrödinger bridges

For general continuous spaces, exact finite-dimensional Sinkhorn recursions are usually unavailable. A notable exception is the Gaussian case. When the marginals are Gaussian and the reference kernel is a linear Gaussian Markov transition, the iterative proportional fitting procedure remains within the Gaussian family and reduces to a finite-dimensional recursion on means and covariances (Akyildiz et al., 2024).

The covariance dynamics can be written in terms of rescaled matrices $\bar A_i = L A_i R^\top$5 through Riccati-type recursions,

$\bar A_i = L A_i R^\top$6

with alternating left/right updates in the full algorithm. The resulting methodology is closely related to the Kalman filter and discrete Riccati difference equations. The convergence analysis proves exponential convergence, and the paper also derives closed-form expressions for Schrödinger bridges, quadratic Schrödinger potentials, and entropic transport maps in the Gaussian setting (Akyildiz et al., 2024).

The fixed-point equation

$\bar A_i = L A_i R^\top$7

admits the closed-form solution

$\bar A_i = L A_i R^\top$8

The associated entropic transport map is randomized: $\bar A_i = L A_i R^\top$9 with explicit $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$0 and $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$1 determined by the Riccati fixed point (Akyildiz et al., 2024).

This Gaussian branch clarifies an important structural point. In the operator viewpoint, the distinction between “matrix Sinkhorn” and “continuous Sinkhorn” is not binary. Gaussian models show that genuinely continuous Schrödinger-bridge iterations can still admit exact finite-dimensional recursions, but the state variables are no longer row and column scaling vectors; they are means, covariances, gains, and conditional Gaussian parameters (Akyildiz et al., 2024).

5. Generalized marginals, differentiation, and optimization interfaces

Several works broaden the operator Sinkhorn paradigm by replacing exact marginal equalities with softer or more general constraints. For generalized optimal partial transport, the updates are written through integral operators

$$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$2

followed by KL prox-divide steps. For TV penalties, the update is

$$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$3

and similarly for $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$4; for partial total variation, min replaces clip. This generalizes classical row/column normalization to nonlinear pointwise proximal operators and connects Sinkhorn to Dykstra’s alternating Bregman projection methods and to linear-programming formulations on augmented spaces (Bai, 2024).

A related but distinct relaxation is unbalanced optimal transport. For entropic regularized UOT, Sinkhorn alternates dual updates and achieves an $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$5-approximate solution in complexity $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$6, improving on the $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$7 complexity stated for balanced OT. The analysis relies on geometric convergence of the dual iterates and on the fact that UOT uses KL penalties rather than hard marginal constraints (Pham et al., 2020). This suggests that relaxing exact marginal feasibility can materially change both the proof techniques and the dependence on $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$8.

In differentiable programming, the Sinkhorn operator is treated as an implicit layer. A unified implicit-differentiation framework computes vector-Jacobian products for arbitrary scalar losses and jointly differentiates with respect to cost matrices and target capacities. The backward pass is based on a structured linear system, has backward cost $$\sum_{i=1}^k \bar A_i \bar A_i^\top = \frac{1}{m}I_m, \qquad \sum_{i=1}^k \bar A_i^\top \bar A_i = \frac{1}{n} I_n.$$9, memory $\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$0, and is independent of the number of forward Sinkhorn iterations $\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$1; the paper also gives explicit error bounds for approximate forward inputs (Eisenberger et al., 2022).

The same alternating-normalization machinery has also been recast as a proximal operator in inverse problems. Generalized Sinkhorn iterations compute the proximal operator of entropy-regularized optimal mass transport, and this proximal block is then embedded in Douglas–Rachford splitting for composite imaging problems. In the quadratic proximal case, the dual update involves the Wright omega function, and the framework is demonstrated for limited-angle computerized tomography with TV and data-fidelity terms (Karlsson et al., 2016).

6. Noncommutative certificates and quantum variants

The operator Sinkhorn idea also appears in explicitly noncommutative settings. For noncommutative rank and shrunk subspaces, the paper on “Shrunk subspaces via operator Sinkhorn iteration” generalizes operator scaling so that the marginal spectra need only be majorized by specified vectors. The resulting MajSinkhorn algorithm alternates minimizations of a capacity functional,

$\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$2

and uses KL projections onto permutahedra. A sufficiently long run yields an approximate shrunk subspace close to the minimum exact shrunk subspace, and the paper gives deterministic polynomial-time algorithms and rounding results (Franks et al., 2022).

This branch is significant because operator Sinkhorn is no longer merely a feasibility algorithm for balanced marginals. It becomes a certificate-producing method: the output can encode the smallest shrunk subspace, with applications to fractional linear matroid matching and weak membership and optimization algorithms for the rank-2 Brascamp–Lieb polytope (Franks et al., 2022).

A more visibly noncommutative variant is the Sinkhorn-type algorithm for quantum permutation groups. Here the iterates are square matrices whose entries are orthogonal projections onto one-dimensional subspaces, subject to “magic unitary” relations

$\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$3

The algorithm alternates row and column operator normalization by applying inverse square roots of row- and column-Gram operators to random initial vectors. For graph symmetry questions, it incorporates the extra relation $\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$4 through a soft-hard interpolation parameter $\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$5, and the output is used experimentally to detect quantum symmetries by testing commutativity of the resulting projections (Nechita et al., 2019).

These noncommutative extensions dispel another common misconception: operator Sinkhorn is not confined to optimal transport. The same alternating normalization pattern appears in operator scaling, invariant theory, combinatorial optimization, and quantum symmetry problems, with the state space ranging from positive definite matrices to projections in finite-dimensional $\Phi(Y)=\sum_{i=1}^k A_i Y A_i^\top$6-algebraic models (Franks et al., 2022, Nechita et al., 2019).