Sinkhorn–Knopp Fixed Point Iterations

• Sinkhorn–Knopp fixed point iterations are an alternating row and column normalization process that transforms strictly positive matrices into a unique doubly stochastic matrix.
• The algorithm converges robustly by contracting the Hilbert projective metric and strictly decreasing a convex divergence measure at each iteration.
• Recent research has refined finite-step termination, complexity bounds, and accelerated variants, expanding its applications in optimal transport, machine learning, and matrix balancing.

A Sinkhorn–Knopp fixed point iteration is an alternating row and column normalization process for positive matrices, central to matrix scaling problems and entropic optimal transport. Given a strictly positive $n\times n$ matrix $A$, the Sinkhorn–Knopp algorithm scales $A$ to a unique diagonally equivalent doubly stochastic matrix—i.e., a matrix with all rows and columns summing to one—provided such a scaling exists. The iterations underpin matrix balancing, optimal transport solvers, bipartite ranking models, and general operator scaling, and converge robustly under minimal positivity assumptions. Recent research has refined finite-step characterization, complexity bounds, acceleration schemes, and functional-analytic perspectives.

1. Algebraic Structure and Fixed-Point Scheme

Given $A \in \mathbb{R}^{n \times n}_{>0}$, define the row- and column-sum operators: $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$ and construct the diagonal scaling matrices: $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$

$$D_c(A) = \mathrm{diag}(c_1(A), ..., c_n(A)), \ c_j(A) = 1/\mathrm{Col}_j(A)$$

The iteration alternates between row and column scaling:

• Column scaling: $A^{(k+1)} = D_c(A^{(k)}) A^{(k)} \$ (if $k$ even)
• Row scaling: $A^{(k+1)} = A^{(k)} D_r(A^{(k)}) \$ (if $A$0 odd)

The process generates a sequence $A$1 converging entrywise to the unique doubly stochastic matrix $A$2 diagonally equivalent to $A$3, characterized by positive diagonal matrices $A$4, $A$5 with $A$6 and $A$7 (Nathanson, 2019, Nathanson, 2018).

2. Convergence Theory and Fixed-Point Characterization

The Sinkhorn–Knopp map is a contraction in the Hilbert projective metric and strictly decreases an associated convex divergence (such as Kullback–Leibler) at each step unless $A$8 is already doubly stochastic. Compactness of the positive orthant and strict monotonicity guarantee entrywise convergence.

The fixed-point equations for the limit $A$9 are: $A$0 or equivalently, there exist unique $A$1 such that

$A$2

Proofs rely on convexity or projective-metric contraction arguments (Nathanson, 2019).

Table: Essential Operators in Sinkhorn–Knopp Iteration

Notation	Definition	Action
$A$3	$A$4	Row-wise normalization
$A$5	$A$6	Column-wise normalization
$A$7	Iterative scaling sequence	Alternating row/column scaling

3. Special Finite-Terminating Families

Although the generic behavior is asymptotic convergence, Nathanson constructed, for every $A$8, an explicit two-parameter family of positive $A$9 matrices that are row-stochastic but not column-stochastic, and become exactly doubly stochastic after a single column normalization.

Construction (abridged): Given $A \in \mathbb{R}^{n \times n}_{>0}$0, $A \in \mathbb{R}^{n \times n}_{>0}$1, parameters $A \in \mathbb{R}^{n \times n}_{>0}$2, $A \in \mathbb{R}^{n \times n}_{>0}$3, $A \in \mathbb{R}^{n \times n}_{>0}$4, define $A \in \mathbb{R}^{n \times n}_{>0}$5, $A \in \mathbb{R}^{n \times n}_{>0}$6. Define $A \in \mathbb{R}^{n \times n}_{>0}$7 so rows $A \in \mathbb{R}^{n \times n}_{>0}$8 or $A \in \mathbb{R}^{n \times n}_{>0}$9 have $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$0; others have $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$1. Then $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$2 is row-stochastic and $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$3 is doubly stochastic.

For $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$4, $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$5, and choice $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$6, the matrix

$$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$7

becomes doubly stochastic after exactly one column normalization (Nathanson, 2019).

Determinant Zero: Every such $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$8 has $$\mathrm{Row}_i(A) = \sum_{j=1}^n a_{ij}, \quad \mathrm{Col}_j(A) = \sum_{i=1}^n a_{ij}$$9. The existence of a nonsingular (invertible) example terminating in a single step remains open.

4. The General Case, Two-by-Two Matrices, and Uniqueness

For strictly positive $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$0 matrices, explicit closed-form solutions characterize the Sinkhorn limit. Every finite-step termination occurs in at most two steps. The $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$1 Sinkhorn limit is: $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$2 Finite-step one-step and two-step terminations are fully classified: up to permutation, the forms attaining a doubly stochastic matrix in exactly 1 or 2 steps are known (Nathanson, 2018). For $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$3, closed-form formulas are generally unavailable.

Any two different diagonal scalings $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$4, $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$5 yielding doubly stochastic matrices must coincide up to a scalar factor, implying uniqueness up to scaling (Nathanson, 2018).

5. Implications for Matrix Scaling, Applications, and Variants

Sinkhorn–Knopp iterations underpin matrix scaling to prescribed marginals, used in optimal transport, machine learning (notably through entropic OT and Schrödinger bridges), and network analysis. In economic complexity, the so-called Fitness–Complexity algorithm coincides, up to normalization, with Sinkhorn–Knopp—both are alternating diagonal scaling systems that minimize a logarithmic barrier potential under linear constraints (Mazzilli et al., 2022).

Constrained Sinkhorn–Knopp schemes extend to structured zero-patterns or entropic-regularized optimal transport with prescribed zeros, replacing positivity with Bregman-projected alternating scaling, yielding unique solutions even with infinite costs or infeasible marginals (Corless et al., 2024).

Recent advances have elucidated phase transitions in iterative complexity tied to combinatorial matrix density; when the normalized density exceeds $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$6, Sinkhorn–Knopp achieves $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$7 iteration count; otherwise, the count can scale as $$D_r(A) = \mathrm{diag}(r_1(A), ..., r_n(A)), \ r_i(A) = 1/\mathrm{Row}_i(A)$$8 in the worst case (He, 13 Jul 2025, He, 4 Apr 2026).

6. Accelerated and Operator Variants

The classical Sinkhorn–Knopp exhibits linear convergence, governed by the spectral gap of the balanced matrix; convergence can stall if this gap closes. Krylov subspace and Arnoldi-type accelerations, building on the nonlinear multiparameter eigenvalue formulation of the scaling problem, enable faster convergence for matrices with clustered singular values near 1 (Aristodemo et al., 2018).

Nonlinear overrelaxation of the SK projection—implemented as geodesic or Cholesky-based SOR schemes (notably in operator scaling for completely positive maps)—can dramatically enhance rates, provided numerical stability is ensured via on-the-fly scaling (Eisenmann et al., 13 Mar 2026, 1711.01851). In ill-conditioned settings, care must be taken to avoid instabilities from direct scaling; empirically, the stabilized schemes reach higher accuracies and achieve speedups for both matrix and operator settings.

7. Concluding Synthesis and Open Questions

Sinkhorn–Knopp fixed point algorithms provide a canonical, robust, and convergent procedure for rescaling positive matrices to prescribed row and column sums—foundational in many theoretical and applied disciplines. Their convergence is generally linear, but families exist exhibiting finite-step termination, especially in structured low-rank or degenerate settings, but only when the determinants vanish. The existence and explicit construction of invertible matrices with finite-step termination remains open beyond trivial or low-dimensional examples (Nathanson, 2019). Recent advances have connected iteration complexity to “bulk” matrix density, clarified phase transitions, and introduced numerically robust acceleration methods, establishing a sharp boundary between the regimes of rapid and slow convergence.

Key references:

• "Matrix scaling limits in finitely many iterations" (Nathanson, 2019)
• "Alternate minimization and doubly stochastic matrices" (Nathanson, 2018)
• "Equivalence between the Fitness-Complexity and the Sinkhorn-Knopp algorithms" (Mazzilli et al., 2022)
• "Phase transition of the Sinkhorn-Knopp algorithm" (He, 13 Jul 2025)
• "On the Efficiency of Sinkhorn-Knopp for Entropically Regularized Optimal Transport" (He, 4 Apr 2026)
• "Accelerating the Sinkhorn-Knopp iteration by Arnoldi-type methods" (Aristodemo et al., 2018)
• "Numerically stable variants of overrelaxation for operator Sinkhorn iteration" (Eisenmann et al., 13 Mar 2026)