SDSN: Scaling Doubly Stochastic Normalization

• SDSN is a family of methods that transforms nonnegative matrices into doubly stochastic matrices, ensuring balanced row and column sums for robust mass conservation.
• It is applied across optimal transport, manifold learning, graph neural networks, and clustering, offering scalable solutions that preserve key spectral properties.
• Design trade-offs in SDSN, including entropy regularization and iteration limits, are crucial to preventing spectral collapse while maintaining informative geometric detail.

Scaling Doubly Stochastic Normalization (SDSN) denotes a family of procedures that transform a nonnegative matrix into a doubly stochastic matrix, typically enforcing

$$A\,\mathbf1=\mathbf1,\qquad A^\top\mathbf1=\mathbf1,$$

or the corresponding balanced-marginal variant used in transport. Across recent work, SDSN appears in matrix scaling, optimal transport, manifold learning, graph neural networks, subspace clustering, visualization, Transformer attention, and graph matching. The common attraction is that doubly stochastic structure provides balanced mass conservation, numerical regularity, and a convex feasible set; the central caveat is that some implementations, especially high-entropy Sinkhorn projections, can induce spectral collapse and loss of expressive detail (Nathanson, 2019, Liu, 5 Jan 2026).

1. Classical matrix scaling and the doubly stochastic constraint

A standard formalization uses the set

$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$

which is the nonnegative row- and column-stochastic feasible set used throughout the recent SDSN literature (Liu, 5 Jan 2026). In the classical matrix-scaling setting, one starts from a strictly positive matrix $A$ and alternates row scaling and column scaling. Nathanson summarizes the corresponding operators $R(A)=X(A)A$ and $C(A)=AY(A)$, where $X(A)$ and $Y(A)$ are diagonal matrices built from inverse row sums and inverse column sums, respectively; the alternating process converges to a unique doubly stochastic matrix $S(A)=XAY$, the Sinkhorn limit, with uniqueness up to the trivial scalar ambiguity $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$. In the symmetric case one may take $Y=X$, so the limit has the form $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$0 (Nathanson, 2019).

This convergence statement is complemented by support conditions in later summaries. For nonnegative matrices with total support or full indecomposability, Sinkhorn–Knopp converges geometrically to the unique doubly stochastic scaling; sparse implementations preserve zeros exactly and can therefore be used on $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$1-NN or graph-derived matrices without densifying them (Lu et al., 2016). Nathanson also emphasizes that explicit Sinkhorn limits are rare in closed form: even among symmetric $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$2 two-value families, some cases reduce to quadratic formulas, others to cubic radicals, and at least one family leads to an octic polynomial (Nathanson, 2019). This establishes SDSN as both an algorithmic procedure and an algebraic object.

Lu et al. describe an alternative “fast” normalization for asymmetric or bipartite affinities. Starting from an $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$3 nonnegative matrix $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$4, one first row-normalizes it to $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$5, then forms

$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$6

The resulting $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$7 is symmetric and doubly stochastic, and no further iteration is required (Lu et al., 2016).

2. Variational formulations: KL projection, entropic transport, Gaussian kernels, and Frobenius regularization

A major modern interpretation of SDSN is variational. In Sinkhorn-based architectures, the projection of a nonnegative kernel $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$8 onto $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$9 is written as

$A$0

or, equivalently, as an entropic optimal transport problem

$A$1

The practical solver is the Sinkhorn iteration, alternating row normalization and column normalization on $A$2 until convergence (Liu, 5 Jan 2026). In this formulation, doubly stochasticity is not merely a constraint; it is the output of an entropy-regularized projection.

In manifold learning, SDSN is imposed on Gaussian affinity matrices. For noisy observations $A$3, one defines

$A$4

and seeks positive diagonal scalings $A$5 so that $A$6 is doubly stochastic, or, in the symmetric form, scalars $A$7 such that $A$8 and $A$9 (Landa et al., 2022). The corresponding population object is a positive continuous function $R(A)=X(A)A$0 solving an integral equation on the manifold. The paper derives concentration of finite-sample scaling factors around this population form under high-dimensional sub-Gaussian noise, and then uses the scaled matrix for robust density estimation, noise inference, clean-distance estimation, and graph Laplacian construction (Landa et al., 2022).

Landa and Cheng extend the same Gaussian-kernel scaling to manifolds with boundary. Their continuous scaling factor $R(A)=X(A)A$1 satisfies, to second order in $R(A)=X(A)A$2, a quadratic equation with coefficients depending on the boundary distance $R(A)=X(A)A$3, boundary mean curvature $R(A)=X(A)A$4, and explicit Gaussian moments $R(A)=X(A)A$5. They further derive a first-order boundary correction in which $R(A)=X(A)A$6, with $R(A)=X(A)A$7 an explicit strictly decreasing convex function of $R(A)=X(A)A$8 (Kohli et al., 2024).

A distinct variational route appears in graph matching. In FRAM, SDSN solves exactly

$R(A)=X(A)A$9

equivalently

$C(A)=AY(A)$0

where $C(A)=AY(A)$1 is the doubly stochastic set. This Frobenius-Regularized Linear Assignment (FRA) formulation is introduced specifically to remove the scale sensitivity of the ordinary Euclidean projection onto $C(A)=AY(A)$2 (Shen et al., 26 Jul 2025). This suggests that SDSN is best understood as a constraint class instantiated by multiple objective functions rather than as a single normalization rule.

3. Computational scaling and algorithmic realizations

Recent SDSN work is largely driven by the cost of exact normalization. Dense Sinkhorn on an $C(A)=AY(A)$3 matrix is intrinsically quadratic per iteration, and this becomes the main systems bottleneck in long-sequence attention or large-graph learning. The literature therefore develops sparse, truncated, low-rank, one-shot, and mixed-precision alternatives.

Setting	Normalized object	Stated scaling
Dense Gaussian-kernel Sinkhorn	$C(A)=AY(A)$4 or $C(A)=AY(A)$5	$C(A)=AY(A)$6 per iteration; usually $C(A)=AY(A)$7 iterations (Landa et al., 2022)
Sparse Sinkhorn / DOSNES scaling	Sparse similarity matrix	$C(A)=AY(A)$8; typically $C(A)=AY(A)$9 for $X(A)$0–$X(A)$1 tolerance (Lu et al., 2016)
DsmNet / DsmNet-compensate	$X(A)$2	$X(A)$3 time, $X(A)$4 memory (Hu et al., 16 Apr 2026)
A-DSSC active-set projection	Doubly stochastic affinity on support $X(A)$5	$X(A)$6 per dual evaluation; near-linear if final affinity has $X(A)$7 nonzeros (Lim et al., 2020)
ESPFormer	Expected sliced transport plan $X(A)$8	$X(A)$9, fully parallel over slices (Shahbazi et al., 11 Feb 2025)
LOTFormer	Low-rank glued coupling	$Y(A)$0 time and memory, with $Y(A)$1 (Shahbazi et al., 27 Sep 2025)
FRAM SDSN	FRA projection	$Y(A)$2; mixed precision gives up to $Y(A)$3 speedup over CPU-FP64 (Shen et al., 26 Jul 2025)

Hu et al. replace an exact inverse $Y(A)$4, whose direct computation is $Y(A)$5, by a truncated Neumann expansion

$Y(A)$6

which can be applied to features through sequential sparse multiplications in $Y(A)$7 time (Hu et al., 16 Apr 2026). Because truncation leaks probability mass, they add a residual correction

$Y(A)$8

thereby restoring exact row-stochasticity while keeping the hard $Y(A)$9-hop cutoff off the diagonal (Hu et al., 16 Apr 2026).

In subspace clustering, the active-set solver for A-DSSC exploits the sparsity of the final affinity. Each restricted dual solve costs $S(A)=XAY$0 rather than $S(A)=XAY$1, and if the final support has $S(A)=XAY$2 nonzeros the overall routine becomes near-linear (Lim et al., 2020). In attention, ESPFormer replaces iterative Sinkhorn with a one-shot aggregation of sliced transport plans, while LOTFormer factorizes the transport through a learned pivot measure of size $S(A)=XAY$3, so that one never materializes the full $S(A)=XAY$4 attention map (Shahbazi et al., 11 Feb 2025, Shahbazi et al., 27 Sep 2025). In FRAM, the SDSN subroutine itself remains $S(A)=XAY$5 per iteration, but the authors place different steps in FP64, TF32, and FP32 to obtain substantial hardware acceleration without changing the converged SDSN solution (Shen et al., 26 Jul 2025).

4. Spectral behavior, effective depth, and the “Homogeneity Trap”

The most explicit spectral analysis is developed for Sinkhorn-projected deep networks. Once a layer’s mixing matrix $S(A)=XAY$6 is doubly stochastic, Perron–Frobenius guarantees spectral radius $S(A)=XAY$7 with eigenvector $S(A)=XAY$8, and all remaining singular values lie in $S(A)=XAY$9. The key quantity is the subdominant singular value

$(X,Y)\sim(\lambda X,\lambda^{-1}Y)$0

which governs contraction on the detail subspace $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$1. Repeated application satisfies $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$2, leading to the spectral effective-depth scale

$(X,Y)\sim(\lambda X,\lambda^{-1}Y)$3

The central claim is that maximum-entropy Sinkhorn projections drive $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$4 toward the uniform barycenter $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$5, forcing $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$6 and thus collapsing effective depth (Liu, 5 Jan 2026).

This collapse has a Fourier interpretation. For a doubly stochastic operator on the complete graph, all nonconstant harmonics lie in the detail subspace and are contracted by at most $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$7. As $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$8 in the entropy maximization, or as the Sinkhorn temperature $(X,Y)\sim(\lambda X,\lambda^{-1}Y)$9 grows, $Y=X$0, so the mean mode survives and all higher-frequency modes are annihilated. The paper terms this behavior the “Homogeneity Trap”: repeated mixing does not enlarge a useful receptive field, but instead removes detail until only the average remains (Liu, 5 Jan 2026).

The same work also analyzes LayerNorm in a noise-dominated regime. Writing the pre-normalized feature as

$Y=X$1

the detail-subspace SNR is

$Y=X$2

Their finite-$Y=X$3 bound states that once $Y=X$4, LayerNorm produces nearly orthogonal outputs from distinct inputs with overwhelming probability; affine rescaling cannot restore the lost direction because it rescales signal and noise equally (Liu, 5 Jan 2026).

Hu et al. obtain a closely related but graph-specific picture. For the exact operator $Y=X$5, repeated application empirically collapses Dirichlet energy to zero, which they identify as over-smoothing. By contrast, $Y=X$6 plateaus too soon because of mass leakage, yielding under-mixing, while the compensated $Y=X$7 maintains a high nonzero energy floor. They also characterize the second-largest eigenvalue of the compensated operator, with $Y=X$8 as $Y=X$9 and $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$00 as $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$01 (Hu et al., 16 Apr 2026). Taken together, these analyses show that doubly stochasticity alone does not determine the usable spectral regime; the construction method and regularization strength do.

5. Geometry, density, boundary inference, and graph diffusion

In manifold inference, doubly stochastic Gaussian kernels are used as a robust alternative to standard row-stochastic normalization. The core result of Landa and Cheng is that, under a compact $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$02-dimensional manifold model with high-dimensional sub-Gaussian noise, the doubly stochastic affinity matrix and its scaling factors concentrate around population forms with explicit finite-sample probabilistic error bounds. From this, they derive a robust density estimator, estimators for pointwise noise magnitude, pointwise signal magnitude, and pairwise clean Euclidean distances, and a family of robust graph Laplacians $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$03 that converge to a Schrödinger-type operator; the choice $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$04 yields the Laplace–Beltrami operator (Landa et al., 2022).

The boundary-estimation extension sharpens the geometric role of the scaling factor. Landa and Cheng show that the continuous analogue $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$05 depends systematically on boundary distance and curvature, and they combine the doubly stochastic kernel with local PCA to estimate a boundary direction $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$06 and detect boundary points by thresholding $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$07. Their simulations show that the resulting boundary-point estimates outperform the standard Gaussian-kernel approach, particularly under noisy conditions (Kohli et al., 2024).

In graph neural networks, Hu et al. derive a different doubly stochastic operator directly from graph topology: $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$08 This $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$09 is doubly stochastic and encodes continuous multi-hop proximity together with strict local centrality. The associated DsmNet and DsmNet-compensate architectures operate in $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$10 time, mitigate over-smoothing by bounding Dirichlet-energy decay, perform strongly on homophilic benchmarks, and can also supply continuous structural encodings for Graph Transformers. The same paper also states the theoretical boundaries of such DSM diffusion on heterophilic topologies, where any low-pass diffusion operator suffers performance drops and DsmNet variants help only marginally (Hu et al., 16 Apr 2026).

6. Clustering, visualization, attention, and assignment matching

In subspace clustering, doubly stochastic normalization is used not merely as post-processing but as part of the learning objective. The feasible set

$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$11

appears in J-DSSC, a convex joint model that learns both a self-expressive representation and a normalized affinity. The paper identifies several consequences of enforcing $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$12: exact Laplacian normalization for symmetric $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$13, row/column balancing, a connectivity guarantee because every row sums to one, and a convex feasible set. A faster approximation, A-DSSC, separates the self-expressive step from the doubly stochastic projection and solves the latter with a sparsity-aware active-set dual method; on GPU, full A-DSSC on MNIST with $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$14 takes less than $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$15 seconds to reach $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$16 clustering accuracy (Lim et al., 2020).

In data visualization, Lu et al. use doubly stochastic normalization to equalize total similarities before running SNE. They report that the doubly stochasticity constraint often leads to embeddings that are approximately spherical, motivating the replacement of a flat embedding space by spheres. In their account, the spherical embedding eliminates the discrepancy between center and periphery and thereby resolves the crowding problem caused by highly imbalanced similarities (Lu et al., 2016).

In Transformer attention, SDSN is used to balance incoming and outgoing token mass. ESPFormer constructs queries and keys as empirical measures, projects them along $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$17 directions, computes one-dimensional transport plans, and aggregates them into

$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$18

The paper states that each slice plan $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$19 is doubly stochastic and that the convex combination $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$20 therefore remains in the transport polytope. Because the slices can be processed fully in parallel, the stated complexity is $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$21, and the reported speedups over Sinkhorn-based attention on long sequences are $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$22–$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$23 (Shahbazi et al., 11 Feb 2025).

LOTFormer attacks the same problem through low-rank entropic transport. With a learnable pivot measure $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$24, it solves two entropic OT subproblems, $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$25 and $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$26, and glues them into

$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$27

Under uniform query and key marginals, the paper verifies $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$28 and $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$29; after rescaling by $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$30, the attention matrix is exactly doubly stochastic. The same factorization yields $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$31 and $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$32 application to values. Empirically, LOTFormer reports a $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$33 average on Long Range Arena with depthwise convolution, $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$34 top-1 on ImageNet-1K with a DeiT-Tiny backbone at $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$35 resolution, and $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$36 BLEU on IWSLT’14 De$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$37En after fine-tuning (Shahbazi et al., 27 Sep 2025).

In graph matching, FRAM uses SDSN as the inner solver for Frobenius-regularized assignment matching. Its alternating-projection scheme splits the problem into $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$38, which enforces row and column sums but allows negatives, and $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$39, which enforces nonnegativity. The resulting SDSN solution is scale-invariant because the input is normalized by $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$40 before solving the FRA problem. The paper also proves an assignment-gap bound $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$41, identifies the barycentric limit as $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$42, and shows convergence to the minimum-Frobenius-norm point of the optimal face as $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$43. With GPU mixed precision, the full FRAM system achieves up to $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$44 speedup over its CPU-FP64 counterpart, with negligible loss in solution accuracy (Shen et al., 26 Jul 2025).

7. Design trade-offs, parameter regimes, and recurrent misconceptions

The most important practical trade-off concerns entropy. In the Sinkhorn-based deep-network analysis, increasing the entropy strength $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$45, raising Sinkhorn temperature $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$46, or letting the number of Sinkhorn iterations $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$47 improves numerical stability but shrinks $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$48 and shortens effective depth. The paper therefore recommends moderate $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$49, monitoring $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$50 against $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$51, and choosing the “elbow” before $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$52 plummets. It also recommends finite Sinkhorn iteration counts, explicitly suggesting early stopping such as $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$53 instead of $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$54, small learnable post-projection scaling that keeps the underlying SNR above the $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$55 threshold, and interleaving DSM blocks with non-DSM expressive layers such as un-normalized self-attention or residual MLPs (Liu, 5 Jan 2026).

Graph diffusion models exhibit an analogous locality-versus-mixing trade-off. In DsmNet, $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$56 controls the transition from local $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$57-hop structure to global diffusion; on homophilic benchmarks such as Cora, CiteSeer, PubMed, Amazon, and Coauthor, $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$58 often maximizes accuracy, while on heterophilic datasets including Chameleon, Squirrel, Texas, Wisconsin, and Cornell, any low-pass diffusion operator suffers performance drops. The same paper notes that residual mass compensation always improves centrality preservation and helps keep node representations distinct in deep layers (Hu et al., 16 Apr 2026). A related misconception is that exact normalization by itself guarantees preservation of informative geometry; the heterophily results show that normalization can still be mismatched to the graph regime.

Kernel-based SDSN carries its own parameter sensitivities. In the manifold and boundary papers, bandwidth is tied to the usual bias-variance and locality-noise balance: $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$59 may be chosen as the median $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$60-th nearest-neighbor distance, tolerances of $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$61–$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$62 are described as adequate for Sinkhorn stopping, and practical numerical stabilization includes log-domain computation, per-row rescaling, or discarding pairs beyond a fixed multiple of $$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$63 when the Gaussian kernel underflows (Kohli et al., 2024, Landa et al., 2022). These recommendations indicate that SDSN is rarely a drop-in normalization; it is a parameterized operator whose spectral, geometric, and numerical behavior must be co-designed with the downstream task.

A recurrent misconception in deep learning is that post-normalization layers can repair whatever information doubly stochastic mixing suppresses. The finite-$$\mathcal{DSM}=\{\,M\in\mathbb R_+^{n\times n}:M\mathbf1=\mathbf1,\;M^\top\mathbf1=\mathbf1\,\},$$64 SNR theorem for LayerNorm directly contradicts that view in noise-dominated regimes, and the GNN diffusion results similarly show that exact normalization can yield over-smoothing unless the operator is truncated or compensated (Liu, 5 Jan 2026, Hu et al., 16 Apr 2026). The broader implication is that the central question in SDSN is not whether to impose doubly stochasticity, but how to do so without collapsing the nontrivial spectrum that carries task-relevant structure.