Iterative Proportional Terminus-Matching (IPTM)

• IPTM is an iterative method that enforces prescribed terminal constraints, such as marginals or endpoints, using alternating rescaling techniques.
• It unifies classical IPFP/Sinkhorn in entropic optimal transport with endpoint prediction strategies in Schrödinger bridge and diffusion-based models.
• Empirical and theoretical studies show that IPTM achieves rapid convergence and Lipschitz stability across applications ranging from dynamic networks to high-dimensional Gaussian bridges.

Iterative Proportional Terminus-Matching (IPTM) denotes an iterative proportional procedure for enforcing prescribed terminal constraints. In the classical entropy-regularized optimal transport and Schrödinger bridge literature, it is the Iterative Proportional Fitting Procedure (IPFP), also known as Sinkhorn’s algorithm, viewed as alternating rescaling so that row and column sums, or more generally endpoint marginals, match prescribed targets (Deligiannidis et al., 2021). In more recent diffusion-based Schrödinger bridge work, IPTM is also a specific reparameterized training objective in which the backward model predicts the initial endpoint $x_0$ from an intermediate state $x_{k+1}$, and the forward model predicts the terminal endpoint $x_N$ from $x_k$ (Tang et al., 25 Aug 2025). Across these usages, “terminus-matching” refers to the enforcement of boundary marginals such as $P(X_0)=\mu_0$ and $P(X_T)=\mu_T$, or, in matrix settings, prescribed row and column totals.

1. Terminology and conceptual scope

The term IPTM is not used in the 2021 stability analysis of IPFP/Sinkhorn, nor in the 2024 IPMF paper, nor in the 2024 dynamic-network paper; in those settings, it is a descriptive name for iterative proportional methods that alternately enforce terminal marginals (Deligiannidis et al., 2021). In that sense, IPTM is exactly IPFP/Sinkhorn: a transport plan $\pi$ is alternately rescaled so that one marginal is matched, then the other, until both termini are matched.

In the 2024 Schrödinger bridge paper on Iterative Proportional Markovian Fitting (IPMF), IPTM corresponds to the IPF component of the unified procedure. The paper states that the exact terminus-matching is performed by the IPF projections $\mathrm{proj}_0$ and $\mathrm{proj}_1$ inside IPMF, while the Markovian projections improve the optimality of the bridge dynamics (Kholkin et al., 2024). In this usage, IPTM is the boundary-matching step embedded within a broader reciprocal/Markovian alternation.

In the 2025 diffusion-model paper, IPTM is elevated from a descriptive label to a named reparameterization alongside Iterative Proportional Mean-Matching (IPMM) and Iterative Proportional Flow-Matching (IPFM) (Tang et al., 25 Aug 2025). There, “terminus” means the endpoint of the trajectory, and the learning target is the endpoint itself rather than the next-step mean or a displacement field.

This suggests that IPTM functions as a unifying label across related but non-identical levels of description: classical marginal matching in entropic transport, endpoint projections in path-space Schrödinger bridge algorithms, and endpoint-regression parameterizations in diffusion-based Schrödinger bridge training.

2. Marginal matching in entropy-regularized optimal transport

In the discrete entropic optimal transport formulation, with marginals $p\in\mathbb{R}^n_+$ and $x_{k+1}$0, the problem is

$x_{k+1}$1

The associated Gibbs kernel is $x_{k+1}$2, and the entropic OT solution has the factorized form

$x_{k+1}$3

Sinkhorn/IPFP updates are

$x_{k+1}$4

with element-wise division. The update $x_{k+1}$5 enforces $x_{k+1}$6, and the update $x_{k+1}$7 enforces $x_{k+1}$8. In this formulation, “terminus-matching” is exactly marginal matching (Deligiannidis et al., 2021).

The continuous counterpart is formulated on compact metric spaces $x_{k+1}$9, with $x_N$0, $x_N$1, and kernel $x_N$2. The minimizer has the scaling representation

$x_N$3

Using the paper’s potentials $x_N$4 and $x_N$5, the half-bridge recursion is

$x_N$6

and the iterate $x_N$7 has density with respect to $x_N$8

$x_N$9

One update enforces the first marginal constraint, the next enforces the second, so the alternating sequence is an iterative terminus-matching procedure in the precise sense of the paper’s synthesis (Deligiannidis et al., 2021).

The significance of this formulation is that IPTM is not an auxiliary heuristic in the classical setting. It is the core computational mechanism by which the entropy-regularized coupling is represented and solved.

3. Endpoint projections in Schrödinger bridges and IPMF

For the Schrödinger bridge problem, the continuous-time objective is KL minimization over path measures with endpoint constraints. With Wiener reference $x_k$0 and endpoint distributions $x_k$1, the problem is

$x_k$2

The optimal process $x_k$3 admits forward and backward diffusion representations, and the static reduction is an entropic OT problem on endpoint couplings $x_k$4 (Kholkin et al., 2024).

In the classical Schrödinger system, the optimal endpoint coupling takes the multiplicative form

$x_k$5

with marginals $x_k$6 and $x_k$7. The corresponding IPF/Sinkhorn iteration alternates

$x_k$8

The 2024 IPMF paper writes this same endpoint matching as path-space projections

$x_k$9

$P(X_0)=\mu_0$0

that is, alternately replacing $P(X_0)=\mu_0$1 by $P(X_0)=\mu_0$2 and $P(X_0)=\mu_0$3 by $P(X_0)=\mu_0$4 while keeping the chain of transition densities (Kholkin et al., 2024).

IPMF extends this by interleaving reciprocal projections and Markovian projections. The practical bidirectional procedure is

$P(X_0)=\mu_0$5

$P(X_0)=\mu_0$6

$P(X_0)=\mu_0$7

$P(X_0)=\mu_0$8

The paper identifies this as

$P(X_0)=\mu_0$9

so the practical heuristic integrates IMF and IPF and is named Iterative Proportional Markovian Fitting (Kholkin et al., 2024).

In the one-dimensional Gaussian case, the IPMF iterates converge exponentially to the static Schrödinger bridge solution. The paper states

$P(X_T)=\mu_T$0

$P(X_T)=\mu_T$1

$P(X_T)=\mu_T$2

with explicit contraction factors $P(X_T)=\mu_T$3. Here the IPF projections preserve $P(X_T)=\mu_T$4 but fix the marginals, while the IMF projections move $P(X_T)=\mu_T$5 without changing endpoints. The paper’s interpretation is that IPMF decreases both endpoint mismatch and dynamical optimality gap (Kholkin et al., 2024).

4. Endpoint-prediction parameterization in diffusion-based Schrödinger bridge training

The 2025 paper introduces IPTM as one of three reparameterizations for solving Schrödinger bridges with score-based generative models. In the paper’s discrete-time setting, the reference path measure is a Gaussian forward chain with transitions

$P(X_T)=\mu_T$6

and the bridge problem is KL minimization with endpoint constraints $P(X_T)=\mu_T$7 and $P(X_T)=\mu_T$8 (Tang et al., 25 Aug 2025).

IPTM replaces next-step prediction by endpoint prediction. Its losses are

$P(X_T)=\mu_T$9

$\pi$0

The backward half-epoch predicts the initial terminus $\pi$1 from $\pi$2, and the forward half-epoch predicts the final terminus $\pi$3 from $\pi$4. The paper states that IPTM is closely related to DDPM’s $\pi$5-parameterization (Tang et al., 25 Aug 2025).

The IPTM predictors are connected to mean-matching parameterizations by

$\pi$6

$\pi$7

The inverse relations are also given: $\pi$8

$\pi$9

The paper further states that IPMM, IPTM, and IPFM are equivalent up to the provided transformations and yield the same fixed point under mild assumptions (Tang et al., 25 Aug 2025).

The endpoint-conditioned Gaussian identities justify the scaling factors. Under $\mathrm{proj}_0$0, the paper gives approximate conditionals with means

$\mathrm{proj}_0$1

This places IPTM directly at the interface between Schrödinger bridge endpoint matching and diffusion-model endpoint prediction.

The same paper introduces initialization with pre-trained SGMs. For IPMM/IPFM the initialization is

$\mathrm{proj}_0$2

and IPTM predictors are then obtained by conversion. The paper’s conclusion is that proper initialization with powerful pre-trained SGMs accelerates and stabilizes convergence (Tang et al., 25 Aug 2025).

5. Quantitative stability and convergence

The central theoretical result in the 2021 analysis is a uniform-in-time stability theorem for IPFP/Sinkhorn. Let $\mathrm{proj}_0$3 be compact metric spaces, let $\mathrm{proj}_0$4, and let $\mathrm{proj}_0$5 be positive and continuous. For marginal pairs $\mathrm{proj}_0$6 and $\mathrm{proj}_0$7, with IPFP iterates $\mathrm{proj}_0$8 and $\mathrm{proj}_0$9, the theorem states

$\mathrm{proj}_1$0

with

$\mathrm{proj}_1$1

The same constant appears in the corollary for the static Schrödinger bridge solution $\mathrm{proj}_1$2:

$\mathrm{proj}_1$3

Thus the entire IPTM/IPFP trajectory, and not merely its limit, is Lipschitz-stable in $\mathrm{proj}_1$4 with respect to perturbations of the marginals (Deligiannidis et al., 2021).

The proof uses the Hilbert–Birkhoff metric on cones of positive functions and positive operators, with contraction ratio at most $\mathrm{proj}_1$5 where $\mathrm{proj}_1$6 is the projective diameter. The paper also gives uniform bounds

$\mathrm{proj}_1$7

and a projective-distance bound

$\mathrm{proj}_1$8

The final $\mathrm{proj}_1$9 estimate follows by testing against Lipschitz functions and controlling integral differences through these bounds (Deligiannidis et al., 2021).

On the algorithmic side, the IPMF paper adds convergence guarantees for the hybrid path-space procedure in Gaussian settings, while the 2025 diffusion paper states that IPTM, IPMM, and IPFM are equivalent up to approximations similar to the standard reverse-time Gaussian approximation, and that the convergence analysis highlights the dependence of each half-epoch on the previous trajectories (Kholkin et al., 2024).

A practical implication stated explicitly in the 2021 paper is that sensitivity does not deteriorate with more iterations. This is relevant when only finitely many scaling rounds are used, as in matrix scaling, approximate Schrödinger bridge solvers, and learned bridge samplers.

6. Adaptations to dynamic network inference

A different application of iterative proportional matching arises in dynamic network inference from marginals. The 2024 network paper does not use the term IPTM, but treats it as an IPF/Sinkhorn-based procedure that, at each time slice, matches terminus marginals, namely row and column sums, and can optionally include an across-time normalization step to match an aggregate adjacency matrix $p\in\mathbb{R}^n_+$0 (Chang et al., 2024).

The basic constraints are

$p\in\mathbb{R}^n_+$1

For the decoupled per-time procedure, IPF updates are

$p\in\mathbb{R}^n_+$2

$p\in\mathbb{R}^n_+$3

If exact enforcement of $p\in\mathbb{R}^n_+$4 is also required, the paper’s synthesis gives a natural third projection: $p\in\mathbb{R}^n_+$5 This is described as a natural extension aligning with alternating Bregman projections onto row, column, and per-edge aggregate constraints (Chang et al., 2024).

The paper supplies a statistical interpretation. For each $p\in\mathbb{R}^n_+$6, with base matrix $p\in\mathbb{R}^n_+$7, it posits

$p\in\mathbb{R}^n_+$8

when $p\in\mathbb{R}^n_+$9, and $x_{k+1}$00 when $x_{k+1}$01. The log-likelihood is

$x_{k+1}$02

while the dual KL-projection potential is

$x_{k+1}$03

The paper’s theorem states that maximizing $x_{k+1}$04 is equivalent to minimizing $x_{k+1}$05, so IPF recovers the MLEs (Chang et al., 2024).

The paper also gives a structure-dependent error bound. With $x_{k+1}$06 and bipartite graph Laplacian $x_{k+1}$07,

$x_{k+1}$08

When IPF fails on sparse data, the paper introduces ConvIPF, a polynomial-time support-editing algorithm based on MAX-FLOW, BLOCKING-SET, and MODIFY-X, designed to restore feasibility under minimal structural changes (Chang et al., 2024).

This application broadens the meaning of “terminus” from endpoint distributions of a stochastic process to time-indexed origin and destination totals in network flows. The underlying mechanism remains iterative proportional enforcement of marginal constraints.

7. Empirical behavior, limitations, and conceptual boundaries

In the 2025 diffusion-model experiments, IPTM shows quantitative gains in both analytic and image-generation settings. For Gaussian bridges, the averaged $x_{k+1}$09 from scratch is reported as $x_{k+1}$10 for $x_{k+1}$11, $x_{k+1}$12 for $x_{k+1}$13, $x_{k+1}$14 for $x_{k+1}$15, and $x_{k+1}$16 for $x_{k+1}$17; with pre-trained initialization, the corresponding values are $x_{k+1}$18, $x_{k+1}$19, $x_{k+1}$20, and $x_{k+1}$21 (Tang et al., 25 Aug 2025). For unpaired image translation at $x_{k+1}$22, the reported FID values are $x_{k+1}$23 for Cat$x_{k+1}$24Dog and $x_{k+1}$25 for Dog$x_{k+1}$26Cat, and the ablation states that initialization reduces IPTM from $x_{k+1}$27 to $x_{k+1}$28 on Cat$x_{k+1}$29Dog and from $x_{k+1}$30 to $x_{k+1}$31 on Dog$x_{k+1}$32Cat (Tang et al., 25 Aug 2025).

The IPMF paper reports related empirical behavior: high-dimensional Gaussians with $x_{k+1}$33 show decreasing forward and reverse KL to the static SB solution across IPMF iterations; on 2D Swiss roll, DSBM and ASBM converge to $x_{k+1}$34 from various starts; and on Colored MNIST and Celeba, models trained under IPMF converge in FID/CMMD while preserving salient features (Kholkin et al., 2024). The paper also states that alternating forward/backward fitting stabilizes training and prevents marginal drift.

The main limitations are also explicit. In the 2021 stability paper, compactness of $x_{k+1}$35, bounded and Lipschitz $x_{k+1}$36, and strict positivity of $x_{k+1}$37 are part of the theorem’s setting, and the constants are stated to be not sharp (Deligiannidis et al., 2021). The same synthesis notes that, when $x_{k+1}$38 is scaled as $x_{k+1}$39, the Lipschitz dependence on marginals blows up as $x_{k+1}$40, consistently with classical OT being only $x_{k+1}$41-Hölder in marginals (Deligiannidis et al., 2021). In the 2025 paper, mismatch of schedules and time indexing between SB training and pre-trained SGMs can degrade performance, and improper noise scheduling can trade off alignment against diversity (Tang et al., 25 Aug 2025). In dynamic networks, sparse support can violate feasibility conditions, and structural zeros are binding unless the support is edited by a method such as ConvIPF (Chang et al., 2024).

A recurrent source of confusion is terminological. IPTM is not a standard classical term in all of the relevant literatures. It is absent from the 2021 IPFP stability paper, absent from the 2024 IPMF paper, and absent from the dynamic-network paper, where it is used only as an interpretive label for IPF/Sinkhorn-style terminus matching. It should also not be confused with the “proportional fair matching” problem on edge-colored bipartite graphs, which introduces $x_{k+1}$42-ProbablyAlmostFair randomized rounding guarantees but does not define IPTM (Duppala et al., 2024).

Taken together, the literature supports a stable core meaning. IPTM is an iterative proportional method for matching termini—marginals, endpoints, or time-slice origin/destination totals. In entropic OT and Schrödinger bridges, it is the classical IPFP/Sinkhorn mechanism. In IPMF, it is the endpoint-projection component that maintains $x_{k+1}$43 and $x_{k+1}$44. In diffusion-based Schrödinger bridge training, it is an endpoint-prediction reparameterization that connects classical proportional fitting to DDPM-style $x_{k+1}$45-prediction and pre-trained SGM initialization.