Iterative Proportional Markovian Fitting (IPMF)

• Iterative Proportional Markovian Fitting (IPMF) is an alternating-projection algorithm that unifies IPF and IMF to enforce both endpoint and Markovian constraints in statistical inference and optimal transport.
• It employs reciprocal-class and Markovian projections to achieve exponential convergence under broad regularity conditions, ensuring reliable marginal matching and computational efficiency.
• IPMF applies to both discrete and continuous settings, offering practical benefits in generative modeling and image-to-image translation through its robust convergence properties.

Iterative Proportional Markovian Fitting (IPMF) is an operator-splitting algorithmic framework that unifies and generalizes classical iterative proportional fitting (IPF) and iterative Markovian fitting (IMF) for solving high-dimensional statistical inference and optimal transport problems, with particular focus on the discrete and continuous Schrödinger bridge models. IPMF alternates two forms of projection operations, one enforcing reciprocal-class constraints (static couplings or endpoint consistency), the other enforcing Markovianity or pathwise constraints, thereby achieving exponential convergence to the unique entropic optimal transport plan under broad regularity and convexity conditions (Sokolov et al., 4 Aug 2025, Silveri et al., 23 Oct 2025, Kholkin et al., 2024, Wiegerinck et al., 2012, Brossard et al., 2016).

1. Mathematical Definition and Foundations

IPMF is fundamentally an alternating-projection algorithm on probability distributions, designed for problems where both Markovian structure and marginal endpoint constraints must be enforced. In the discrete-time Schrödinger bridge setting, given a finite state space $\mathcal{X}$ with prescribed marginals $\mu, \nu \in \mathcal{P}(\mathcal{X})$ at times $t_0=0$ and $t_{N+1}=1$, and a reference Markovian path kernel $q(x_{t_1:t_N}|x_0,x_{N+1})$ with full support, the SB problem seeks

$$\min_{p \in \Pi_{N+2}(\mu, \nu)} \operatorname{KL}(p \| q)$$

where $$\Pi_{N+2}(\mu, \nu) = \{ p \in \mathcal{P}(\mathcal{X}^{N+2}) : p(x_0) = \mu(x_0),\ p(x_{N+1}) = \nu(x_{N+1}) \}$$ and $\operatorname{KL}(\cdot\|\cdot)$ denotes the Kullback–Leibler divergence.

The unique minimizer $p^\ast$, under mild positivity assumptions, possesses a Markov structure matching the support of $q$. The IPMF procedure generates a sequence $\mu, \nu \in \mathcal{P}(\mathcal{X})$0 by alternating two Bregman (KL) projections:

• Markov projection: $\mu, \nu \in \mathcal{P}(\mathcal{X})$1 projects onto the set fixing all two-point marginals, enforcing Markovianity.
• Reciprocal/endpoints projection: $\mu, \nu \in \mathcal{P}(\mathcal{X})$2 projects onto the set fixing the joint initial-final distribution, enforcing endpoint marginals.

Closed-form update formulas are: $\mu, \nu \in \mathcal{P}(\mathcal{X})$3

$\mu, \nu \in \mathcal{P}(\mathcal{X})$4

(Sokolov et al., 4 Aug 2025, Kholkin et al., 2024)

A similar two-step alternating-projection structure persists in continuous path-space, with projections over reciprocal path measures (reference bridges) and Markovian diffusions (Silveri et al., 23 Oct 2025, Kholkin et al., 2024).

2. Relationship to IPF, IMF, and Generalizations

IPMF unifies three historically distinct iterative schemes:

• Iterative Proportional Fitting (IPF): Also known as Sinkhorn, alternates KL projections to enforce marginal constraints in couplings, with the update $\mu, \nu \in \mathcal{P}(\mathcal{X})$5 (Brossard et al., 2016).
• Iterative Markovian Fitting (IMF): Alternates projections onto reciprocal and Markovian path constraints. Each iterate remains Markovian and matches endpoints (Silveri et al., 23 Oct 2025, Sokolov et al., 4 Aug 2025).
• IPMF: Views IMF itself as an alternating KL-projection procedure, with constraint sets generalized to the reciprocal class of the reference $\mu, \nu \in \mathcal{P}(\mathcal{X})$6 and the set of Markov path measures. This framework admits both a pathwise dynamical and a classical marginal-based interpretation, depending on context (Kholkin et al., 2024, Silveri et al., 23 Oct 2025).

Generalizations to more complex graphical models, such as chain factor graphs, are realized by cluster-wise multiplicative updates, retaining closed-form step similar to classical IPF (Wiegerinck et al., 2012).

3. Convergence and Explicit Rates

A central theoretical advance is the establishment of explicit exponential convergence rates for IPMF iterations under natural regularity conditions:

For the discrete-time finite state SB problem, the contraction factor is (Sokolov et al., 4 Aug 2025): $\mu, \nu \in \mathcal{P}(\mathcal{X})$7 implying

$\mu, \nu \in \mathcal{P}(\mathcal{X})$8

provided all reference kernels and marginals are strictly positive.

In the continuous setting, with reference measure $\mu, \nu \in \mathcal{P}(\mathcal{X})$9 (Langevin dynamics) and marginals $t_0=0$0 on $t_0=0$1,

• For strongly log-concave marginals and a reference bridge having sufficient regularity and convexity, the rate is $t_0=0$2 with explicit dependence on convexity and time horizon, and exponential contraction in KL divergence (Silveri et al., 23 Oct 2025).
• For weakly log-concave marginals, a more involved rate $t_0=0$3 is given, but still guarantees exponential decay (Silveri et al., 23 Oct 2025).

These results crucially rely on a new contraction inequality for the Markovian projection operator in KL, coupled with orthogonality of constraint directions in the function space.

4. Algorithmic Structure and Practical Implementation

IPMF (finite-state or continuous) is implementable by two-phase iterations, each a KL projection onto an affine constraint set:

Discrete-time pseudocode (finite-state case) (Sokolov et al., 4 Aug 2025):

$t_0=0$4

Path-space (continuous) pseudocode (Silveri et al., 23 Oct 2025):

$t_0=0$5

In high-dimensional or continuous cases, drifts can be parameterized by neural networks (as in DSBM), and projections approximated via regression losses (Silveri et al., 23 Oct 2025).

5. Applications and Empirical Properties

IPMF provides a robust paradigm for Schrödinger bridge inference in settings ranging from discrete tables to high-dimensional generative modeling, unifying entropic optimal transport, measure-valued diffusion matching, and graphical model estimation.

Key properties:

• Closed-form or efficiently computable updates in discrete and graphical-chain settings (Wiegerinck et al., 2012, Sokolov et al., 4 Aug 2025).
• Monotonic KL decrease, requiring no tuning of step size or regularization (Wiegerinck et al., 2012, Sokolov et al., 4 Aug 2025).
• Empirical stability and reliable marginal matching in unpaired image-to-image translation and other generative tasks, with a tunable trade-off between content preservation and generation quality by adjusting the number of IPMF steps (Kholkin et al., 2024).

In image translation, as the number of IPMF projections increases, FID improves but MSE degrades, thereby interpolating between stylization fidelity and photorealism—a property absent in single-projection (IPF only or IMF only) methods (Kholkin et al., 2024).

6. Theoretical Significance and Interpretability

IPMF can be characterized as a block-coordinate descent on the KL divergence over convex sets defined by the imposed constraints, with each block being a Bregman projection. The geometric proof for exponential contraction leverages the strong convexity of the functional and orthogonality between affine constraint directions (Sokolov et al., 4 Aug 2025).

In the continuous path-space case, Markovian projection contraction is established using Talagrand transport inequalities, Girsanov's pathwise KL formula, and Lipschitz properties of the reference bridge. This underlies the rigorous exponential convergence theorems and closes the gap between practical generative matching and entropic optimal transport theory (Silveri et al., 23 Oct 2025).

7. Extensions, Limitations, and Future Directions

IPMF generalizes Sinkhorn’s IPF to Markov and path-space constraints, with extension to chain factor graphs and nonparametric path inference (Wiegerinck et al., 2012, Kholkin et al., 2024). The explicit rate theorems clarify circumstances in which slow convergence or instability may arise, for instance, when reference kernels or marginals are poorly conditioned.

Practical implementation in high-dimensional, deep-learning frameworks (e.g., DSBM) depends on the ability to fit drifts accurately. While the theory guarantees contraction if regression errors are controlled, the quantitative effect of inexact projection or limited sample budgets remains a topic for further research (Silveri et al., 23 Oct 2025).

Empirical performance demonstrates that bidirectional alternation of forward and backward projections, as prescribed by IPMF, can mitigate drift error accumulation and stabilize training in adversarial and diffusion models (Kholkin et al., 2024).

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References

• (Sokolov et al., 4 Aug 2025) Exponential convergence rate for Iterative Markovian Fitting
• (Silveri et al., 23 Oct 2025) Exponential Convergence Guarantees for Iterative Markovian Fitting
• (Kholkin et al., 2024) Diffusion & Adversarial Schrödinger Bridges via Iterative Proportional Markovian Fitting
• (Wiegerinck et al., 2012) IPF for Discrete Chain Factor Graphs
• (Brossard et al., 2016) Iterated proportional fitting procedure and infinite products of stochastic matrices