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arxiv: 2604.06078 · v1 · submitted 2026-04-07 · 🧮 math.OC · cs.SY· eess.SY

A proximal approach to the Schr\"odinger bridge problem with incomplete information and application to contamination tracking in water networks

Michele Mascherpa , Victor Moln\"o , Carsten Skovmose Kalles{\o}e , Johan Karlsson This is my paper

Pith reviewed 2026-05-10 18:49 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY
keywords Schrödinger bridgepartial marginalsentropic proximal schemewater distribution networkscontamination trackingoptimal transportdualityobservability
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The pith

An entropic proximal scheme solves the discrete Schrödinger bridge problem from partial marginal observations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper addresses the discrete Schrödinger bridge problem when only incomplete observations of the starting and ending distributions are available. Because the resulting optimization lacks strict convexity, classical Sinkhorn iterations do not apply directly. The authors therefore introduce an entropic proximal algorithm that remains computationally scalable, together with duality results that characterize the optimal plans and an observability condition that identifies when the solution is unique. The method is demonstrated on the task of reconstructing contamination spread inside a water distribution network from sensor readings taken only at selected nodes.

Core claim

The central claim is that the Schrödinger bridge problem with partial marginals admits a dual formulation whose solutions can be recovered by an entropic proximal scheme; this scheme converges to the optimal transport plan, and an observability condition on the measured locations determines uniqueness of that plan.

What carries the argument

The entropic proximal scheme, which alternates proximal steps with entropy regularization to enforce the observed marginal constraints while keeping the iterates feasible.

If this is right

  • The duality framework supplies a certificate for optimality that does not require recovering the full primal plan.
  • Uniqueness holds precisely when the sensor placement satisfies the stated observability condition.
  • The same proximal iteration can be used for any linear marginal constraints that leave the problem convex but not strictly convex.
  • Sensor data alone suffice to estimate the full space-time contamination field inside the network.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same proximal construction could be applied to other optimal-transport problems that become non-strictly convex once some marginals are replaced by linear measurements.
  • In network settings the observability condition gives a concrete rule for placing the minimal number of sensors needed to guarantee a unique reconstruction.
  • Because the method is iterative and matrix-free, it extends naturally to time-varying or large-scale networks without storing the full transport plan.

Load-bearing premise

The proximal iterations still converge to a meaningful optimum even though the problem is not strictly convex.

What would settle it

Apply the algorithm to the laboratory water network with a known injected contaminant at a chosen node and check whether the reconstructed concentrations at unmonitored nodes match the laboratory measurements within experimental error.

Figures

Figures reproduced from arXiv: 2604.06078 by Carsten Skovmose Kalles{\o}e, Johan Karlsson, Michele Mascherpa, Victor Moln\"o.

Figure 1
Figure 1. Figure 1: Illustration of water network modeling. Each pipe at [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Experimental setup at SWIL. A pipe unite (left), [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: Concentration of salt over time in the reconstructed solution for the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison between measured (solid) and reconstructed (dashed) pollutant mass signals for the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Water network for the contaminated pipe scenario. Water flows according to the arrows, from the tanks to the consumers. The widths of the segments are scaled to the pipe diameters, but their lengths do not correspond to the actual pipe lengths. place, with a quantitative mismatch in (J4, J3). The method indeed expects a uniform mixing in node J4, suggesting a signal closer to (J4, J5) then what it actually… view at source ↗
Figure 7
Figure 7. Figure 7: Concentration of salt over time in the reconstructed solution for the [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison between measured (solid) and reconstructed (dashed) pollutant mass signals for the [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
read the original abstract

In this work, we study a discrete Schr\"odinger bridge problem with partial marginal observations. A main difficulty compared to the classical Schr\"odinger bridge formulation is that our problem is not strictly convex and standard Sinkhorn-type methods cannot be directly applied. To address this issue, we propose a scalable computational method based on an entropic proximal scheme. Furthermore, we develop a framework for this problem that includes duality results, characterization of the optimal solutions, and an observability condition that determines when the optimal solution is unique. We validate the method on the problem of estimating contamination in a water distribution network, where the partial marginals correspond to measured pollutant concentrations at the sensor locations. The experiments were conducted on a laboratory-scale water distribution network.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies the discrete Schrödinger bridge problem with partial marginal observations, which lacks strict convexity and thus precludes direct application of Sinkhorn-type algorithms. It introduces an entropic proximal scheme as a scalable solver, develops duality results together with a characterization of optimal solutions, and states an observability condition that guarantees uniqueness. The approach is validated on contamination tracking in a laboratory-scale water distribution network, where partial marginals are supplied by sensor measurements of pollutant concentrations.

Significance. If the duality framework and convergence properties are established, the work supplies a theoretically grounded method for a practically relevant class of optimal-transport problems with incomplete data. The observability condition and the water-network application illustrate how the theory translates to network estimation tasks; the laboratory experiments provide concrete empirical support for the computational scheme.

major comments (2)
  1. [§3] §3 (entropic proximal scheme) and the convergence statement following the duality results: because the problem is explicitly not strictly convex, the proof that the proximal iterates converge to a minimizer (and, when the observability condition holds, to the unique minimizer) must be supplied in full. The current sketch does not address the case of multiple optima or the interaction between the proximal parameter and the observability matrix; without this, the claim that the scheme “recovers the optimal solution” remains unsubstantiated.
  2. [§4] Theorem on uniqueness via the observability condition (likely §4): the condition is stated to determine uniqueness, yet it is not shown that the proximal scheme is guaranteed to select the unique solution when the condition holds, nor that it remains stable when the condition fails. This link is load-bearing for both the theoretical framework and the water-network experiments.
minor comments (2)
  1. [Notation and §2] Notation for the partial marginals and the observability matrix should be introduced once and used consistently; several symbols appear to be redefined between the duality section and the algorithm description.
  2. [Experiments] The experimental section would benefit from a table reporting iteration counts, proximal-parameter sensitivity, and a direct comparison against a baseline (e.g., projected gradient or ADMM) on the same network instances.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive major comments. We agree that the convergence analysis and its connection to uniqueness require fuller treatment and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3 (entropic proximal scheme) and the convergence statement following the duality results: because the problem is explicitly not strictly convex, the proof that the proximal iterates converge to a minimizer (and, when the observability condition holds, to the unique minimizer) must be supplied in full. The current sketch does not address the case of multiple optima or the interaction between the proximal parameter and the observability matrix; without this, the claim that the scheme “recovers the optimal solution” remains unsubstantiated.

    Authors: We agree that a complete proof is needed. The manuscript currently contains only a sketch relying on standard proximal-point convergence results for convex problems. In the revision we will supply the full argument in an expanded §3 (or dedicated appendix). The proof will show that the iterates converge to a minimizer of the original problem for any positive proximal parameter, explicitly handling the non-strictly-convex case by working with the subdifferential and the duality framework already developed. When the observability condition holds, uniqueness of the limit follows directly from the characterization of optimal solutions; we will add the corresponding corollary. The role of the proximal parameter will be clarified: convergence holds independently of its specific positive value, while its magnitude affects only the speed and numerical conditioning. revision: yes

  2. Referee: [§4] Theorem on uniqueness via the observability condition (likely §4): the condition is stated to determine uniqueness, yet it is not shown that the proximal scheme is guaranteed to select the unique solution when the condition holds, nor that it remains stable when the condition fails. This link is load-bearing for both the theoretical framework and the water-network experiments.

    Authors: We acknowledge that the algorithmic implication of the uniqueness theorem must be stated explicitly. In the revised manuscript we will insert a corollary in §4 establishing that, whenever the observability condition is satisfied, every limit point of the proximal sequence is the unique optimal solution. When the condition fails we will prove that the sequence still converges to some minimizer (not necessarily unique) and add a short stability discussion. For the water-network application we will include additional numerical diagnostics (e.g., variation of recovered contamination maps over different proximal-parameter choices and sensor configurations) to illustrate practical robustness even in the non-unique regime. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation is self-contained via new proximal scheme and internal duality/observability results

full rationale

The paper introduces an entropic proximal scheme to handle the non-strictly-convex discrete Schrödinger bridge problem with partial observations, then derives duality results, optimal-solution characterizations, and an observability condition for uniqueness entirely within the present framework. No load-bearing step reduces to a fitted input renamed as prediction, a self-citation chain, or an ansatz smuggled from prior author work; the water-network experiments serve only as validation and do not substitute for or circularly define the mathematical claims. The derivation therefore stands on its own stated assumptions and constructions without reducing to its inputs by definition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the convergence properties of the entropic proximal scheme for a non-strictly convex problem and on the validity of the observability condition for uniqueness; these are not derived from first principles in the abstract but assumed to hold for the discrete setting and the water-network application.

axioms (2)
  • domain assumption Entropic regularization yields a well-behaved proximal mapping even when the original problem is not strictly convex
    Invoked to justify replacing standard Sinkhorn methods with the proposed proximal scheme.
  • domain assumption The observability condition on sensor locations determines uniqueness of the optimal solution
    Stated as part of the developed framework without further derivation in the abstract.

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