Debiasing optimal transport: classical and entropic

Introduction to the show: ident: Mathematics. The day in mathematics: new theorems, new proofs, and what they open.

Ada: We're looking at the paper: "Debiasing optimal transport: classical and entropic".

Emmy: Here's the gist of the paper. We study the notion of debiasability for cost functions arising in optimal transport, focusing on entropic regularization across different regimes to establish sufficient conditions for debiasability The main result shows that a debiasable ground cost lifts to a debiasable cost on measures under specific conditions Characterization of Debiasable Costs The paper introduces the concept of debiability for a symmetric cost function c, defined by the inequality c(x, y) ⩾ 1/2c(x, x)+ 1/2c(y, y) This is characterized by an inf-representation c(x, y) = infz∈Z ψ(x, z) + ψ(y, z), which is equivalent to-c being tropically positive semidefinite The authors detail various examples of debiasable costs and their corresponding inf-representations, such as the squared Euclidean distance.

Ada: They also establish operations that preserve debiasability, including sums and infima over parameters Debiasing Optimal Transport for Epsilon = 0 For the classical optimal transport case where epsilon = 0, the paper provides a direct proof of debiability without relying on an inf-representation This proof shows that if c is debiable, then OTc is debiable Furthermore, it establishes that OTc(strictly) debiasable implies c is strictly debiasable Debiasing Entropic (U)OT for Epsilon in (0, +∞) When dealing with entropic optimal transport where epsilon ∈ (0,+∞), the paper investigates sufficient conditions such as negative definiteness of the ground cost or continuity and positive definiteness of the induced kernel The authors show that for a debiasable ground cost c, OTϵc is debiable provided certain conditions hold, including when c behaves like a squared Euclidean distance or when c is continuous and negative definite They construct novel barycentric formulas for entropic optimal transport involving an intermediate measure ηt and a potential z∗t Entropic Interpolation and Geometric Paths The paper extends the inf-representations to include time parameter t ∈ (0, 1) to construct canonical entropic interpolation curves This leads to an intermediate measure ηt that coincides with the marginals of the Schrödinger bridge between µ and ν The resulting curve ηt provides a purely static, variational, barycentric formulation of the dynamical Schrödinger bridge This construction demonstrates that entropic interpolation induces fundamentally distinct geometric paths across different spaces Debiasing Entropic OT for Epsilon = +∞ For the limit case epsilon = +∞, the paper shows that c is negative definite if and only if OT∞c is debiable for all relevant measures In this regime, OT∞c(µ,ν) can be written as a sum of potentials over the Hilbert space Hc The strict debiability of S∞c is equivalent to the kernel k(x, y) being integrally strictly positive definite on M+(X) Barycentric Formulation for Entropic Interpolation The authors derive a generalized barycentric formulation for entropic interpolation, showing that the optimal intermediate potential z∗t is the convex combination of the optimally tilted measures in the flat geometry of Hk This geometric path in Hk contrasts with the primal interpolant ηt which defines a flow of probability measures over X Where the Gains Come From The gains in understanding arise from establishing an inf-representation characterization for debiable costs This representation allows for the decomposition of optimal transport problems into simpler variational forms, such as those involving OTψ(µ,η) + OTψ(ν,η) The explicit computation for the squared Euclidean cost demonstrates how these representations recover known results like the midpoint formula Related Works The paper references several related works covering various aspects of optimal transport and kernel methods, including

FSV+19: for Sinkhorn divergences and

LLM+25: for the Riemannian geometry of Sinkhorn divergences The work connects concepts from optimal transport to reproducing kernels and minimax theorems in compact convex-nonconcave settings It also relates entropic interpolation to the Schrödinger bridge problem The paper concludes by showing that the two perspectives—the flow of probability measures and the straight segment in Hk—provide distinct geometric paths Appendix A: Proof of Theorem 2.1 The appendix provides a detailed proof for the characterization of debiable costs through an inf-representation This proof shows that the required lower semicontinuity holds under appropriate conditions on c Appendix B: Useful Lemmas The appendix contains useful lemmas, such as Lemma B.2 for commuting infimum and integral, which is crucial for deriving the chain rule decomposition of KL divergence This lemma is used to show that the entropic interpolation coincides with the Schrödinger bridge The proof of Lemma B.3 involves complex gluing constructions of transport plans The proof of Theorem 4.6 relies on the chain rule for entropy and the structure of the inf-representation The final argument in Theorem 4.

Paper summary: Ada: So we're looking at this paper called "Debiasing optimal transport: classical and entropic".

Emmy: It looks like they're diving into whether certain cost functions used in optimal transport have a property called debiasability.

Ada: They define debiasable costs as symmetric functions where the cost between two points is greater than or equal to half the cost of going from a point to itself plus half the cost from another point to itself, which sounds like it's about some kind of convexity or structure on the costs themselves.

Emmy: Exactly.

Ada: And they characterize this debiasability by saying it has an inf-representation, meaning you can write that cost between any two points as an infimum over some set Z and some function psi, which is basically a generalization of how we define distances in geometry Ada It connects this idea to something called tropically positive semidefinite for the negative of the cost function.

Kolya: So what they claim is that this structure, this debiasability, lifts from just the ground cost function c itself to the actual optimal transport problem on measures under specific conditions Emmy They are looking at three regimes: classical optimal transport where epsilon is zero, entropic optimal transport with epsilon greater than zero, and then they look at the maximum mean discrepancy case where epsilon is infinity Ada

Ada: That's right. For the classical case, epsilon equals zero, they provide a direct proof that if the ground cost c is debiable, then the optimal transport problem OTc on measures is also debiable Kolya They even show that if you have strictly debiasable optimal transport, then the original ground cost c has to be strictly debiasable Emmy

Kolya: That's interesting because it means we don't just have a property of the raw numbers; it persists when you move to the actual transport plans between probability measures Ada But then they get into entropic optimal transport where epsilon is in the range (zero, plus infinity) Emmy Here, they look for sufficient conditions like whether the ground cost c is negative definite or if some induced kernel has certain properties Ada

Paper summary: Emmy: They construct these new barycentric formulas for entropic optimal transport using an intermediate measure ηt and a potential z star t as the regularization parameter t goes from zero to one Kolya This construction links the static, variational formulation of the Schrödinger bridge problem to the dynamics of probability measures Ada

Ada: So, what they're showing here is that entropic interpolation creates these canonical curves ηt which act like a flow of probability measures over the space X Emmy It suggests that this process induces fundamentally different geometric paths compared to just looking at the static setup Kolya

Ada: And then there's the limit case when epsilon is infinity, which they call OT infinity c where it relates to the maximum mean discrepancy Emmy In this specific regime, they find a strong condition: if c is negative definite, then OT infinity c on any measures will be debiasable Kolya

Ada: The paper makes sense that the core idea is using this inf-representation to break down these complex optimal transport problems into simpler variational forms Emmy They say this representation lets you decompose things into sums of terms like OT psi over µeta plus OT psi over nueta Kolya And they show how this decomposition helps compute things like the midpoint formula for the squared Euclidean cost Ada

Emmy: It’s powerful because it moves us from just solving one big problem to solving smaller ones based on that representation, which is what allows them to recover known results when you test it on something concrete like squared Euclidean distance Kolya But they admit that this isn't true in general for all examples, which is a fair caveat Ada

Ada: So, looking at the whole thing, the paper "Debiasing optimal transport: classical and entropic" seems to be establishing the necessary conditions for when debiasability holds across these different types of regularized optimal transport problems Emmy It’s important because it gives us a framework to understand when we can expect certain structural properties in how probability measures are transported Kolya

Paper summary: Emmy: The authors explore the connection between entropic interpolation, which is this smooth path ηt, and the geometric paths found in the Hilbert space Hc, showing they are actually very different ways of describing movement Ada It’s a subtle point that connects analysis with geometry Kolya

Kolya: I just wonder about how robust these conditions are when you move away from the specific examples like squared Euclidean distance Ada The paper shows how these representations work for those cases, but does it hold up if the cost function is more complicated than a simple power of a distance Emmy

Ada: The authors do establish operations that preserve debiasability, like sums and infima over parameters, which gives us some tools to check if a new cost inherits this property Kolya It’s about finding the right algebraic structure to maintain the debiasability when you change things up Emmy

Ada: So, in conclusion for this paper on "Debiasing optimal transport: classical and entropic", we have a characterization of what makes a cost function debiasable through that inf-representation Emmy It covers classical, entropic, and the maximum mean discrepancy cases with different conditions for each Kolya The authors show that having a debiasable ground cost c doesn't automatically guarantee it for the transport problem on measures, but they provide specific sufficient conditions like negative definiteness when epsilon is high Ada

Emmy: The main implication is that we now have a way to decompose these problems using these representations, which helps us compute things like the midpoint formula and gives us a geometric path through entropic interpolation Kolya It’s about connecting the flow of probability measures to paths in a Hilbert space Ada

Ada: They show how these two perspectives, the flow and the straight segment in Hc, give distinct geometric paths for entropic interpolation Emmy It's a different way of thinking about how things move between distributions Kolya

Conclusion: Ada: So we've seen how these cost functions can be debiased, moving from classical optimal transport to entropic versions.

Emmy: And the core of this work is showing exactly what conditions make that debiasing happen in those different regimes.

Ada: The title itself, "Debiasing optimal transport: classical and entropic," tells you they're covering two major areas.

Emmy: Right. It's about looking at how these measures move between each other under different rules—classical versus entropic—and finding the necessary ingredients for that movement to be structured in a certain way.

Ada: The authors use an inf-representation, which is this way of writing the cost between two points as a minimum over some set and some function.

Kolya: So you're saying that if you can write it like that, then the transport problem actually has this debiasing property, regardless of what the specific measures are?

Emmy: Not exactly. It's not just about writing it down; they show how that representation lets them break down those big optimal transport problems into simpler pieces.

Ada: They link this to geometric paths in a Hilbert space, which is this abstract mathematical space where things are represented by functions or vectors.

Kolya: That Hilbert space thing, does it actually help us understand the actual flow of probability measures on our physical systems?

Emmy: It shows two different ways to look at it. you have the flow of measures itself, and you have these straight segments in that Hilbert space.

Ada: And they point out that these two paths are fundamentally different, which is a key result for understanding the geometry involved.

Kolya: So what does this mean for applying it? Is this something we can use when we're dealing with more complex, non-smooth systems?