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Where the rigor comes from

This page is not here for academic vanity. When someone commissions a model they will have to defend to a committee, a regulator or a court, it is reasonable to want to know how deep the foundation goes under the people building it. Here it is, with the equations, and with links to the original documents for anyone who wants to check.

Doctoral thesis · Founder of JoaFin

Four problems in optimal transportation

Degree
PhD in Mathematics · University of Toronto · 2024
Advisor
Advisor: Robert J. McCann

Optimal transport studies how to move one distribution of mass onto another at least possible cost. It is a theory living at the intersection of analysis, geometry and probability, and it turns out to be the natural language for a surprising number of applied problems: volatility surface calibration, resource allocation, matching in markets, and domain adaptation in machine learning. The thesis attacks four distinct problems with that tool.

inf⁡π∈Π(μ,ν) ∫X×Yc(x,y) dπ(x,y)\inf_{\pi \in \Pi(\mu,\nu)}\ \int_{X \times Y} c(x,y)\,d\pi(x,y)
dp(μ,ν)=(inf⁡π∈Π(μ,ν)∫M×Md(x,y)p dπ(x,y))1/pd_p(\mu,\nu) = \left( \inf_{\pi \in \Pi(\mu,\nu)} \int_{M \times M} d(x,y)^p \, d\pi(x,y) \right)^{1/p}
ℓ(μ0,μ1)q=sup⁡{∫01 ⁣ ⁣∫M1q(gab(x) vtavtb)q/2dμt(x) dt},∂tμt+∇M ⁣⋅(vtμt)=0\ell(\mu_0,\mu_1)^q = \sup \left\{ \int_0^1\!\!\int_M \frac{1}{q}\bigl( g_{ab}(x)\, v_t^a v_t^b \bigr)^{q/2} d\mu_t(x)\,dt \right\}, \qquad \partial_t \mu_t + \nabla_M \!\cdot (v_t \mu_t) = 0
Read the thesis (PDF)
Institution
University of Toronto
Year
2024
Advisor
Robert J. McCann
Problems
4, across 4 distinct fields
Papers
2, single-authored

The four problems

In plain language, with the reason each one matters outside academia.

01

The non-smooth relativistic Schrödinger problem

The Schrödinger problem is the entropically regularized version of optimal transport: instead of a deterministic map, one looks for the law of the Brownian bridge interpolating two distributions. The thesis extends the formulation to synthetic Lorentzian geometries, where no heat kernel exists and bridges cease to be Markovian. It constructs Lévy-like processes emulating Brownian bridges and recovers a partial version of entropic convergence via large deviation principles.

Ent⁡(ν∣μ)=∫Mh ⁣(dνdμ)dμ,h(x)=xlog⁡x−x+1\operatorname{Ent}(\nu \mid \mu) = \int_M h\!\left( \frac{d\nu}{d\mu} \right) d\mu, \qquad h(x) = x\log x - x + 1
min⁡π ∈ Γ⪯(μ0,μ1)Ent⁡(π∣r)\min_{\pi \,\in\, \Gamma_{\preceq}(\mu_0,\mu_1)} \operatorname{Ent}(\pi \mid r)
Γ⪯(μ0,μ1)={π∈P(M2)  :  π(M≤2)=1,  Proj1#π=μ0,  Proj2#π=μ1}\Gamma_{\preceq}(\mu_0,\mu_1) = \bigl\{ \pi \in \mathcal{P}(M^2) \;:\; \pi\bigl(M^2_{\leq}\bigr) = 1, \ \ \mathrm{Proj}_{1\#}\pi = \mu_0, \ \ \mathrm{Proj}_{2\#}\pi = \mu_1 \bigr\}

Ent is relative entropy, r the reference measure, and Γ⪯(μ₀,μ₁) the set of causal transport plans: the condition π(M²≤) = 1 requires all mass to move into the causal future. That is what makes the problem physical, and what breaks the classical theory.

Entropic regularization · Chapter 1 of the thesis

-2-1012-2-1012xyT
  • ε = 0.5
  • ε = 0.12
  • ε = 0.03
  • T (ε → 0 limit)
Each curve is the barycentric projection of the entropic plan, solved by Sinkhorn iteration for a different ε. As ε decreases the curves converge to the Monge map T. The marginal constraints hold to machine precision, so the convergence you see is the real thing rather than an illustration.
02

The aggregation equation via minimizing movements

The thesis proves small-time existence of measure-valued solutions to the aggregation equation on compact Riemannian manifolds for non-regular interaction potentials, using the minimizing movement (JKO) scheme. The technical difficulty is non-differentiability of the potential on the cut locus, resolved through the propagation properties of geodesic interpolations.

∂tμt−∇M ⁣⋅(μt ∇M(W∗μt))=0,W(x,y)=h(d(x,y)2)\partial_t \mu_t - \nabla_M \!\cdot \bigl( \mu_t \, \nabla_M (W * \mu_t) \bigr) = 0, \qquad W(x,y) = h\bigl(d(x,y)^2\bigr)
μk+1τ∈arg min⁡ρ ∈ P(M){EW(ρ)+12τ d2 2(ρ,μkτ)}\mu^{\tau}_{k+1} \in \operatorname*{arg\,min}_{\rho \,\in\, \mathcal{P}(M)} \left\{ E_W(\rho) + \frac{1}{2\tau}\, d_2^{\,2}\bigl(\rho, \mu^{\tau}_k\bigr) \right\}
0<T<δμ2L,δμ=inf⁡x,y∈spt(μ)(x′,y′)∈Cut{d(x,x′)+d(y,y′)}0 < T < \frac{\delta_\mu}{2L}, \qquad \delta_\mu = \inf_{\substack{x,y \,\in\, \mathrm{spt}(\mu) \\ (x',y') \,\in\, \mathrm{Cut}}} \bigl\{ d(x,x') + d(y,y') \bigr\}

W is the interaction potential, depending only on Riemannian distance; τ is the time step; d₂ the Wasserstein-2 metric; δ_μ the distance from the support to the cut locus and L the Lipschitz constant of W. The bound on T is explicit — that is an error bound, not a footnote.

Gradient flow · Chapter 2 of the thesis

0246-2-1012tposition
Lagrangian form of the same equation: each particle follows the velocity field v = −∇(W ∗ μ), with W(x,y) = h(d(x,y)²) and h(r) = r/(1+r), integrated by fourth-order Runge–Kutta. Attraction is local, so the population concentrates into clusters rather than a single point. The energy E_W decreases along the flow, which is the proof that this really is the gradient flow it claims to be.
03

Measure pre-conditioning in machine learning

What happens to a learning model when the data it trains on is slightly modified? The chapter shows that if the modification is done in a particular way there is Γ-convergence to the original model — via a two-sided analogue of Fatou's lemma — with applications to domain adaptation in transfer learning and to Wasserstein generative adversarial networks.

arg min⁡f∈C Eπ[L(f(X),Y)]vs.arg min⁡f∈C Eπn[L(f(X),Y)]\operatorname*{arg\,min}_{f \in \mathcal{C}} \ \mathbf{E}_{\pi}\bigl[ L(f(X), Y) \bigr] \qquad \text{vs.} \qquad \operatorname*{arg\,min}_{f \in \mathcal{C}} \ \mathbf{E}_{\pi_n}\bigl[ L(f(X), Y) \bigr]
lim inf⁡n→∞Eπn[L(fn(X),Y)] ≥ Eπ[L(f(X),Y)] ≥ lim sup⁡j→∞Eπj[L(fj(X),Y)]\liminf_{n \to \infty} \mathbf{E}_{\pi_n}\bigl[ L(f_n(X), Y) \bigr] \ \geq \ \mathbf{E}_{\pi}\bigl[ L(f(X), Y) \bigr] \ \geq \ \limsup_{j \to \infty} \mathbf{E}_{\pi_j}\bigl[ L(f_j(X), Y) \bigr]

πₙ is the measure associated with the sample and π the true one; L the loss and f the model. The two inequalities are a two-sided Fatou lemma, and they are exactly the condition needed to conclude that minimizers converge.

04

A generalization of Roy's model

Roy's model describes how a labor force partitions across occupations. The thesis generalizes Siow's version by treating occupational choice as a constraint rather than a consequence, which lets the problem be rewritten in an analytically useful form and removes the assumption that the separating function is linear. The nonlinear model is more realistic and still admits economically meaningful conclusions.

sup⁡(ϕ, π, w, μ) ∈ C{∫F(k,μ(k)) dHϕ}\sup_{(\phi,\,\pi,\,w,\,\mu) \,\in\, \mathcal{C}} \left\{ \int F\bigl(k, \mu(k)\bigr) \, dH^{\phi} \right\}
F(c,d)+F(a,b)−F(a,d)−F(c,b)>0whenever a<c, b<dF(c,d) + F(a,b) - F(a,d) - F(c,b) > 0 \qquad \text{whenever } a < c, \ b < d
∫R ⁣ ⁣∫−∞ϕ(u)R(u,v) dv du=12\int_{\mathbb{R}} \!\! \int_{-\infty}^{\phi(u)} R(u,v) \, dv \, du = \frac{1}{2}

F is the production function, strictly supermodular; π and w the wage schedules for each role; φ = w⁻¹ ∘ π the separating function; R the skill density and H^φ the occupational distribution φ induces. The last condition says the separation splits the population into halves.

Labor partition · Chapter 4 of the thesis

12341234kskey rolesecondary roleφ
  • nonlinear φ (this model)
  • linear separation (earlier models)
Each dot is a worker with a pair of skills (k, s). The separating function φ marks indifference between the two roles, and in equilibrium it cuts the population exactly in half. Both curves satisfy that condition — each was placed by bisection until half the mass sat below it — and they still partition people differently. That difference is what gets lost by assuming the separation is linear.

Publications

Both papers are single-authored, written during the doctorate.

How this reaches client work

The connection is not metaphorical. Arbitrage-free local volatility calibration is posed as an optimal transport problem, and it is day-to-day work on the team inside a derivatives analytics platform. Resource allocation and matching in markets are transport problems. Domain adaptation in machine learning is a transport problem. Nodal price forecasting, by contrast, is not — which is why other tools are used there. The judgment about which to use is exactly what this kind of training gives you.

Worth saying plainly: optimal transport is one tool in the workshop, not the workshop. The machine learning chapter is the root of how we treat statistics and predictive models; the Roy chapter, of how we treat economic and identification problems. Most consulting work is settled with statistics, time series, extreme values and optimization — not with optimal transport.

What research leaves behind is not a list of techniques but a habit: insist the problem is well posed before trying to solve it, and know precisely what can and cannot be asserted about a solution. That is what turns into models that hold up under scrutiny.

inf⁡T#μ = ν ∫Xc(x, T(x)) dμ(x)\inf_{T_\#\mu \,=\, \nu}\ \int_X c\bigl(x,\,T(x)\bigr)\,d\mu(x)

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