Optimal transport for operations research, risk & planning: classical methods

Every claim on this page comes from the lab's sourced research notes (checked 2026-10-05).

Purpose. Sourced notes for a Research Lab learning page: classical statistics and machine-learning methods built on Wasserstein spaces / optimal transport (OT), for operations research, risk, and business planning (forecasting, budgeting, capacity, inventory, credit/market risk, scenario planning, challenger models).

House rules followed.

• No neural networks or large pretrained models in the core toolkit. Neural mentions are labelled EXCEPTION (two, both minor and excluded; see §13).

• Papers from 2020+ preferred; older work is tagged background.

• Every paper below was checked against arXiv (API metadata + abstract), Crossref/DOI, or the publisher/CRAN page on 2026-10-05. Summaries come from the abstract or full text actually read. Anything not verified is tagged UNVERIFIED or listed in §16.

• Works are cited by title, venue, year, link only — no researcher names, no personal framing.

§0. One-screen brief

Why OT matters for planners. Planning and risk work already lives in distributions (demand scenarios, P&L distributions, forecast fans, return densities). OT gives three practical things:

1. A geometry for distributions that respects "how far mass must move" — a forecast off by 5 units is closer than one off by 50, unlike KL or total variation, which saturate once supports stop overlapping.

2. A robustness language (Wasserstein distributionally robust optimization, DRO): optimize against every distribution within distance $\varepsilon$ of the data. For many problems this becomes a convex program — often a regularized version of what you already run.

3. Cheap 1-D closed forms: in one dimension OT is sorting/quantiles. $W_2$ averaging = quantile averaging; distribution regression = regression on quantile functions. Most planning variables are 1-D or low-D.

Nine families (rating = potential for OR/risk/planning):

§10 covers the tools and statistical rates underneath everything; §11.5 is the Small-N toolkit (N<100); §11 is a computed worked example (including N=10/30); §12 ranks five project ideas.

§0.5 Core objects (used throughout)

Wasserstein distance. For $\mu,\nu$ on a metric space with finite $p$-th moments,

where $\Pi(\mu,\nu)$ is the set of couplings (joint laws with marginals $\mu,\nu$).

One dimension (the planner's workhorse). With quantile functions $F^{-1},G^{-1}$:

Duality ($p=1$, Kantorovich–Rubinstein). $W_1(\mu,\nu)=\sup_{\mathrm{Lip}(f)\le1}\big(\mathbb E_\mu f-\mathbb E_\nu f\big)$ — the reason Wasserstein DRO behaves like Lipschitz regularization.

Entropic OT / Sinkhorn background. Add $\varepsilon\,\mathrm{KL}(\pi\,\|\,\mu\otimes\nu)$ to the transport cost; solve by alternating scalings $u\leftarrow a/(Kv)$, $v\leftarrow b/(K^\top u)$ with $K=e^{-C/\varepsilon}$. Cost $O(n^2)$ per iteration versus $O(n^3\log n)$ for exact OT. Reference text: Computational Optimal Transport — Foundations and Trends in Machine Learning, 2019 — arXiv:1803.00567.

Barycenter. $\bar\mu=\arg\min_\nu\sum_k\lambda_k W_2^2(\nu,\mu_k)$. In 1-D: $\bar F^{-1}=\sum_k\lambda_kF_k^{-1}$ (average the quantiles).

Gaussian / Gelbrich formula. $W_2^2\big(\mathcal N(m_1,\Sigma_1),\mathcal N(m_2,\Sigma_2)\big)=\|m_1-m_2\|^2+\operatorname{tr}\!\big(\Sigma_1+\Sigma_2-2(\Sigma_2^{1/2}\Sigma_1\Sigma_2^{1/2})^{1/2}\big)$. For non-Gaussian laws with these moments, the right side (the Gelbrich distance) is a lower bound on $W_2$.

§1. Wasserstein DRO for decisions High

Core idea. You have $N$ observations $\hat\xi_1,\dots,\hat\xi_N$ of an uncertain quantity (demand, renewable output, returns). Sample average approximation (SAA) optimizes against the empirical law $\hat P_N$ and overfits when $N$ is small. Wasserstein DRO hedges against every law within $W_p$-distance $\varepsilon$:

Key dual (type-1, strong duality).

For piecewise-linear losses and polyhedral support this is a linear program (background paper below). If $\ell(x,\cdot)$ is $L$-Lipschitz and the adversary has room to move mass, the worst case is $\mathbb E_{\hat P_N}[\ell]+L\varepsilon$ — the decision does not change, only the certified cost does (our worked example reproduces this exactly). Type-2 ($W_2$) balls, or binding support bounds, are what actually move decisions.

Regularization view. For linear/logistic models, Wasserstein DRO reproduces norm-regularized estimators (square-root lasso, regularized logistic regression), with the ball radius playing the role of the penalty and giving it a statistical meaning.

Verified papers.

• background Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations — Mathematical Programming, 2018 — arXiv:1505.05116. Finite convex (often LP) reformulations + finite-sample guarantees; mean-risk portfolio example.

• background Distributionally robust stochastic optimization with Wasserstein distance — Mathematics of Operations Research (accepted, per arXiv) — arXiv:1604.02199. Strong duality in great generality; structure of worst-case distributions.

• Distributionally Robust Optimization (survey) — Acta Numerica 34 (2025) 579–804 — arXiv:2411.02549, DOI 10.1017/S0962492924000084. The reference to keep open.

• Finite-sample guarantees for Wasserstein distributionally robust optimization: breaking the curse of dimensionality — arXiv 2020 — arXiv:2009.04382. With a properly chosen radius, WDRO balances empirical loss against the loss's Lipschitz/gradient norm; guarantees free of the $N^{-1/d}$ curse.

• Wasserstein distributionally robust optimization and variation regularization — arXiv (v3 2020) — arXiv:1712.06050. General "variation regularization" theory; examples include multi-item newsvendor and portfolio selection.

• Nash equilibria, regularization and computation in optimal transport-based distributionally robust optimization — arXiv 2023 (v4 2025) — arXiv:2303.03900. Robustification ≈ variation/Lipschitz regularization even for non-metric costs; nature's equilibrium distribution sits on "deceptive adversarial samples".

• background Regularization via mass transportation — arXiv 2017 — arXiv:1710.10016; background Robust Wasserstein profile inference and applications to machine learning — Journal of Applied Probability, 2019 — arXiv:1610.05627. Regularized regression/classification as WDRO; profile inference picks the radius without cross-validation.

• Confidence regions in Wasserstein distributionally robust estimation — arXiv (v4 2021) — arXiv:1906.01614. Asymptotic normality of DRO estimators; DRO regularizes via the loss derivative.

• Distributionally robust learning (monograph) — arXiv 2021 — arXiv:2108.08993. DR linear regression, grouped predictors, multi-output, DR regression + nearest-neighbour decisions.

• Residuals-based distributionally robust optimization with covariate information — arXiv 2020 (v2 2022) — arXiv:2012.01088. Fit any regression, then put a Wasserstein (or other) ambiguity set on the residuals for the decision; size it by cross-validation. The cleanest bridge from a forecast model to a robust plan.

• Data-driven chance constrained programs over Wasserstein balls — arXiv 2018 (v3 2022) — arXiv:1809.00210. Exact mixed-integer conic (with 1-/∞-norms, MILP) reformulations.

• On distributionally robust chance constrained programs with Wasserstein distance — arXiv 2018 — arXiv:1806.07418. DR chance constraint ⇔ CVaR-constrained program; big-M and big-M-free formulations.

• A data-driven distributionally robust newsvendor model with a Wasserstein ambiguity set — Journal of the Operational Research Society 72(8), 2021 — DOI 10.1080/01605682.2020.1746203. Closed-form order and worst-case distribution for order $p\in[1,\infty)$; CVaR variant.

• Distributionally robust newsvendor with moment constraints — arXiv 2020 — arXiv:2010.16369. Wasserstein + moment constraints; auto-sales case study.

• On affine policies for Wasserstein distributionally robust unit commitment — IEEE CDC 2022 (per arXiv) — arXiv:2203.15333. Reformulation whose size does not grow with sample size; 6- and 24-bus tests.

• Outlier-robust Wasserstein DRO — NeurIPS 2023 — arXiv:2311.05573. Handles geometric perturbation and an $\varepsilon$-fraction of corrupted data (total-variation contamination).

• Sinkhorn distributionally robust optimization — arXiv 2021 (v5 2025) — arXiv:2109.11926. Entropic ball; convex dual; stochastic mirror descent.

• background Wasserstein distributionally robust Kalman filtering — NeurIPS 2018 — arXiv:1809.08830. Robust MMSE over a $W_2$ ball of Gaussians; tractable convex program; Frank–Wolfe filter. Code: github.com/sorooshafiee/WKF (MATLAB). Relevant to state-space nowcasting.

Software. POT (pythonot.github.io; POT: Python Optimal Transport — JMLR 22(78), 2021 — jmlr.org/papers/v22/20-451.html); skwdro — scikit-learn-style Wasserstein DRO estimators plus a PyTorch wrapper (skwdro: a library for Wasserstein distributionally robust machine learning — arXiv:2410.21231; PyPI skwdro; docs skwdro.readthedocs.io); CVXPY / Pyomo / python-mip / SciPy-HiGHS for the LP/SOCP reformulations (our example uses SciPy HiGHS).

Data needs. Tens to thousands of roughly i.i.d. observations (or residuals after a forecast model); a sensible ground metric (scale features before using $\|\cdot\|$); known support bounds help.

Failure modes. (i) Radius choice dominates results; cross-validation is noisy at small $N$ (seen in our example). (ii) Type-1 with slack support leaves decisions unchanged. (iii) Too large a radius → absurdly conservative decisions (example: order jumps to the support edge). (iv) Serial dependence breaks i.i.d. guarantees. (v) Outliers inflate $W$ distances — use the outlier-robust variant.

Potential: HIGH. Mature theory, LP/SOCP reformulations, libraries, and a direct mapping to safety stock, budget contingency, capacity buffers and reserve sizing. Honest caveat: in simple 1-D problems the average out-of-sample gain over SAA is often small (see §11); the main value is a defensible certificate and protection when data are scarce.

Small-N callout (N<100). Radius choice dominates — and cross-validation is actively harmful at N≤30 in our newsvendor Monte Carlo (§11: CV win rate vs SAA = 0.30 at N=10, 0.23 at N=30). Prefer: (i) type-2 balls when the decision must move; (ii) √N variation radii or RWPI/Wasserstein-projection radii instead of concentration balls in d>1; (iii) residuals-based DRO after a simple forecast. Full recipes → §11.5.

§2. Mean–covariance (Gelbrich) robust risk — the standout result High (standout)

Core idea. Practitioners trust means and covariances more than full return distributions. Put a ball in (mean, covariance) space using the Gelbrich distance (= $W_2$ between Gaussians):

Key result (Mean-covariance robust risk measurement — arXiv 2021, v3 2025 — arXiv:2112.09959; full text read). For any law-invariant, translation-invariant, positively homogeneous risk measure (VaR, CVaR, spectral, distortion, mean–std) and a "stable" structural set (all distributions, symmetric, log-concave, Gaussian…), the worst-case risk of portfolio $w$ over the Gelbrich ball of radius $\rho$ is in closed form:

• $\alpha$ is a scalar depending only on the risk measure and structure. From the paper: CVaR or VaR at tail level $\beta$ with no structure, $\alpha=\sqrt{(1-\beta)/\beta}$; CVaR under Gaussianity, $\alpha=\phi(\Phi^{-1}(1-\beta))/\beta$; mean–std with aversion $\beta$, $\alpha=\beta$.

• So robust CVaR/VaR portfolio choice = Markowitz + an explicit $\ell_2$ penalty whose weight is tied to a statistical radius (a second-order cone program).

• As $\rho\to\infty$ the optimum tends to the equal-weight portfolio.

• With sample mean/covariance of sub-Gaussian returns, $\rho=(c_1+c_2\log(1/\eta))/\sqrt N$ gives a $(1-\eta)$ confidence ball — a dimension-free rate.

• The paper gives the worst-case mean and covariance in closed form — ready-made stress scenarios for a given portfolio.

• Gelbrich risk upper-bounds the corresponding Wasserstein-ball risk, with equality for Gaussian reference laws.

Related verified papers.

• background Distributionally robust mean-variance portfolio selection with Wasserstein distances — arXiv 2018 — arXiv:1802.04885. DR Markowitz reduces to empirical variance + regularizer; data-driven radius.

• Portfolio optimisation within a Wasserstein ball — arXiv 2020 (v3 2022) — arXiv:2012.04500. Improve on a benchmark's risk profile while staying within a Wasserstein ball of its terminal wealth; isotonic projections.

Software. NumPy + CVXPY (SOCP).

Data needs. Return history long enough for a stable $\hat\Sigma$ (the paper notes any mean/covariance estimator with $O(1/\sqrt N)$ error works, e.g. shrinkage).

Failure modes. The penalty is isotropic $\|w\|_2$ (ignores factor structure unless you use the paper's Mahalanobis variant); heavy tails weaken sub-Gaussian calibration; very large $\rho$ just gives $1/N$.

Potential: HIGH. Usable on an ETF panel immediately: allocation, a radius with meaning, and stress scenarios from one formula.

Small-N callout (N<100). This is the standout small-sample portfolio tool: ρ ∼ 1/√N is dimension-free. At N≲20 prefer 1/N or heavy shrinkage (large ρ → equal weight, which is the method's own limit). At N≈30–80 with ≤10–15 assets and a shrinkage covariance, Gelbrich-robust CVaR is the default over empirical Wasserstein balls in asset dimension. Details → §11.5.

§3. Distributional regression and density time series in Wasserstein space High

Core idea. Treat a whole distribution (income distribution, cross-section of returns, intraday load profile, distribution of store sales) as the response — or even the predictor. In 1-D, $W_2$ space is isometric to a convex subset of $L^2(0,1)$ via the quantile function, so you regress or autoregress quantile functions and project back onto non-decreasing functions.

Formulas.

• Fréchet regression (global): $m_\oplus(x)=\arg\min_{\nu}\ \mathbb E\big[s(x,X)\,W_2^2(\nu,Y)\big]$ with linear-regression-type weights $s(x,X)$. In 1-D the solution is a weighted average of quantile functions, projected to be monotone.

• Wasserstein autoregression (WAR) on density time series: map $Y_t$ to log-maps / tangent vectors $T_t=F_t^{-1}\circ F_\oplus-\mathrm{id}$ around the Fréchet mean $F_\oplus$, then fit an AR model in that (Hilbert) tangent space, e.g. $T_t=\beta\,T_{t-1}+\varepsilon_t$; map forecasts back.

• Distribution-on-distribution via OT maps: $Y_t\approx (T)_\#Y_{t-1}$ with $T$ a monotone map — isotonic regression in 1-D.

Verified papers.

• Fréchet regression for random objects with Euclidean predictors — Annals of Statistics 47(2), 2019 — DOI 10.1214/17-AOS1624, arXiv:1608.03012. background

• Wasserstein regression — Journal of the American Statistical Association, 2021 — DOI 10.1080/01621459.2021.1956937, arXiv:2006.09660. Distributions as predictors and responses via tangent-space linear models.

• Wasserstein autoregressive models for density time series — Journal of Time Series Analysis 43(1), 2022 — DOI 10.1111/jtsa.12590, arXiv:2006.12640. Applications include stock-return cross-sections; WAR beats competitors on some but not all datasets — honest mixed evidence.

• Autoregressive optimal transport models — arXiv 2021 (v4; journal not confirmed) — arXiv:2105.05439. Intrinsic autoregressive models in the space of OT maps: regress transport maps (barycenter→current, or between consecutive distributions) on each other, moving along Wasserstein geodesics.

• Distribution-on-distribution regression via optimal transport maps — arXiv 2021 (Biometrika per arXiv comment) — arXiv:2104.09418. Regression map estimated by isotonic regression in 1-D.

• Short-term forecasting with optimal transport — Journal of the Royal Statistical Society Series A 189(2), 2026 — DOI 10.1093/jrsssa/qnaf036. Uses a Sinkhorn-divergence step inside a dynamic factor model with EM for mixed-frequency nowcasting of US GDP — evidence that OT ideas plug into standard nowcasting machinery without neural nets.

Software. R frechet (CRAN; global/local Fréchet regression for distributions); R WRI (CRAN; Wasserstein regression and inference, including WARp for autoregression); Python: NumPy quantile grids + scipy.optimize.isotonic_regression / scikit-learn IsotonicRegression.

Data needs. A panel of distributions: either raw samples per period (≥ ~30 per period to estimate quantiles) or histograms; tens to hundreds of periods for time series.

Failure modes. Only 1-D responses are easy; multivariate responses lose the quantile isometry. Tangent-space linearization breaks for distributions far from the mean. Short histories → overfitting. Quantile crossing must be repaired by projection.

Potential: HIGH. Direct use for forecasting entire distributions of planning variables (store-level demand cross-sections, regional growth, sector return dispersion), and a natural home for nowcasting the shape of the cross-section, not just its mean.

§4. Barycenters, forecast combination, scenario generation and reduction High

Core idea A — consensus forecasts. Linear pooling (averaging densities) inflates dispersion and can create spurious modes. The $W_2$ barycenter averages quantiles: location-scale shape is preserved and the result is unimodal when inputs are. In 1-D:

Entropic regularization of the barycenter widens it — a single knob to undo over-confidence.

Core idea B — scenario reduction. Replace $N$ scenarios by $m\ll N$ weighted ones, minimizing $W_p(\hat P_N,Q_m)$ (a $k$-means-like problem for $p=2$; $k$-medians for $p=1$), or minimizing a problem-driven distance that measures what matters for the decision.

Core idea C — multistage. Scenario trees should be close in the nested distance (OT respecting information flow — see adapted Wasserstein, §8).

Verified papers.

• Probability forecast combination via entropy regularized Wasserstein distance — Entropy 22(9):929, 2020 — DOI 10.3390/e22090929. Barycenter combination of probability forecasts; in 1-D equals quantile averaging; entropic regularization tunes dispersion; US inflation application.

• Scenario reduction revisited: fundamental limits and guarantees — Mathematical Programming, 2022 (online 2018) — DOI 10.1007/s10107-018-1269-1, arXiv:1701.04072. Worst-case approximation limits of Wasserstein scenario reduction; MILP and heuristics with guarantees.

• Optimization-based scenario reduction for data-driven two-stage stochastic optimization — Operations Research 71(4), 2023 — DOI 10.1287/opre.2022.2265. Problem-dependent distance; competitive with only ~1–2% of the original scenarios.

• Problem-driven scenario clustering in stochastic optimization — arXiv 2021 — arXiv:2106.11717. Cluster scenarios by their effect on decisions, not by raw distance.

• Entropic regularization of the nested distance — arXiv 2021 — arXiv:2107.09864. Sinkhorn-style computation for multistage scenario-tree distances.

Software. POT (ot.lp.barycenter, ot.bregman.barycenter, ot.emd2_1d); scikit-learn KMeans (as an approximate $W_2$ scenario reducer); Pyomo for the downstream stochastic program.

Data needs. Several probabilistic forecasts as quantiles or samples (combination); a scenario set plus the decision model (reduction).

Failure modes. The barycenter is narrower than the linear pool — if inputs disagree because of genuine uncertainty, that is under-dispersion (our example: P10–P90 width 54 vs 68). In our example the barycenter was closer in $W_2$ to only one of three inputs than the pool was — it is a different compromise, not uniformly "closer". Distance-based reduction ignores which scenarios drive costs (use problem-driven variants).

Potential: HIGH. Cheap, interpretable consensus for challenger-forecast panels; scenario reduction is standard OR practice with OT foundations.

Small-N callout (N<100). Quantile averaging needs no large N — each input can be a thin-history model or a judgmental fan. Scenario reduction from N≈50 scenarios down to m=5–15 (problem-driven) is exactly the scarce-data move for multistage LPs. → §11.5.

§5. OT for causal inference and policy evaluation Medium–High

Core idea. Policy effects on planning metrics are often distributional (a pricing change shifts the tail of order sizes, not just the mean). OT offers: (i) synthetic controls for whole distributions — choose weights so a weighted $W_2$ barycenter of control units' distributions matches the treated unit before treatment; (ii) OT-based matching/balancing weights; (iii) changes-in-changes / nonlinear difference-in-differences via transport maps.

Verified papers.

• Distributional synthetic controls — Econometrica 91(3), 2023 — DOI 10.3982/ECTA18260, arXiv:2001.06118.

• An optimal transport approach to estimating causal effects via nonlinear difference-in-differences — arXiv 2021 — arXiv:2108.05858. Generalizes changes-in-changes to multivariate outcomes; reanalysis of a classic minimum-wage dataset finds substitution from full- to part-time labour.

• Optimal transport weights for causal inference — arXiv 2021 — arXiv:2109.01991. Balancing weights that minimize an OT distance between treated and control covariate laws.

• Optimal transport for counterfactual estimation: a method for causal inference — arXiv 2023 — arXiv:2301.07755.

Software. R DiSCos (CRAN; distributional synthetic controls); POT for custom weights.

Data needs. Unit-level micro-data per period (e.g. order sizes per region per month); a pool of untreated units; several pre-periods.

Failure modes. Few control units → weights unstable; pre-period fit can be good by chance (use placebo/permutation inference); spillovers violate assumptions.

Potential: MEDIUM–HIGH. Valuable when evaluating planning policies (pricing, assortment, process changes) where the mean hides the story; less central than DRO for day-to-day risk.

§6. Multivariate quantiles, OT ranks, OT conformal prediction, multivariate VaR regions Medium (conformal: speculative-but-promising)

Core idea. There is no natural ordering in $\mathbb R^d$, so "the 95% quantile of a vector of P&L lines" is ill-defined. Center-outward quantiles define it via the OT map $\mathbf F_\pm$ pushing the data law to the uniform law on the unit ball $\mathbb S_d$:

Quantile regions $\mathbb Q_\pm(\tau)$ are nested, distribution-free with probability content $\tau$, and capture non-elliptical shapes — a principled multivariate VaR / stress region. Pairing OT ranks with split-conformal calibration gives multivariate prediction regions with finite-sample coverage.

Verified papers.

• Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach — Annals of Statistics 49(2), 2021 — DOI 10.1214/20-AOS1996 (preprint arXiv:1806.01238, earlier title).

• Center-outward quantiles and the measurement of multivariate risk — arXiv 2019 — arXiv:1912.04924. Smooth approximations of empirical center-outward transports, giving new empirical multivariate risk measures (via the convex potential or the volumes of quantile regions).

• Multivariate rank-based distribution-free nonparametric testing using measure transportation — arXiv 2019 (journal not confirmed) — arXiv:1909.08733.

• Optimal transport-based conformal prediction — arXiv 2025 — arXiv:2501.18991.

• Multivariate conformal prediction using optimal transport — arXiv 2025 — arXiv:2502.03609.

Software. POT (ot.emd to a uniform grid on the ball) is enough to build empirical center-outward ranks. No maintained dedicated package was verified (§16).

Data needs. Hundreds+ multivariate observations; grid sizes grow with $d$, so practical for $d\le 5$ or so.

Failure modes. Curse of dimensionality in the OT map; computational cost $O(n^3)$ exact; region estimates jagged at small $n$; 2025 conformal work is new and lightly tested.

Potential: MEDIUM (risk regions), speculative-but-promising (OT conformal bands for multi-line P&L or joint demand).

§7. Drift, change-point and regime detection Medium–High

Core idea. Monitor $W_p(\hat P_{\text{recent}},\hat P_{\text{reference}})$ for a feature, residual or return distribution. In 1-D it is a quantile comparison — cheap, interpretable in the units of the variable ("the distribution has moved by 3.2 units of mass-weighted distance"), and sensitive to shape, not just mean.

Wasserstein $k$-means clusters rolling windows of returns into regimes using $W_p$ between empirical laws as the dissimilarity and barycenters as centroids.

Verified papers.

• Monitoring for a change point in a sequence of distributions — Annals of Statistics 49(4), 2021, 2271–2291 (journal record verified; no arXiv ID found).

• Clustering market regimes using the Wasserstein distance — arXiv 2021 — arXiv:2110.11848. Wasserstein $k$-means on empirical return distributions; model-free; compared in the paper against a more traditional (moment-based) clustering approach.

• background Optimal transport based change point detection and time series segment clustering — arXiv 2019 — arXiv:1911.01325.

• Multivariate goodness-of-fit tests based on Wasserstein distance — arXiv 2020 — arXiv:2003.06684.

Software. scipy.stats.wasserstein_distance; POT ot.wasserstein_1d, ot.sliced_wasserstein_distance; R transport (CRAN 0.15-4).

Data needs. A reference window and streaming data; for regimes, rolling windows (e.g. 20–60 days) of returns.

Failure modes. Thresholds need permutation or bootstrap calibration; serial dependence inflates false alarms; high-dimensional $W$ needs slicing; regime labels can be unstable across reruns.

Potential: MEDIUM–HIGH. Directly usable as model-risk drift monitors and as a regime switch feeding allocation or DRO radius choice.

§8. Robust risk measures, stress testing, martingale OT, adapted Wasserstein Medium–High (pricing: Medium)

Core idea A — Wasserstein-robust risk. Report $\sup_{W_p(Q,\hat P)\le\varepsilon}\rho_Q(L)$ for a risk measure $\rho$ (VaR, CVaR, distortion). For distortion risk measures there are closed or near-closed forms; first-order sensitivity is informally (informal, see paper for exact conditions)

which turns stress testing into "which loss gradients matter most".

Core idea B — martingale OT (MOT). Model-free price bounds for path-dependent claims given observed option marginals: optimize over couplings that are martingales,

On a grid this is an LP.

Core idea C — adapted Wasserstein. Ordinary $W_p$ between path laws ignores information flow, so two processes can be close in $W_p$ but induce very different optimal-stopping or hedging values. Adapted (causal / nested) distances fix this — the right distance for robustness of multistage decisions.

Verified papers.

• Robust distortion risk measures — arXiv 2022 (v3; journal not confirmed) — arXiv:2205.08850.

• Sensitivity analysis of Wasserstein distributionally robust optimization problems — Proceedings of the Royal Society A, 2021 — DOI 10.1098/rspa.2021.0176, arXiv:2006.12022.

• Computational methods for martingale optimal transport problems — arXiv 2017 — arXiv:1710.07911. background

• Robust pricing and hedging of options on multiple assets and its numerics — arXiv 2019 ("forthcoming in SIAM Journal on Financial Mathematics" per arXiv comment) — arXiv:1909.03870. Multi-marginal MOT; two numerical routes — discretization + linear programming on the primal (classical, recommended here) and penalization + deep neural networks on the dual (EXCEPTION mention, not recommended).

• Adapted Wasserstein distances and stability in mathematical finance — Finance and Stochastics 24, 2020 — arXiv:1901.07450.

• Estimating processes in adapted Wasserstein distance — arXiv 2020 (journal not confirmed) — arXiv:2002.07261.

Software. POT and CVXPY/HiGHS for MOT LPs on grids; NumPy for Wasserstein-robust CVaR.

Data needs. Loss/P&L samples (robust risk); option-implied marginals (MOT); path data (adapted).

Failure modes. Bounds can be wide and uninformative; MOT grids explode with assets/periods; adapted distances are computationally heavy beyond a few periods.

Potential: MEDIUM–HIGH for Wasserstein-robust risk reporting and stress ranking; MEDIUM for MOT pricing outside derivatives desks; adapted distances matter for multistage scenario trees.

§9. Matching, demand inversion, Gromov–Wasserstein, forecast reconciliation Medium / Speculative

Core idea A — OT as assignment/demand model. Discrete OT is the assignment/transportation LP planners already use (workers→shifts, stores→DCs). Entropic OT corresponds to logit-type matching and gives demand inversion in discrete choice.

Core idea B — Gromov–Wasserstein (GW). Match two datasets whose items live in different feature spaces by comparing internal distance structures:

Fused GW adds a feature cost. Use: aligning SKU hierarchies after a reorganization, mapping fund universes across data vendors.

Core idea C — reconciliation. Hierarchical forecasts must add up. No published OT-based probabilistic reconciliation was found (§16). An open idea: project an incoherent joint forecast onto the coherent subspace by minimizing $W_2$, which for Gaussians reduces to Gelbrich-distance projection.

Verified papers.

• The unreasonable effectiveness of optimal transport in economics — arXiv 2021 — arXiv:2107.04700.

• Yogurts choose consumers? Estimation of random-utility models via two-sided matching — Review of Economic Studies 89(6), 2022 — arXiv:2111.13744.

• background Fused Gromov–Wasserstein distance for structured objects: theoretical foundations and mathematical properties — arXiv 2018 — arXiv:1811.02834.

• background Sliced Gromov–Wasserstein — NeurIPS 2019 — arXiv:1905.10124.

• Context (not OT): Probabilistic forecast reconciliation: properties, evaluation and score optimisation — European Journal of Operational Research 306(2), 2023 — DOI 10.1016/j.ejor.2022.07.040; Hierarchical forecasting with robust reconciliation — arXiv 2025 — arXiv:2510.20383.

Software. POT ot.gromov.gromov_wasserstein, ot.gromov.fused_gromov_wasserstein, ot.emd, ot.sinkhorn.

Failure modes. GW is non-convex (local optima; symmetric structures are ambiguous); entropic matching depends on temperature; reconciliation idea untested. EXCEPTION excluded: neural-embedding entity alignment is out of scope.

Potential: MEDIUM (matching/assignment, demand inversion), SPECULATIVE (GW entity alignment, OT reconciliation).

§10. Tools and statistical rates that matter in practice

Sample complexity — the curse of dimensionality. For an empirical measure $\hat P_N$ of a law in $\mathbb R^d$ ($d\ge3$), $\mathbb E\,W_p(\hat P_N,P)\asymp N^{-1/d}$: with $d=10$, halving the error needs $2^{10}\approx 1000\times$ more data. Practical consequences:

• 1-D and low-D problems are where OT is statistically cheap ($N^{-1/2}$ in 1-D).

• Entropic OT / Sinkhorn divergences converge at $N^{-1/2}$ (constants exponential in $d/\varepsilon$).

• Sliced Wasserstein (average of 1-D projections) has dimension-free rates.

• WDRO guarantees can avoid the curse with suitably chosen radii (§1, arXiv:2009.04382), and the Gelbrich ball has an explicit dimension-free $1/\sqrt N$ radius (§2).

Verified references.

• background Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance — arXiv 2017 — arXiv:1707.00087.

• background Sample complexity of Sinkhorn divergences — arXiv 2018 — arXiv:1810.02733.

• background Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem — arXiv 2019 — arXiv:1905.11882.

• Statistical and topological properties of sliced probability divergences — NeurIPS 2020 — arXiv:2003.05783.

• Statistical optimal transport (lecture notes) — arXiv 2024 — arXiv:2407.18163.

Small-N callout (N<100). The curse bites hard above d≈3; at scarce data impose structure (1-D, Gelbrich, residuals) or use task-local √N radii (Gao OR 2023; RWPI) — never a raw N^{-1/d} concentration ball as the decision radius. Exact OT is computationally free at N≤100; prefer ot.emd / 1-D formulas over Sinkhorn (entropic bias can dominate sampling noise). Full toolkit → §11.5.

When to use what.

§11. Worked example (computed; Python with NumPy, SciPy-HiGHS, POT 0.9.7)

Script: worked_example.py (seed 20261005). Exact output in worked_example_output.txt.

Part A — Wasserstein-DRO newsvendor versus SAA

Setup. Holding cost $h=1$, backorder cost $b=3$ (critical ratio $b/(h+b)=0.75$). True demand LogNormal($\ln 100$, 0.6), support $[0,600]$ for the type-1 LP. Cost $\ell(x,\xi)=h(x-\xi)^++b(\xi-x)^+$. Oracle order $x^*=149.88$, oracle expected cost $105.504$.

One sample of $N=10$ demands: 42.8, 58.1, 60.4, 71.2, 76.3, 86.9, 88.1, 96.6, 148.3, 209.5. SAA order (empirical 75% quantile) $=96.56$; in-sample cost 68.601; true expected cost 131.423 (25% above the oracle).

Type-1 ($W_1$) DRO — exact LP reformulation of the dual in §1:

Reading: the certificate is exactly SAA in-sample cost $+\ \varepsilon\max(h,b)=68.601+3\varepsilon$, and the order does not move — the Lipschitz-regularization effect from §1. Only when the radius is large enough for the adversary to exploit the support cap does the order jump (to an absurdly conservative 450). With unbounded support the order would equal SAA for every $\varepsilon$ (stated from the dual; the run uses a compact support).

Type-2 ($W_2$) DRO — via the strong dual (penalty $\lambda\varepsilon^2$, quadratic transport cost):

Here the order moves smoothly toward the oracle and on this (unlucky, low) sample true cost falls from 131.4 to 113.0.

Monte Carlo (300 independent datasets; true cost evaluated exactly; regret vs oracle 105.504):

CV picked $\varepsilon\in\{0,5,10,20,40\}$ with frequencies {98, 58, 56, 50, 38} at $N=10$ and {100, 101, 49, 41, 9} at $N=30$ ("win" counts strict improvements; ties count as non-wins).

Honest takeaways.

1. Small radii give a small average gain (≈0.3 percentage points of regret at $N=10$) and essentially none at $N=30$.

2. Cross-validated radius selection did worse than SAA here — radius choice is the practical bottleneck, consistent with §1's failure modes.

3. DRO's clearest value is (a) a certified worst-case cost and (b) protection on bad samples (the single-sample table).

4. Type-1 DRO leaves 1-D newsvendor decisions unchanged unless support bounds bind.

Part B — 1-D $W_2$ barycenter of three demand-scenario forecasts

Inputs (weights 1/3 each): A "bottom-up" $\mathcal N(100,10)$; B "sales" $\mathcal N(130,25)$; C bimodal $0.5\,\mathcal N(80,8)+0.5\,\mathcal N(140,8)$. Barycenter by POT's LP barycenter on a grid, checked against direct quantile averaging.

Quantile-average check: P10/P50/P90 = 85.9 / 113.1 / 140.3, identical to the LP (its mean, 113.08, differs only because the grid truncates the extreme 0.05% tails).

$W_2$ distance from each input to the consensus: A 17.87 (barycenter) vs 21.62 (pool); B 17.57 vs 17.29; C 11.46 vs 10.86.

Reading. Same mean; the barycenter is unimodal and tighter (P10–P90 width 54 vs 68), while the pool inherits C's two modes and widens. The barycenter is not uniformly "closer" to the inputs (only to A here). If disagreement between forecasters reflects real uncertainty, the barycenter is under-dispersed — tune with entropic regularization or CRPS-fit weights (§4).

§11.5 Small-N toolkit (N<100)

When this applies

Weekly/monthly demand, short ETF windows, residual series after a simple forecast, or any planning panel with N < 100 observations (often N ≈ 10–50). Complements §1–§2 and §10; numbers below lean on §11 and small_n_radius_output.txt.

Ranking — what to use first on a small panel

Radius recipes

A. Variation / finite-sample √N radius (Finite-sample guarantees for WDRO… — Operations Research 2023 — DOI 10.1287/opre.2022.2326, arXiv:2009.04382). Set $\varepsilon_N=\tilde O(N^{-1/2})$ to balance empirical loss against loss variation (Lipschitz / gradient norm), avoiding the $N^{-1/d}$ curse. Practical 1-D proxy after scaling:

Report a small grid $c\in\{0.5,1,2,4\}$ or freeze $c$ on a pre-sample — do not re-CV every period at $N\le 30$.

B. RWPI / Wasserstein projection (background Robust Wasserstein profile inference… — JAP 2019 — arXiv:1610.05627; refinement Wasserstein Projection Tests… — arXiv 2024 — arXiv:2408.11753). For estimating-equation problems, take $\delta$ as a $(1-\alpha)$-quantile of the profile / WP statistic (limiting $\chi^2$-type law). Rates are typically $N^{-1}$ or $N^{-1/2}$ depending on loss/cost — parametric, not $N^{-1/d}$. At $N\approx 30$–$80$, Bartlett corrections in arXiv:2408.11753 improve quantile accuracy for published confidence statements.

C. Concentration coverage of $P$ (background Esfahani–Kuhn — arXiv:1505.05116): $\varepsilon_N(\beta)\asymp(\log(C/\beta)/N)^{1/\max\{d,2\}}$. Use for theory or $d=1$; not as the default decision radius in $d>1$ at small $N$.

D. Bootstrap frontier philosophy (Calibration of DRO… — Operations Research 2021 — arXiv:1711.06565; $\phi$-divergence setting): open-loop 90% radii are often too conservative; bootstrap a robust mean–variance frontier. Transfers as a calibration mindset to Wasserstein radii.

Recipes by sample size

Evidence from this lab

• §11 Monte Carlo (LogNormal demand, $h=1,b=3$, $R=300$): at $N=10$, best fixed $W_2$ $\varepsilon$ cuts mean regret by ~0.3 pp vs SAA; at $N=30$, essentially no average gain. CV-picked $\varepsilon$ is worse than SAA (win rates 0.30 / 0.23). Type-1 leaves the order unchanged until support binds.
• small_n_radius_compare.py ($R=200$, same truth; small_n_radius_output.txt):
  • Best average costs: fixed $\varepsilon\approx 5$–$10$ at $N=10$–$20$; by $N=30$–$80$ SAA is already hard to beat on average.
  • Gao-style $\hat\varepsilon_N(c)=c\,\hat\sigma/\sqrt N$ with MAD-scale $\hat\sigma$: at $N=10$, mean $\hat\varepsilon(0.5)\approx 8.2$ (competitive); $c\ge 2$ over-conserves (regret 11–18%). Scale $c$ to the loss/ground-metric units — raw $c=1$ is not universal.
  • Leave-one-out CV over $\{0,5,10,20,40\}$: win rate vs SAA $=0.45$ ($N=10$), $0.30$ ($N=20$), $0.33$ ($N=30$) — confirms CV hurts at scarce data.
• External finite-sample newsvendor (How Big Should Your Data Really Be?… — Management Science 2023 — DOI 10.1287/mnsc.2023.4725, arXiv:2107.02742): tens of samples already make SAA strong on relative regret; SAA regret can be non-monotone in $N$; a convex combination of consecutive order statistics is minimax-optimal. Read: at $N\approx 10$–$50$, SAA is a serious baseline — DRO’s job is certificate + bad-sample protection, not large average wins.

Drift monitors at short windows

Use 1-D $W_p$ or sliced $W$ on rolling windows of length 20–60. Calibrate alarms by permutation, not naive $n$-out-of-$n$ bootstrap (Minimax-Optimal Two-Sample Test with Sliced Wasserstein — arXiv:2510.27498; bootstrap inconsistency background arXiv:1610.03287). Covariance barycenters: Generalized bootstrap in the Bures–Wasserstein space — arXiv:2111.12612.

Ambiguity + misspecification (non-stationary short series)

Newsvendor under Ambiguity and Misspecification — arXiv:2405.07008: mean–variance ambiguity with an optimal-transport misspecification penalty; closed-form order; finite-sample bound splitting estimation error vs distribution shift. Relevant when $N$ is small and the environment may drift.

Software note for scarce data

At $N\le 100$, prefer exact 1-D / LP OT (numpy quantiles, scipy.stats.wasserstein_distance, POT ot.emd2_1d / ot.emd) and HiGHS/CVXPY duals. Sinkhorn is unnecessary and can bias more than sampling noise. skwdro still needs an external radius policy (§11.5 A–B).

§12. Five ranked "brave but within reason" project ideas

1. Gelbrich-robust CVaR allocation and stress report for an ETF panel. (Small-N friendly — preferred when rolling windows are 30–80 weeks.)

• Hypothesis: the closed-form Gelbrich-robust CVaR portfolio (§2) with $\rho=c/\sqrt N$ improves out-of-sample CVaR and turnover versus Markowitz and Ledoit–Wolf shrinkage, at similar return.

• Method: SOCP $\min_w -\hat\mu^\top w+\alpha\sqrt{w^\top\hat\Sigma w}+\rho\sqrt{1+\alpha^2}\|w\|_2$; calibrate $c$ on a pre-sample; produce worst-case mean/covariance as stress scenarios each rebalance. Benchmarks: Markowitz, Ledoit–Wolf, $1/N$, ridge-penalized with CV radius, $W_1$-DRO mean-CVaR.

• Data: daily/weekly ETF total returns (10+ years), rolling windows.

• Success: lower realized 95% CVaR and max drawdown with Sharpe within noise; lower turnover.

• Falsification: no out-of-sample CVaR gain versus Ledoit–Wolf over rolling windows (block-bootstrap test), or gains vanish once $\rho$ is fixed ex ante.

• Effort: 3–5 days.

2. Quantile-averaging consensus for challenger forecasts + residual-based Wasserstein buffers. (Works with short histories if each challenger is low-parameter; size residual balls with §11.5 √N rules, not LOO.)

• Hypothesis: a CRPS-weighted 1-D $W_2$ barycenter of challenger forecasts beats the linear pool and the best single model on CRPS, and a residuals-based $W$-DRO buffer (§1, arXiv:2012.01088) sizes budget contingencies at lower cost for the same breach rate than SAA quantiles.

• Method: quantile grids per model; weights by rolling CRPS; entropic widening knob; residual ambiguity set with radius by time-series CV.

• Data: historical monthly actuals and forecasts by line/region (≥ 36 months).

• Success: CRPS gain, Diebold–Mariano significant; buffer cost at target breach rate below SAA.

• Falsification: DM test versus linear pool not significant, or buffer cost ≥ SAA at matched coverage.

• Effort: 4–7 days.

3. Distributional nowcasting of cross-sections (sector returns, regional growth).

• Hypothesis: forecasting the whole cross-sectional distribution via Wasserstein autoregression / transport-map AR (§3) beats a random walk and a moments-only AR in $W_2$ error, and its quantiles add information to mean nowcasts.

• Method: quantile-function panel; WAR (R WRI) and isotonic transport-map AR; optionally OT-based mixed-frequency steps in a factor model as in the JRSS-A 2026 paper.

• Data: monthly cross-sections of ETF/sector returns or regional indicators, 15+ years.

• Success: lower out-of-sample $W_2$ and CRPS over rolling 10-year windows.

• Falsification: no improvement over random walk in $W_2$ error out of sample.

• Effort: 5–8 days.

4. Wasserstein regime and drift monitor.

• Hypothesis: WK-means regimes (§7) plus sequential $W$ change-point alarms anticipate risk-model breakdowns and can switch the DRO radius in projects 1–2.

• Method: rolling 1-D $W$ on returns/residuals; sliced $W$ for multivariate; permutation-calibrated thresholds.

• Data: the same panels as 1–3.

• Success: earlier alarms than a trailing-volatility rule at equal false-alarm rate; radius switching improves CVaR.

• Falsification: adds nothing beyond a trailing-vol threshold.

• Effort: 4–6 days.

5. Distributional synthetic control for planning-policy evaluation.

• Hypothesis: distributional synthetic controls (§5) reveal tail effects of a policy (pricing, process change) that mean-based DiD misses.

• Method: R DiSCos; placebo-in-space permutation inference.

• Data: unit-level distributions (orders, tickets, lead times) for treated and ≥ 10 control units, several pre-periods.

• Success: good pre-fit; effects visible in quantiles not means.

• Falsification: placebo rank $p>0.1$.

• Effort: 3–5 days per case.

Small-N add-on (project-shaped). Freeze a √N / RWPI radius policy for a 1-D newsvendor or capacity buffer on a real series with N∈[10,80]; falsify if out-of-sample cost ≥ SAA at matched certificate level over a rolling chart. Effort: 2–4 days. See §11.5.

Also considered. OT-based reconciliation of hierarchical probabilistic forecasts (open research — §9); a Wasserstein-robust Kalman filter inside a dynamic-factor nowcast (§1); OT conformal regions for multi-line P&L (§6); Gromov–Wasserstein alignment of fund or SKU universes across vendors (§9).

§13. Is anything truly "amazing"?

• Yes — one result. The Gelbrich mean–covariance closed form (§2): one formula turns any of VaR/CVaR/spectral/mean-std into a robust, convex, regularized portfolio rule with a dimension-free $1/\sqrt N$ radius, equal weights as the limit, and closed-form worst-case moments for stress reporting. Rare combination of rigor and immediate usability.

• Quietly excellent. 1-D quantile averaging as forecast combination (§4) and distributional synthetic controls (§5): simple, interpretable, hard to beat as baselines.

• Promising but early. OT-based multivariate conformal prediction (2025, §6).

• Not amazing on its own. Type-1 $W$-DRO for 1-D inventory: our example shows decisions often equal SAA and gains are small; value is the certificate.

• Neural exceptions. None needed. The only neural mentions are flagged EXCEPTION (a neural dual in one MOT paper; neural-embedding entity alignment) and are excluded from recommendations.

§14. Glossary

• Coupling / transport plan $\pi$ — a joint distribution with given marginals; "who sends how much mass where".

• Wasserstein distance $W_p$ — minimum average cost (distance$^p$) of moving one distribution onto another.

• Quantile function $F^{-1}$ — inverse CDF; in 1-D, $W_2$ = $L^2$ distance between quantile functions.

• Kantorovich–Rubinstein duality — $W_1$ equals the largest mean difference over 1-Lipschitz functions.

• Entropic OT / Sinkhorn — OT plus an entropy penalty; solved by alternating scaling; fast and smooth.

• Sinkhorn divergence — debiased entropic OT cost; a valid discrepancy between distributions.

• Sliced Wasserstein — average of 1-D Wasserstein distances over random projections.

• Barycenter — Fréchet mean of distributions under $W_2$; in 1-D, average quantiles.

• Linear pool — mixture of forecast densities (averaging probabilities, not quantiles).

• Gelbrich distance — $W_2$ between Gaussians with given means/covariances; lower bound on $W_2$ for general laws with those moments.

• SAA — sample average approximation: optimize against the empirical distribution.

• DRO — distributionally robust optimization: optimize against the worst distribution in an ambiguity set.

• Ambiguity set / ball radius $\varepsilon,\rho$ — set of plausible distributions; radius sets conservatism.

• Type-1 / type-2 WDRO — ambiguity balls using $W_1$ (Lipschitz-like regularization) or $W_2$ (variance-like regularization).

• Certificate — the worst-case objective value; an upper bound that holds with stated confidence when the radius is calibrated.

• CVaR — expected loss in the worst $\beta$ fraction of outcomes.

• Distortion risk measure — risk measure computed by reweighting quantiles with a distortion function (VaR, CVaR are special cases).

• Fréchet regression — regression where the response is a metric-space object (e.g. a distribution).

• Tangent space / log map — linearization of $W_2$ space around a reference distribution; allows linear models.

• Center-outward quantile/rank — multivariate quantile defined by the OT map to the uniform law on the unit ball.

• Conformal prediction — calibration method giving finite-sample coverage for prediction sets.

• Scenario reduction — replacing many scenarios by few weighted ones close in a probability metric.

• Nested distance / adapted Wasserstein — OT distance between stochastic processes respecting information flow.

• Martingale OT — OT restricted to martingale couplings; model-free price bounds.

• Gromov–Wasserstein — OT comparing internal distance structures of two datasets in different spaces.

• CRPS — continuous ranked probability score; for 1-D forecasts it equals the integrated pinball loss.

• Markov kernel — a map from inputs to probability distributions; the object studied on the companion page.

RWPI / Wasserstein profile — smallest transport-cost perturbation of the empirical law that restores a moment/estimating equation; used to size DRO radii without CV.

Wasserstein projection (WP) statistic — empirical projection distance onto a moment-constrained set; basis for tests and radius calibration (Bartlett-correctable).

√N variation radius — ambiguity radius of order $N^{-1/2}$ from finite-sample WDRO theory balancing fit vs loss variation.

Order-statistic policy — inventory/quantile rule that orders the sample and picks (or randomizes between) one or two consecutive ranks; SAA is the $\lceil qN\rceil$-th order statistic.

§15. Sources (title — venue, year — link)

Foundations and surveys

1. Computational Optimal Transport — Foundations and Trends in Machine Learning, 2019 — https://arxiv.org/abs/1803.00567 background

2. Distributionally Robust Optimization — Acta Numerica 34, 2025 — https://doi.org/10.1017/S0962492924000084 (arXiv:2411.02549)

3. Statistical optimal transport (lecture notes) — arXiv, 2024 — https://arxiv.org/abs/2407.18163

4. POT: Python Optimal Transport — Journal of Machine Learning Research 22(78), 2021 — https://jmlr.org/papers/v22/20-451.html

5. skwdro: a library for Wasserstein distributionally robust machine learning — arXiv, 2024 — https://arxiv.org/abs/2410.21231

§1 Wasserstein DRO

6. Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations — Mathematical Programming, 2018 — https://arxiv.org/abs/1505.05116 background

7. Distributionally robust stochastic optimization with Wasserstein distance — Mathematics of Operations Research (accepted per arXiv) — https://arxiv.org/abs/1604.02199 background

8. Finite-sample guarantees for Wasserstein distributionally robust optimization: breaking the curse of dimensionality — Operations Research, 2023 — https://doi.org/10.1287/opre.2022.2326 (arXiv:2009.04382)

9. Wasserstein distributionally robust optimization and variation regularization — arXiv, 2017/2020 — https://arxiv.org/abs/1712.06050

10. Nash equilibria, regularization and computation in optimal transport-based distributionally robust optimization — arXiv, 2023 — https://arxiv.org/abs/2303.03900

11. Regularization via mass transportation — arXiv, 2017 — https://arxiv.org/abs/1710.10016 background

12. Robust Wasserstein profile inference and applications to machine learning — Journal of Applied Probability, 2019 — https://arxiv.org/abs/1610.05627 background

13. Confidence regions in Wasserstein distributionally robust estimation — arXiv, 2019/2021 — https://arxiv.org/abs/1906.01614

14. Distributionally robust learning — arXiv monograph, 2021 — https://arxiv.org/abs/2108.08993

15. Residuals-based distributionally robust optimization with covariate information — arXiv, 2020 — https://arxiv.org/abs/2012.01088

16. Data-driven chance constrained programs over Wasserstein balls — arXiv, 2018/2022 — https://arxiv.org/abs/1809.00210

17. On distributionally robust chance constrained programs with Wasserstein distance — arXiv, 2018 — https://arxiv.org/abs/1806.07418

18. A data-driven distributionally robust newsvendor model with a Wasserstein ambiguity set — Journal of the Operational Research Society 72(8), 2021 — https://doi.org/10.1080/01605682.2020.1746203

19. Distributionally robust newsvendor with moment constraints — arXiv, 2020 — https://arxiv.org/abs/2010.16369

20. On affine policies for Wasserstein distributionally robust unit commitment — IEEE CDC 2022 (per arXiv) — https://arxiv.org/abs/2203.15333

21. Outlier-robust Wasserstein DRO — NeurIPS, 2023 — https://arxiv.org/abs/2311.05573

22. Sinkhorn distributionally robust optimization — arXiv, 2021 — https://arxiv.org/abs/2109.11926

23. Wasserstein distributionally robust Kalman filtering — NeurIPS, 2018 — https://arxiv.org/abs/1809.08830 background

• 23a. Finite-Sample Guarantees for Wasserstein Distributionally Robust Optimization: Breaking the Curse of Dimensionality — Operations Research, 2023 — https://doi.org/10.1287/opre.2022.2326 (arXiv:2009.04382) (venue upgrade; was arXiv-only as item 8)

• 23b. How Big Should Your Data Really Be? Data-Driven Newsvendor: Learning One Sample at a Time — Management Science, 2023 — https://doi.org/10.1287/mnsc.2023.4725 (arXiv:2107.02742)

• 23c. Calibration of Distributionally Robust Empirical Optimization Models — Operations Research 69(5), 2021 — https://arxiv.org/abs/1711.06565 ($\phi$-divergence; bootstrap frontier calibration lesson)

• 23d. Newsvendor under Ambiguity and Misspecification — arXiv, 2024 — https://arxiv.org/abs/2405.07008

• 23e. Wasserstein Projection Tests: Higher-Order Asymptotics and Connections to Empirical Likelihood — arXiv, 2024 (v2 2026) — https://arxiv.org/abs/2408.11753 (Bartlett corrections for WP/RWPI quantiles)

§2 Mean–covariance / portfolios

24. Mean-covariance robust risk measurement — arXiv, 2021 (v3 2025); SSRN https://doi.org/10.2139/ssrn.3990847 — https://arxiv.org/abs/2112.09959

25. Distributionally robust mean-variance portfolio selection with Wasserstein distances — arXiv, 2018 — https://arxiv.org/abs/1802.04885 background

26. Portfolio optimisation within a Wasserstein ball — arXiv, 2020 — https://arxiv.org/abs/2012.04500

§3 Distributional regression / density time series / nowcasting

27. Fréchet regression for random objects with Euclidean predictors — Annals of Statistics 47(2), 2019 — https://doi.org/10.1214/17-AOS1624 background

28. Wasserstein regression — Journal of the American Statistical Association, 2021 — https://doi.org/10.1080/01621459.2021.1956937

29. Wasserstein autoregressive models for density time series — Journal of Time Series Analysis 43(1), 2022 — https://doi.org/10.1111/jtsa.12590

30. Autoregressive optimal transport models — arXiv, 2021 — https://arxiv.org/abs/2105.05439

31. Distribution-on-distribution regression via optimal transport maps — arXiv, 2021 (to appear in Biometrika per arXiv) — https://arxiv.org/abs/2104.09418

32. Short-term forecasting with optimal transport — Journal of the Royal Statistical Society Series A 189(2), 2026 — https://doi.org/10.1093/jrsssa/qnaf036

§4 Barycenters / scenarios

33. Probability forecast combination via entropy regularized Wasserstein distance — Entropy 22(9):929, 2020 — https://doi.org/10.3390/e22090929

34. Scenario reduction revisited: fundamental limits and guarantees — Mathematical Programming — https://doi.org/10.1007/s10107-018-1269-1

35. Optimization-based scenario reduction for data-driven two-stage stochastic optimization — Operations Research 71(4), 2023 — https://doi.org/10.1287/opre.2022.2265

36. Problem-driven scenario clustering in stochastic optimization — arXiv, 2021 — https://arxiv.org/abs/2106.11717

37. Entropic regularization of the nested distance — arXiv, 2021 — https://arxiv.org/abs/2107.09864

§5 Causal

38. Distributional synthetic controls — Econometrica 91(3), 2023 — https://doi.org/10.3982/ECTA18260

39. An optimal transport approach to estimating causal effects via nonlinear difference-in-differences — arXiv, 2021 — https://arxiv.org/abs/2108.05858

40. Optimal transport weights for causal inference — arXiv, 2021 — https://arxiv.org/abs/2109.01991

41. Optimal transport for counterfactual estimation: a method for causal inference — arXiv, 2023 — https://arxiv.org/abs/2301.07755

§6 Multivariate quantiles / conformal

42. Distribution and quantile functions, ranks and signs in dimension d: a measure transportation approach — Annals of Statistics 49(2), 2021 — https://doi.org/10.1214/20-AOS1996

43. Center-outward quantiles and the measurement of multivariate risk — arXiv, 2019 — https://arxiv.org/abs/1912.04924

44. Multivariate rank-based distribution-free nonparametric testing using measure transportation — arXiv, 2019 — https://arxiv.org/abs/1909.08733

45. Optimal transport-based conformal prediction — arXiv, 2025 — https://arxiv.org/abs/2501.18991

46. Multivariate conformal prediction using optimal transport — arXiv, 2025 — https://arxiv.org/abs/2502.03609

§7 Drift / regimes / tests

47. Monitoring for a change point in a sequence of distributions — Annals of Statistics 49(4), 2021, 2271–2291 — journal record (no arXiv ID found)

48. Clustering market regimes using the Wasserstein distance — arXiv, 2021 — https://arxiv.org/abs/2110.11848

49. Optimal transport based change point detection and time series segment clustering — arXiv, 2019 — https://arxiv.org/abs/1911.01325 background

50. Multivariate goodness-of-fit tests based on Wasserstein distance — arXiv, 2020 — https://arxiv.org/abs/2003.06684

• 50a. Minimax-Optimal Two-Sample Test with Sliced Wasserstein — arXiv, 2025 — https://arxiv.org/abs/2510.27498

• 50b. Generalized bootstrap in the Bures–Wasserstein space — arXiv, 2021 (v3 2026) — https://arxiv.org/abs/2111.12612

• 50c. Inference for empirical Wasserstein distances on finite spaces — Annals of Statistics (per arXiv), 2017 — https://arxiv.org/abs/1610.03287 background

§8 Risk / pricing / processes

51. Robust distortion risk measures — arXiv, 2022 — https://arxiv.org/abs/2205.08850

52. Sensitivity analysis of Wasserstein distributionally robust optimization problems — Proceedings of the Royal Society A, 2021 — https://doi.org/10.1098/rspa.2021.0176

53. Computational methods for martingale optimal transport problems — arXiv, 2017 — https://arxiv.org/abs/1710.07911 background

54. Robust pricing and hedging of options on multiple assets and its numerics — arXiv, 2019 (forthcoming SIAM J. Financial Mathematics per arXiv) — https://arxiv.org/abs/1909.03870

55. Adapted Wasserstein distances and stability in mathematical finance — Finance and Stochastics 24, 2020 — https://arxiv.org/abs/1901.07450

56. Estimating processes in adapted Wasserstein distance — arXiv, 2020 — https://arxiv.org/abs/2002.07261

§9 Matching / GW / reconciliation

57. The unreasonable effectiveness of optimal transport in economics — arXiv, 2021 — https://arxiv.org/abs/2107.04700

58. Yogurts choose consumers? Estimation of random-utility models via two-sided matching — Review of Economic Studies 89(6), 2022 — https://arxiv.org/abs/2111.13744

59. Fused Gromov–Wasserstein distance for structured objects: theoretical foundations and mathematical properties — arXiv, 2018 — https://arxiv.org/abs/1811.02834 background

60. Sliced Gromov–Wasserstein — NeurIPS, 2019 — https://arxiv.org/abs/1905.10124 background

61. Probabilistic forecast reconciliation: properties, evaluation and score optimisation — European Journal of Operational Research 306(2), 2023 — https://doi.org/10.1016/j.ejor.2022.07.040 (context, not OT)

62. Hierarchical forecasting with robust reconciliation — arXiv, 2025 — https://arxiv.org/abs/2510.20383 (context, not OT)

§10 Rates

63. Sharp asymptotic and finite-sample rates of convergence of empirical measures in Wasserstein distance — arXiv, 2017 — https://arxiv.org/abs/1707.00087 background

64. Sample complexity of Sinkhorn divergences — arXiv, 2018 — https://arxiv.org/abs/1810.02733 background

65. Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem — arXiv, 2019 — https://arxiv.org/abs/1905.11882 background

66. Statistical and topological properties of sliced probability divergences — NeurIPS, 2020 — https://arxiv.org/abs/2003.05783

Software (verified to exist)

67. POT (Python Optimal Transport) — https://pythonot.github.io — functions used/cited: ot.wasserstein_1d, ot.emd2_1d, ot.emd, ot.emd2, ot.sinkhorn2, ot.sliced_wasserstein_distance, ot.lp.barycenter, ot.bregman.barycenter, ot.gromov.gromov_wasserstein, ot.gromov.fused_gromov_wasserstein

68. skwdro — https://skwdro.readthedocs.io (PyPI skwdro)

69. R transport (CRAN, 0.15-4) — https://CRAN.R-project.org/package=transport

70. R frechet (CRAN) — https://CRAN.R-project.org/package=frechet

71. R WRI (CRAN; Wasserstein regression and inference, includes WARp) — https://CRAN.R-project.org/package=WRI

72. R DiSCos (CRAN; distributional synthetic controls) — https://CRAN.R-project.org/package=DiSCos

73. WKF (MATLAB code for Wasserstein-robust Kalman filtering) — https://github.com/sorooshafiee/WKF

74. SciPy (scipy.stats.wasserstein_distance, scipy.optimize.linprog with HiGHS), CVXPY, Pyomo, python-mip — standard open-source solvers/modelling layers.

§16. Verification notes and open uncertainties

• Checked 2026-10-05 against arXiv API metadata/abstracts, Crossref/DOI landing pages, CRAN pages and JMLR. The mean–covariance paper (§2) was read in full; the others from abstracts (plus key sections where noted). Small-N addendum papers (items 23a–23e, 50a–50c) verified the same day via arXiv API + Crossref (MS 2023 DOI for 23b; OR 2023 DOI for 8/23a).

• Small-N section (§11.5) distilled from small_n_addendum.md; radius comparison numbers from small_n_radius_output.txt / small_n_radius_compare.py.

• Not found / could not verify:

  • Any paper on OT-based hierarchical forecast reconciliation — none found; §9 presents it as an open idea.
  • A credit-portfolio-specific 2020+ Wasserstein stress-testing paper — not found; §8 covers generic Wasserstein-robust risk measures and sensitivity instead.
  • A maintained package for center-outward ranks/quantiles — none verified; build with POT.

• No arXiv version found (verified via journal/DOI instead): items 18, 35, 47, 61 and 32.

• Venues taken only from arXiv comments (not independently confirmed): items 7 (MOR), 20 (CDC 2022), 31 (Biometrika), 54 (SIAM J. Financial Mathematics). Journal of item 24 not confirmed (arXiv + SSRN only). Items 30, 44, 51, 56 carry no confirmed journal.

• Informal content: the first-order sensitivity formula in §8 is a heuristic summary — see item 52 for exact conditions. Fréchet-regression weights in §3 are described qualitatively.

• Correction vs a common citation: the Wasserstein autoregressive density time-series model (item 29) is often misattributed; the JTSA paper and arXiv record are the authoritative source.

• Neural content: none recommended. EXCEPTION mentions only: the deep-network dual route in item 54, and neural-embedding entity alignment (excluded from §9).

• Worked example (§11) is fully reproducible from worked_example.py; numbers in this file are copied from worked_example_output.txt.

Companion page: Optimal transport on kernels: learning guide