Optimal transport on kernels: learning guide

Every claim on this page comes from the lab's sourced research notes; math checked against sourced notes and the dissertation (Quant, 2026-10-05).

Revised 2026-10-05: dissertation addendum folded in (definitions and theorems quoted from the dissertation).

Companion page. Optimal transport for operations research, risk & planning: classical methods (classical methods learning guide).

One-screen brief and research arc

What the area is. A bridge between classical Bayesian / statistical inference and modern generative / representation learning, built by treating distribution-valued maps as first-class objects and putting metrics from optimal transport on them.

Research arc (thematic, not biographical).

1. Foundations (2023): Markov kernels valued in Wasserstein spaces; integration and disintegration; Bayesian inverse maps; generalized Bayes law; proved convergence of the expected update under stated hypotheses (Proposition 5.7 and Theorem 5.10). Bayesian Inference for Markov Kernels Valued in Wasserstein Spaces, PhD dissertation, Arizona State University, 2023. Dissertation (2023-10-24 public copy, 57 pp.) read for this revision via the addendum.
   Basics: Disintegration

   Any joint law $\gamma$ on $X\times Y$ can be written as a marginal on $X$ times a family of conditionals on $Y$ given $x$ (or the other way around). That family of conditionals is a Markov kernel. Disintegration is the measure-theoretic version of “conditioning.” The dissertation builds an integration theory so this statement sits inside an $L^p$/Wasserstein calculus, not only a $\sigma$-algebra calculus.

2. unverified Latent-space OT dependence (2024 talk): Sliced Wasserstein dependence as a training objective for autoencoders; Bayesian reading of what latent statistics mean after decoding (SIAM Conference on Mathematics of Data Science, 2024 abstract).
   Basics: Sliced Wasserstein

   Full $W_p$ in high dimension is costly. Sliced Wasserstein averages one-dimensional $W_p$ after projecting both measures onto many random lines. It is a Monte Carlo approximation used for training objectives such as Wasserstein correlation (WaCo) in the TMLR paper. It is not the same as Sinkhorn (entropic OT), though both are common computational shortcuts.

3. Post-training invariance (2025–2026): Augmented encoders formalized as Markov–Wasserstein kernels; two OT-based losses (Markov–Wasserstein / MaWa minimization; Wasserstein correlation / WaCo maximization) that add augmentation invariance to a frozen pretrained encoder via a tiny adapter, without wrecking clean-feature geometry. Paper: Post-Training Augmentation Invariance, arXiv:2505.11702 (v3 dated 2026-06-08), published in Transactions on Machine Learning Research (2026); OpenReview forum id Z4uUwU6zRe. Earlier arXiv title/version: Invariant Representations via Wasserstein Correlation Maximization.
   Basics: Markov–Wasserstein metric

   Once each value $K(x)$ lives in a Wasserstein space, two kernels $K$ and $L$ can be compared by averaging $W_p(K(x),L(x))$ over the input measure $\mu$:

   $$\mathrm{MW}_p(K,L)=\Bigl(\int_X W_p\bigl(K(x),L(x)\bigr)^p\,\mu(dx)\Bigr)^{1/p}.$$

   So MW is “Wasserstein, then $L^p$ over $x$.” It measures how different two conditional-distribution-valued maps are, not how different two pushforward laws are.

Three-sentence summary of “OT on kernels”. A Markov kernel is a map $x \mapsto K(x) \in \mathcal{P}(Y)$ (a conditional distribution). Putting the Wasserstein metric $W_p$ on $\mathcal{P}_p(Y)$ and an $L^p$-type integral metric on the space of such maps yields Markov–Wasserstein kernels, which support both structural (categorical / Bayesian) and quantitative (convergence, approximation) questions. Downstream ML work uses the same objects to define augmented encoders and OT-based losses that enforce augmentation invariance while approximately preserving metric structure of pretrained features.

Deep dive — optimal transport on Markov kernels

B.1 Intuition first

Function approximation vs distribution approximation. The recorded talk opens by contrasting:

• Function approximation: given pairs $(x_i, y_i)$, find $f: X \to Y$. Methods (harmonic analysis, RKHS kernels, GPs, nets) differ, but the problem statement is uniform.

• Distribution approximation: parameter estimation for PDEs, fitting mixture models, probabilistic programming (distributions as data structures), generative models (learn a law on images so you can sample / condition). Problems look less unified.

Unifying move. Study distribution-valued functions $F: X \to \mathcal{P}(Y)$ — i.e. Markov kernels — but equip $X$, $\mathcal{P}(Y)$, and the space of kernels with metrics so one can ask quantitative questions (rates, stability, convergence), not only measure-theoretic ones. Optimal transport supplies the metric on probability measures.

What this is not. It is not “OT between RKHS kernel matrices,” not MMD-as-primary-object (though MMD appears as a contrast in the later paper’s HSIC experiments), and not Sinkhorn training of couplings as the main algorithm in the 2023 talk. Sinkhorn / entropic OT appear in the broader OT-ML literature the paper cites; the core objects here are Wasserstein geometry on the value space of Markov kernels.

B.2 Standard OT objects (background used throughout)

Assume Polish metric spaces (complete, separable) with Borel $\sigma$-algebras.

Finite $p$-th moment. $\mu \in \mathcal{P}_p(X)$ if $\int_X d_X^p(x_0,x)\,\mu(dx) \lt \infty$ for some (hence any) $x_0$.

Couplings. $\Pi(\mu,\nu) = \{\pi \in \mathcal{P}(X\times Y): \pi^1=\mu,\ \pi^2=\nu\}$.

Wasserstein-$p$ distance (paper / talk background):

The space $(\mathcal{P}_p(X), W_p)$ is itself Polish if $(X,d_X)$ is.

Sliced Wasserstein. Used computationally for Wasserstein correlation in the ML paper: project measures to 1D lines, average 1D $W_p$. Denoted $SW_p$ in the paper.

Product / independence. Wasserstein dependence compares a joint $\pi$ to the product of its marginals $\pi^1\otimes\pi^2$ in $W_p$ on the product space.

B.3 Markov kernels and Markov–Wasserstein kernels

Markov kernel (classical). A measurable map $K: X \to \mathcal{P}(Y)$, equivalently a Markov kernel $K(dy\mid x)$. Example from the talk: $x \mapsto \mathcal{N}(f(x), \sigma^2)$ for a deterministic $f$.

Dissertation definitions. Def. 2.17 requires $F(-\mid x)$ to be a probability measure for every $x$, and $x\mapsto F(E\mid x)$ measurable for every measurable $E$. Def. 2.18 defines $\mathcal{K}_\mu^p(X,Y)$ by the finite-moment condition, for some (hence any) $y_0\in Y$:

Def. 2.20: a coupling form due to Patterson (dissertation reference [45]). Here $\Pi(F,G)$ consists of kernels with fiberwise couplings of $F_x,G_x$. Lemma 2.22 identifies it with the $L^p$ form:

$L^p$ maps into a metric space (Post-Training Augmentation Invariance, Def. 2.3). For a metric measure space $(X,d_X,\mu_X)$ and Polish $(Y,d_Y)$,

on (equivalence classes of) measurable $f: X\to Y$ with finite $p$-moment.

Markov–Wasserstein kernel. Specialize $Y$ to a Wasserstein space $\mathcal{P}_p(Z)$ with $d_Y = W_p$. Elements of $L^p_{\mu_X}\big(X,\mathcal{P}_p(Z)\big)$ are Markov–Wasserstein kernels. The metric is written $\mathrm{MW}_p$:

Interpretation: distance between two conditional-distribution-valued maps, averaged in $L^p$ over the input measure.

Talk claim (foundations). These objects form a “generalized distribution-valued $L^p$ space,” convenient for distribution-approximation problems in Bayesian inference, generative modeling, and probabilistic programming. Categorical structure (probability monads / monad structure) connects to PPL semantics; metrics enable classical statistical questions simultaneously.

B.4 Integration, disintegrations, Bayesian inverses (talk / dissertation line)

The dissertation addendum supplies the following definitions and proved statements.

1. Markov–Lebesgue integral. Def. 3.1: a countably simple kernel has the form $F=\sum_{j=1}^\infty\nu_j1_{E_j}$, where the measurable sets $E_j$ partition $X$ and $\nu_j\in\mathcal{P}_p(Y)$. Def. 3.2 integrates a countably simple $L^p_\mu$ kernel as a mixture:

   $$\int_X F(x)\,\mu(dx):=\sum_{j=1}^\infty\mu(E_j)\nu_j.$$

   Def. 3.4 calls a sequence weakly Cauchy in the $L^p$ metric if, for every $\varepsilon\gt0$, eventually for all $n,m$,

   $$\int_X W_p(F_n(x),F_m(x))^p\,\mu(dx)\lt\varepsilon.$$

   Theorem 3.9 approximates every $F\in L^p_\mu(X,\mathcal{P}_p(Y))$ uniformly in $W_p$ by countably simple $L^p$ kernels forming a weakly Cauchy sequence. Def. 3.11 defines the general integral using any such sequence:

   $$\int_X F(x)\,\mu(dx):=\lim_{n\to\infty}\int_X F_n(x)\,\mu(dx).$$

   Theorem 3.12: equivalence with iterated integrals, for $\varphi\in C_b(Y)$:

   $$\begin{aligned} &\int_Y\varphi(y)\left(\int_X F(x)\,\mu(dx)\right)(dy)\\ &\qquad=\int_X\left(\int_Y\varphi(y)\,F_x(dy)\right)\mu(dx). \end{aligned}$$

   Theorem 3.16 ($L^p$ disintegration). For $\gamma\in\mathcal{P}_p(X\times Y)$ with marginals $\mu,\nu$, there are almost-surely unique $F\in L^p_\mu(X,\mathcal{P}_p(Y))$ and $G\in L^p_\nu(Y,\mathcal{P}_p(X))$ such that

   $$\begin{aligned} \int_X(\delta_x\otimes F_x)\,\mu(dx)&=\gamma\\ &=\int_Y(G_y\otimes\delta_y)\,\nu(dy). \end{aligned}$$

   These are literal equalities of Markov–Lebesgue integrals: informal notation made literal. Also $\nu=\int_X F\,d\mu$ and $\mu=\int_Y G\,d\nu$.

   Basics: Dirac and pushforward

   The Dirac $\delta_x$ is the measure that puts all mass at a single point $x$. The pushforward $f_\sharp\mu$ is the law of $f(X)$ when $X\sim\mu$—transport the mass of $\mu$ along $f$. Many generative models are pushforwards of a simple latent prior through a map $F_\theta$.

2. Isometry and monad: separate results. Theorem 2.25 gives, for a metric measure space $(X,d_X,\mu)$, Polish $Y$, and $1\le p\lt\infty$, the isometric isomorphism

   $$\begin{aligned} &(\mathcal{K}_\mu^p(X,Y),\mathrm{MW}_p)\\ &\qquad\cong(L^p_\mu(X,\mathcal{P}_p(Y)),d_{L^p}). \end{aligned}$$

   Theorem 4.10: $(P_p,\delta,m)$ is a monad on $\mathbf{Met}$, the category of complete separable metric spaces with Lipschitz morphisms. This is the $p$-Kantorovich monad; $p=1$ is the classical Kantorovich monad. The monad/Kleisli package alone does not give a categorical account of Bayesian inversion, because the inverse depends on the prior (pp. 36–37).

3. Bayesian inverse (Def. 5.1). For $F\in L^p_\mu(X,\mathcal{P}_p(Y))$ and $\nu=F\odot\mu$, $F_\mu^\dagger\in L^p_\nu(Y,\mathcal{P}_p(X))$ is the $\nu$-almost-surely unique kernel satisfying

   $$\begin{aligned} &\int_X(\delta_x\otimes F(x))\,\mu(dx)\\ &\quad=\int_Y(F_\mu^\dagger(y)\otimes\delta_y)\,(F\odot\mu)(dy). \end{aligned}$$

   Existence follows from Theorem 3.16; the inverse depends on $\mu$. An approximate inverse (§6.1.1) is a kernel $G:Y\to\mathcal{P}_p(X)$ close to $F_\mu^\dagger$ in the MW metric, meaning small

   $$\int_Y W_{p,d_X}(G(y),F_\mu^\dagger(y))^p\,(F\odot\mu)(dy).$$

4. Generalized Bayes law (Theorem 5.4). Let $F:X\to\mathcal{P}_p(Y)$ be continuous and $L^p$-integrable with respect to a prior $\mu_0\in\mathcal{P}_p(X)$. For data $y_1,\ldots,y_n\in\mathrm{supp}(F\odot\mu_0)$, not necessarily distinct, sequential and batch updates agree:

   $$\begin{aligned} \mu_n&=F_{\mu_{n-1}}^\dagger(y_n)\\ &=(F_{\mu_0}^{\otimes n})^\dagger(y_1,\ldots,y_n). \end{aligned}$$

   For every $E\in\Sigma_X$, the ball-ratio formula (9) is

   $$\mu_n(E)=\lim_{s\to0} \frac{\int_E\prod_{k=1}^n F_x(B_s(y_k))\,\mu_0(dx)} {\int_X\prod_{k=1}^n F_x(B_s(y_k))\,\mu_0(dx)}. \tag{9}$$

   Here $B_s(y_k)$ is an open ball of radius $s$. The proof uses countably-simple-kernel approximation (Theorem 3.9), discrete-style Bayes on the approximations, and passage to the limit. It gives a simpler proof under more restrictive continuity hypotheses than Jost et al. (2021).

5. Convergence of the expected update is proved in the dissertation. Def. 5.5 defines

   $$\begin{aligned} \bar\mu_1&=F_{\mu_0}^\dagger\odot\tau,\\ \bar\mu_n&=F_{\bar\mu_{n-1}}^\dagger\odot\tau\quad(n\ge2). \end{aligned}$$

   Def. 5.6 sets $F_\tau(\mu)=(F_\mu^\dagger)_\sharp\tau\in\mathcal{P}_p^2(X)$; the expected posterior is $E_X\circ F_\tau$.

   Proposition 5.7. If $F$ is Lipschitz, $\tau\in F\odot\mathcal{P}_p(X)$, and $F_\tau$ is Lipschitz with constant $0\lt K\lt1$, the expected update converges from any prior to a unique fixed point $\mu^*$ of $E_X\circ F_\tau$, with $F\odot\mu^*=\tau$. No explicit rate is stated. The text leaves determining when the strict contraction hypothesis holds in practice for further work.

   Theorem 5.10. Let $F$ be Lipschitz and regular (Def. 5.9), let $\tau$ and $\mu_0$ have compact supports, and assume

   $$\mathrm{supp}(\tau)\subseteq\mathrm{supp}(F_x) \quad\text{for all }x\in\mathrm{supp}(\mu_0).$$

   Regularity requires joint continuity of $\mathrm{ev}_F(x,s,y)=F_x(B_s(y))$ on $X\times[0,1]\times Y$, and continuously varying bi-Lipschitz contraction maps (Def. 5.8): the radius maps are bi-Lipschitz, differentiable at zero, with derivative jointly continuous in $(x,y)$. Then

   $$\bar\mu:=\lim_{n\to\infty}\bar\mu_n\in\mathcal{P}_p(X), \qquad F_{\bar\mu}^\dagger\odot\tau=\bar\mu.$$

   Theorem 5.10 gives no convergence rate. Its conclusion is an expected-update fixed point. The July 2023 talk predated the write-up: its “90–95% of the way” remark is historical, superseded by Proposition 5.7 and Theorem 5.10.

   Open points (p. 43, §6.1). Exact (non-expected) updates and recovery/identification questions remain open: when does $F_\mu^\dagger\odot\tau=\mu$ imply $F\odot\mu=\tau$, and when is the fixed point a $W_p$-projection of $\tau$ onto the Kleisli image?

B.5 Wasserstein dependence and Wasserstein correlation (ML paper)

Wasserstein dependence (Post-Training…, Def. 2.1), order $p$:

OT analog of mutual information’s $D_{\mathrm{KL}}(\pi\,\|\,\pi^1\otimes\pi^2)$, but with $W_p$ instead of KL.

Wasserstein correlation (Def. 2.2) — normalized to avoid trivial “spread latent codes to infinity”:

where $\pi^1_D$ is the diagonal coupling of the first marginal with itself (self-variance), similarly $\pi^2_D$. Bounded in $[0,1]$ under the paper’s setup.

Sliced versions $\mathrm{SD}_p$, $\mathrm{SC}_p$: replace $W_p$ by $SW_p$. Empirical estimate: joint empirical measure of pairs $(x_i, E_\theta(x_i))$; approximate product by coordinate shuffling (paper cites prior screening work (2023) for the shuffle trick).

Remark (paper Remark 3.5). WaCo maximization alone is a poor generic InfoMax stand-in: it pushes toward approximate (local) isometries / structure preservation, which prevents the semantic clustering contrastive methods want — and that is exactly why it suits post-training invariance (do not scramble a pretrained latent geometry).

B.6 Augmented encoders (the ML “kernel” object of study)

Augmentation. Measurable $t: A\times X\to X$ with parameter law $\nu$ on $A$. Collections $T=\{t_j\}$, weights $w_j$, parameter laws $\nu_j$.

Augmented encoder (paper Eq. 10). For deterministic encoder $E: X\to Z$,

Empirically: for each $x_i$, draw $s$ augmented codes (paper often uses $s=3$) and form a discrete mixture. This is explicitly a Markov–Wasserstein kernel $X\to\mathcal{P}_p(Z)$.

Collision-aware invariance. Exact invariance $E(t(a,x))=E(x)$ for all $a$ can force representational collapse when augmentations create collisions across distinct data points. Paper defines unique / collision-free augmentations $\mathrm{UAug}_{\mu_X}(x,t)$ and $(t,\mu_X)$-invariance only on those (Defs. 3.1–3.3).

Post-training target (Def. 3.4). Given frozen pretrained $F: X\to Z$ and admissible maps $\mathcal{V}$ (e.g. isometries), an adapter $E: Z\to W$ is $(t,\mu_X,F,V)$-invariant if $E\circ F$ is $(t,\mu_X)$-invariant and $(E\circ F)(x)=(V\circ F)(x)$ on $\mathrm{supp}(\mu_X)$ for some $V\in\mathcal{V}$. Goal: add invariants without corrupting the clean latent point cloud.

Collision rate (Def. 3.2). Fraction of augmented features whose nearest clean neighbor in latent space has a different class label. After rigid alignment (orthogonal + translation), residual collision rate predicts whether MaWa optimization is feasible (low → success; high, as on supervised ResNet50 in the paper → MaWa collapse).

B.7 Algorithms / losses

Algorithm A — Markov–Wasserstein (MaWa) minimization / anchored MSE

Practical batch loss (Eq. 21). Freeze $F$. Train $E_\theta: Z\to Z$. For batch $\{x_i\}_{i=1}^N$ and $s$ augments $a_i^k$ per point:

• Term 1: anchor — stay near identity on clean features.

• Term 2: invariance — pull augmented features toward the clean feature of the same $x_i$ (not toward each other in a collapse-prone way).

OT derivation (Eqs. 22–25). Include identity as a trivial augmentation with weights $(\frac{1}{s+1},\frac{s}{s+1})$. Then

and uniqueness of couplings to Diracs reduces $W_2^2$ pointwise to squared Euclidean terms, recovering the batch loss.

When it works. Low aligned collision rate in $F$’s latent space (self-supervised backbones in the paper). Fails / collapses when residual collisions are high (supervised ResNet50 example in the paper).

Algorithm B — Wasserstein correlation (WaCo) maximization

Post-training form (Eq. 27):

From-scratch / dim-reduction form (Eq. 26): replace $F(x)$ by $x$ and allow $E_\theta: X\to Z$ with $Z$ lower-dimensional.

Practice: maximize sliced correlation $\mathrm{SC}_2$ on the empirical joint that mixes clean and augmented pairs (Eqs. 28–29). Optional AE reconstruction upper-bounds $W_p$ by an $L^2$ decode term (Eqs. 30–32).

Properties vs MaWa. Competitive invariance; supports dimensionality reduction; more robust when MaWa’s identity-anchoring fights high collision rates; looser isometry (lower $R^2$ of distance–distance scatter than MaWa in reported tables).

Baselines that fail the post-training goal (empirically in the paper)

• SimCLR-style contrastive loss on adapter outputs: can improve some invariance numbers but corrupts pairwise distance structure ($R^2$ near 0; huge NRMSD).

• HSIC maximization (kernel dependence in the RKHS sense): similarly corrupts geometry; included precisely as an alternative dependence measure.

B.8 Relation to standard OT, Sinkhorn, MMD, kernel methods

B.9 Key empirical results (sourced from Post-Training Augmentation Invariance)

Setting: STL-10; freeze pretrained $F$; train one-hidden-layer MLP adapter; classify with linear / nonlinear heads trained on clean data, evaluate on augmented data.

DINOv2 / DINO features, rotations (Table 1, NC + MaWa): clean accuracy $\approx 98.6\%$; rotated $\approx 94.7\%$ vs end-to-end without adapter $\approx 71.2\%$.

Noise: MaWa NC $\approx 86.2\%$ vs EC $\approx 58.2\%$.

Structure preservation (Tables 2–3, MaWa): near-isometry on clean pairs — e.g. rotation $R^2\approx 0.96$, NRMSD$\approx 0.02$; noise $R^2\approx 0.98$, NRMSD$\approx 0.01$.

SimCLR / HSIC: poor structure metrics (e.g. rotation SCLR $R^2\approx 0.04$).

Collision rates (Tables 9–11): DINOv2 aligned CR $\approx 0\%$ (rotation) / $0.6\%$ (noise); SwAV low after alignment; supervised ResNet50 residual CR $\approx 13\%$ (rotation) / $42\%$ (noise) — MaWa collapses; WaCo still somewhat useful.

Abstract headline numbers (also in abstract): $\sim 94\%$ vs $71\%$ rotation; $\sim 86\%$ vs $58\%$ noise — match the detailed tables (minor rounding).

B.10 What it buys you / limitations (from paper §5)

Buys.

• Add task-relevant invariants (rotation, noise, affine, crop) without fine-tuning foundation weights.

• Preserve clean geometry → downstream probes on clean data stay strong.

• Tiny adapters (1-hidden MLP) suffice when collision structure cooperates.

• Conceptual unity: same MW-kernel language for Bayesian foundations and SSL adapters.

Limits.

• Sample efficiency / convergence slows as augmentation parameter dimension grows.

• MaWa structurally limited by high residual collision rates.

• WaCo theory still thin (paper leaves rigorous sliced-vs-full WC maximization justification to future work).

• Composite vs separate augmentations: training on $\{t_1,\dots,t_m\}$ separately need not yield invariance to $t_2\circ t_1$ unless composites are included.

• Expected-update convergence is proved in Proposition 5.7 and Theorem 5.10; Theorem 5.10 gives no rate, and exact updates and recovery questions remain open (§6.1).

B.11 Worked toy example (computed for these notes; illustrative, not from the paper)

Setup. Input support $X=\{-1,0,1\}$ with $\mu_X$ uniform. Let $K(x)=\mathcal{N}(x, 0.5^2)$ and $L(x)=\mathcal{N}(0.5\,x,\ 0.5^2)$ in $1$D. For $1$D Gaussians, $W_2^2(\mathcal{N}(m_1,s^2),\mathcal{N}(m_2,s^2))=(m_1-m_2)^2$.

Moral. MW distance is an input-averaged Wasserstein distance between the kernels’ output laws. MaWa training is gradient descent on an empirical MW objective that forces an augmented encoder toward a Dirac kernel $\delta_F$ anchored on clean features.

Recorded-talk summary

Metadata. A recorded seminar talk (CodEx seminar, 2023); talk date 2023-07-18; Upload 2023-10-16; duration 3060 s (~51 min); English captions available. Abstract on the CodEx site matches the dissertation theme: introduce MW kernels; integration theory; approximate Bayesian inverses; convergence for a generalized Bayes law.

unverified Transcript source. Cleaned transcript from captions. Auto-captions are noisy (“Wasserstein” → “Buster scene / Vosserstein”, “Bayes” → “Asian / phase”, “Lebesgue” → “vague”, etc.). Below uses corrected technical terms; bracket uncertainty where captions fail.

Timestamped outline

Talk takeaways to internalize

1. The “kernel” is a conditional probability object, metricized by OT.
2. Integration theory is doing real work (disintegration + Bayes), not just category theory aesthetics.
3. Generalized Bayes is designed so approximation of kernels recovers the textbook discrete update — a template for numerical schemes.
4. The July 2023 talk predated the write-up. Expected-update convergence is proved in Proposition 5.7 and Theorem 5.10 under their stated hypotheses; Theorem 5.10 gives no rate. Exact updates and recovery questions remain open.
5. Explicit future path already points at generative modeling and universal approximation in the MW-kernel category — later WaCo/MaWa work is a different but compatible applied branch (representation invariance), not a contradiction.

Other lines of work

1. M.A. thesis (2016). Functorial Results for C*-algebras of Higher-Rank Graphs, Arizona State University, 2016. Earlier pure-math work in operator algebras and categorical graph algebras, not OT.

2. SIMDA MURI challenge write-up (2022). Problem 2: Lagrangian Observation Mapping, SIMDA MURI, 2022. OT registration of satellite sea-ice thickness / Lagrangian observations; a remote-sensing challenge text, not a journal article.

3. Dartmouth ACMS seminar (2024). Wasserstein Dependence and Representation Learning, Applied and Computational Mathematics Seminar, Dartmouth College, November 2024. A thematic companion to the SIAM item; no separate paper under this title located.

4. PhD dissertation (2023). Bayesian Inference for Markov Kernels Valued in Wasserstein Spaces, Arizona State University, 2023. Dissertation (2023-10-24 public copy, 57 pp.) read for this revision via the addendum.

5. SIAM MDS 2024 — title conflict. Predicting in Latent Space with Wasserstein Dependence-Maximizing Autoencoders per the SIAM meetings listing; Maximizing Multiview Wasserstein Dependence in Autoencoders per later public listings. SIAM Conference on Mathematics of Data Science, Atlanta, October 2024. Both titles are retained without reconciliation. The theme is Wasserstein-dependence autoencoders and latent prediction; full slides remain unverified.

6. Post-Training Augmentation Invariance (TMLR 2026; arXiv:2505.11702). Primary readable source for MaWa / WaCo / augmented encoders / experiments. OpenReview: https://openreview.net/forum?id=Z4uUwU6zRe. PDF: https://arxiv.org/pdf/2505.11702. Earlier public title/version: Invariant Representations via Wasserstein Correlation Maximization (arXiv html v1).

7. Epidemiology side paper (distinct research line). To mask or not to mask: Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic, Infectious Disease Modelling 5 (2020) 248–255; arXiv:2004.03251. Compartmental epidemic model; mask effectiveness × coverage; nonlinear impact on deaths/hospitalizations. Not OT / MW-kernel work — include only as “other published line,” not as part of the OT-on-kernels deep dive.

8. Code artifacts. The paper points to repositories implementing augmentation invariance / WaCo experiments.

Timeline of the research line

Discussion questions

Technical (grounded)

1. In one sentence, what is a Markov–Wasserstein kernel, and how does $\mathrm{MW}_p$ differ from simply computing $W_p$ between two pushforward laws $K_\sharp\mu$ and $L_\sharp\mu$?
2. Why put Wasserstein structure on the values of a Markov kernel rather than using total variation or KL between $K(x)$ and $L(x)$ inside the $L^p$ integral?
3. Walk through how the Markov–Lebesgue integral is supposed to clarify the disintegration theorem — what is the new conceptual claim versus the classical iterated-integral/disintegration statement?
4. State the generalized Bayes update in the continuous MW-kernel setting, and show how a simple-kernel approximation recovers discrete Bayes.
5. Which hypotheses in Proposition 5.7 and Theorem 5.10 restrict their applicability: contraction of $F_\tau$, regularity, compact supports, and the support condition? What recovery questions remain open?
6. Derive why couplings to a Dirac reduce MaWa’s $\mathrm{MW}_2^2$ objective to anchored squared Euclidean loss (Eqs. 24–25).
7. Contrast Wasserstein correlation with mutual information and with HSIC: what geometry does each promote in latent space?
8. Why include the identity map as a trivial augmentation with weight $1/(s+1)$ in MaWa, and what happens to collapse if one instead minimizes $\|E(t(a,x))-E(x)\|^2$ without an anchor?
9. Explain collision rate (raw vs rigidly aligned). Why does aligned collision rate predict MaWa success across DINOv2 / SwAV / supervised ResNet50 in the paper?
10. For WaCo, what is still missing for a theorem that “maximizing sliced WC implies maximizing WC,” analogous to InfoNCE bounds for MI?

Direction / applications

11. Where should the foundational Bayes-convergence theory and the post-training adapter methods meet computationally (particle methods, sliced projections, dual potentials)?
12. The paper flags noise invariance → adversarial robustness without adversarial training. What threat model would make that claim precise?
13. How should composite augmentations ($t_2\circ t_1$) be scheduled versus separate $t_j$ to get invariance to realistic corruptions?

Bridges to quant finance / nowcasting / OT portfolios (discussion prompts)

14. Macro / nowcasting: OT nowcasting (the lab's DFM × Sinkhorn track) transports mass between cross-sectional economic distributions over time (Sinkhorn-type couplings). How would one phrase a Markov kernel $x_t \mapsto \pi(\cdot\mid x_t)$ whose values live in $\mathcal{P}_p$ of cross-sections — and when is $\mathrm{MW}_p$ a better calibration loss than a single $W_p$ between consecutive barycenters?
15. ETF / allocation panels: Treat each rebalance date’s weight vector (or return kernel across scenarios) as a measure. Could a frozen factor encoder $F$ plus a MaWa/WaCo adapter enforce invariance to “augmentations” that matter in markets (currency quoting conventions, corporate actions, mild microstructure noise) without destroying the clean factor geometry used for allocation?
16. Sinkhorn vs WaCo: Entropic OT (Sinkhorn) yields a coupling between two fixed measures. WaCo maximizes dependence of a joint induced by a parameterized kernel. When is each the right tool for stress-testing a portfolio across scenario measures?
17. Stability: The talk emphasizes stability of Bayesian inverses under perturbations of prior/forward map. Map that language onto posterior robustness of a nowcasting kernel when the observation channel (data revisions) is misspecified.

Terms to know

• Markov kernel vs RKHS kernel

• $\mathcal{P}_p$, $W_p$, coupling, dual transport formulation

• Sliced Wasserstein

• Disintegration

• Wasserstein dependence / correlation

• Markov–Wasserstein metric $\mathrm{MW}_p$

• Augmented encoder

• Post-training $(t,\mu_X,F,V)$-invariance

• Collision rate (aligned)

• MaWa vs WaCo vs SimCLR vs HSIC

Glossary

• Markov kernel. Measurable map $K: X\to\mathcal{P}(Y)$ (conditional distributions).

• Wasserstein space. $(\mathcal{P}_p(Y), W_p)$.

• Markov–Wasserstein (MW) kernel. Markov kernel viewed as an $L^p$ map into a Wasserstein space; metric $\mathrm{MW}_p$.

• Markov–Lebesgue integral. Defs. 3.1–3.2 integrate countably simple kernels as $\sum_j\mu(E_j)\nu_j$. Def. 3.4 specifies weakly Cauchy sequences; Theorem 3.9 supplies uniform-$W_p$, weakly Cauchy approximations. Def. 3.11 takes their integral limit; Theorem 3.12 identifies it with iterated integrals. See B.4 for formulas.

• Disintegration. Theorem 3.16: an $L^p$ joint law equals Markov–Lebesgue integrals of conditional kernels against either marginal, literally; see B.4 for the equality.

• Bayesian inverse map. Def. 5.1: the prior-dependent inverse is a disintegration of the joint law; §6.1.1 defines an approximate inverse by small integrated $W_p^p$ error. See B.4 for formulas.

• Wasserstein dependence $\mathrm{WD}_p$. $W_p$ distance from a joint to the product of its marginals.

• Wasserstein correlation $\mathrm{WC}_p$. Variance-normalized dependence in $[0,1]$.

• Sliced Wasserstein / SC. Monte Carlo projections used for tractable WaCo training.

• Augmented encoder $E^T$. Mixture/kernel of encodings of random augmentations of $x$.

• Post-training augmentation invariance. Add invariants with frozen $F$ via adapter $E_\theta$ while preserving clean latent geometry up to admissible $V$.

• MaWa loss. Anchored MSE = empirical $\mathrm{MW}_2^2$ to $\delta_F$.

• WaCo loss. Maximize (sliced) WC of the joint induced by the augmented encoder.

• Collision rate. Frequency that an augmented feature’s nearest clean neighbor has another class; aligned version after rigid registration.

• HSIC. RKHS dependence criterion used as a failing baseline for the post-training geometry goal.

• Sinkhorn distances. Entropic OT couplings (related literature; not the MaWa closed form).

Sources

Papers / abstracts / venues

• Bayesian Inference for Markov Kernels Valued in Wasserstein Spaces, PhD dissertation, Arizona State University, 2023 — dissertation (read for this revision). Source: dissertation addendum; public copy dated 2023-10-24, 57 pp. Supplies the definitions, theorems, and corrections in B.3–B.4.

• Post-Training Augmentation Invariance, arXiv:2505.11702 (v3 2026-06-08); Transactions on Machine Learning Research, 2026; OpenReview https://openreview.net/forum?id=Z4uUwU6zRe; PDF https://arxiv.org/pdf/2505.11702. Primary source for §B.5–B.10.

• Earlier arXiv html version titled Invariant Representations via Wasserstein Correlation Maximization: https://arxiv.org/html/2505.11702v1.

• unverified Predicting in Latent Space with Wasserstein Dependence-Maximizing Autoencoders, SIAM MDS 2024 abstract: https://meetings.siam.org/sess/dsp_talk.cfm?p=139135.

• To mask or not to mask…, Infectious Disease Modelling (2020); arXiv:2004.03251.

Talk sources

• A recorded seminar talk (CodEx seminar, 2023): seminar abstract, metadata, captions, and cleaned transcript.

Background references

• Topics in Optimal Transportation (book).

• Sinkhorn distances (2013).

• HSIC (2005).

• SimCLR (2020).

Verification gaps

1. Dissertation PDF now read via the addendum. Definitions and theorems are verified from that source. Remaining gaps: no rate for Theorem 5.10; exact (non-expected) updates and recovery/identification questions remain open (p. 43, §6.1).
2. Dissertation statements supersede the noisy caption summaries: Theorem 2.25 is the isometry, Theorem 4.10 is the monad result, and Theorem 5.4 is generalized Bayes.
3. unverified SIAM 2024 talk: abstract only; title conflict between the SIAM listing and later public listings; no full slides verified.
4. unverified Whether arXiv:2505.11702 fully supersedes the SIAM dependence-AE story is inferred, not author-confirmed.
5. unverified No attempt was made to re-run the paper’s STL-10 experiments; numbers are quoted from the paper’s tables/abstract.