Block-Wise Control of Wasserstein Distances

• Block-wise control of Wasserstein distances is the decomposition of transport costs over partitioned domains, allowing independent mass adjustments and transport among blocks.
• This approach employs generalized Wasserstein metrics, duality principles, and projection algorithms to transform high-dimensional optimal transport problems into decoupled, manageable sub-problems.
• Applications span decentralized multi-agent control, high-dimensional statistical learning, and noncommutative operator algebras, offering improved computational efficiency and robust system analysis.

Block-wise control of Wasserstein distances refers to the decomposition, quantification, and manipulation of Wasserstein costs across partitioned domains, subspaces, or algebraic components (blocks) in optimal transport problems. This concept arises naturally in both classical and noncommutative settings, with importance in areas such as multi-agent control, distributed resource management, high-dimensional statistical learning, and the geometry of operator algebras. Block-wise control enables the separation of transport costs over subsystems or data blocks, affording both theoretical insight and computational leverage in complex or high-dimensional domains.

1. Generalized Wasserstein Distances and Block Decomposition

The generalized Wasserstein distance $W_p^{a,b}$ was introduced to address a major intrinsic limitation of classical Wasserstein distances $W_p$: the restriction to pairs of measures with equal total mass. In block-wise control settings—where different “blocks” (subsystems) can independently create or destroy mass—this is inadequate.

For measures $\mu$, $\nu$ on a domain and parameters $a, b > 0$, $p \ge 1$, the generalized Wasserstein distance is defined by

$W_p^{a,b}(\mu,\nu) = [ T^{a,b}(\mu, \nu) ]^{1/p},$

where

$$T^{a,b}(\mu,\nu) = \inf \left\{ a(|\mu - \tilde\mu| + |\nu - \tilde\nu|)^p + b\, W_p^p(\tilde\mu, \tilde\nu) \right\},$$

with the infimum over intermediary measures $\tilde\mu \leq \mu$, $\tilde\nu \leq \nu$.

This decomposition reflects a “block” structure: the first term quantifies mass variations (creation/destruction per block), whereas the second quantifies the conventional mass transport (between matched parts). For $p = 2$, the generalized Benamou–Brenier formula gives a dynamic action functional separating the costs, incorporating a source term $h$ for local mass control: $$B^{a,b}[\mu, v, h] = a^2 \int_0^1\left( \int |h_t| \right)^p dt + b^2 \int_0^1\int |v_t(x)|^2 d\mu_t(x) dt.$$ Block-wise controllers in distributed resource networks, for example, can then solve optimization problems minimizing

$$\sum_\text{blocks} \bigl[ a^2 \text{(mass creation/removal)} + b^2 \text{(transport energy)} \bigr],$$

transforming the global problem into summations over “block” costs, each controlled by the respective actions for mass adjustment and block-to-block transport.

2. Duality Structures and the Flat Metric

The Kantorovich–Rubinstein duality for the Wasserstein $W_1$ distance is classically stated as

$$W_1(\mu, \nu) = \sup\left\{ \int f\, d(\mu - \nu) : f\in \text{Lip}_1(\mathbb{R}^d) \right\},$$

where $\text{Lip}_1$ denotes the class of Lipschitz functions with constant $\leq 1$. In the generalized setting, for $p = 1$ and $a = b = 1$,

$$W_1^{1,1}(\mu, \nu) = \sup\left\{ \int f\, d(\mu - \nu) : \|f\|_\infty \leq 1, \mathrm{Lip}(f) \leq 1 \right\},$$

which exactly equals the flat metric $\|\mu - \nu\|_\text{flat}$. In block-wise control, this duality is significant: for complex domains partitioned into blocks, choosing appropriate test functions $f$ that are block-separating (or localized) provides direct tools for bounding distances, error estimates, or stability criteria of decentralized controllers.

Duality transforms a possibly high-dimensional minimization over transport plans into a maximization over test functions, which is advantageous for both analysis and computation, especially when performance guarantees are required for each block independently.

3. Algorithms for Block-Wise (Subspace or Fibre-Wise) Wasserstein Control

Block-wise decompositions are computationally realized in several forms, all translating the global OT problem into a collection of coupled or decoupled lower-dimensional problems.

a) Subspace Robust Wasserstein Distances

In high dimensions, the Subspace Robust Wasserstein (SRW) distance leverages projections. The max–min and min–max variants have the form: $$P_k(\mu,\nu) = \sup_{E\in G_k} W(P_{E\#}\mu, P_{E\#}\nu),\qquad S_k(\mu,\nu) = \inf_{\pi\in\Pi(\mu,\nu)} \sup_{E\in G_k} \left[ \int \|P_E(x-y)\|^2 d\pi \right]^{1/2},$$ where $G_k$ is the Grassmannian of $k$-dimensional subspaces. Through efficient entropic regularization and projection-based updates (e.g., via Dykstra or Frank–Wolfe algorithms), the computation of $S_k$ and $P_k$ is tractable, allowing one to control the “block”-indexed transport cost and focus on most discriminative subspaces.

b) Noncommutative Block-Wise Control via Disintegration

In noncommutative geometry (e.g., tracial $W^*$-algebras), $L^2$-Wasserstein distances are “disintegrated” into a family of fibre-wise problems. For an algebra $A \cong C(X)\otimes M_n(\mathbb{C})$, the global Wasserstein metric is decomposed as

$$W_2(P,Q) = \int_X W_{2,x}\Big(O_{p(x)}^2P(x), O_{p(x)}^2 Q(x)\Big)\, dv_p(x),$$

where $W_{2,x}$ are block (fibre)-wise Wasserstein distances in the (finite-dimensional) fibres. Mean curvature and entropy regularity at the global level are controlled via the essential infimum over the fibre-wise (block) curvatures.

This structure is critical, since tractability and sharp estimates for curvature, transport plans, or regularity are often available in each block.

4. Statistical and Computational Aspects of Block-Wise Control

Statistically, additive and subadditive properties of Wasserstein distances underpin their suitability for block-wise control: $$W_2^2\left( \bigotimes_{i=1}^n \mu_i,\, \bigotimes_{i=1}^n \nu_i \right) = \sum_{i=1}^n W_2^2(\mu_i, \nu_i).$$ This property allows both error bounds and performance guarantees to decompose over blocks, ensuring global control via local (block-specific) controls.

For computational efficiency, slicing (SW, SGW), orthogonal Monte Carlo projections, and control variate estimators all efficiently approximate the Wasserstein cost by aggregating one-dimensional or block-wise computations. Such techniques are especially valuable for variance reduction and fast convergence in high-dimensional or distributed settings.

In projection-based OT (including sliced and min-SWGG schemes), identification and control over the “most significant blocks” (directions, subspaces) is achieved via bilevel optimization or variance-reduced gradient estimators, yielding block-wise accuracy with linear or sublinear complexity per block.

5. Applications Across Distributed, High-Dimensional, and Noncommutative Systems

Block-wise control is central in a variety of applications:

• Control of distributed systems: Each region or subsystem (block) can have independent sources or sinks, and transport (communication, migration) occurs between blocks. Controllers minimize joint action:

$$\min \sum_\text{blocks} \left[ a^2 (\text{mass adjustment}) + b^2 (\text{transport cost}) \right]$$

subject to generalized continuity equations.

• Statistical learning on block-partitioned data: Additivity and subadditivity enable parallelization, robustness to block-specific perturbations, and error control via block-level deviations.
• Noncommutative metric geometry: Disintegration theorems (for $C^*$-algebras) show that global transport can be assembled from fibre (block) contributions, a principle essential to extending classical OT theory to operator algebraic models.
• Shape, graph, and structured data analysis: Multi-marginal and block-fused GW transport uses block coupling costs $c_{ij}$ to match parts (blocks) with fidelity, enabling, for instance, the construction of barycenters reflecting block-wise structure.

6. Theoretical and Practical Limitations

While block-wise Wasserstein control offers substantial flexibility, several challenges arise:

• Increased computational complexity: Particularly for large numbers of blocks or intricate source terms (e.g., control of sources and sinks), the dimensionality and coupling of block-wise problems can become severe.
• Parameter selection: Weights (e.g., $a$, $b$ in $W_p^{a,b}$), block partitioning, and regularization parameters must be calibrated to balance mass adjustment and transport, often requiring data-driven methods or model-specific analysis.
• Regularity assumptions: Derivations of Benamou–Brenier type formulas and noncommutative extensions can demand strong smoothness or uniform continuity for the relevant vector fields or derivations, posing limitations in discontinuous or highly localized control scenarios.

7. Summary and Outlook

Block-wise control of Wasserstein distances encompasses a broad range of methodologies enabling the decomposition, efficient computation, and robust estimation of Wasserstein costs in partitioned, networked, or high-dimensional domains. Generalized Wasserstein distances, dynamic and dual formulations (Benamou–Brenier, Kantorovich–Rubinstein duality), projection/subspace-based strategies, and noncommutative disintegration form the foundational techniques underpinning this area.

Through these tools, one can flexibly manage mass transport, creation, and deletion at the block level, design decentralized and robust controllers, and develop scalable algorithms for complex or structured data analysis. Continuing developments focus on enhanced computational scalability, deeper theoretical guarantees for regularity and curvature, and the design of block-adaptive algorithms in both classical and operator-algebraic OT frameworks.