Dynamic OAT in Sample–Velocity Space

• Dynamic OAT is a framework that jointly models sample and velocity spaces using second-order dynamics to optimize trajectories.
• It extends classical Optimal Transport by incorporating acceleration cost, enabling precise interpolation and transport in phase space via cubic trajectory minimizers.
• Its applications in super-resolution imaging, dynamic tomography, and generative modeling highlight improved recovery guarantees and computational efficiency.

Dynamic Optimal Acceleration Transport (OAT) in Sample–Velocity Space (also referred to as phase space) concerns the joint modeling, analysis, and optimization of trajectories and distributions in the product space of physical position and velocity. This framework enables simultaneous estimation, interpolation, and transport of sample–velocity pairs under dynamical measurement models and generalizes classical Optimal Transport (OT) theory by incorporating second-order (acceleration) control. Dynamic OAT underpins advanced methods in super-resolution imaging, dynamic tomographic reconstruction, and next-generation generative modeling via flow matching, by directly leveraging the geometry of sample–velocity space (Alberti et al., 2018, Yue et al., 29 Sep 2025).

1. Mathematical Formulation of Dynamic OAT in Phase Space

The dynamic OAT problem is posed in the product space $X \times V$, where $X \subseteq \mathbb{R}^d$ is the domain of samples (e.g., spatial positions) and $V \subseteq \mathbb{R}^d$ is the velocity space. Given two probability measures $\mu_0, \mu_1 \in P(X \times V)$, the objective is to transport $\mu_0$ to $\mu_1$ along curves in $(x, v)$-space governed by deterministic, second-order dynamics $({\dot{x}}=v,\, {\dot{v}}=a)$, optimizing an action functional:

$$S[\mu, a] = \int_0^1 \int_{X \times V} \frac{1}{2} \mu(x, v, t) \|a(x, v, t)\|^2 \, dx \, dv \, dt$$

subject to the Vlasov continuity equation:

$$\partial_t \mu + v \cdot \nabla_x \mu + \nabla_v \cdot (a \mu) = 0,\quad \mu(\cdot, \cdot, 0) = \mu_0,\, \mu(\cdot, \cdot, 1) = \mu_1.$$

The minimal action $A_2^2(\mu_0, \mu_1)$ gives the optimal squared acceleration transport cost (Yue et al., 29 Sep 2025).

This generalizes static OT: in the limit where velocities are fixed ($a\equiv 0$), this reduces to Benamou–Brenier geodesics.

2. Kantorovich-Type Reformulation and Action Cost Structure

Dynamic OAT admits a static coupling reformulation. The minimal action equals

$$A_2^2(\mu_0, \mu_1) = \min_{\pi \in \Pi(\mu_0, \mu_1)} \mathbb{E}_{(z_0, z_1)\sim\pi} [ c_{A^2}(z_0, z_1) ]$$

with $z_i = (x_i, v_i)$ and pairwise cost

$$c_{A^2}(z_0, z_1) = 12 \Bigl\| \frac{x_1-x_0}{T} - \frac{v_0 + v_1}{2} \Bigr\|^2 + \|v_1 - v_0\|^2.$$

This cost enforces alignment between endpoint displacement and averaged velocity, penalizing both deviation from constant-velocity transport and velocity mismatch (acceleration) (Yue et al., 29 Sep 2025).

3. Trajectory Optimality, Cubic Interpolation, and Flow Straightness

For endpoint pairs $(x_0, v_0), (x_1, v_1)$, the deterministic OAT problem

$$\min_{x(\cdot), v(\cdot)} \frac{1}{2} \int_0^1 \|\ddot{x}(t)\|^2 dt,\;\text{with}\; x(0)=x_0, x(1)=x_1, v(0)=v_0, v(1)=v_1$$

has solutions that are coordinatewise cubic polynomials, determined by the four boundary conditions. The trajectory is straight (i.e., the cubic traces a line segment) if and only if $v_0$ and $v_1$ are collinear with $x_1 - x_0$. The necessary and sufficient condition for flow straightness is constant velocity direction and parallel acceleration (Yue et al., 29 Sep 2025). Constant-velocity transport ($v_0 = v_1$) minimizes both action and total displacement, recovering the first-order OT geodesic.

4. OAT in Inverse Problems: Dynamic Super-resolution and Tomography

Dynamic OAT provides a rigorous framework for the recovery of positions and velocities of moving sources from time-resolved tomographic or imaging data. A canonical application is dynamic optoacoustic tomography (OAT) with moving point absorbers (spikes), whose initial pressure field at time $t$ is modeled by

$$p_t(x) = \sum_{j=1}^N a_j \delta(x - (x_j + v_j t)),$$

with unknown $(x_j, v_j, a_j)$. After beamforming and sampling, one acquires spatiotemporal linear measurements $y_{\ell, k} = \langle \phi_\ell, p_{t_k} \rangle$, which can be re-expressed as

$$y_{\ell, k} = \langle \Phi_{\ell, k}, \mu \rangle,\quad \Phi_{\ell, k}(x, v) = \phi_\ell(x + kT v),$$

where $\mu = \sum_j a_j \delta((x, v) - (x_j, v_j))$ is an atomic measure in phase space (Alberti et al., 2018).

Simultaneous recovery is performed via continuous-domain total-variation minimization (atomic-norm minimization)

$$\min_{\mu \in M(X \times V)} \|\mu\|_{TV}\;\; \text{s.t.}\; G\mu = y,$$

with the dual problem producing certificates for exact and stable recovery, subject to static separation and no “ghost trajectories” (Alberti et al., 2018).

5. Algorithms, Complexity, and Numerical Aspects

Dynamiс OAT in sample–velocity space admits several practical algorithms, including:

• Discretization of $X\times V$ and solution via large-scale linear programming (basis pursuit).
• Continuous-domain solvers, notably conditional gradient (Frank–Wolfe) methods, supporting efficient recovery without explicit discretization.
• For measurement operators with Fourier structure, FFTs accelerate convolutional computations, e.g., dual polynomial evaluation (Alberti et al., 2018).

Complexity scales with the number of measurements, iterations, and grid resolution in discretized solvers. For dynamic flow matching in generative modeling, OAT–Flow Matching (OAT-FM) leverages efficient OT solvers (e.g., Sinkhorn, LP) and alternates lower-level coupling updates with upper-level velocity-field minimization. The computational cost per minibatch remains $O(B^2 \log B)$ for batch size $B$ (Yue et al., 29 Sep 2025).

6. Applications in Imaging and Generative Modeling

Imaging

Dynamic OAT phase-space approaches have been validated on ultrafast ultrasound localization microscopy. There, the methodology replaces the measurement kernel with clutter-filtered Gaussian point spread functions and recovers super-resolved vessel positions and flow velocities from M-mode data. Results demonstrate simultaneous super-resolved localization (beyond the diffraction limit) and velocity estimation, outperforming two-step static+tracking pipelines (Alberti et al., 2018).

In dynamic OAT, practical adaptations involve depth-varying PSFs, accommodation of heterogeneous speed-of-sound via ray-based models, and technical constraints to avoid ghost trajectories (Alberti et al., 2018).

Generative Modeling

OAT has been adopted in the flow matching paradigm for generative modeling. OAT-FM fine-tunes pretrained flow-matching models by optimizing over endpoint couplings in sample–velocity space to minimize second-order costs. The two-phase OAT-FM paradigm—standard flow matching followed by OAT-FM fine-tuning—consistently improves metrics such as FID, NFE, and $W_2^2$ distance in benchmarks including CIFAR-10 and ImageNet, while preventing distribution drift since off-distribution samples are not required (Yue et al., 29 Sep 2025).

7. Theoretical Guarantees and Recovery Conditions

Dynamic OAT in phase space supports precise recovery guarantees:

• Exact recovery: If, at all time frames, moving source positions are separated by at least a constant over spatial bandwidth and there are no ghost trajectories, the atomic-norm minimization exactly recovers position–velocity–amplitude tuples (Alberti et al., 2018).
• Stability: In the presence of bounded noise, the error in the recovered measure is controlled by a power-law in the spatial super-resolution factor, with graceful degradation (Alberti et al., 2018).
• Flow straightness and minimizers: The unique minimizer of the cubic trajectory variational problem is characterized by the straightness condition. OAT-FM loss provides provable lower bounds to the second-order transport cost, and equality holds iff the velocity is constant along the path (Yue et al., 29 Sep 2025).

OAT thus generalizes first-order OT, bridges static/dynamic inverse problems and learning, and yields efficient, theoretically grounded algorithms for challenging sample–velocity inference tasks.