Trajectory-Dependent Diffusion Distances

• Trajectory-dependent diffusion distance is a geometric metric that quantifies how random walk trajectories traverse physical domains and data graphs based on their sequential history.
• It generalizes classical metrics like the mean-chord-length by accounting for process dynamics through inhomogeneous, multi-view diffusion operators.
• The approach provides practical insights for imaging, transport engineering, and manifold learning, validated via theoretical analysis and experimental studies.

Trajectory-dependent diffusion distance refers to a class of geometric quantities that measure how diffusion processes traverse a domain or a data set, with the specific metric depending on the history or sequence (trajectory) of random walk steps—either in continuous physical domains or on data graphs constructed from multiple data views. Two principal lines of research have developed rigorous, trajectory-dependent diffusion distance notions: (1) the geometric average path length for exit trajectories in physical domains, independent of microscopic transport properties (Blanco et al., 2019), and (2) the multi-view diffusion distances in data manifolds, where the generative random walk process chains together heterogeneous diffusion operators in a prescribed trajectory (Debaussart-Joniec et al., 1 Dec 2025).

1. Geometric Trajectory-Dependent Diffusion Distance in Physical Domains

In the context of physical diffusion, the trajectory-dependent diffusion distance is captured by the mean length ⟨L⟩ of a purely diffusive random walk from a point of first entry to a point of first exit on a bounded domain Ω with boundary Σ. Remarkably, for isotropic, uniform incidence, ⟨L⟩ is invariant under arbitrary internal heterogeneity of the mean free path λ(x) and scattering phase function p(u_s, u_i) (assuming pure scattering and no absorption):

$\langle L \rangle = \frac{4\,\Omega}{\Sigma}$

Here, Ω is the volume (or in 2D, area S), and Σ is the boundary surface area (or in 2D, perimeter P), generalizing the classical mean-chord-length result from ballistic (straight-line) transport to arbitrary diffusive systems (Blanco et al., 2019). This invariance holds under the following precise modeling assumptions:

• Isotropic, cosine-law incidence at each boundary point.
• Constant velocity motion between scattering events.
• Arbitrarily heterogeneous λ(x) and scattering kernel p, but with strictly conservative (no absorption) dynamics.
• Entrance events at statistical equilibrium, ensuring loss of memory of initial conditions.
• First-exit length is the length from entry to exit, with no multiple boundary crossings considered.

This average length—a trajectory-dependent diffusion distance—characterizes the effective geometric “thickness” or “leakiness” of the domain for escaping random walks, independent of microscopic details. In two dimensions, the result is ⟨L⟩ = π S / P, matching the chord-length formula.

2. Two-way Absorption Argument and Derivation

The derivation proceeds by embedding the domain Ω in a larger, statistically equilibrated medium with uniform intensity f₀, then introducing a weak, uniform absorber (μₐ):

1. Volume Integral Form: The total absorption rate A is given by integrating the local rate μₐ v (4π f₀) over Ω, yielding $A = 4\pi v f_0 \mu_a \Omega$.
2. Boundary-Flux Form: Under isotropic incidence, the incoming flux is π v f₀ Σ, and the probability of absorption along a length L is 1 − e^{-μₐL}. Integrating over path-length distribution p(L):

$$A = \pi v f_0 \Sigma \int [1-e^{-\mu_a L}]\,p(L)\,dL$$

1. Equate and Take Limit: Equating the two forms and letting μₐ → 0, the ratio yields

$\langle L \rangle = \frac{4\Omega}{\Sigma}$

This invariance property is a generalization of classical results from transport theory, and Monte Carlo simulations confirm its validity even as path statistics (e.g., λ) are varied (Blanco et al., 2019).

3. Trajectory-Dependent Diffusion Distance in Multi-View Data via Inhomogeneous Random Walks

In data analysis, trajectory-dependent diffusion distance arises in the context of multi-view diffusion geometry. Let $P = \{P^{(1)}, ..., P^{(V)}\}$ be a collection of row-stochastic random-walk operators derived from V different data views over N points. A multi-view diffusion trajectory (MDT) of length t is defined as a sequence:

$\tau = (P_1, P_2, ..., P_t),\quad P_i \in P$

The associated trajectory-dependent diffusion operator is the left product:

$P^{(\tau)} = P_t \cdots P_1$

For fixed τ, the trajectory-dependent diffusion distance between points i and j is:

$$d_\tau^2(i, j) = \sum_{k=1}^N \frac{1}{\pi_\tau(k)} [P^{(\tau)}_{ik} - P^{(\tau)}_{jk}]^2$$

where π_τ is the unique stationary distribution of P^{(\tau)} (assuming aperiodicity and irreducibility of the constituent P^{(v)}).

The embedding derived from the leading singular vectors of P^{(\tau)} preserves the diffusion distance:

$\|\Psi_\tau(i) - \Psi_\tau(j)\|_2 = d_\tau(i, j)$

This framework generalizes diffusion maps to allow for inhomogeneous, trajectory-dependent Markov processes, fusing structural information from multiple views in a precise, tunable sequence (Debaussart-Joniec et al., 1 Dec 2025).

4. Theoretical Properties and Metric Interpretation

The trajectory-dependent diffusion distance d_τ has the following theoretical guarantees (Debaussart-Joniec et al., 1 Dec 2025):

• Metric Property: d_τ is a true distance on the set of points for any fixed τ.
• Stationarity and Ergodicity: If each P^{(v)} is irreducible and aperiodic, P^{(\tau)} maintains these properties, ensuring a unique π_τ and ergodicity of the inhomogeneous chain.
• Spectrum and Embedding: The singular values of P^{(\tau)} need not decay monotonically in t, reflecting inhomogeneity. The “singular entropy” curve is used to select the embedding dimension.
• Probabilistic and Geometric Views: P^{(\tau)}_{ik} is the probability of reaching node k after following τ from node i. d_τ(i, j) quantifies how similarly pairs i, j “diffuse” throughout the full data graph when traversing the prescribed trajectory.

5. Practical Computation and Experimental Validation

Computation of trajectory-dependent diffusion distances proceeds as follows (Debaussart-Joniec et al., 1 Dec 2025):

1. Compute a kernel (e.g., Gaussian) for each view and normalize to a stochastic matrix P^{(v)}.
2. Choose a trajectory τ of views (or convex combinations).
3. Form the operator P^{(\tau)} by matrix multiplication.
4. Compute the singular value decomposition (SVD) of P^{(\tau)} and retain leading components.
5. Embed the data and compute pairwise Euclidean distances in this embedding, yielding d_τ(i, j).

This procedure enables manifold learning and clustering tasks, allowing systematic variation or learning of τ to optimize representation. Experimental results on both synthetic and real data sets demonstrate that various MDT strategies (e.g., random, convex, beam-searched, directly optimized in convex hull) achieve or surpass state-of-the-art multi-view diffusion and clustering performance, often surpassing classical single-view or alternating diffusion methods (Debaussart-Joniec et al., 1 Dec 2025).

6. Applications and Interpretational Significance

In physical systems, the geometric trajectory-dependent diffusion distance D_diff = 4Ω/Σ gives a direct estimator for average residence times or absorption yields, useful for imaging and transport engineering scenarios without recourse to detailed Monte Carlo simulation (Blanco et al., 2019).

In multi-view data analysis, MDT-based trajectory-dependent diffusion distances enable:

• Flexible uncovering of joint and consensus structure across views.
• Tunable locality or globality of embeddings depending on τ.
• Quantitative comparison and evaluation of diffusion-based approaches via stochastic and optimized MDTs.
• Robustness to heterogeneous or noisy modalities, as some MDTs yield near-optimal geometry even under random sampling (Debaussart-Joniec et al., 1 Dec 2025).

7. Connections to Classical Results and Generalizations

The invariance ⟨L⟩ = 4Ω/Σ in physical diffusion generalizes the ballistic, mean-chord-length formula (Case-Zweifel) to arbitrary, conservative random walks (Blanco et al., 2019). In multi-view data, trajectory-dependent diffusion distances generalize the single-view, time-homogeneous diffusion map framework by considering inhomogeneous chains, wherein the stepwise dynamics are parameterized by the trajectory itself.

A plausible implication is that both physical and data-driven versions of trajectory-dependent diffusion distance encode a fundamental geometric summary of the interplay between process dynamics (random walk, Markov chain) and ambient structural constraints (domain geometry or data manifold topology). This aligns both lines of research as geometric-statistical invariants emergent from the interaction of process and structure.