Dimension-Decaying Diffusion Process

• Dimension-decaying diffusion process is a stochastic system that progressively reduces the effective state space by absorption at boundaries, with key implications for probability theory and machine learning.
• The process utilizes singular drift and modified diffusion coefficients to ensure that once it reaches lower-dimensional faces, the dynamics lock into reduced systems, as evidenced by scaling limits from condensing zero-range processes.
• Algorithmic implementations in generative models decompose signals into orthogonal subspaces, iteratively attenuating less significant components to reduce computational cost and improve synthesis quality.

A dimension-decaying diffusion process is a stochastic system characterized by progressive reductions in the effective dimension of its state space as it evolves, typically via absorption at boundaries which transforms the dynamics onto lower-dimensional manifolds. This phenomenon has rigorous realization as the scaling limit of condensing zero-range processes on finite sets, direct connections to spectral dimension in multi-scale spacetimes, and practical implications for generative models on data manifolds. Mathematical formulations, generator structures, absorption mechanisms, and manifold-based interpretations unify this class of processes and illuminate their significance in probability theory, statistical physics, and machine learning.

1. Mathematical Formulation and State-Space Structure

Let $S$ be a finite index set with cardinality $p$, and consider the simplex

$$\Sigma = \left\{ (x_1,\ldots,x_p) \in \mathbb{R}^p_+ : \sum_{i \in S} x_i = 1 \right\}$$

as the state space of the process. For each nonempty subset $A \subset S$, define the face

$$\Sigma_A = \left\{ x \in \Sigma : x_j = 0\ \forall j \notin A,\ \sum_{i \in A} x_i = 1 \right\},$$

which partitions the boundary $\partial\Sigma$ into lower-dimensional simplices. The process evolves on $\Sigma$, but upon reaching any face $\Sigma_A$, it remains confined to it and continues as a diffusion process with modified parameters in the lower-dimensional simplex $\Sigma_A$ (Beltrán et al., 6 Jan 2026).

2. Diffusion Generator and Absorption Dynamics

The generator $L$ for the dimension-decaying diffusion has both singular drift and reversible diffusion terms:

$$(L F)(x) = \nabla_{b(x)} F(x) + \frac{1}{2}\sum_{i,j \in S} a_{ij} \, \partial_{x_i} \partial_{x_j} F(x),$$

where $$b_i(x) = b \cdot 1_{x_i > 0} \cdot \frac{m_i}{x_i} v_i$$ for each $i \in S$, with $v_i$ encoding the Markov jump rates $r(i,j)$, and $a_{ij}$ the diffusion matrix coefficients. The drift term diverges as $x_i \to 0$, driving the process toward the boundary. Once the process reaches a face $\Sigma_A$, all coordinates $x_j$ for $j \notin A$ remain zero and the generator restricts to the induced diffusion on $\Sigma_A$ with its own drift $b_j^A(x)$ and diffusion $a_{jk}^A$ (Beltrán et al., 6 Jan 2026). Absorption is guaranteed by positive super-harmonic functions constructed for each $A$, ensuring finite expected hitting time for the boundary.

3. Iterative Dimension Reduction and Absorption Cascade

Each time the process is absorbed on a face $\Sigma_A$, it is locked to evolve on $\Sigma_A$ with transition rates and generator adapted to the subset $A$. This reduction iterates: after each absorption, the surviving coordinate set $B_n$ shrinks, and almost surely, after a finite number $n_0 \le p$ of absorptions, only a singleton vertex remains—i.e., one coordinate $x_j=1$ and the others vanish. At this stage, the process becomes trapped and ceases further evolution (Beltrán et al., 6 Jan 2026). Uniform bounds on absorption times follow from super-harmonic arguments.

4. Scaling Limit of Condensing Zero-Range Processes

Dimension-decaying diffusion is realized as the macroscopic (diffusive time-scale) limit of condensing zero-range interacting particle systems. Labels $i \in S$ represent sites: the empirical density process

$X_t^N = \frac{\eta_N(tN^2)}{N} \in \Sigma_N$

(where $\eta_N$ encodes occupancy counts) converges to dimension-decaying diffusion $X$ as $N \to \infty$, provided jump rate functions satisfy $g_i(n)/m_i \to 1 + \frac{b}{n}$ for $b=1$. The convergence is verified via tightness, generator expansion in the bulk, and boundary control through super-harmonic functions and extended martingale problem domains (Beltrán et al., 6 Jan 2026).

5. Intrinsic Dimensionality in Generative Diffusion Models

Diffusion processes operating on high-dimensional data can exhibit effective dimension decay by virtue of the manifold hypothesis: if $M \subset \mathbb{R}^D$ is an embedded $d$-dimensional submanifold supporting the data, then as noise accumulates (e.g., in Gaussian convolution with standard deviation $e^\delta$, $\delta \to -\infty$), the local intrinsic dimension (LID) can be read off as

$$\lim_{\delta \to -\infty} \partial_\delta \log q_\delta(x) = d - D,$$

where $q_\delta(x)$ is the density under convolution. For diffusion models, LID estimation via the FLIPD algorithm leverages this rate of change, encapsulating local manifold complexity. As noise dominates, the process marginal escapes the true $d$-dimensional support and approaches the ambient dimension $D$—effectively decaying the support dimension. This provides both geometric interpretation and diagnostics for training generative models (Leung et al., 25 Jun 2025).

6. Spectral Dimension Flow in Multiscale Spacetimes

Diffusion equations on multiscale (fractional or multifractional) spacetimes exhibit running spectral dimension $d_S(\sigma)$, defined via the return probability as

$$d_S(\sigma) = -2 \frac{d \ln \mathcal{P}(\sigma)}{d \ln \sigma} = D \frac{d \ln \ell^2(\sigma)}{d \ln \sigma},$$

where $\ell^2(\sigma)$ is the scale-dependent Gaussian width of the heat kernel (Calcagni et al., 2013). In regimes with binomial two-scale measure, the process transitions from an ultraviolet plateau $D(2-\alpha)$ to the topological dimension $D$ in the infrared, corresponding to physical transitions from fractal-like geometry to ordinary diffusion. The underlying Langevin stochastic process ensures proper normalization and probabilistic consistency, resolving previous ambiguities in spectral dimension computation.

7. Algorithmic Realizations and Practical Acceleration

In machine learning, dimension-decaying (or dimensionality-varying) diffusion processes are operationalized by decomposing input signals (such as images) into orthogonal subspaces, with successive attenuation of inconsequential components. For example, in the DVDP paradigm, signal evolution proceeds through a sequence of linear downsampling operators $D_k$ applied at turning points $T_k$, which eliminate low signal-to-noise components and reduce the effective dimension. When such components approach pure noise, they are dropped with negligible information loss, yielding substantial reductions in computational cost and superior synthesis quality at high resolution (Zhang et al., 2022). Training and sampling are accelerated, and theoretical error bounds confirm the validity of discarding attenuated coordinates.

8. Theoretical and Practical Significance

Dimension-decaying diffusion processes rigorously intertwine stochastic analysis, spectral geometry, and machine learning. The successive absorption mechanism on the simplex manifests as iterative loss of system degrees of freedom, underpinned by singular drift and adapted reversible diffusion. In generative models and statistical estimation on manifolds, effective control or estimation of local dimension is both theoretically justified and practically significant: it guides architectural choices, noise scheduling, and the design of dimension-aware diffusion algorithms. In physical and geometric contexts, these processes serve as probes of fractality, multiscaling, and the interplay between geometry and stochastic dynamics.