2.1 Notation
Let \( \mathbf{x}_t \in \mathbb{R}^{D} \) denote the observation at time \( t \). We use a superscript \( (\tau) \) for the compression coordinate, where \( \tau=1 \) is data space and \( \tau=0 \) is the compressed space. Therefore, \( \mathbf{x}_{t}^{(\tau)} \) denotes the state at compression level \( \tau\in[0,1] \) and time \( t \), so that \( \mathbf{x}_{t}^{(1)}=\mathbf{x}_t \). In the following, we assume that data are sampled at times \( t = k\Delta t \), \( k\in \mathbb{Z} \), though our approach can accommodate non-uniform spacing. Our goal is to learn a pair of flow fields: a compressive flow \( \mathbf{u}_{\boldsymbol{\phi}} \) that transports states along \( \tau \) from a low-dimensional generative subspace to high-dimensional data space, and a dynamical flow \( \mathbf{v}_{\boldsymbol{\theta}} \) that transports states through time within each \( \tau \).
2.2 Encoder
For training a flow-matching model, we require a mapping, called the coupling, for pairing points in the target (data) distribution \( p_{\tau=1}(\mathbf{x}) \) with those in the source (latent) distribution \( p_{\tau = 0}(\mathbf{x}) \) (Lipman et al., 2024). Here, rather than fixing the form of the source distribution, we instead choose a deterministic coupling that enforces dimension reduction:
\[\begin{equation} \mathbf{x}_t^{(0)}=\mathbf{b}+\mathbf{LD}^{1/2}\cdot \boldsymbol{\mu}_{\boldsymbol{\psi}}\left(\mathbf{x}_t^{(1)}\right), \label{eqn:encoder} \end{equation}\]where \( \mathbf{b}\in\mathbb{R}^{D} \) is a bias term, \( \mathbf{L}\in\mathbb{R}^{D\times D} \) has orthonormal columns (e.g., parameterized via non-pivoted QR), \( \mathbf{D}=\mathrm{diag}(d_1,\ldots,d_{D}) \) with \( d_i>0 \), and \( \boldsymbol{\mu}_{\boldsymbol{\psi}}(\mathbf{x})\in\mathbb{R}^{D} \) is a parameterized nonlinear mapping.
This parameterization separates source distribution orientation and spread: \( \mathbf{L} \) defines the axes of the linear subspace in ambient space, and \( \mathbf{D} \) controls per-coordinate scale, since sample energy \( \lVert\mathbf{x}_t^{(0)}-\mathbf{b}\rVert_2^2=\boldsymbol{\mu}_{\boldsymbol{\psi}}(\mathbf{x}_t^{(1)})^\top \mathbf{D}\,\boldsymbol{\mu}_{\boldsymbol{\psi}}(\mathbf{x}_t^{(1)}) \) when \( \mathbf{L} \) has orthonormal columns. Moreover, when \( \mathbf{D} \) is low-rank, the source points \( \mathbf{x}_t^{(0)} \) lie in a low-dimensional subspace. Our goal is to learn an encoding map that effectively minimizes the rank of \( \mathbf{D} \) while maintaining the predictive accuracy of both compressive and dynamical flows.
In practice, to stabilize training and remove the scale ambiguity between \( \mathbf{D} \) and \( \boldsymbol{\mu}_{\boldsymbol{\psi}}(\mathbf{x}) \), we normalize the encoder features dimension-wise (Ioffe & Szegedy, 2015; Ba et al., 2016). That is, if \( \boldsymbol{\mu}^{\mathrm{raw}}_{\boldsymbol{\psi}}(\mathbf{x})\in\mathbb{R}^{D} \) denotes the raw feature output, we set
\[\begin{equation} \big[\boldsymbol{\mu}_{\boldsymbol{\psi}}(\mathbf{x})\big]_i = \frac{\big[\boldsymbol{\mu}^{\mathrm{raw}}_{\boldsymbol{\psi}}(\mathbf{x})\big]_i - m_i}{\sigma_i}, \qquad i=1,\ldots,D, \label{eqn:mu_norm} \end{equation}\]where \( \mathbf{m}\in\mathbb{R}^{D} \) and \( \boldsymbol{\sigma}\in\mathbb{R}^{D} \) are the running (EMA) estimates of the coordinate-wise mean and standard deviation of the \( \boldsymbol{\mu}^{\mathrm{raw}}_{\boldsymbol{\psi}}(\mathbf{x}) \), respectively.
Thus each coordinate of \( \boldsymbol{\mu}_{\boldsymbol{\psi}}(\mathbf{x}) \) has approximately zero mean and unit variance over the data distribution. We additionally cap the per-sample \( \ell_2 \) norm of the raw features to prevent rare large activations from destabilizing the running statistics and downstream flow fitting (Pascanu et al., 2013).
Lastly, and critically for latent space reproducibility, the combination of our parameterization Equation \( \eqref{eqn:encoder} \), the deterministic non-pivoted QR parameterization for \( \mathbf{L} \), the normalization Equation \( \eqref{eqn:mu_norm} \), and nested dropout (Section 3.1), which enforces an ordered, prefix-based notion of coordinate importance, renders the encoder identifiable up to a sign for each component.
2.3 Compressive Flow
At any fixed time \( t \), the compression/generation axis connects a compressed representation to a point in the data. Flow matching learns a marginal probability path by integrating over conditional probability paths (bridges) that connect endpoints drawn from the coupling. Given a data point \( \mathbf{x}_t^{(1)} \) and its image under the encoder (Equation \( \eqref{eqn:encoder} \)), \( \mathbf{x}_t^{(0)} \), we define the linear \( \tau \)-bridge as
\[\begin{equation} \mathbf{x}_t^{(\tau)} = (1-\tau)\,\mathbf{x}_t^{(0)} + \tau\,\mathbf{x}_t^{(1)}, \label{eqn:tau_bridge} \end{equation}\]and learn a compression velocity field
\[\begin{equation*} \mathbf{u}_{\boldsymbol{\phi}}:\ (\mathbf{x}_t^{(\tau)},\tau)\ \mapsto\ \partial_\tau \mathbf{x} \in \mathbb{R}^{D}. \end{equation*}\]Note that while the bridge Equation \( \eqref{eqn:tau_bridge} \) is constructed using the encoder endpoint \( \mathbf{x}_t^{(0)} \), the learned flow \( \mathbf{u}_{\boldsymbol{\phi}} \) defines a distinct projection to \( \tau=0 \) via integration from \( \mathbf{x}_t^{(1)} \). We denote the data at compressive level \( \tau \), obtaining via compressive flow integration, by \( \mathbf{\tilde{x}}_t^{(\tau)} \).
The flow-based endpoint (\( \mathbf{\tilde{x}}_t^{(0)} \)) can differ from the encoder endpoint (\( \mathbf{x}_t^{(0)} \)), effectively rearranging the encoder geometry at \( \tau=0 \). In practice, flow matching tends to produce near-straight \( \tau \) paths and local neighborhoods in data space are transported smoothly, while the marginal distribution at \( \tau=0 \) is preserved by the transport (Lipman et al., 2023; Liu et al., 2023).
2.4 Dynamical Flow
Unlike the compressive flow bridge Equation \( \eqref{eqn:tau_bridge} \), the dynamical bridge requires a coupling between pairs of points in data space and their source representations. Additionally, because we want the dynamical flow \( \mathbf{v}_{\boldsymbol{\theta}} \) to exist at every compression level \( \tau \), the corresponding linear bridge must involve a double interpolation (Figure 2). More specifically, given a fixed compression level \( \tau\in[0,1] \), we first interpolate along the compression dimension for each data point as in Equation \( \eqref{eqn:tau_bridge} \), yielding start and end points \( \mathbf{x}^{(\tau)}_{k\Delta t} \) and \( \mathbf{x}^{(\tau)}_{(k+1)\Delta t} \). We then perform a second, dynamical interpolation between these points for \( s \in [0, 1] \):
\[\begin{equation} \mathbf{x}_{t}^{(\tau)} = (1-s)\,\mathbf{x}_{k\Delta t}^{(\tau)} + s\,\mathbf{x}_{(k+1)\Delta t}^{(\tau)}. \label{eqn:t_bridge} \end{equation}\]The dynamical flow \( \mathbf{v}_{\boldsymbol{\theta}}\) then models the instantaneous dynamical velocity,
\[\begin{equation*} \mathbf{v}_{\boldsymbol{\theta}}:\ (\mathbf{x}_{t}^{(\tau)},\tau,s,\mathbf{x}_{\mathrm{hist}}^{(\tau)}) \ \mapsto\ \partial_t \mathbf{x} \in \mathbb{R}^{D}. \end{equation*}\]Here, in addition to the bridge variables \( \tau \) and \( s \), we have allowed a potential dependence on the lag-\( h \) history at compression level \( \tau \), \( \mathbf{x}_{\mathrm{hist}}^{(\tau)}=\Big(\mathbf{x}_{(k-h)\Delta t}^{(\tau)},\ldots,\mathbf{x}_{(k-1)\Delta t}^{(\tau)},\mathbf{x}_{k\Delta t}^{(\tau)}\Big) \) (Zhang et al., 2024).
