Sparsity for dynamic inverse problems on Wasserstein curves with bounded variation

TL;DR

The article develops a dynamic inverse problem framework with a Wasserstein-1 regularizer acting on BV curves in time, allowing jumps and sparse temporal structure. It proves the existence of a sparse minimizer consisting of at most $m$ Dirac-atom trajectories with BV evolution, and furnishes a representer theorem to guarantee sparsity. An FC-GCG method is derived and discretized for grid-free computation of BV trajectories, with a detailed insertion/coefficients/pruning scheme. Numerical experiments validate the ability to recover sparse and jumpy ground truths, even under noise, demonstrating the approach's effectiveness for dynamic inverse problems in Wasserstein space.

Abstract

We investigate a dynamic inverse problem using a regularization which implements the so-called Wasserstein-$1$ distance. It naturally extends well-known static problems such as lasso or total variation regularized problems to a (temporally) dynamic setting. Further, the decision variables, realized as BV curves, are allowed to exhibit discontinuities, in contrast to the design variables in classical optimal transport based regularization techniques. We prove the existence and a characterization of a sparse solution. Further, we use an adaption of the fully-corrective generalized conditional gradient method to experimentally justify that the determination of BV curves in the Wasserstein-$1$ space is numerically implementable.

Figures (9)

• Figure 1: Example of a shortest càdlàg curve $\bar{\rho}:[0,1]\to\mathcal{W}_1(\Omega)$ matching given data (displayed in gray) at time points $t=0,0.2,\ldots,1$, where $\Omega\subset\mathbb{R}$. In the one-dimensional case, Wasserstein-$1$ transport is well-understood (by the linearity of the transportation cost combined with the fact that each particle can only move in two directions), see San. For example, we have $\bar{\rho}_t=\delta_{x_t}$ for $t\in[0,0.3]$, $\bar{\rho}_t=\frac{1}{2}\delta_{x_t}+\frac{1}{2}\delta_{y}$ for $t\in[0.3,0.4)$, and $\bar{\rho}_t=(2\mathcal{L}(\ell_t))^{-1}\mathcal{L}\mathbin{}\ell_t+\frac{1}{2}\delta_{z}$ for $t\in(0.4,0.6)$. Here $\mathcal{L}\mathbin{}\ell_t$ denotes the restriction of one-dimensional Lebesgue measure $\mathcal{L}=\mathcal{L}^1$ to line segment $\ell_t$ (which would be replaced by the one-dimensional Hausdorff measure if $n>1$). Note that the jumps at $t=0.4$ and $t=0.6$ are càdlàg (in particular, the right-hand limits correspond to the given data). At $t=1$ the curve $\bar{\rho}$ jumps to data consisting of a line segment and a Dirac mass. In this example, each particle (with infinitesimal small mass) moves linearly with respect to $t$ (which does not need to be satisfied), waits, or jumps.
• Figure 2: Sketch of the (piecewise constant) BV curve $\mu^k$ with $k=k(4,3)$ (\ref{['convExamp']}).
• Figure 3: Lipschitz curves $\gamma_i$ and $\tilde{\gamma}_i$ from \ref{['SuperExamp']}.
• Figure 4: Discretization of the ground truth $\bar{\mu}^\dagger$ defined by \ref{['eq:gt1', 'eq:gt2']}. The legend shows the weights associated with the càdlàg curves $\delta_{\bar{\gamma}_i^\dagger}$.
• Figure 5: Reconstruction of the discretized ground truth from \ref{['eq:gt']}. Note that it is faithful to the ground truth in the sense that it recovers the jump of $\bar{\gamma}^\dagger_3$. The effect of regularization is observable in the attenuated weights and decreased variation of each curve (in particular, close to $t = 0$ and $t=1$).

Theorems & Definitions (46)

• Definition 1.1: Admissible BV curves, weight functional
• Remark 1.2: Structure of $\mathcal{A}$
• Definition 1.3: Regularizer for BV curves
• Proposition 1.5: Existence
• Remark 1.6: Stability
• Example 1.7: $K$ with values in infinite-dimensional Hilbert space
• Proposition 1.8: Extremal points of $L_c^-(\mathcal{R}_{\alpha,\beta})$
• Theorem 1.10: Existence of sparse solution
• Example 1.11: $K$ with values in finite-dimensional Hilbert space
• Definition 2.1: Variation