Projection-based distances replace one difficult high-dimensional transport problem by a family of low-dimensional ones. Their appeal is mathematical as well as computational: one-dimensional transport is explicit, empirical rates improve, and the Cramér--Wold theorem preserves identifiability when all directions are retained.
Sliced Wasserstein distance
Definition. For \(\theta\in\mathbb S^{d-1}\), let \(P_\theta(x)=\theta^{\mathsf T}x\), and let \(\sigma\) be normalized surface measure on the sphere. Define \[ \operatorname{SW}_p^p(\mu,\nu) =\int_{\mathbb S^{d-1}} W_p^p(P_{\theta\#}\mu,P_{\theta\#}\nu)\,\mathrm{d}\sigma(\theta). \] The max-sliced distance is \[ \operatorname{MSW}_p(\mu,\nu) =\sup_{\theta\in\mathbb S^{d-1}} W_p(P_{\theta\#}\mu,P_{\theta\#}\nu). \]
Metric property. \(\operatorname{SW}_p\) and \(\operatorname{MSW}_p\) are metrics on \(\mathcal P_p(\mathbb R^d)\).
Proof. Symmetry and nonnegativity are immediate. Minkowski's inequality in \(L^p(\mathbb S^{d-1})\), combined with the triangle inequality for each one-dimensional \(W_p\), proves the sliced triangle inequality; the max version follows by taking suprema. If \(\operatorname{SW}_p(\mu,\nu)=0\), projected laws agree for almost every \(\theta\). Continuity of characteristic functions in \(\theta\) extends equality to every direction. The Cramér--Wold theorem then gives \(\mu=\nu\). The max version is immediate.
Comparison with ordinary Wasserstein distance
Projection contraction. \[ \operatorname{SW}_p(\mu,\nu) \le\operatorname{MSW}_p(\mu,\nu) \le W_p(\mu,\nu). \]
Proof. The first inequality is \(L^p\)-average at most supremum. Since \(P_\theta\) is \(1\)-Lipschitz, the pushforward of any coupling gives \[ W_p(P_{\theta\#}\mu,P_{\theta\#}\nu)\le W_p(\mu,\nu). \] Take the supremum over \(\theta\).
A reverse inequality requires restrictions such as compact support or regularity, and its constant generally deteriorates with dimension. Projection distances are therefore weaker quantitatively even though they identify the law.
Quantile representation
Every projected problem is one-dimensional: \[ \operatorname{SW}_p^p(\mu,\nu) = \int_{\mathbb S^{d-1}}\int_0^1 |F_{\mu,\theta}^{-1}(u)-F_{\nu,\theta}^{-1}(u)|^p \,\mathrm{d}u\,\mathrm{d}\sigma(\theta). \] Here \(F_{\mu,\theta}\) is the distribution function of \(\theta^{\mathsf T}X\). This formula replaces a high-dimensional coupling by sorting along directions.
Identifiability through the Radon transform
If \(\mu\) has density \(f\), the density of \(P_{\theta\#}\mu\) is the Radon transform \[ \mathcal Rf(\theta,t) =\int_{\{x:\theta^{\mathsf T}x=t\}}f(x)\,\mathrm{d}\mathcal H^{d-1}(x). \] The Fourier-slice theorem states that the one-dimensional Fourier transform in \(t\) equals \(\widehat f(s\theta)\). Thus all projected laws determine the full characteristic function and hence the original distribution.
Empirical behavior
Reduction to one-dimensional rates. For the empirical measure \(\widehat\mu_n\), \[ \mathbb E\operatorname{SW}_p^p(\widehat\mu_n,\mu) = \int_{\mathbb S^{d-1}} \mathbb E W_p^p(P_{\theta\#}\widehat\mu_n,P_{\theta\#}\mu) \,\mathrm{d}\sigma(\theta). \] Any one-dimensional bound that is uniform in \(\theta\) therefore transfers directly to the sliced distance without the \(n^{-p/d}\) full-dimensional quantization barrier.
Proof. Projection commutes with empirical measures: \[ P_{\theta\#}\widehat\mu_n =\frac1n\sum_i\delta_{\theta^{\mathsf T}X_i}. \] The identity then follows from Tonelli's theorem.
The constants may still depend on dimension through projected moments or through uniformity over directions. Slicing mitigates the geometric curse of matching but does not make high-dimensional statistics entirely dimension free.
The sliced Wasserstein process
For inference, one studies the random field \[ \theta\longmapsto W_p(P_{\theta\#}\widehat\mu_n,P_{\theta\#}\mu). \] Uniform weak convergence over \(\theta\) requires empirical-process control of projected distribution or quantile functions. Continuous functionals of this process yield limit laws for average-sliced, max-sliced, and randomly sliced statistics. Max-slicing is nonsmooth because it also estimates the maximizing direction.
Monte Carlo approximation over directions
Random slicing. For independent uniform directions \(\theta_1,\ldots,\theta_L\), \[ \widehat{\operatorname{SW}}_{p,L}^p =\frac1L\sum_{\ell=1}^L W_p^p(P_{\theta_\ell\#}\mu,P_{\theta_\ell\#}\nu) \] is unbiased for \(\operatorname{SW}_p^p\). If the integrand has finite variance, its directional Monte Carlo error is \(O_{\mathbb P}(L^{-1/2})\).
With empirical measures there are two independent errors: sample error in the projected laws and numerical integration error over directions. Their budgets should be balanced rather than conflated.
Projection-robust transport
Definition. For \(1\le k\le d\), define \[ \operatorname{PRW}_{p,k}(\mu,\nu) = \sup_{E\in\operatorname{Gr}(k,d)} W_p(P_{E\#}\mu,P_{E\#}\nu), \] where \(P_E\) is orthogonal projection and \(\operatorname{Gr}(k,d)\) is the Grassmannian. At \(k=1\) this is max-sliced transport; at \(k=d\) it is ordinary \(W_p\).
The parameter \(k\) interpolates statistical tractability and geometric fidelity. Optimization over \(E\) is nonconvex, and a data-dependent maximizing subspace contributes additional uncertainty.
Gaussian example
Projected Gaussians. If \(\mu=N(m_0,\Sigma_0)\), \(\nu=N(m_1,\Sigma_1)\), then \[ W_2^2(P_{\theta\#}\mu,P_{\theta\#}\nu) = \{\theta^{\mathsf T}(m_0-m_1)\}^2 + \left( \sqrt{\theta^{\mathsf T}\Sigma_0\theta} - \sqrt{\theta^{\mathsf T}\Sigma_1\theta} \right)^2. \] Slicing averages this explicit directional discrepancy; max-slicing searches for the direction where mean or variance differs most.
Limitations
Different directions are not coupled consistently by separate one-dimensional optimal plans, so sliced geodesics need not correspond to a single flow in \(\mathbb R^d\). A discrepancy concentrated in a small set of directions can be diluted by averaging. Finally, using finitely many random directions makes the computed quantity a random approximation to the metric. These are structural differences, not merely implementation details.