Existing differentially private (DP) synthetic data techniques focus on preserving low‑dimensional marginals, leaving high‑dimensional global structure to chance. Methods that aim for global accuracy typically suffer an exponential cost in the data dimension $d$. This work introduces SPLICE, a DP synthetic data algorithm that retains a strong global distribution guarantee even in high dimensions and achieves state‑of‑the‑art practical performance. SPLICE measures noisy sliced Chebyshev polynomial moments of the target distribution and fits a distribution to match these moments, extending recent 1‑D polynomial approaches. We prove that for a dataset of $n$ examples, SPLICE returns a private distribution whose sliced Wasserstein‑1 distance to the target is $\tilde{O}(1/\sqrt{n})$ for constant privacy parameters, a bound independent of $d$. Along the way we establish a moment‑matching theorem of independent interest: for a distribution on the unit ball, matching the first $O(1/\alpha)$ Chebyshev moments along $\tilde{O}(d/\alpha^2)$ random directions guarantees an $\alpha$ sliced Wasserstein error. The implementation leverages tools from tensor decomposition literature and consistently outperforms prior DP synthetic data methods on extensive benchmarks.
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