{"ID":2827651,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.16768","arxiv_id":"2512.16768","title":"On The Hidden Biases of Flow Matching Samplers","abstract":"Flow matching (FM) constructs continuous-time ODE samplers by prescribing probability paths between a base distribution and a target distribution. In this note, we study FM through the lens of finite-sample plug-in estimation. In addition to replacing population expectations by sample averages, one may replace the target distribution itself by a finite-sample surrogate, ranging from the empirical measure to a smoothed estimator. This viewpoint yields a natural hierarchy of empirical FM models. For affine conditional flows, we derive the exact empirical minimizer and identify a smoothed plug-in regime in which the terminal law is exactly a kernel-mixture estimator. This plug-in perspective clarifies several coupled finite-sample biases of empirical FM. First, replacing the target law by a finite-sample surrogate changes the statistical target. Second, the empirical minimizer is generally not a gradient field, even when each conditional flow is. Third, a fixed empirical marginal path does not determine a unique particle dynamics: one may add extra vector fields whose probability flux has zero divergence without changing the marginal path. For Gaussian affine conditional paths, we give explicit families of such flux-null corrections. Finally, the source distribution provides a primary mechanism controlling upper tails of kinetic energy. In particular, Gaussian bases yield exponential upper-tail bounds for instantaneous and integrated kinetic energies, whereas polynomially tailed bases yield corresponding polynomial upper-tail bounds.","short_abstract":"Flow matching (FM) constructs continuous-time ODE samplers by prescribing probability paths between a base distribution and a target distribution. In this note, we study FM through the lens of finite-sample plug-in estimation. In addition to replacing population expectations by sample averages, one may replace the targ...","url_abs":"https://arxiv.org/abs/2512.16768","url_pdf":"https://arxiv.org/pdf/2512.16768v3","authors":"[\"Soon Hoe Lim\"]","published":"2025-12-18T17:02:11Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"math.PR\"]","methods":"[]","has_code":false}