arXiv subjects

Jiaping Yang

Publications and source records attributed to Jiaping Yang.

7 recordsLinked to original sources

arXiv2026

Wasserstein Metric Normalization of Projective Cycle Spaces

Let $X$ be an irreducible reduced projective subvariety of a Chow variety. On a generic smooth parameter locus, normal motions of resolved components define a quadratic Wasserstein metric, even when the cycles are singular, reducible, or carry multiplicities. Its metric completion is canonically $X^ν$: resolution fibers collapse, whereas distinct normalization branches over the same Chow cycle remain separated. The completed metric admits an exact ambient action formula. For degree-$d$ hypersurfaces, the inner $W_q$ geometry is compact and induces the projective topology for every $1\le q\le2$. For $d\ge2$, the uniform Hölder exponent $1/d$ is attained and sharp for each such $q$; when $d=1$, the comparison is Lipschitz. Elliptic quartics furnish an explicit model of the boundary branching.

math.AG↗

arXiv2026

Gaussian-Restricted Barycenters for KL-Unbalanced Optimal Transport: Variational Theory and Fixed-Point Convergence

We study Gaussian-restricted barycenters for quadratic two-sided Kullback--Leibler unbalanced optimal transport with independent marginal penalties and no coupling entropy. Exact profiling of the barycenter mass reduces the problem to a smooth Gaussian shape functional with endogenous Gibbs weights. We establish global attainment, derive the stationary moment equations, and construct a reverse-KL majorization--minimization (MM) iteration whose full sequence converges from every nondegenerate Gaussian initialization to a stationary fixed point. The diagonal second variation induces a parallel-sum tensor coupling the Bures--Wasserstein and Fisher--Rao metrics; its finite-mass extension admits a radial cone representation. Under common penalty scaling, global minimizers converge to a Gaussian Wasserstein barycenter with effective weights; for sufficiently large penalties, the minimizer is unique and admits a first-order analytic expansion. In the small-penalty regime, the distance of every global minimizer to the compact maximizer set of a weighted Chernoff affinity functional vanishes with respect to the mean--covariance parameter distance. Numerical experiments illustrate MM descent, local contraction, and the two penalty limits.

math.OC↗

arXiv2026

High-Dimensional Spectral Limits for Gaussian KL-Unbalanced Optimal Transport

We study high-dimensional random-matrix limits of Gaussian Kullback--Leibler unbalanced optimal transport (KL-UOT). Under equal marginal penalties, the covariance action admits an exact log-determinant representation in terms of a nonlinear ridge product, together with a positive-semidefinite extension that remains finite at arbitrary aspect ratios. For independent real Wishart samples, strong asymptotic freeness gives the limiting free multiplicative convolution and almost-sure Hausdorff convergence of the ridge-product spectrum; independent Haar orientations yield the corresponding first-order limit for deformed populations. In the symmetric nonsingular identity-Wishart model, we derive an explicit $η$-transform and a low-degree algebraic equation that select the physical branch and determine the support interval, square-root edges, and extreme-eigenvalue limits. We further obtain all-aspect one-sample Marchenko--Pastur limits under finite fourth moments, real-Gaussian Bai--Silverstein fluctuations for $c<1$, and a joint random-matrix/penalty limit showing that sample-covariance noise produces the critical scale $τ_p\asymp p$.

math.PR↗

arXiv2026

General Divergence Regularized Optimal Transport: Sample Complexity and Central Limit Theorems

We study empirical divergence-regularized optimal transport on arbitrary Polish spaces. For bounded continuous costs and dual-regular conjugates $ψ\in C^1(\mathbb R)$, we establish a dimension-free $n^{-1/2}$ bound for the empirical transport value using dual interpolation, Hoeffding projection, and bounded-difference arguments. For the asymptotic theory, we isolate the population identifiability condition needed for stability and give sufficient conditions for population uniqueness. In particular, active-set identifiability implies uniqueness of the canonical population potentials, and connectedness of either population support is sufficient. This condition requires neither compactness nor Euclidean structure and covers quadratic regularization, whose conjugate is $C^1$ but not $C^2$. We also give finite-support and large-regularization criteria. Under a bounded Lipschitz cost, the empirical potentials are stable, the transport value is asymptotically linear, and one- and two-sample central limit theorems hold with population centering. This includes noncompact and infinite-dimensional Polish state spaces.

math.ST↗

arXiv2026

Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments

Gaussian smoothing has emerged as an effective technique for reducing the sample complexity of optimal transport. In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost $T_p^{(σ)}(μ,ν)=W_p(μ*γ_σ,ν*γ_σ)^p$ on $\R^d$. For fixed smoothing and finite polynomial moments $M_{q_μ}(μ)<\infty$, $M_{q_ν}(ν)<\infty$, with $q_μ,q_ν>p$, we establish upper bounds in probability of order $ρ_{q_μ,p,d}(m)+ρ_{q_ν,p,d}(n)$. Here $ρ_{q,p,d}(N)=N^{-(q-p)/(q+d)}$ for $p d+2p$. This order also holds in expectation under $q_μ,q_ν\ge2p$. When the smoothed population distance is positive, the cost bound yields this rate for the distance itself. For $p>1$ and $q_μ,q_ν>d+2p$, we also derive a first-order expansion, a separated two-sample central limit theorem, and a sample-splitting variance estimator.

math.ST↗

arXiv2026

Closed Forms for Gaussian Kullback--Leibler Unbalanced Optimal Transport without Coupling Entropy

We obtain an explicit solution for the static Kullback--Leibler (KL) unbalanced optimal transport problem between finite non-degenerate Gaussian measures with quadratic cost, two independent positive marginal relaxation parameters, and no entropy penalty on the coupling. The minimizer is a scaled Wasserstein coupling between two adjusted Gaussian marginals and is supported on an affine graph; in entropic Gaussian unbalanced transport, by contrast, the optimal plan is non-degenerate on the product space. The covariance map is the unique positive definite solution of a Riccati equation and admits a principal-square-root representation. Compared with the known equal-penalty Gaussian Hellinger--Kantorovich endpoint, the result treats the asymmetric two-sided Kullback--Leibler relaxation and gives the modified marginals, joint minimizer, value, and a direct quadratic KL-dual certificate. The large-relaxation limit recovers the Gaussian Wasserstein cost for equal masses.

math.OC↗

arXiv2026

Bures Geodesics and Restricted Barycenters for Kronecker Positive Definite Matrices

We study the extrinsic Bures--Wasserstein geometry of the determinant-normalized Kronecker model $\mcK_n=\{V\ot U:U,V\in\Sp^n,\ \det U=1\}\subset\Sp^{n^2}$, asking when the ambient Bures geodesic between two Kronecker positive definite matrices can remain in this lower-dimensional model. Local membership near an endpoint is shown to be equivalent to membership of the whole segment, and this happens exactly in the one-factor cases: either $U_1=U_0$ or $V_1$ is a positive scalar multiple of $V_0$. Consequently, any endpoint pair not confined to these one-factor alternatives leaves the model immediately. The criterion is expressed by a partial-trace residual. In fixed commuting charts it becomes an equivalent rank-one square-root profile and yields computable departure diagnostics. We also obtain exact formulas for two restricted barycenter problems: fixed commuting-coordinate slices, solved by Perron singular vectors, and one-factor subfamilies, reduced to standard Bures--Wasserstein barycenters on $\Sp^n$.

math.AG↗