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 $\eta$-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 $\tau_p\asymp p$.
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.
We prove explicit upper bounds for the number of eigenvalues of a one-dimensional time-frequency localization operator in an open transition window. An abstract residual-energy principle combines a rank-N approximant with the Hilbert-Schmidt energy of its remainder and retains the strict integer correction imposed by the open threshold. For the Fourier kernel, centered least-squares polynomials, optimized moments, and Chebyshev-Bessel truncations improve or match the diameter-Taylor certificate; on interval blocks the moment envelope gains the exact factor 1/(2N+1). Finite measurable partitions yield best-polynomial matrix and residual-energy bounds, together with two-level threshold allocations that exploit spatial or Fourier-side orthogonality and remain stable under large empty gaps. A complementary defect channel gives a componentwise variational Schatten envelope for finite interval unions. For arbitrary finite-measure hard windows, exact cross-boundary and symmetric-difference formulas give strict transition counting and a certified near-one cluster that can be subtracted from direct counts. We also treat integrable soft masks, a root-sum-square core-tail decomposition, and unbounded windows controlled by moments. An explicit unbounded finite-measure set has all polynomial moments and infinite fractional translation perimeter for every order in (0,1], yet admits an O(log(1/epsilon)/log log(1/epsilon)) certificate. Finally, kernels with finitely many bounded separated phase interactions and finite separated amplitude rank admit factorial, least-squares, Chebyshev-Bessel, span-compressed, and anisotropic-degree bounds. For bounded hard Fourier windows, the final hybrid is no larger than the corresponding earlier certificates proved here; no universal comparison with all regular-domain estimates is asserted.
We introduce an intrinsic spectral sparsity model for nonparametric density estimation on compact connected Riemannian manifolds. Instead of penalizing coefficients in an arbitrarily chosen Laplace--Beltrami eigenbasis, we group each complete eigenspace and measure the Hilbert norm of its spectral component. The resulting block-variation space is basis independent and isometry invariant. We establish its structural, atomic, and nonlinear approximation properties and clarify its relation to Sobolev, Besov, and coefficientwise spectral $\ell^1$ classes. We then construct a coordinate-free block-shrinkage estimator and prove a nonasymptotic signal-dependent $L^2$-oracle inequality that adapts to the unknown set of detectable eigenspaces. Under polynomial spectral growth, the risk theory separates the number of spectral blocks from their multiplicities and exhibits two regimes: one driven by a single high-dimensional eigenspace and the other by cumulative spectral complexity. Under matching spectral-growth and nondegeneracy assumptions, corresponding minimax lower bounds show that this multiplicity dependence is intrinsic, with sharp consequences for spheres and the rotation group $SO(3)$. Finally, we develop a positive, normalized, block-penalized exponential spectral sieve for log-densities and derive likelihood oracle inequalities together with expected Kullback--Leibler, Hellinger, and $L^2$ risk bounds. The resulting framework provides a geometry-respecting theory of sparse density estimation that remains invariant under changes of eigenbasis.
We study simultaneous inference for maxima of canonical order-two $U$-statistics in high dimension. Degeneracy makes quadratic fluctuations leading, so ordinary Gaussian calibration can fail even after exact variance normalization. We show that the appropriate general target is a joint signed Gaussian quadratic chaos and establish a general approximation result that permits indefinite kernels. The general anti-concentration bound is too crude for high-dimensional inference, and we obtain sharper bounds under additional spectral structure. We also identify a phase transition from a non-Gaussian signed-chaos maximum to its covariance-matched Gaussian counterpart driven by the effective rank. For feasible inference, we propose a Gaussian multiplier bootstrap that avoid estimating eigensystems, and establish its validity. Two applications and extensive numerical simulations further illustrate the scope and practical performance of the proposed framework.
We study Gaussian approximation for probability measures obtained by normalizing one-dimensional profile functions, with one-dimensional marginals of Euclidean balls as the principal example. We first establish quantitative concentration estimates near the set of maximizers. For profiles with a unique nondegenerate maximizer, we give sufficient conditions under which centering at the maximizer and rescaling according to the local quadratic approximation of the logarithm of the profile yield densities that converge in $L^1(\mathbb{R})$ to the standard Gaussian density. We then specialize to one-dimensional marginals of Euclidean balls in $\mathbb{R}^n$. Writing $N=n-1$, we consider two standardizations of the marginal distribution: one determined by the logarithmic curvature at the maximizer, and the other by the exact standard deviation. For each standardization, we identify the first-order correction, of order $N^{-1}$, to the standard Gaussian density in $L^1(\mathbb{R})$. These expansions also determine the corresponding first-order corrections to the probabilities of symmetric intervals and the leading terms of the total variation distances from the standard Gaussian distribution. In particular, under exact-variance standardization, the total variation distance is asymptotic to an explicit positive constant times $N^{-1}$. Consequently, the previously known $O(N^{-1})$ bound for approximation by the Gaussian distribution with the same variance is sharp in order.
Bo-Si Chen, Yen-Chang Huang· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
