arXiv subjects

Ho Yun

Publications and source records attributed to Ho Yun.

8 recordsLinked to original sources

arXiv2026

Kernel Embedding for Operator-Valued Measures and Its Application to Quantum Tomography

This paper introduces the Quantum Covariance Embedding, which embeds Positive Operator-Valued Measures into a tensor product of a Reproducing Kernel Hilbert Space and the quantum state space via a tensorized Bochner integral. This construction induces the Quantum Maximum Discrepancy that metrizes the space of quantum measurements. Applying this framework to Quantum State Tomography, we reformulate density estimation as a tensorized kernel regression, enabling optimal inference without the basis-dependent sparsity constraints that restrict existing methods. We develop a unified geometric design theory for quantum Gram superoperators, establishing that Unitary Designs are strictly E-optimal experimental designs and thus statistically superior to Pauli observables. For general structure-free estimation, we derive the exact minimax lower bound and prove that our tensorized estimators achieve this optimal rate. Furthermore, we introduce the QUAntum Regression with Kernels (QUARK) estimator to accommodate the spectral geometry of physical implementations, deriving central limit theorem and concentration inequalities. To facilitate practical estimation, we establish the exactness of trace-preserving projections and demonstrate efficient estimation under mutually unbiased bases via the fast Walsh-Hadamard transform.

math.ST↗

arXiv2025

Gaussian Optimal Transport Beyond Brenier's Theorem

We explore the geometry of the Bures-Wasserstein space for potentially degenerate Gaussian measures on a separable Hilbert space. In this general setting, the optimal transport map is formally the subgradient of a convex function that is infinite almost everywhere, rendering conventional duality-based variational methods ineffective. We overcome this analytical barrier by exploiting a constructive operator-theoretic approach. Our central result proves that the Kantorovich problem for any pair of Gaussian measures reduces to a Monge problem; that is, an optimal transport map exists in at least one direction between two measures. This reduction allows for a complete characterization with explicit formulas for all optimal (potentially unbounded) Monge transport map and Kantorovich couplings, as well as establishing their uniqueness. Furthermore, we provide a full description of the convex set of geodesics between degenerate measures, revealing a rich geometric structure where the classical McCann interpolants arise as the extreme points. We apply these findings to construct transport maps for Gaussian processes and introduce a novel framework for Wasserstein barycenters based on random Green's operators.

math.FA↗

arXiv2025

Spectral Shrinkage of Gaussian Entropic Optimal Transport

We present a functional calculus treatment of Entropic Optimal Transport (EOT) between Gaussian measures on separable Hilbert spaces, providing a unified framework that handles infinite-dimensional degeneracy. By leveraging the notion of proper alignment and the Schur complement, we reveal that the Gaussian EOT solution operates as a precise \textit{spectral shrinkage}: the optimal coupling is uniquely determined by contracting the spectrum of the correlation operator via a universal scalar function. This geometric insight facilitates an algorithmic shift from iterative fixed-point schemes (e.g., Sinkhorn) to direct algebraic computation, enabling efficient multi-scale analysis, where a single spectral decomposition allows for the exact evaluation of entropic costs across arbitrary regularization parameters $\varepsilon > 0$ at negligible additional cost. Furthermore, we investigate the asymptotic behavior as $\varepsilon \downarrow 0$ in settings where the unregularized Optimal Transport problem admits non-unique solutions. We establish a selection principle that the regularized limit converges to the most diffusive optimal coupling --characterized as the centroid of the convex set of optimal Kantorovich plans. This demonstrates that in degenerate regimes, the entropic limit systematically rejects deterministic Monge solutions (extremal points) in favor of the optimal solution with minimal Hilbert-Schmidt correlation, effectively filtering out spurious correlations in the null space. Finally, we derive stability bounds and convergence rates, recovering established parametric rates ($\varepsilon \log(1/\varepsilon)$) in finite dimensions while identifying distinct non-parametric rates dependent on spectral decay in infinite-dimensional settings.

math.OC↗

arXiv2025

Low-Dose Tomography of Random Fields and the Problem of Continuous Heterogeneity

We consider the problem of nonparametric estimation of the conformational variability in a population of related structures, based on low-dose tomography of a random sample of representative individuals. In this context, each individual represents a random perturbation of a common template and is imaged noisily and discretely at but a few projection angles. Such problems arise in the cryo Electron Microscopy of structurally heterogeneous biological macromolecules. We model the population as a random field, whose mean captures the typical structure, and whose covariance reflects the heterogeneity. We show that consistent estimation is achievable with as few as two projections per individual, and derive uniform convergence rates reflecting how the various parameters of the problem affect statistical efficiency, and their trade-offs. Our analysis formulates the domain of the forward operator to be a reproducing kernel Hilbert space, where we establish representer and Mercer theorems tailored to question at hand. This allows us to exploit pooling estimation strategies central to functional data analysis, illustrating their versatility in a novel context. We provide an efficient computational implementation using tensorized Krylov methods and demonstrate the performance of our methodology by way of simulation.

math.ST↗

arXiv2025

Fast and Cheap Krylov-Based Covariance Smoothing

We introduce the Tensorized-and-Restricted Krylov (TReK) method, a simple and efficient algorithm for estimating covariance tensors with large observational sizes. TReK extends the conjugate gradient method to incorporate range restrictions, enabling its use in a variety of covariance smoothing applications. By leveraging matrix-level operations, it achieves significant improvements in both computational speed and memory cost, improving over existing methods by an order of magnitude. TReK ensures finite-step convergence in the absence of rounding errors and converges fast in practice, making it well-suited for large-scale problems. The algorithm is also highly flexible, supporting a wide range of forward and projection tensors.

stat.CO↗

arXiv2023

Computerized Tomography and Reproducing Kernels

The X-ray transform is one of the most fundamental integral operators in image processing and reconstruction. In this article, we revisit the formalism of the X-ray transform by considering it as an operator between Reproducing Kernel Hilbert Spaces (RKHS). Within this framework, the X-ray transform can be viewed as a natural analogue of Euclidean projection. The RKHS framework considerably simplifies projection image interpolation, and leads to an analogue of the celebrated representer theorem for the problem of tomographic reconstruction. It leads to methodology that is dimension-free and stands apart from conventional filtered back-projection techniques, as it does not hinge on the Fourier transform. It also allows us to establish sharp stability results at a genuinely functional level (i.e. without recourse to discretization), but in the realistic setting where the data are discrete and noisy. The RKHS framework is versatile, accommodating any reproducing kernel on a unit ball, affording a high level of generality. When the kernel is chosen to be rotation-invariant, explicit spectral representations can be obtained, elucidating the regularity structure of the associated Hilbert spaces. Moreover, the reconstruction problem can be solved at the same computational cost as filtered back-projection.

math.FA↗

arXiv2023

Redheffer: Trig to Quantum Error Bounds

In the existing literature, the Redheffer inequality is typically proven using mathematical induction. In this short paper, we present a straightforward proof of this inequality by leveraging trigonometric substitution. We then extend the Redheffer inequality by introducing an exponent factor, aiming for the sharpest possible refinement. Notably, when the exponent is 2, our findings have implications for quantum error correction in the context of quantum phase estimation.

quant-ph↗

arXiv2022

Exponential Concentration for Geometric-Median-of-Means in Non-Positive Curvature Spaces

In Euclidean spaces, the empirical mean vector as an estimator of the population mean is known to have polynomial concentration unless a strong tail assumption is imposed on the underlying probability measure. The idea of median-of-means tournament has been considered as a way of overcoming the sub-optimality of the empirical mean vector. In this paper, to address the sub-optimal performance of the empirical mean in a more general setting, we consider general Polish spaces with a general metric, which are allowed to be non-compact and of infinite-dimension. We discuss the estimation of the associated population Frechet mean, and for this we extend the existing notion of median-of-means to this general setting. We devise several new notions and inequalities associated with the geometry of the underlying metric, and using them we study the concentration properties of the extended notions of median-of-means as the estimators of the population Frechet mean. We show that the new estimators achieve exponential concentration under only a second moment condition on the underlying distribution, while the empirical Frechet mean has polynomial concentration. We focus our study on spaces with non-positive Alexandrov curvature since they afford slower rates of convergence than spaces with positive curvature. We note that this is the first work that derives non-asymptotic concentration inequalities for extended notions of the median-of-means in non-vector spaces with a general metric.

math.ST↗