arXiv subjects

Victor M. Panaretos

Publications and source records attributed to Victor M. Panaretos.

At least 19 recordsLinked to original sources

arXiv2026

Kernel Embeddings and the Separation of Measure Phenomenon

We prove that kernel covariance embeddings lead to information-theoretically perfect separation of distinct continuous probability distributions. In statistical terms, we establish that testing for the \emph{equality} of two non-atomic (Borel) probability measures on a locally compact uncountable Polish space is \emph{equivalent} to testing for the \emph{singularity} between two centered Gaussian measures on a reproducing kernel Hilbert space. The corresponding Gaussians are defined via the notion of kernel covariance embedding of a probability measure, and the Hilbert space is that generated by the embedding kernel. Distinguishing singular Gaussians is structurally simpler from an information-theoretic perspective than non-parametric two-sample testing, particularly in complex or high-dimensional domains. This is because singular Gaussians are supported on essentially separate and affine subspaces. Our proof leverages the classical Feldman-Hájek dichotomy, and shows that even a small perturbation of a continuous distribution will be maximally magnified through its Gaussian embedding. This ``separation of measure phenomenon'' appears to be a blessing of infinite dimensionality, by means of embedding, with the potential to inform the design of efficient inference tools in considerable generality. The elicitation of this phenomenon also appears to crystallize, in a precise and simple mathematical statement, a core mechanism underpinning the empirical effectiveness of kernel methods.

stat.ML↗

arXiv2026

Inference for Functional Data under Markov Constraints

Smoothness has long been the dominant form of parsimony in functional data analysis, to the point of occasionally being conflated with the very notion of functional data. However, many core inferential tasks depend on the inverse covariance, where sparsity--rather than smoothness--emerges as the more natural structural constraint. In this paper, we explore Markovianity as an alternative to smoothness. Focusing on the Gaussian case as a central motivating setting, we exploit the fact that Markovianity induces a shape constraint on the covariance kernel. Building on this observation, we introduce a Markov transform of the empirical covariance together with a corresponding estimator that enforces the Markov structure. The estimator is adaptive and requires no regularity of the underlying covariance beyond continuity. In simulation experiments, it is seen to improve prediction performance even under model misspecification. Unlike smoothness-based assumptions, Markovianity is falsifiable. To assess its validity, we further propose a novel and computationally efficient test for the Markov property based on a new characterization of continuous graphical structure.

stat.ME↗

arXiv2025

Likelihood Ratio Tests by Kernel Gaussian Embedding

We propose a novel kernel-based nonparametric two-sample test, employing the combined use of kernel mean and kernel covariance embedding. Our test builds on recent results showing how such combined embeddings map distinct probability measures to mutually singular Gaussian measures on the kernel's RKHS. Leveraging this ``separation of measure phenomenon", we construct a test statistic based on the relative entropy between the Gaussian embeddings, in effect the likelihood ratio. The likelihood ratio is specifically tailored to detect equality versus singularity of two Gaussians, and satisfies a ``$0/\infty$" law, in that it vanishes under the null and diverges under the alternative. To implement the test in finite samples, we introduce a regularised version, calibrated by way of permutation. We prove consistency, establish uniform power guarantees under mild conditions, and discuss how our framework unifies and extends prior approaches based on spectrally regularized MMD. Empirical results on synthetic and real data demonstrate remarkable gains in power compared to state-of-the-art methods, particularly in high-dimensional and weak-signal regimes.

stat.ML↗

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↗

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↗

arXiv2024

Extreme Points of Spectrahedra

We consider the problem of characterizing extreme points of the convex set of positive linear operators on a possibly infinite-dimensional Hilbert space under linear constraints. We show that even perturbations of points in such sets admit what resembles a Douglas factorization. Using this result, we prove that an operator is extreme iff a corresponding set of linear operators is dense in the space of trace-class self-adjoint operators with range contained in the closure of the range of that operator. If the number of constraints is finite, we show that the extreme point must be of low-rank relative to the number of constraints and derive a purely rank-based characterization of the extreme points. In the finite-dimensional setting, our results lead to a remarkably simple characterization of the elliptope, that is, the set of correlation matrices, in terms of the Hadamard product which allows us to characterize the set of matrices which constitute the equality case of the Hadamard rank inequality when the involved matrices are equal and positive semi-definite. We illustrate the importance of our results using examples from statistics and quantum mechanics.

math.OC↗

arXiv2024

Large Sample Theory for Bures-Wasserstein Barycentres

We establish a strong law of large numbers and a central limit theorem in the Bures-Wasserstein space of covariance operators -- or equivalently centred Gaussian measures -- over a general separable Hilbert space. Specifically, we show that empirical barycentre sequences indexed by sample size are almost certainly relatively compact, with accumulation points comprising population barycentres. We give a sufficient regularity condition for the limit to be unique. When the limit is unique, we also establish a central limit theorem under a refined pair of moment and regularity conditions. Finally, we prove strong operator convergence of the empirical optimal transport maps to their population counterparts. Though our results naturally extend finite-dimensional counterparts, including associated regularity conditions, our techniques are distinctly different owing to the functional nature of the problem in the general setting. A key element is the characterisation of compact sets in the Bures-Wasserstein topology that reflects an ordered Heine-Borel property of the Bures-Wasserstein space.

math.PR↗

arXiv2024

Statistical Inference for Bures-Wasserstein Flows

We develop a statistical framework for conducting inference on collections of time-varying covariance operators (covariance flows) over a general, possibly infinite dimensional, Hilbert space. We model the intrinsically non-linear structure of covariances by means of the Bures-Wasserstein metric geometry. We make use of the Riemmanian-like structure induced by this metric to define a notion of mean and covariance of a random flow, and develop an associated Karhunen-Loève expansion. We then treat the problem of estimation and construction of functional principal components from a finite collection of covariance flows, observed fully or irregularly. Our theoretical results are motivated by modern problems in functional data analysis, where one observes operator-valued random processes -- for instance when analysing dynamic functional connectivity and fMRI data, or when analysing multiple functional time series in the frequency domain. Nevertheless, our framework is also novel in the finite-dimensions (matrix case), and we demonstrate what simplifications can be afforded then. We illustrate our methodology by means of simulations and data analyses.

stat.ME↗

arXiv2024

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↗

arXiv2024

Transportation of Measure Regression in Higher Dimensions

We present an optimal transport framework for performing regression when both the covariate and the response are probability distributions on a compact Euclidean subset $Ω\subset\mathbb{R}^d$, where $d>1$. Extending beyond compactly supported distributions, this method also applies when both the predictor and responses are Gaussian distributions on $\mathbb{R}^d$. Our approach generalizes an existing transportation-based regression model to higher dimensions. This model postulates that the conditional Fréchet mean of the response distribution is linked to the covariate distribution via an optimal transport map. We establish an upper bound for the rate of convergence of a plug-in estimator. We propose an iterative algorithm for computing the estimator, which is based on DC (Difference of Convex Functions) Programming. In the Gaussian case, the estimator achieves a parametric rate of convergence, and the computation of the estimator simplifies to a finite-dimensional optimization over positive definite matrices, allowing for an efficient solution. The performance of the estimator is demonstrated in a simulation study.

math.ST↗

arXiv2023

Continuously Indexed Graphical Models

Let $X = \{X_{u}\}_{u \in U}$ be a real-valued Gaussian process indexed by a set $U$. It can be thought of as an undirected graphical model with every random variable $X_{u}$ serving as a vertex. We characterize this graph in terms of the covariance of $X$ through its reproducing kernel property. Unlike other characterizations in the literature, our characterization does not restrict the index set $U$ to be finite or countable, and hence can be used to model the intrinsic dependence structure of stochastic processes in continuous time/space. Consequently, this characterization is not in terms of the zero entries of an inverse covariance. This poses novel challenges for the problem of recovery of the dependence structure from a sample of independent realizations of $X$, also known as structure estimation. We propose a methodology that circumvents these issues, by targeting the recovery of the underlying graph up to a finite resolution, which can be arbitrarily fine and is limited only by the available sample size. The recovery is shown to be consistent so long as the graph is sufficiently regular in an appropriate sense. We derive corresponding convergence rates and finite sample guarantees. Our methodology is illustrated by means of a simulation study and two data analyses.

math.ST↗

arXiv2023

Nonparametric Estimation for SDE with Sparsely Sampled Paths: an FDA Perspective

We consider the problem of nonparametric estimation of the drift and diffusion coefficients of a Stochastic Differential Equation (SDE), based on $n$ independent replicates $\left\{X_i(t)\::\: t\in [0,1]\right\}_{1 \leq i \leq n}$, observed sparsely and irregularly on the unit interval, and subject to additive noise corruption. By sparse we intend to mean that the number of measurements per path can be arbitrary (as small as two), and remain constant with respect to $n$. We focus on time-inhomogeneous SDE of the form $dX_t = μ(t)X_t^αdt + σ(t)X_t^βdW_t$, where $α\in \{0,1\}$ and $β\in \{0,1/2,1\}$, which includes prominent examples such as Brownian motion, Ornstein-Uhlenbeck process, geometric Brownian motion, and Brownian bridge. Our estimators are constructed by relating the local (drift/diffusion) parameters of the diffusion to their global parameters (mean/covariance, and their derivatives) by means of an apparently novel Partial Differential Equation (PDE). This allows us to use methods inspired by functional data analysis, and pool information across the sparsely measured paths. The methodology we develop is fully non-parametric and avoids any functional form specification on the time-dependency of either the drift function or the diffusion function. We establish almost sure uniform asymptotic convergence rates of the proposed estimators as the number of observed curves $n$ grows to infinity. Our rates are non-asymptotic in the number of measurements per path, explicitly reflecting how different sampling frequency might affect the speed of convergence. Our framework suggests possible further fruitful interactions between FDA and SDE methods in problems with replication.

math.ST↗

arXiv2023

The Positive-Definite Completion Problem

We study the positive-definite completion problem for kernels on a variety of domains and prove results concerning the existence, uniqueness, and characterization of solutions. In particular, we study a special solution called the canonical completion which is the reproducing kernel analogue of the determinant-maximizing completion known to exist for matrices. We establish several results concerning its existence and uniqueness, which include algebraic and variational characterizations. Notably, we prove the existence of a canonical completion for domains which are equivalent to the band containing the diagonal. This corresponds to the existence of a canonical extension in the context of the classical extension problem of positive-definite functions, which can be understood as the solution to an abstract Cauchy problem in a certain reproducing kernel Hilbert space.

math.FA↗

arXiv2023

The Functional Graphical Lasso

We consider the problem of recovering conditional independence relationships between $p$ jointly distributed Hilbertian random elements given $n$ realizations thereof. We operate in the sparse high-dimensional regime, where $n \ll p$ and no element is related to more than $d \ll p$ other elements. In this context, we propose an infinite-dimensional generalization of the graphical lasso. We prove model selection consistency under natural assumptions and extend many classical results to infinite dimensions. In particular, we do not require finite truncation or additional structural restrictions. The plug-in nature of our method makes it applicable to any observational regime, whether sparse or dense, and indifferent to serial dependence. Importantly, our method can be understood as naturally arising from a coherent maximum likelihood philosophy.

stat.ME↗

arXiv2023

On Distributional Autoregression and Iterated Transportation

We consider the problem of defining and fitting models of autoregressive time series of probability distributions on a compact interval of $\mathbb{R}$. An order-$1$ autoregressive model in this context is to be understood as a Markov chain, where one specifies a certain structure (regression) for the one-step conditional Fréchet mean with respect to a natural probability metric. We construct and explore different models based on iterated random function systems of optimal transport maps. While the properties and interpretation of these models depend on how they relate to the iterated transport system, they can all be analyzed theoretically in a unified way. We present such a theoretical analysis, including convergence rates, and illustrate our methodology using real and simulated data. Our approach generalises or extends certain existing models of transportation-based regression and autoregression, and in doing so also provides some additional insights on existing models.

stat.ME↗

arXiv2023

A Karhunen-Loève Theorem for Random Flows in Hilbert spaces

We develop a generalisation of Mercer's theorem to operator-valued kernels in infinite dimensional Hilbert spaces. We then apply our result to deduce a Karhunen-Loève theorem, valid for mean-square continuous Hilbertian functional data, i.e. flows in Hilbert spaces. That is, we prove a series expansion with uncorrelated coefficients for square-integrable random flows in a Hilbert space, that holds uniformly over time.

math.PR↗

arXiv2022

Transportation-Based Functional ANOVA and PCA for Covariance Operators

We consider the problem of comparing several samples of stochastic processes with respect to their second-order structure, and describing the main modes of variation in this second order structure, if present. These tasks can be seen as an Analysis of Variance (ANOVA) and a Principal Component Analysis (PCA) of covariance operators, respectively. They arise naturally in functional data analysis, where several populations are to be contrasted relative to the nature of their dispersion around their means, rather than relative to their means themselves. We contribute a novel approach based on optimal (multi)transport, where each covariance can be identified with a a centred Gaussian process of corresponding covariance. By means of constructing the optimal simultaneous coupling of these Gaussian processes, we contrast the (linear) maps that achieve it with the identity with respect to a norm-induced distance. The resulting test statistic, calibrated by permutation, is seen to distinctly outperform the state-of-the-art, and to furnish considerable power even under local alternatives. This effect is seen to be genuinely functional, and is related to the potential for perfect discrimination in infinite dimensions. In the event of a rejection of the null hypothesis stipulating equality, a geometric interpretation of the transport maps allows us to construct a (tangent space) PCA revealing the main modes of variation. As a necessary step to developing our methodology, we prove results on the existence and boundedness of optimal multitransport maps. These are of independent interest in the theory of transport of Gaussian processes. The transportation ANOVA and PCA are illustrated on a variety of simulated and real examples.

stat.ME↗

arXiv2022

Functional Estimation of Anisotropic Covariance and Autocovariance Operators on the Sphere

We propose nonparametric estimators for the second-order central moments of possibly anisotropic spherical random fields, within a functional data analysis context. We consider a measurement framework where each random field among an identically distributed collection of spherical random fields is sampled at a few random directions, possibly subject to measurement error. The collection of random fields could be i.i.d. or serially dependent. Though similar setups have already been explored for random functions defined on the unit interval, the nonparametric estimators proposed in the literature often rely on local polynomials, which do not readily extend to the (product) spherical setting. We therefore formulate our estimation procedure as a variational problem involving a generalized Tikhonov regularization term. The latter favours smooth covariance/autocovariance functions, where the smoothness is specified by means of suitable Sobolev-like pseudo-differential operators. Using the machinery of reproducing kernel Hilbert spaces, we establish representer theorems that fully characterize the form of our estimators. We determine their uniform rates of convergence as the number of random fields diverges, both for the dense (increasing number of spatial samples) and sparse (bounded number of spatial samples) regimes. We moreover demonstrate the computational feasibility and practical merits of our estimation procedure in a simulation setting, assuming a fixed number of samples per random field. Our numerical estimation procedure leverages the sparsity and second-order Kronecker structure of our setup to reduce the computational and memory requirements by approximately three orders of magnitude compared to a naive implementation would require.

math.ST↗