arXiv subjects

Mattes Mollenhauer

Publications and source records attributed to Mattes Mollenhauer.

17 recordsLinked to original sources

arXiv2026

Asymptotic e-processes

We investigate the concept of an asymptotic e-process, which is a doubly-indexed stochastic process $(E_{m,n})_{m,n\in\mathbb{N}}$ that possesses, asymptotically for an approximation index $m\to\infty$, the properties of an e-process along a monitoring time index $n$. This constitutes the first in-depth study of this recently introduced concept, which is relevant in asymptotic sequential anytime-valid inference. Our theory is motivated by practical applications in sequential hypothesis testing, in which e-variables and e-processes can only be constructed approximately from observations due to model misspecification or estimation errors. Technically, asymptotic e-processes satisfy an asymptotic version of Ville's inequality, which bounds excursion probabilities of $(E_{m,n})_{m,n\in\mathbb{N}}$ uniformly over $n$ up to a monitoring time horizon $r_m$. We show the necessity of allowing for finite values of $r_m$, recovering truly anytime-valid guarantees asymptotically if $r_m\to\infty$. We derive various properties of asymptotic e-processes, and study their connections to asymptotic supermartingales. We also investigate general methods for their construction such as calibration, the cumulative product of asymptotic e-variables, and the monitoring an of an e-process that depends on an estimated parameter. The latter construction constitutes a generalization of a recent approach within the context of asymptotic post-hoc inference.

math.ST↗

arXiv2026

Float8@2bits: Entropy Coding Enables Data-Free Model Compression

Post-training compression is currently divided into two contrasting regimes. On the one hand, fast, data-free, and model-agnostic methods (e.g., NF4 or HQQ) offer maximum accessibility but suffer from functional collapse at extreme bit-rates below 4 bits. On the other hand, techniques leveraging calibration data or extensive recovery training achieve superior fidelity but impose high computational constraints and face uncertain robustness under data distribution shifts. We introduce EntQuant, a framework that unites the advantages of these distinct paradigms. By matching the performance of data-dependent methods with the speed and universality of data-free techniques, EntQuant enables practical utility in the extreme compression regime. Our method decouples numerical precision from storage cost via entropy coding, compressing a 70B parameter model in less than 10 minutes. We demonstrate that EntQuant does not only achieve state-of-the-art results on standard evaluation sets and models, but also retains functional performance on more complex benchmarks with instruction-tuned models, all at modest inference overhead.

cs.LG↗

arXiv2025

Fuk-Nagaev inequality in smooth Banach spaces: Optimum bounds for distributions of heavy-tailed martingales

We derive a Fuk-Nagaev inequality for the maxima of norms of martingale sequences in smooth Banach spaces which allow for a finite number of higher conditional moments. The bound is obtained by combining an optimization approach for a Chernoff bound due to Rio (2017) with a classical bound for moment generating functions of smooth Banach space norms by Pinelis (1994). Our result improves comparable infinite-dimensional bounds in the literature by removing unnecessary centering terms and giving precise constants. As an application, we propose a McDiarmid-type bound for vector-valued functions which allow for a uniform bound on their conditional higher moments.

math.PR↗

arXiv2025

Choose Your Model Size: Any Compression of Large Language Models Without Re-Computation

The adoption of Foundation Models in resource-constrained environments remains challenging due to their large size and inference costs. A promising way to overcome these limitations is post-training compression, which aims to balance reduced model size against performance degradation. This work presents Any Compression via Iterative Pruning (ACIP), a novel algorithmic approach to determine a compression-performance trade-off from a single stochastic gradient descent run. To achieve parameter efficiency, we use an SVD-reparametrization of linear layers and iteratively prune their singular values with a sparsity-inducing penalty. Importantly, the pruning order of the parameters is used to derive a global score map that allows compressing a model to any target size without re-computation. We evaluate ACIP on a large selection of open-weight LLMs and downstream tasks, demonstrating state-of-the-art results compared to existing factorization-based compression methods. We also show that ACIP seamlessly complements common quantization-based compression techniques.

cs.LG↗

arXiv2025

Regularized least squares learning with heavy-tailed noise is minimax optimal

This paper examines the performance of ridge regression in reproducing kernel Hilbert spaces in the presence of noise that exhibits a finite number of higher moments. We establish excess risk bounds consisting of subgaussian and polynomial terms based on the well known integral operator framework. The dominant subgaussian component allows to achieve convergence rates that have previously only been derived under subexponential noise - a prevalent assumption in related work from the last two decades. These rates are optimal under standard eigenvalue decay conditions, demonstrating the asymptotic robustness of regularized least squares against heavy-tailed noise. Our derivations are based on a Fuk-Nagaev inequality for Hilbert-space valued random variables.

cs.LG↗

arXiv2024

Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm

We present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between $L_2$ and the hypothesis space, which we consider as a vector-valued reproducing kernel Hilbert space. These rates allow to treat the misspecified case in which the true regression function is not contained in the hypothesis space. We combine standard assumptions on the capacity of the hypothesis space with a novel tensor product construction of vector-valued interpolation spaces in order to characterize the smoothness of the regression function. Our upper bound not only attains the same rate as real-valued kernel ridge regression, but also removes the assumption that the target regression function is bounded. For the lower bound, we reduce the problem to the scalar setting using a projection argument. We show that these rates are optimal in most cases and independent of the dimension of the output space. We illustrate our results for the special case of vector-valued Sobolev spaces.

stat.ML↗

arXiv2024

Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problem

We consider the problem of learning a linear operator $θ$ between two Hilbert spaces from empirical observations, which we interpret as least squares regression in infinite dimensions. We show that this goal can be reformulated as an inverse problem for $θ$ with the feature that its forward operator is generally non-compact (even if $θ$ is assumed to be compact or of $p$-Schatten class). However, we prove that, in terms of spectral properties and regularisation theory, this inverse problem is equivalent to the known compact inverse problem associated with scalar response regression. Our framework allows for the elegant derivation of dimension-free rates for generic learning algorithms under Hölder-type source conditions. The proofs rely on the combination of techniques from kernel regression with recent results on concentration of measure for sub-exponential Hilbertian random variables. The obtained rates hold for a variety of practically-relevant scenarios in functional regression as well as nonlinear regression with operator-valued kernels and match those of classical kernel regression with scalar response.

math.ST↗

arXiv2024

Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms

We study theoretical properties of a broad class of regularized algorithms with vector-valued output. These spectral algorithms include kernel ridge regression, kernel principal component regression, various implementations of gradient descent and many more. Our contributions are twofold. First, we rigorously confirm the so-called saturation effect for ridge regression with vector-valued output by deriving a novel lower bound on learning rates; this bound is shown to be suboptimal when the smoothness of the regression function exceeds a certain level. Second, we present the upper bound for the finite sample risk general vector-valued spectral algorithms, applicable to both well-specified and misspecified scenarios (where the true regression function lies outside of the hypothesis space) which is minimax optimal in various regimes. All of our results explicitly allow the case of infinite-dimensional output variables, proving consistency of recent practical applications.

stat.ML↗

arXiv2023

Optimal Rates for Regularized Conditional Mean Embedding Learning

We address the consistency of a kernel ridge regression estimate of the conditional mean embedding (CME), which is an embedding of the conditional distribution of $Y$ given $X$ into a target reproducing kernel Hilbert space $\mathcal{H}_Y$. The CME allows us to take conditional expectations of target RKHS functions, and has been employed in nonparametric causal and Bayesian inference. We address the misspecified setting, where the target CME is in the space of Hilbert-Schmidt operators acting from an input interpolation space between $\mathcal{H}_X$ and $L_2$, to $\mathcal{H}_Y$. This space of operators is shown to be isomorphic to a newly defined vector-valued interpolation space. Using this isomorphism, we derive a novel and adaptive statistical learning rate for the empirical CME estimator under the misspecified setting. Our analysis reveals that our rates match the optimal $O(\log n / n)$ rates without assuming $\mathcal{H}_Y$ to be finite dimensional. We further establish a lower bound on the learning rate, which shows that the obtained upper bound is optimal.

stat.ML↗

arXiv2023

On the concentration of subgaussian vectors and positive quadratic forms in Hilbert spaces

In these notes, we investigate the tail behaviour of the norm of subgaussian vectors in a Hilbert space. The subgaussian variance proxy is given as a trace class operator, allowing for a precise control of the moments along each dimension of the space. This leads to useful extensions and analogues of known Hoeffding-type inequalities and deviation bounds for positive random quadratic forms. We give a straightforward application in terms of a variance bound for the regularisation of statistical inverse problems.

math.PR↗

arXiv2023

Nonparametric approximation of conditional expectation operators

Given the joint distribution of two random variables $X,Y$ on some second countable locally compact Hausdorff space, we investigate the statistical approximation of the $L^2$-operator defined by $[Pf](x) := \mathbb{E}[ f(Y) \mid X = x ]$ under minimal assumptions. By modifying its domain, we prove that $P$ can be arbitrarily well approximated in operator norm by Hilbert-Schmidt operators acting on a reproducing kernel Hilbert space. This fact allows to estimate $P$ uniformly by finite-rank operators over a dense subspace even when $P$ is not compact. In terms of modes of convergence, we thereby obtain the superiority of kernel-based techniques over classically used parametric projection approaches such as Galerkin methods. This also provides a novel perspective on which limiting object the nonparametric estimate of $P$ converges to. As an application, we show that these results are particularly important for a large family of spectral analysis techniques for Markov transition operators. Our investigation also gives a new asymptotic perspective on the so-called kernel conditional mean embedding, which is the theoretical foundation of a wide variety of techniques in kernel-based nonparametric inference.

math.ST↗

arXiv2022

Kernel Autocovariance Operators of Stationary Processes: Estimation and Convergence

We consider autocovariance operators of a stationary stochastic process on a Polish space that is embedded into a reproducing kernel Hilbert space. We investigate how empirical estimates of these operators converge along realizations of the process under various conditions. In particular, we examine ergodic and strongly mixing processes and obtain several asymptotic results as well as finite sample error bounds. We provide applications of our theory in terms of consistency results for kernel PCA with dependent data and the conditional mean embedding of transition probabilities. Finally, we use our approach to examine the nonparametric estimation of Markov transition operators and highlight how our theory can give a consistency analysis for a large family of spectral analysis methods including kernel-based dynamic mode decomposition.

math.PR↗

arXiv2021

Optimal Reaction Coordinates: Variational Characterization and Sparse Computation

Reaction Coordinates (RCs) are indicators of hidden, low-dimensional mechanisms that govern the long-term behavior of high-dimensional stochastic processes. We present a novel and general variational characterization of optimal RCs and provide conditions for their existence. Optimal RCs are minimizers of a certain loss function and reduced models based on them guarantee very good approximation of the long-term dynamics of the original high-dimensional process. We show that, for slow-fast systems, metastable systems, and other systems with known good RCs, the novel theory reproduces previous insight. Remarkably, the numerical effort required to evaluate the loss function scales only with the complexity of the underlying, low-dimensional mechanism, and not with that of the full system. The theory provided lays the foundation for an efficient and data-sparse computation of RCs via modern machine learning techniques.

math.DS↗

arXiv2020

Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spaces

Reproducing kernel Hilbert spaces (RKHSs) play an important role in many statistics and machine learning applications ranging from support vector machines to Gaussian processes and kernel embeddings of distributions. Operators acting on such spaces are, for instance, required to embed conditional probability distributions in order to implement the kernel Bayes rule and build sequential data models. It was recently shown that transfer operators such as the Perron-Frobenius or Koopman operator can also be approximated in a similar fashion using covariance and cross-covariance operators and that eigenfunctions of these operators can be obtained by solving associated matrix eigenvalue problems. The goal of this paper is to provide a solid functional analytic foundation for the eigenvalue decomposition of RKHS operators and to extend the approach to the singular value decomposition. The results are illustrated with simple guiding examples.

math.FA↗

arXiv2019

Kernel Conditional Density Operators

We introduce a novel conditional density estimation model termed the conditional density operator (CDO). It naturally captures multivariate, multimodal output densities and shows performance that is competitive with recent neural conditional density models and Gaussian processes. The proposed model is based on a novel approach to the reconstruction of probability densities from their kernel mean embeddings by drawing connections to estimation of Radon-Nikodym derivatives in the reproducing kernel Hilbert space (RKHS). We prove finite sample bounds for the estimation error in a standard density reconstruction scenario, independent of problem dimensionality. Interestingly, when a kernel is used that is also a probability density, the CDO allows us to both evaluate and sample the output density efficiently. We demonstrate the versatility and performance of the proposed model on both synthetic and real-world data.

cs.LG↗

arXiv2019

Kernel methods for detecting coherent structures in dynamical data

We illustrate relationships between classical kernel-based dimensionality reduction techniques and eigendecompositions of empirical estimates of reproducing kernel Hilbert space (RKHS) operators associated with dynamical systems. In particular, we show that kernel canonical correlation analysis (CCA) can be interpreted in terms of kernel transfer operators and that it can be obtained by optimizing the variational approach for Markov processes (VAMP) score. As a result, we show that coherent sets of particle trajectories can be computed by kernel CCA. We demonstrate the efficiency of this approach with several examples, namely the well-known Bickley jet, ocean drifter data, and a molecular dynamics problem with a time-dependent potential. Finally, we propose a straightforward generalization of dynamic mode decomposition (DMD) called coherent mode decomposition (CMD). Our results provide a generic machine learning approach to the computation of coherent sets with an objective score that can be used for cross-validation and the comparison of different methods.

math.DS↗

arXiv2018

On Hyperparameter Search in Cluster Ensembles

Quality assessments of models in unsupervised learning and clustering verification in particular have been a long-standing problem in the machine learning research. The lack of robust and universally applicable cluster validity scores often makes the algorithm selection and hyperparameter evaluation a tough guess. In this paper, we show that cluster ensemble aggregation techniques such as consensus clustering may be used to evaluate clusterings and their hyperparameter configurations. We use normalized mutual information to compare individual objects of a clustering ensemble to the constructed consensus of the whole ensemble and show, that the resulting score can serve as an overall quality measure for clustering problems. This method is capable of highlighting the standout clustering and hyperparameter configuration in the ensemble even in the case of a distorted consensus. We apply this very general framework to various data sets and give possible directions for future research.

stat.ML↗