In this manuscript, we propose an efficient manifold denoiser based on landmark diffusion and optimal shrinkage under the complicated high dimensional noise and compact manifold setup. It is flexible to handle several setups, including the high ambient space dimension with a manifold embedding that occupies a subspace of high or low dimensions, and the noise could be colored and dependent. A systematic comparison with other existing algorithms on both simulated and real datasets is provided. This manuscript is mainly algorithmic and we report several existing tools and numerical results. Theoretical guarantees and more comparisons will be reported in the official paper of this manuscript.

We introduce a new mean-field ODE and corresponding interacting particle systems for sampling from an unnormalized target density or Bayesian posterior. The interacting particle systems are gradient-free, available in closed form, and only require the ability to sample from the reference density and compute the (unnormalized) target-to-reference density ratio. The mean-field ODE is obtained by solving a Poisson equation for a velocity field that transports samples along the geometric mixture of the two densities, which is the path of a particular Fisher-Rao gradient flow. We employ a reproducing kernel Hilbert space ansatz for the velocity field, which makes the Poisson equation tractable and enables us to discretize the resulting mean-field ODE over finite samples, as a simple interacting particle system. The mean-field ODE can be additionally be derived from a discrete-time perspective as the limit of successive linearizations of the Monge-Amp\`ere equations within a framework known as sample-driven optimal transport. We demonstrate empirically that our interacting particle systems can produce high-quality samples from distributions with varying characteristics.

A compression function is a map that slims down an observational set into a subset of reduced size, while preserving its informational content. In multiple applications, the condition that one new observation makes the compressed set change is interpreted that this observation brings in extra information and, in learning theory, this corresponds to misclassification, or misprediction. In this paper, we lay the foundations of a new theory that allows one to keep control on the probability of change of compression (which maps into the statistical "risk" in learning applications). Under suitable conditions, the cardinality of the compressed set is shown to be a consistent estimator of the probability of change of compression (without any upper limit on the size of the compressed set); moreover, unprecedentedly tight finite-sample bounds to evaluate the probability of change of compression are obtained under a generally applicable condition of preference. All results are usable in a fully agnostic setup, i.e., without requiring any a priori knowledge on the probability distribution of the observations. Not only these results offer a valid support to develop trust in observation-driven methodologies, they also play a fundamental role in learning techniques as a tool for hyper-parameter tuning.

Principal component analysis (PCA), along with its extensions to manifolds and outlier contaminated data, have been indispensable in computer vision and machine learning. In this work, we present a unifying formalism for PCA and its variants, and introduce a framework based on the flags of linear subspaces, \ie a hierarchy of nested linear subspaces of increasing dimension, which not only allows for a common implementation but also yields novel variants, not explored previously. We begin by generalizing traditional PCA methods that either maximize variance or minimize reconstruction error. We expand these interpretations to develop a wide array of new dimensionality reduction algorithms by accounting for outliers and the data manifold. To devise a common computational approach, we recast robust and dual forms of PCA as optimization problems on flag manifolds. We then integrate tangent space approximations of principal geodesic analysis (tangent-PCA) into this flag-based framework, creating novel robust and dual geodesic PCA variations. The remarkable flexibility offered by the 'flagification' introduced here enables even more algorithmic variants identified by specific flag types. Last but not least, we propose an effective convergent solver for these flag-formulations employing the Stiefel manifold. Our empirical results on both real-world and synthetic scenarios, demonstrate the superiority of our novel algorithms, especially in terms of robustness to outliers on manifolds.

Deep learning models, such as wide neural networks, can be conceptualized as nonlinear dynamical physical systems characterized by a multitude of interacting degrees of freedom. Such systems in the infinite limit, tend to exhibit simplified dynamics. This paper delves into gradient descent-based learning algorithms, that display a linear structure in their parameter dynamics, reminiscent of the neural tangent kernel. We establish this apparent linearity arises due to weak correlations between the first and higher-order derivatives of the hypothesis function, concerning the parameters, taken around their initial values. This insight suggests that these weak correlations could be the underlying reason for the observed linearization in such systems. As a case in point, we showcase this weak correlations structure within neural networks in the large width limit. Exploiting the relationship between linearity and weak correlations, we derive a bound on deviations from linearity observed during the training trajectory of stochastic gradient descent. To facilitate our proof, we introduce a novel method to characterise the asymptotic behavior of random tensors.

Characterizing the distribution of high-dimensional statistical estimators is a challenging task, due to the breakdown of classical asymptotic theory in high dimension. This paper makes progress towards this by developing non-asymptotic distributional characterizations for approximate message passing (AMP) -- a family of iterative algorithms that prove effective as both fast estimators and powerful theoretical machinery -- for both sparse and robust regression. Prior AMP theory, which focused on high-dimensional asymptotics for the most part, failed to describe the behavior of AMP when the number of iterations exceeds $o\big({\log n}/{\log \log n}\big)$ (with $n$ the sample size). We establish the first finite-sample non-asymptotic distributional theory of AMP for both sparse and robust regression that accommodates a polynomial number of iterations. Our results derive approximate accuracy of Gaussian approximation of the AMP iterates, which improves upon all prior results and implies enhanced distributional characterizations for both optimally tuned Lasso and robust M-estimator.

Estimating a covariance matrix and its associated principal components is a fundamental problem in contemporary statistics. While optimal estimation procedures have been developed with well-understood properties, the increasing demand for privacy preservation introduces new complexities to this classical problem. In this paper, we study optimal differentially private Principal Component Analysis (PCA) and covariance estimation within the spiked covariance model. We precisely characterize the sensitivity of eigenvalues and eigenvectors under this model and establish the minimax rates of convergence for estimating both the principal components and covariance matrix. These rates hold up to logarithmic factors and encompass general Schatten norms, including spectral norm, Frobenius norm, and nuclear norm as special cases. We introduce computationally efficient differentially private estimators and prove their minimax optimality, up to logarithmic factors. Additionally, matching minimax lower bounds are established. Notably, in comparison with existing literature, our results accommodate a diverging rank, necessitate no eigengap condition between distinct principal components, and remain valid even if the sample size is much smaller than the dimension.

Topological Data Analysis (TDA) has emerged as a transformative approach to extract meaningful information from complex datasets, offering a lens through which to understand the data's underlying structure. Unlike traditional data analysis methods that rely on geometric or statistical measures, TDA employs tools from both computational geometry and algebraic topology to study the topological features inherent in datasets. In the context of cosmology, where the distribution of matter exhibits complex and interconnected patterns, TDA becomes a valuable tool for uncovering the underlying cosmic topology. The cosmic web, encompassing galaxies, intergalactic gas, and dark matter, exhibits an organized tendency to form structures such as galaxy clusters, filaments (thread-like structures that connect galaxy clusters), and walls, surrounded by low-density void regions (Colberg et al. [2008], Van de Weygaert and Platen [2011], Cautun et al. [2014], Wilding et al. [2021]). Within this cosmic context, large galaxy clusters aggregate into more extensive formations referred to as filaments or superclusters of galaxies Kelesis et al. [2022].

Bayesian optimization has become a powerful tool for safe online optimization of systems, due to its high sample efficiency and noise robustness. For further speed-up reduced physical models of the system can be incorporated into the optimization to accelerate the process, since the models are able to offer an approximation of the actual system, and sampling from them is significantly cheaper. The similarity between model and reality is represented by additional hyperparameters and learned within the optimization process. Safety is an important criteria for online optimization methods like Bayesian optimization, which has been addressed by recent literature, which provide safety guarantees under the assumption of known hyperparameters. However, in practice this is not applicable. Therefore, we extend the robust Gaussian process uniform error bounds to meet the multi-task setting, which involves the calculation of a confidence region from the hyperparameter posterior distribution utilizing Markov chain Monte Carlo methods. Then, using the robust safety bounds, Bayesian optimization is applied to safely optimize the system while incorporating measurements of the models. Simulations show that the optimization can be significantly accelerated compared to other state-of-the-art safe Bayesian optimization methods depending on the fidelity of the models.

Gaussian process state-space models (GPSSMs) are a versatile and principled family of nonlinear dynamical system models. However, existing variational learning and inference methods for GPSSMs often necessitate optimizing a substantial number of variational parameters, leading to inadequate performance and efficiency. To overcome this issue, we propose incorporating the ensemble Kalman filter (EnKF), a well-established model-based filtering technique, into the variational inference framework to approximate the posterior distribution of latent states. This utilization of EnKF can effectively exploit the dependencies between latent states and GP dynamics, while eliminating the need for parameterizing the variational distribution, thereby significantly reducing the number of variational parameters. Moreover, we show that our proposed algorithm allows straightforward evaluation of an approximated evidence lower bound (ELBO) in variational inference via simply summating multiple terms with readily available closed-form solutions. Leveraging automatic differentiation tools, we hence can maximize the ELBO and train the GPSSM efficiently. We also extend the proposed algorithm to accommodate an online setting and provide detailed algorithmic analyses and insights. Extensive evaluation on diverse real and synthetic datasets demonstrates the superiority of our EnKF-aided variational inference algorithms in terms of learning and inference performance compared to existing methods.