Regression with non-Euclidean responses -- e.g., probability distributions, networks, symmetric positive-definite matrices, and compositions -- has become increasingly important in modern applications. In this paper, we propose deep Fréchet neural networks (DFNNs), an end-to-end deep learning framework for predicting non-Euclidean responses -- which are considered as random objects in a metric space -- from Euclidean predictors. Our method leverages the representation-learning power of deep neural networks (DNNs) to the task of approximating conditional Fréchet means of the response given the predictors, the metric-space analogue of conditional expectations, by minimizing a Fréchet risk. The framework is highly flexible, accommodating diverse metrics and high-dimensional predictors. We establish a universal approximation theorem for DFNNs, advancing the state-of-the-art of neural network approximation theory to general metric-space-valued responses without making model assumptions or relying on local smoothing. Empirical studies on synthetic distributional and network-valued responses, as well as a real-world application to predicting employment occupational compositions, demonstrate that DFNNs consistently outperform existing methods.

We study nonparametric regression and classification for path-valued data. We introduce a functional Nadaraya-Watson estimator that combines the signature transform from rough path theory with local kernel regression. The signature transform provides a principled way to encode sequential data through iterated integrals, enabling direct comparison of paths in a natural metric space. Our approach leverages signature-induced distances within the classical kernel regression framework, achieving computational efficiency while avoiding the scalability bottlenecks of large-scale kernel matrix operations. We establish finite-sample convergence bounds demonstrating favorable statistical properties of signature-based distances compared to traditional metrics in infinite-dimensional settings. We propose robust signature variants that provide stability against outliers, enhancing practical performance. Applications to both synthetic and real-world data - including stochastic differential equation learning and time series classification - demonstrate competitive accuracy while offering significant computational advantages over existing methods.

Analysis and processing of data is a vital part of our modern society and requires vast amounts of computational resources. To reduce the computational burden, compressing and approximating data has become a central topic. We consider the approximation of labeled data samples, mathematically described as site-to-value maps between finite metric spaces. Within this setting, we identify the discrete modulus of continuity as an effective data-intrinsic quantity to measure regularity of site-to-value maps without imposing further structural assumptions. We investigate the consistency of the discrete modulus of continuity in the infinite data limit and propose an algorithm for its efficient computation. Building on these results, we present a sample based approximation theory for labeled data. For data subject to statistical uncertainty we consider multilevel approximation spaces and a variant of the multilevel Monte Carlo method to compute statistical quantities of interest. Our considerations connect approximation theory for labeled data in metric spaces to the covering problem for (random) balls on the one hand and the efficient evaluation of the discrete modulus of continuity to combinatorial optimization on the other hand. We provide extensive numerical studies to illustrate the feasibility of the approach and to validate our theoretical results.

We define a stochastic variant of the proximal point algorithm in the general setting of nonlinear (separable) Hadamard spaces for approximating zeros of the mean of a stochastically perturbed monotone vector field and prove its convergence under a suitable strong monotonicity assumption, together with a probabilistic independence assumption and a separability assumption on the tangent spaces. As a particular case, our results transfer previous work by P. Bianchi on that method in Hilbert spaces for the first time to Hadamard manifolds. Moreover, our convergence proof is fully effective and allows for the construction of explicit rates of convergence for the iteration towards the (unique) solution both in mean and almost surely. These rates are moreover highly uniform, being independent of most data surrounding the iteration, space or distribution. In that generality, these rates are novel already in the context of Hilbert spaces. Linear nonasymptotic guarantees under additional second-moment conditions on the Yosida approximates and special cases of stochastic convex minimization are discussed.

Diversity is a key concept in representation learning, referring to the relative abundance and distinctiveness of model outputs.

We explore combining dropout with robust training methods and obtain better generalization.

Online structured prediction, including online classification as a special case, is the task of sequentially predicting labels from input features. Therein the surrogate regret -- the cumulative excess of the target loss (e.g., 0-1 loss) over the surrogate loss (e.g., logistic loss) of the fixed best estimator -- has gained attention, particularly because it often admits a finite bound independent of the time horizon $T$. However, such guarantees break down in non-stationary environments, where every fixed estimator may incur the surrogate loss growing linearly with $T$. We address this by proving a bound of the form $F_T + C(1 + P_T)$ on the cumulative target loss, where $F_T$ is the cumulative surrogate loss of any comparator sequence, $P_T$ is its path length, and $C > 0$ is some constant. This bound depends on $T$ only through $F_T$ and $P_T$, often yielding much stronger guarantees in non-stationary environments. Our core idea is to synthesize the dynamic regret bound of the online gradient descent (OGD) with the technique of exploiting the surrogate gap. Our analysis also sheds light on a new Polyak-style learning rate for OGD, which systematically offers target-loss guarantees and exhibits promising empirical performance. We further extend our approach to a broader class of problems via the convolutional Fenchel--Young loss. Finally, we prove a lower bound showing that the dependence on $F_T$ and $P_T$ is tight.