Random Fourier features provide a way to tackle large-scale machine learning problems with kernel methods. Their slow Monte Carlo convergence rate has motivated the research of deterministic Fourier features whose approximation error can decrease exponentially in the number of basis functions. However, due to their tensor product extension to multiple dimensions, these methods suffer heavily from the curse of dimensionality, limiting their applicability to one, two or three-dimensional scenarios. In our approach we overcome said curse of dimensionality by exploiting the tensor product structure of deterministic Fourier features, which enables us to represent the model parameters as a low-rank tensor decomposition. We derive a monotonically converging block coordinate descent algorithm with linear complexity in both the sample size and the dimensionality of the inputs for a regularized squared loss function, allowing to learn a parsimonious model in decomposed form using deterministic Fourier features.We demonstrate by means of numerical experiments how our low-rank tensor approach obtains the same performance of the corresponding nonparametric model, consistently outperforming random Fourier features.

Markov chain Monte Carlo (MCMC) is an established approach for uncertainty quantification and propagation in scientific applications. A key challenge in applying MCMC to scientific domains is computation: the target density of interest is often a function of expensive computations, such as a high-fidelity physical simulation, an intractable integral, or a slowly-converging iterative algorithm. Thus, using an MCMC algorithms with an expensive target density becomes impractical, as these expensive computations need to be evaluated at each iteration of the algorithm. Multi-fidelity MCMC algorithms combine models of varying fidelities in order to obtain an approximate target density with lower computational cost. In this paper, we describe a class of asymptotically exact multi-fidelity MCMC algorithms for the setting where a sequence of models of increasing fidelity can be computed that approximates the expensive target density of interest. We take a pseudo-marginal MCMC approach for multi-fidelity inference that utilizes a cheaper, randomized-fidelity unbiased estimator of the target fidelity constructed via random truncation of a telescoping series of the low-fidelity sequence of models.

This paper studies the fundamental problem of learning energy-based model (EBM) in the latent space of the generator model. Learning such prior model typically requires running costly Markov Chain Monte Carlo (MCMC). Instead, we propose to use noise contrastive estimation (NCE) to discriminatively learn the EBM through density ratio estimation between the latent prior density and latent posterior density. However, the NCE typically fails to accurately estimate such density ratio given large gap between two densities. To effectively tackle this issue and further learn more expressive prior model, we develop the adaptive multi-stage density ratio estimation which breaks the estimation into multiple stages and learn different stages of density ratio sequentially and adaptively. The latent prior model can be gradually learned using ratio estimated in previous stage so that the final latent space EBM prior can be naturally formed by product of ratios in different stages.

Approximate Bayesian inference estimates descriptors of an intractable target distribution - in essence, an optimization problem within a family of distributions. For example, Langevin dynamics (LD) extracts asymptotically exact samples from a diffusion process because the time evolution of its marginal distributions constitutes a curve that minimizes the KL-divergence via steepest descent in the Wasserstein space. Parallel to LD, Stein variational gradient descent (SVGD) similarly minimizes the KL, albeit endowed with a novel Stein-Wasserstein distance, by deterministically transporting a set of particle samples, thus de-randomizes the stochastic diffusion process. We propose de-randomized kernel-based particle samplers to all diffusion-based samplers known as MCMC dynamics. Following previous work in interpreting MCMC dynamics, we equip the Stein-Wasserstein space with a fiber-Riemannian Poisson structure, with the capacity of characterizing a fiber-gradient Hamiltonian flow that simulates MCMC dynamics.

Tensor decomposition is an important tool for multiway data analysis. In practice, the data is often sparse yet associated with rich temporal information. Existing methods, however, often under-use the time information and ignore the structural knowledge within the sparsely observed tensor entries. To overcome these limitations and to better capture the underlying temporal structure, we propose Dynamic EMbedIngs fOr dynamic Tensor dEcomposition (DEMOTE). We develop a neural diffusion-reaction process to estimate dynamic embeddings for the entities in each tensor mode.

DBSCAN is a popular density-based clustering algorithm. It computes the \epsilon -neighborhood graph of a dataset and uses the connected components of the high-degree nodes to decide the clusters. However, the full neighborhood graph may be too costly to compute with a worst-case complexity of O(n 2) . In this paper, we propose a simple variant called SNG-DBSCAN, which clusters based on a subsampled \epsilon -neighborhood graph, only requires access to similarity queries for pairs of points and in particular avoids any complex data structures which need the embeddings of the data points themselves. The runtime of the procedure is O(sn 2), where s is the sampling rate. We show under some natural theoretical assumptions that s \approx \log n/n is sufficient for statistical cluster recovery guarantees leading to an O(n\log n) complexity.

We address the problem of learning to control an unknown nonlinear dynamical system through sequential interactions. Motivated by high-stakes applications in which mistakes can be catastrophic, such as robotics and healthcare, we study situations where it is possible for fast sequential learning to occur. Fast sequential learning is characterized by the ability of the learning agent to incur logarithmic regret relative to a fully-informed baseline. We demonstrate that fast sequential learning is achievable in a diverse class of continuous control problems where the system dynamics depend smoothly on unknown parameters, provided the optimal control policy is persistently exciting. Additionally, we derive a regret bound which grows with the square root of the number of interactions for cases where the optimal policy is not persistently exciting. Our results provide the first regret bounds for controlling nonlinear dynamical systems depending nonlinearly on unknown parameters. We validate the trends our theory predicts in simulation on a simple dynamical system.

We introduce a technique to enhance the reliability of gravitational wave parameter estimation results produced by machine learning. We develop two independent machine learning models based on the Vision Transformer to estimate effective spin and chirp mass from spectrograms of gravitational wave signals from binary black hole mergers. To enhance the reliability of these models, we utilize attention maps to visualize the areas our models focus on when making predictions. This approach enables demonstrating that both models perform parameter estimation based on physically meaningful information. Furthermore, by leveraging these attention maps, we demonstrate a method to quantify the impact of glitches on parameter estimation. We show that as the models focus more on glitches, the parameter estimation results become more strongly biased. This suggests that attention maps could potentially be used to distinguish between cases where the results produced by the machine learning model are reliable and cases where they are not.

Parameter estimation is a fundamental challenge in machine learning, crucial for tasks such as neural network weight fitting and Bayesian inference. This paper focuses on the complexity of estimating translation $\boldsymbol{\mu} \in \mathbb{R}^l$ and shrinkage $\sigma \in \mathbb{R}_{++}$ parameters for a distribution of the form $\frac{1}{\sigma^l} f_0 \left( \frac{\boldsymbol{x} - \boldsymbol{\mu}}{\sigma} \right)$, where $f_0$ is a known density in $\mathbb{R}^l$ given $n$ samples. We highlight that while the problem is NP-hard for Maximum Likelihood Estimation (MLE), it is possible to obtain $\varepsilon$-approximations for arbitrary $\varepsilon > 0$ within $\text{poly} \left( \frac{1}{\varepsilon} \right)$ time using the Wasserstein distance.

The fast fashion industry's insatiable demand for new styles and rapid production cycles has led to a significant environmental burden. Overproduction, excessive waste, and harmful chemicals have contributed to the negative environmental impact of the industry. To mitigate these issues, a paradigm shift that prioritizes sustainability and efficiency is urgently needed. Integrating learning-based predictive analytics into the fashion industry represents a significant opportunity to address environmental challenges and drive sustainable practices. By forecasting fashion trends and optimizing production, brands can reduce their ecological footprint while remaining competitive in a rapidly changing market. However, one of the key challenges in forecasting fashion sales is the dynamic nature of consumer preferences. Fashion is acyclical, with trends constantly evolving and resurfacing. In addition, cultural changes and unexpected events can disrupt established patterns. This problem is also known as New Fashion Products Performance Forecasting (NFPPF), and it has recently gained more and more interest in the global research landscape. Given its multidisciplinary nature, the field of NFPPF has been approached from many different angles. This comprehensive survey wishes to provide an up-to-date overview that focuses on learning-based NFPPF strategies. The survey is based on the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) methodological flow, allowing for a systematic and complete literature review. In particular, we propose the first taxonomy that covers the learning panorama for NFPPF, examining in detail the different methodologies used to increase the amount of multimodal information, as well as the state-of-the-art available datasets. Finally, we discuss the challenges and future directions.