We prove an upper bound on the covering number of real algebraic varieties, images of polynomial maps and semialgebraic sets. The bound remarkably improves the best known general bound by Yomdin-Comte, and its proof is much more straightforward. As a consequence, our result gives new bounds on the volume of the tubular neighborhood of the image of a polynomial map and a semialgebraic set, where results for varieties by Lotz and Basu-Lerario are not directly applicable. We apply our theory to three main application domains. Firstly, we derive a near-optimal bound on the covering number of low rank CP tensors. Secondly, we prove a bound on the sketching dimension for (general) polynomial optimization problems. Lastly, we deduce generalization error bounds for deep neural networks with rational or ReLU activations, improving or matching the best known results in the literature.

The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. In this study, we delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on probability spaces. Our particular focus centers on the Wasserstein distance function, which serves as a relevant example. In contrast to the existing body of literature focused on approximating efficiently pointwise evaluations, we chart a new course to define functional approximants by adopting three machine learning-based approaches: 1. Solving a finite number of optimal transport problems and computing the corresponding Wasserstein potentials. 2. Employing empirical risk minimization with Tikhonov regularization in Wasserstein Sobolev spaces. 3. Addressing the problem through the saddle point formulation that characterizes the weak form of the Tikhonov functional's Euler-Lagrange equation. As a theoretical contribution, we furnish explicit and quantitative bounds on generalization errors for each of these solutions. In the proofs, we leverage the theory of metric Sobolev spaces and we combine it with techniques of optimal transport, variational calculus, and large deviation bounds. In our numerical implementation, we harness appropriately designed neural networks to serve as basis functions. These networks undergo training using diverse methodologies. This approach allows us to obtain approximating functions that can be rapidly evaluated after training. Consequently, our constructive solutions significantly enhance at equal accuracy the evaluation speed, surpassing that of state-of-the-art methods by several orders of magnitude.

The advent of predictive methodologies has catalyzed the emergence of data-driven decision support across various domains. However, developing models capable of effectively handling input time series data presents an enduring challenge. This study presents novel preference learning approaches to multiple criteria sorting problems in the presence of temporal criteria. We first formulate a convex quadratic programming model characterized by fixed time discount factors, operating within a regularization framework. To enhance scalability and accommodate learnable time discount factors, we introduce a novel monotonic Recurrent Neural Network (mRNN). It is designed to capture the evolving dynamics of preferences over time while upholding critical properties inherent to MCS problems, including criteria monotonicity, preference independence, and the natural ordering of classes. The proposed mRNN can describe the preference dynamics by depicting marginal value functions and personalized time discount factors along with time, effectively amalgamating the interpretability of traditional MCS methods with the predictive potential offered by deep preference learning models. Comprehensive assessments of the proposed models are conducted, encompassing synthetic data scenarios and a real-case study centered on classifying valuable users within a mobile gaming app based on their historical in-app behavioral sequences. Empirical findings underscore the notable performance improvements achieved by the proposed models when compared to a spectrum of baseline methods, spanning machine learning, deep learning, and conventional multiple criteria sorting approaches.

Spectral computed tomography based on a photon-counting detector (PCD) attracts more and more attentions since it has the capability to provide more accurate identification and quantitative analysis for biomedical materials. The limited number of photons within narrow energy bins leads to imaging results of low signal-noise ratio. The existing supervised deep reconstruction networks for CT reconstruction are difficult to address these challenges because it is usually impossible to acquire noise-free clinical images with clear structures as references. In this paper, we propose an iterative deep reconstruction network to synergize unsupervised method and data priors into a unified framework, named as Spectral2Spectral. Our Spectral2Spectral employs an unsupervised deep training strategy to obtain high-quality images from noisy data in an end-to-end fashion. The structural similarity prior within image-spectral domain is refined as a regularization term to further constrain the network training. The weights of neural network are automatically updated to capture image features and structures within the iterative process. Three large-scale preclinical datasets experiments demonstrate that the Spectral2spectral reconstructs better image quality than other the state-of-the-art methods.

Comparing different AutoML frameworks is notoriously challenging and often done incorrectly. We introduce an open and extensible benchmark that follows best practices and avoids common mistakes when comparing AutoML frameworks. We conduct a thorough comparison of 9 well-known AutoML frameworks across 71 classification and 33 regression tasks. The differences between the AutoML frameworks are explored with a multi-faceted analysis, evaluating model accuracy, its trade-offs with inference time, and framework failures. We also use Bradley-Terry trees to discover subsets of tasks where the relative AutoML framework rankings differ. The benchmark comes with an open-source tool that integrates with many AutoML frameworks and automates the empirical evaluation process end-to-end: from framework installation and resource allocation to in-depth evaluation. The benchmark uses public data sets, can be easily extended with other AutoML frameworks and tasks, and has a website with up-to-date results.

Advances in artificial intelligence (AI) are fueling a new paradigm of discoveries in natural sciences. Today, AI has started to advance natural sciences by improving, accelerating, and enabling our understanding of natural phenomena at a wide range of spatial and temporal scales, giving rise to a new area of research known as AI for science (AI4Science). Being an emerging research paradigm, AI4Science is unique in that it is an enormous and highly interdisciplinary area. Thus, a unified and technical treatment of this field is needed yet challenging. This work aims to provide a technically thorough account of a subarea of AI4Science; namely, AI for quantum, atomistic, and continuum systems. These areas aim at understanding the physical world from the subatomic (wavefunctions and electron density), atomic (molecules, proteins, materials, and interactions), to macro (fluids, climate, and subsurface) scales and form an important subarea of AI4Science. A unique advantage of focusing on these areas is that they largely share a common set of challenges, thereby allowing a unified and foundational treatment. A key common challenge is how to capture physics first principles, especially symmetries, in natural systems by deep learning methods. We provide an in-depth yet intuitive account of techniques to achieve equivariance to symmetry transformations. We also discuss other common technical challenges, including explainability, out-of-distribution generalization, knowledge transfer with foundation and large language models, and uncertainty quantification. To facilitate learning and education, we provide categorized lists of resources that we found to be useful. We strive to be thorough and unified and hope this initial effort may trigger more community interests and efforts to further advance AI4Science.

In the last years, one of the fields of artificial intelligence that has been investigated the most is nature-inspired computing. The research done on this specific topic showcases the interest that sparks in researchers and practitioners, who put their focus on this paradigm because of the adaptability and ability of nature-inspired algorithms to reach high-quality outcomes on a wide range of problems. In fact, this kind of methods has been successfully applied to solve real-world problems in heterogeneous fields such as medicine, transportation, industry, or software engineering. Our main objective with this paper is to describe a tool based on nature-inspired computing for solving a specific software engineering problem. The problem faced consists of optimizing Infrastructure as Code deployment configurations. For this reason, the name of the system is IaC Optimizer Platform. A prototypical version of the IOP was described in previous works, in which the functionality of this platform was introduced. With this paper, we take a step forward by describing the final release of the IOP, highlighting its main contribution regarding the current state-of-the-art, and justifying the decisions made on its implementation. Also, we contextualize the IOP within the complete platform in which it is embedded, describing how a user can benefit from its use. To do that, we also present and solve a real-world use case.

Building 3D geometric maps of man-made spaces is a well-established and active field that is fundamental to computer vision and robotics. However, considering the evolving nature of built environments, it is essential to question the capabilities of current mapping efforts in handling temporal changes. In addition, spatiotemporal mapping holds significant potential for achieving sustainability and circularity goals. Existing mapping approaches focus on small changes, such as object relocation or self-driving car operation; in all cases where the main structure of the scene remains fixed. Consequently, these approaches fail to address more radical changes in the structure of the built environment, such as geometry and topology. To this end, we introduce the Nothing Stands Still (NSS) benchmark, which focuses on the spatiotemporal registration of 3D scenes undergoing large spatial and temporal change, ultimately creating one coherent spatiotemporal map. Specifically, the benchmark involves registering two or more partial 3D point clouds (fragments) from the same scene but captured from different spatiotemporal views. In addition to the standard pairwise registration, we assess the multi-way registration of multiple fragments that belong to any temporal stage. As part of NSS, we introduce a dataset of 3D point clouds recurrently captured in large-scale building indoor environments that are under construction or renovation. The NSS benchmark presents three scenarios of increasing difficulty, to quantify the generalization ability of point cloud registration methods over space (within one building and across buildings) and time. We conduct extensive evaluations of state-of-the-art methods on NSS. The results demonstrate the necessity for novel methods specifically designed to handle large spatiotemporal changes. The homepage of our benchmark is at http://nothing-stands-still.com.

In this paper, we study the computation of the rate-distortion-perception function (RDPF) for a multivariate Gaussian source under mean squared error (MSE) distortion and, respectively, Kullback-Leibler divergence, geometric Jensen-Shannon divergence, squared Hellinger distance, and squared Wasserstein-2 distance perception metrics. To this end, we first characterize the analytical bounds of the scalar Gaussian RDPF for the aforementioned divergence functions, also providing the RDPF-achieving forward "test-channel" realization. Focusing on the multivariate case, we establish that, for tensorizable distortion and perception metrics, the optimal solution resides on the vector space spanned by the eigenvector of the source covariance matrix. Consequently, the multivariate optimization problem can be expressed as a function of the scalar Gaussian RDPFs of the source marginals, constrained by global distortion and perception levels. Leveraging this characterization, we design an alternating minimization scheme based on the block nonlinear Gauss-Seidel method, which optimally solves the problem while identifying the Gaussian RDPF-achieving realization. Furthermore, the associated algorithmic embodiment is provided, as well as the convergence and the rate of convergence characterization. Lastly, for the "perfect realism" regime, the analytical solution for the multivariate Gaussian RDPF is obtained. We corroborate our results with numerical simulations and draw connections to existing results.

We give a damped proximal augmented Lagrangian method (DPALM) for solving problems with a weakly-convex objective and convex linear/nonlinear constraints. Instead of taking a full stepsize, DPALM adopts a damped dual stepsize to ensure the boundedness of dual iterates. We show that DPALM can produce a (near) $\vareps$-KKT point within $O(\vareps^{-2})$ outer iterations if each DPALM subproblem is solved to a proper accuracy. In addition, we establish overall iteration complexity of DPALM when the objective is either a regularized smooth function or in a regularized compositional form. For the former case, DPALM achieves the complexity of $\widetilde{\mathcal{O}}\left(\varepsilon^{-2.5} \right)$ to produce an $\varepsilon$-KKT point by applying an accelerated proximal gradient (APG) method to each DPALM subproblem. For the latter case, the complexity of DPALM is $\widetilde{\mathcal{O}}\left(\varepsilon^{-3} \right)$ to produce a near $\varepsilon$-KKT point by using an APG to solve a Moreau-envelope smoothed version of each subproblem. Our outer iteration complexity and the overall complexity either generalize existing best ones from unconstrained or linear-constrained problems to convex-constrained ones, or improve over the best-known results on solving the same-structured problems. Furthermore, numerical experiments on linearly/quadratically constrained non-convex quadratic programs and linear-constrained robust nonlinear least squares are conducted to demonstrate the empirical efficiency of the proposed DPALM over several state-of-the art methods.