We present an algorithm for computing melting points by autonomously learning from coexistence simulations in the NPT ensemble. Given the interatomic interaction model, the method makes decisions regarding the number of atoms and temperature at which to conduct simulations, and based on the collected data predicts the melting point along with the uncertainty, which can be systematically improved with more data. We demonstrate how incorporating physical models of the solid-liquid coexistence evolution enhances the algorithm's accuracy and enables optimal decision-making to effectively reduce predictive uncertainty. To validate our approach, we compare the results of 20 melting point calculations from the literature to the results of our calculations, all conducted with same interatomic potentials. Remarkably, we observe significant deviations in about one-third of the cases, underscoring the need for accurate and reliable algorithms for materials property calculations.
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Parameter identification problems in partial differential equations (PDEs) consist in determining one or more unknown functional parameters in a PDE. Here, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Recent asymptotic theoretical guarantees establishing posterior consistency and convergence rates are reviewed and expanded upon. An implementation of the associated posterior-based inference is provided, and illustrated via a numerical simulation study where two different discretisation strategies are devised. The reproducible code is available at: //github.com/MattGiord.
We propose novel optimal and parameter-free algorithms for computing an approximate solution with small (projected) gradient norm. Specifically, for computing an approximate solution such that the norm of its (projected) gradient does not exceed $\varepsilon$, we obtain the following results: a) for the convex case, the total number of gradient evaluations is bounded by $O(1)\sqrt{L\|x_0 - x^*\|/\varepsilon}$, where $L$ is the Lipschitz constant of the gradient and $x^*$ is any optimal solution; b) for the strongly convex case, the total number of gradient evaluations is bounded by $O(1)\sqrt{L/\mu}\log(\|\nabla f(x_0)\|/\epsilon)$, where $\mu$ is the strong convexity modulus; and c) for the nonconvex case, the total number of gradient evaluations is bounded by $O(1)\sqrt{Ll}(f(x_0) - f(x^*))/\varepsilon^2$, where $l$ is the lower curvature constant. Our complexity results match the lower complexity bounds of the convex and strongly cases, and achieve the above best-known complexity bound for the nonconvex case for the first time in the literature. Moreover, for all the convex, strongly convex, and nonconvex cases, we propose parameter-free algorithms that do not require the input of any problem parameters. To the best of our knowledge, there do not exist such parameter-free methods before especially for the strongly convex and nonconvex cases. Since most regularity conditions (e.g., strong convexity and lower curvature) are imposed over a global scope, the corresponding problem parameters are notoriously difficult to estimate. However, gradient norm minimization equips us with a convenient tool to monitor the progress of algorithms and thus the ability to estimate such parameters in-situ.
In semi-supervised learning, the prevailing understanding suggests that observing additional unlabeled samples improves estimation accuracy for linear parameters only in the case of model misspecification. This paper challenges this notion, demonstrating its inaccuracy in high dimensions. Initially focusing on a dense scenario, we introduce robust semi-supervised estimators for the regression coefficient without relying on sparse structures in the population slope. Even when the true underlying model is linear, we show that leveraging information from large-scale unlabeled data improves both estimation accuracy and inference robustness. Moreover, we propose semi-supervised methods with further enhanced efficiency in scenarios with a sparse linear slope. Diverging from the standard semi-supervised literature, we also allow for covariate shift. The performance of the proposed methods is illustrated through extensive numerical studies, including simulations and a real-data application to the AIDS Clinical Trials Group Protocol 175 (ACTG175).
Feature selection is critical in machine learning to reduce dimensionality and improve model accuracy and efficiency. The exponential growth in feature space dimensionality for modern datasets directly results in ambiguous samples and redundant features, which can severely degrade classification accuracy. Quantum machine learning offers potential advantages for addressing this challenge. In this paper, we propose a novel method, quantum support vector machine feature selection (QSVMF), integrating quantum support vector machines with multi-objective genetic algorithm. QSVMF optimizes multiple simultaneous objectives: maximizing classification accuracy, minimizing selected features and quantum circuit costs, and reducing feature covariance. We apply QSVMF for feature selection on a breast cancer dataset, comparing the performance of QSVMF against classical approaches with the selected features. Experimental results show that QSVMF achieves superior performance. Furthermore, The Pareto front solutions of QSVMF enable analysis of accuracy versus feature set size trade-offs, identifying extremely sparse yet accurate feature subsets. We contextualize the biological relevance of the selected features in terms of known breast cancer biomarkers. This work highlights the potential of quantum-based feature selection to enhance machine learning efficiency and performance on complex real-world data.
Embedding graphs in continous spaces is a key factor in designing and developing algorithms for automatic information extraction to be applied in diverse tasks (e.g., learning, inferring, predicting). The reliability of graph embeddings directly depends on how much the geometry of the continuous space matches the graph structure. Manifolds are mathematical structure that can enable to incorporate in their topological spaces the graph characteristics, and in particular nodes distances. State-of-the-art of manifold-based graph embedding algorithms take advantage of the assumption that the projection on a tangential space of each point in the manifold (corresponding to a node in the graph) would locally resemble a Euclidean space. Although this condition helps in achieving efficient analytical solutions to the embedding problem, it does not represent an adequate set-up to work with modern real life graphs, that are characterized by weighted connections across nodes often computed over sparse datasets with missing records. In this work, we introduce a new class of manifold, named soft manifold, that can solve this situation. In particular, soft manifolds are mathematical structures with spherical symmetry where the tangent spaces to each point are hypocycloids whose shape is defined according to the velocity of information propagation across the data points. Using soft manifolds for graph embedding, we can provide continuous spaces to pursue any task in data analysis over complex datasets. Experimental results on reconstruction tasks on synthetic and real datasets show how the proposed approach enable more accurate and reliable characterization of graphs in continuous spaces with respect to the state-of-the-art.
Unsupervised deep learning approaches have recently become one of the crucial research areas in imaging owing to their ability to learn expressive and powerful reconstruction operators even when paired high-quality training data is scarcely available. In this chapter, we review theoretically principled unsupervised learning schemes for solving imaging inverse problems, with a particular focus on methods rooted in optimal transport and convex analysis. We begin by reviewing the optimal transport-based unsupervised approaches such as the cycle-consistency-based models and learned adversarial regularization methods, which have clear probabilistic interpretations. Subsequently, we give an overview of a recent line of works on provably convergent learned optimization algorithms applied to accelerate the solution of imaging inverse problems, alongside their dedicated unsupervised training schemes. We also survey a number of provably convergent plug-and-play algorithms (based on gradient-step deep denoisers), which are among the most important and widely applied unsupervised approaches for imaging problems. At the end of this survey, we provide an overview of a few related unsupervised learning frameworks that complement our focused schemes. Together with a detailed survey, we provide an overview of the key mathematical results that underlie the methods reviewed in the chapter to keep our discussion self-contained.
Missing data is a common problem in practical settings. Various imputation methods have been developed to deal with missing data. However, even though the label is usually available in the training data, the common practice of imputation usually only relies on the input and ignores the label. In this work, we illustrate how stacking the label into the input can significantly improve the imputation of the input. In addition, we propose a classification strategy that initializes the predicted test label with missing values and stacks the label with the input for imputation. This allows imputing the label and the input at the same time. Also, the technique is capable of handling data training with missing labels without any prior imputation and is applicable to continuous, categorical, or mixed-type data. Experiments show promising results in terms of accuracy.
This study evaluates four fracture simulation methods, comparing their computational expenses and implementation complexities within the Finite Element (FE) framework when employed on multiphase materials. Fracture methods considered encompass the Cohesive Zone Model (CZM) using zero-thickness cohesive interface elements (CIEs), the Standard Phase-Field Fracture (SPFM) approach, the Cohesive Phase-Field fracture (CPFM) approach, and an innovative hybrid model. The hybrid approach combines the CPFM fracture method with the CZM, specifically applying the CZM within the interface zone. The finite element model studied is characterized by three specific phases: Inclusions, matrix, and interface zone. The thorough assessment of these modeling techniques indicates that the CPFM approach stands out as the most effective computational model provided that the thickness of the interface zone is not significantly smaller than that of the other phases. In materials like concrete the interface thickness is notably small when compared to other phases. This leads to the hybrid model standing as the most authentic finite element model, utilizing CIEs within the interface to simulate interface debonding. A significant finding from this investigation is that the CPFM method is in agreement with the hybrid model when the interface zone thickness is not excessively small. This implies that the CPFM fracture methodology may serve as a unified fracture approach for multiphase materials, provided the interface zone's thickness is comparable to that of the other phases. In addition, this research provides valuable insights that can advance efforts to fine-tune material microstructures. An investigation of the influence of the interface material properties, morphological features and spatial arrangement of inclusions showes a pronounced effect of these parameters on the fracture toughness of the material.
Linear regression and classification methods with repeated functional data are considered. For each statistical unit in the sample, a real-valued parameter is observed over time under different conditions. Two regression methods based on fusion penalties are presented. The first one is a generalization of the variable fusion methodology based on the 1-nearest neighbor. The second one, called group fusion lasso, assumes some grouping structure of conditions and allows for homogeneity among the regression coefficient functions within groups. A finite sample numerical simulation and an application on EEG data are presented.
The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting. We survey recent theoretical progress that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behavior of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favorable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
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