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Quantum computers are believed to have the ability to process huge data sizes which can be seen in machine learning applications. In these applications, the data in general is classical. Therefore, to process them on a quantum computer, there is a need for efficient methods which can be used to map classical data on quantum states in a concise manner. On the other hand, to verify the results of quantum computers and study quantum algorithms, we need to be able to approximate quantum operations into forms that are easier to simulate on classical computers with some errors. Motivated by these needs, in this paper we study the approximation of matrices and vectors by using their tensor products obtained through successive Schmidt decompositions. We show that data with distributions such as uniform, Poisson, exponential, or similar to these distributions can be approximated by using only a few terms which can be easily mapped onto quantum circuits. The examples include random data with different distributions, the Gram matrices of iris flower, handwritten digits, 20newsgroup, and labeled faces in the wild. And similarly, some quantum operations such as quantum Fourier transform and variational quantum circuits with a small depth also may be approximated with a few terms that are easier to simulate on classical computers. Furthermore, we show how the method can be used to simplify quantum Hamiltonians: In particular, we show the application to randomly generated transverse field Ising model Hamiltonians. The reduced Hamiltonians can be mapped into quantum circuits easily and therefore can be simulated more efficiently.

Statistical wisdom suggests that very complex models, interpolating training data, will be poor at predicting unseen examples.Yet, this aphorism has been recently challenged by the identification of benign overfitting regimes, specially studied in the case of parametric models: generalization capabilities may be preserved despite model high complexity.While it is widely known that fully-grown decision trees interpolate and, in turn, have bad predictive performances, the same behavior is yet to be analyzed for Random Forests (RF).In this paper, we study the trade-off between interpolation and consistency for several types of RF algorithms. Theoretically, we prove that interpolation regimes and consistency cannot be achieved simultaneously for several non-adaptive RF.Since adaptivity seems to be the cornerstone to bring together interpolation and consistency, we study interpolating Median RF which are proved to be consistent in the interpolating regime. This is the first result conciliating interpolation and consistency for RF, highlighting that the averaging effect introduced by feature randomization is a key mechanism, sufficient to ensure the consistency in the interpolation regime and beyond.Numerical experiments show that Breiman's RF are consistent while exactly interpolating, when no bootstrap step is involved.We theoretically control the size of the interpolation area, which converges fast enough to zero, giving a necessary condition for exact interpolation and consistency to occur in conjunction.

We investigate a simple objective for nonlinear instrumental variable (IV) regression based on a kernelized conditional moment restriction (CMR) known as a maximum moment restriction (MMR). The MMR objective is formulated by maximizing the interaction between the residual and the instruments belonging to a unit ball in a reproducing kernel Hilbert space (RKHS). First, it allows us to simplify the IV regression as an empirical risk minimization problem, where the risk functional depends on the reproducing kernel on the instrument and can be estimated by a U-statistic or V-statistic. Second, based on this simplification, we are able to provide the consistency and asymptotic normality results in both parametric and nonparametric settings. Lastly, we provide easy-to-use IV regression algorithms with an efficient hyper-parameter selection procedure. We demonstrate the effectiveness of our algorithms using experiments on both synthetic and real-world data.

Bayesian posterior distributions arising in modern applications, including inverse problems in partial differential equation models in tomography and subsurface flow, are often computationally intractable due to the large computational cost of evaluating the data likelihood. To alleviate this problem, we consider using Gaussian process regression to build a surrogate model for the likelihood, resulting in an approximate posterior distribution that is amenable to computations in practice. This work serves as an introduction to Gaussian process regression, in particular in the context of building surrogate models for inverse problems, and presents new insights into a suitable choice of training points. We show that the error between the true and approximate posterior distribution can be bounded by the error between the true and approximate likelihood, measured in the $L^2$-norm weighted by the true posterior, and that efficiently bounding the error between the true and approximate likelihood in this norm suggests choosing the training points in the Gaussian process surrogate model based on the true posterior.

In day-ahead electricity markets based on uniform marginal pricing, small variations in the offering and bidding curves may substantially modify the resulting market outcomes. In this work, we deal with the problem of finding the optimal offering curve for a risk-averse profit-maximizing generating company (GENCO) in a data-driven context. In particular, a large GENCO's market share may imply that her offering strategy can alter the marginal price formation, which can be used to increase profit. We tackle this problem from a novel perspective. First, we propose a optimization-based methodology to summarize each GENCO's step-wise supply curves into a subset of representative price-energy blocks. Then, the relationship between the market price and the resulting energy block offering prices is modeled through a Bayesian linear regression approach, which also allows us to generate stochastic scenarios for the sensibility of the market towards the GENCO strategy, represented by the regression coefficient probabilistic distributions. Finally, this predictive model is embedded in the stochastic optimization model by employing a constraint learning approach. Results show how allowing the GENCO to deviate from her true marginal costs renders significant changes in her profits and the market marginal price. Furthermore, these results have also been tested in an out-of-sample validation setting, showing how this optimal offering strategy is also effective in a real-world market contest.

Multiple Instance Learning (MIL) is a weakly supervised learning paradigm that is becoming increasingly popular because it requires less labeling effort than fully supervised methods. This is especially interesting for areas where the creation of large annotated datasets remains challenging, as in medicine. Although recent deep learning MIL approaches have obtained state-of-the-art results, they are fully deterministic and do not provide uncertainty estimations for the predictions. In this work, we introduce the Attention Gaussian Process (AGP) model, a novel probabilistic attention mechanism based on Gaussian Processes for deep MIL. AGP provides accurate bag-level predictions as well as instance-level explainability, and can be trained end-to-end. Moreover, its probabilistic nature guarantees robustness to overfitting on small datasets and uncertainty estimations for the predictions. The latter is especially important in medical applications, where decisions have a direct impact on the patient's health. The proposed model is validated experimentally as follows. First, its behavior is illustrated in two synthetic MIL experiments based on the well-known MNIST and CIFAR-10 datasets, respectively. Then, it is evaluated in three different real-world cancer detection experiments. AGP outperforms state-of-the-art MIL approaches, including deterministic deep learning ones. It shows a strong performance even on a small dataset with less than 100 labels and generalizes better than competing methods on an external test set. Moreover, we experimentally show that predictive uncertainty correlates with the risk of wrong predictions, and therefore it is a good indicator of reliability in practice. Our code is publicly available.

Fr\'echet global regression is extended to the context of bivariate curve stochastic processes with values in a Riemannian manifold. The proposed regression predictor arises as a reformulation of the standard least-squares parametric linear predictor in terms of a weighted Fr\'echet functional mean. Specifically, in our context, in this reformulation, the Euclidean distance is replaced by the integrated quadratic geodesic distance. The regression predictor is then obtained from the weighted Fr\'echet curve mean, lying in the time-varying geodesic submanifold, generated by the regressor process components involved in the time correlation range. The regularized Fr\'echet weights are computed in the time-varying tangent spaces. The weak-consistency of the regression predictor is proved. Model selection is also addressed. A simulation study is undertaken to illustrate the performance of the spherical curve variable selection algorithm proposed in a multivariate framework.

Neural Posterior Estimation methods for simulation-based inference can be ill-suited for dealing with posterior distributions obtained by conditioning on multiple observations, as they tend to require a large number of simulator calls to learn accurate approximations. In contrast, Neural Likelihood Estimation methods can handle multiple observations at inference time after learning from individual observations, but they rely on standard inference methods, such as MCMC or variational inference, which come with certain performance drawbacks. We introduce a new method based on conditional score modeling that enjoys the benefits of both approaches. We model the scores of the (diffused) posterior distributions induced by individual observations, and introduce a way of combining the learned scores to approximately sample from the target posterior distribution. Our approach is sample-efficient, can naturally aggregate multiple observations at inference time, and avoids the drawbacks of standard inference methods.

Generation of graphs is a major challenge for real-world tasks that require understanding the complex nature of their non-Euclidean structures. Although diffusion models have achieved notable success in graph generation recently, they are ill-suited for modeling the structural information of graphs since learning to denoise the noisy samples does not explicitly capture the graph topology. To tackle this limitation, we propose a novel generative process that models the topology of graphs by predicting the destination of the process. Specifically, we design the generative process as a mixture of diffusion processes conditioned on the endpoint in the data distribution, which drives the process toward the probable destination. Further, we introduce new training objectives for learning to predict the destination, and discuss the advantages of our generative framework that can explicitly model the graph topology and exploit the inductive bias of the data. Through extensive experimental validation on general graph and 2D/3D molecular graph generation tasks, we show that our method outperforms previous generative models, generating graphs with correct topology with both continuous and discrete features.

This work derives methods for performing nonparametric, nonasymptotic statistical inference for population means under the constraint of local differential privacy (LDP). Given bounded observations $(X_1, \dots, X_n)$ with mean $\mu^\star$ that are privatized into $(Z_1, \dots, Z_n)$, we present confidence intervals (CI) and time-uniform confidence sequences (CS) for $\mu^\star$ when only given access to the privatized data. To achieve this, we introduce a nonparametric and sequentially interactive generalization of Warner's famous ``randomized response'' mechanism, satisfying LDP for arbitrary bounded random variables, and then provide CIs and CSs for their means given access to the resulting privatized observations. For example, our results yield private analogues of Hoeffding's inequality in both fixed-time and time-uniform regimes. We extend these Hoeffding-type CSs to capture time-varying (non-stationary) means, and conclude by illustrating how these methods can be used to conduct private online A/B tests.