Neural network approaches to approximate the ground state of quantum hamiltonians require the numerical solution of a highly nonlinear optimization problem. We introduce a statistical learning approach that makes the optimization trivial by using kernel methods. Our scheme is an approximate realization of the power method, where supervised learning is used to learn the next step of the power iteration. We show that the ground state properties of arbitrary gapped quantum hamiltonians can be reached with polynomial resources under the assumption that the supervised learning is efficient. Using kernel ridge regression, we provide numerical evidence that the learning assumption is verified by applying our scheme to find the ground states of several prototypical interacting many-body quantum systems, both in one and two dimensions, showing the flexibility of our approach.

Neural networks (NN) have become almost ubiquitous with image classification, but in their standard form produce point estimates, with no measure of confidence. Bayesian neural networks (BNN) provide uncertainty quantification (UQ) for NN predictions and estimates through the posterior distribution. As NN are applied in more high-consequence applications, UQ is becoming a requirement. BNN provide a solution to this problem by not only giving accurate predictions and estimates, but also an interval that includes reasonable values within a desired probability. Despite their positive attributes, BNN are notoriously difficult and time consuming to train. Traditional Bayesian methods use Markov Chain Monte Carlo (MCMC), but this is often brushed aside as being too slow. The most common method is variational inference (VI) due to its fast computation, but there are multiple concerns with its efficacy. We apply and compare MCMC- and VI-trained BNN in the context of target detection in hyperspectral imagery (HSI), where materials of interest can be identified by their unique spectral signature. This is a challenging field, due to the numerous permuting effects practical collection of HSI has on measured spectra. Both models are trained using out-of-the-box tools on a high fidelity HSI target detection scene. Both MCMC- and VI-trained BNN perform well overall at target detection on a simulated HSI scene. This paper provides an example of how to utilize the benefits of UQ, but also to increase awareness that different training methods can give different results for the same model. If sufficient computational resources are available, the best approach rather than the fastest or most efficient should be used, especially for high consequence problems.

The evolutionary distance between two species is proportional to some (unknown) function of the time of divergence from their common ancestor. One way to estimate this time is by comparing the underlying macromolecular sequences that cascade the information of accumulated evolutionary changes across DNA RNA Proteins (sequence structure function). Since the introduction of the molecular evolutionary clock by Zuckerkandl and Pauling (1965) to perform phylogenetic studies, several statistical models have been proposed to estimate the divergence of extant sequences from common ancestors, and to correlate the estimates of time from other sources of information (e.g., fossil records) when they exist (Sarich and Wilson, 1967). Such divergence time estimates require reliable statistical models of DNA/RNA/Proteins macromolecules (Bromham and Penny, 2003). For protein amino acid sequences, several statistical models have been proposed to explain sequence variation as a function of time. The point accepted mutation (PAM) matrix of Dayhoff et al. (1978) was the first successful model to explain the mutability of amino acid sequences. PAM is a stochastic (Markov) matrix defined in PAM (time) units where PAM-1 is a Markov matrix that embodies a 1% expected change to the amino acids. Subsequent studies highlighted the importance of incorporating evolutionary time-dependent substitution and gap models as an elegant way to model the divergent relationships of proteins (Holmes, 1998; Gonnet et al., 1992). The recent approach of Sumanaweera et al. (2022) derives a unified statistical model for quantifying the evolution of pairs of protein sequences

Traditional deep learning (DL) models are powerful classifiers, but many approaches do not provide uncertainties for their estimates. Uncertainty quantification (UQ) methods for DL models have received increased attention in the literature due to their usefulness in decision making, particularly for high-consequence decisions. However, there has been little research done on how to evaluate the quality of such methods. We use statistical methods of frequentist interval coverage and interval width to evaluate the quality of credible intervals, and expected calibration error to evaluate classification predicted confidence. These metrics are evaluated on Bayesian neural networks (BNN) fit using Markov Chain Monte Carlo (MCMC) and variational inference (VI), bootstrapped neural networks (NN), Deep Ensembles (DE), and Monte Carlo (MC) dropout. We apply these different UQ for DL methods to a hyperspectral image target detection problem and show the inconsistency of the different methods' results and the necessity of a UQ quality metric. To reconcile these differences and choose a UQ method that appropriately quantifies the uncertainty, we create a simulated data set with fully parameterized probability distribution for a two-class classification problem. The gold standard MCMC performs the best overall, and the bootstrapped NN is a close second, requiring the same computational expense as DE. Through this comparison, we demonstrate that, for a given data set, different models can produce uncertainty estimates of markedly different quality. This in turn points to a great need for principled assessment methods of UQ quality in DL applications.

Decision-focused learning (DFL) has recently emerged as a powerful approach for predict-then-optimize problems by customizing a predictive model to a downstream optimization task. However, existing end-to-end DFL methods are hindered by three significant bottlenecks: model mismatch error, sample average approximation error, and gradient approximation error. Model mismatch error stems from the misalignment between the model's parameterized predictive distribution and the true probability distribution. Sample average approximation error arises when using finite samples to approximate the expected optimization objective. Gradient approximation error occurs as DFL relies on the KKT condition for exact gradient computation, while most methods approximate the gradient for backpropagation in non-convex objectives. In this paper, we present DF2 -- the first \textit{distribution-free} decision-focused learning method explicitly designed to address these three bottlenecks. Rather than depending on a task-specific forecaster that requires precise model assumptions, our method directly learns the expected optimization function during training. To efficiently learn the function in a data-driven manner, we devise an attention-based model architecture inspired by the distribution-based parameterization of the expected objective. Our method is, to the best of our knowledge, the first to address all three bottlenecks within a single model. We evaluate DF2 on a synthetic problem, a wind power bidding problem, and a non-convex vaccine distribution problem, demonstrating the effectiveness of DF2.

Conditional Independence (CI) graph is a special type of a Probabilistic Graphical Model (PGM) where the feature connections are modeled using an undirected graph and the edge weights show the partial correlation strength between the features. Since the CI graphs capture direct dependence between features, they have been garnering increasing interest within the research community for gaining insights into the systems from various domains, in particular discovering the domain topology. In this work, we propose algorithms for performing knowledge propagation over the CI graphs. Our experiments demonstrate that our techniques improve upon the state-of-the-art on the publicly available Cora and PubMed datasets.

Human-Robot Interaction (HRI) becomes more and more important in a world where robots integrate fast in all aspects of our lives but HRI applications depend massively on the utilized robotic system as well as the deployment environment and cultural differences. Because of these variable dependencies it is often not feasible to use a data-driven approach to train a model for human intent recognition. Expert systems have been proven to close this gap very efficiently. Furthermore, it is important to support understandability in HRI systems to establish trust in the system. To address the above-mentioned challenges in HRI we present an adaptable python library in which current state-of-the-art Models for context recognition can be integrated. For Context-Based Intention Recognition a two-layer Bayesian Network (BN) is used. The bayesian approach offers explainability and clarity in the creation of scenarios and is easily extendable with more modalities. Additionally, it can be used as an expert system if no data is available but can as well be fine-tuned when data becomes available.

Large skew-t factor copula models are attractive for the modeling of financial data because they allow for asymmetric and extreme tail dependence. We show that the copula implicit in the skew-t distribution of Azzalini and Capitanio (2003) allows for a higher level of pairwise asymmetric dependence than two popular alternative skew-t copulas. Estimation of this copula in high dimensions is challenging, and we propose a fast and accurate Bayesian variational inference (VI) approach to do so. The method uses a conditionally Gaussian generative representation of the skew-t distribution to define an augmented posterior that can be approximated accurately. A fast stochastic gradient ascent algorithm is used to solve the variational optimization. The new methodology is used to estimate copula models for intraday returns from 2017 to 2021 on 93 U.S. equities. The copula captures substantial heterogeneity in asymmetric dependence over equity pairs, in addition to the variability in pairwise correlations. We show that intraday predictive densities from the skew-t copula are more accurate than from some other copula models, while portfolio selection strategies based on the estimated pairwise tail dependencies improve performance relative to the benchmark index.

Spike trains data find a growing list of applications in computational neuroscience, imaging, streaming data and finance. Machine learning strategies for spike trains are based on various neural network and probabilistic models. The probabilistic approach is relying on parametric or nonparametric specifications of the underlying spike generation model. In this paper we consider the two-class statistical classification problem for a class of spike train data characterized by nonparametrically specified intensity functions. We derive the optimal Bayes rule and next form the plug-in nonparametric kernel classifier. Asymptotical properties of the rules are established including the limit with respect to the increasing recording time interval and the size of a training set. In particular the convergence of the kernel classifier to the Bayes rule is proved. The obtained results are supported by a finite sample simulation studies.

Multi-task learning is frequently used to model a set of related response variables from the same set of features, improving predictive performance and modeling accuracy relative to methods that handle each response variable separately. Despite the potential of multi-task learning to yield more powerful inference than single-task alternatives, prior work in this area has largely omitted uncertainty quantification. Our focus in this paper is a common multi-task problem in neuroimaging, where the goal is to understand the relationship between multiple cognitive task scores (or other subject-level assessments) and brain connectome data collected from imaging. We propose a framework for selective inference to address this problem, with the flexibility to: (i) jointly identify the relevant covariates for each task through a sparsity-inducing penalty, and (ii) conduct valid inference in a model based on the estimated sparsity structure. Our framework offers a new conditional procedure for inference, based on a refinement of the selection event that yields a tractable selection-adjusted likelihood. This gives an approximate system of estimating equations for maximum likelihood inference, solvable via a single convex optimization problem, and enables us to efficiently form confidence intervals with approximately the correct coverage. Applied to both simulated data and data from the Adolescent Brain Cognitive Development (ABCD) study, our selective inference methods yield tighter confidence intervals than commonly used alternatives, such as data splitting. We also demonstrate through simulations that multi-task learning with selective inference can more accurately recover true signals than single-task methods.