Modern data-driven control applications call for flexible nonlinear models that are amenable to principled controller synthesis and realtime feedback. Many nonlinear dynamical systems of interest are control affine. We propose two novel classes of nonlinear feature representations which capture control affine structure while allowing for arbitrary complexity in the state dependence. Our methods make use of random features (RF) approximations, inheriting the expressiveness of kernel methods at a lower computational cost. We formalize the representational capabilities of our methods by showing their relationship to the Affine Dot Product (ADP) kernel proposed by Casta\~neda et al. (2021) and a novel Affine Dense (AD) kernel that we introduce. We further illustrate the utility by presenting a case study of data-driven optimization-based control using control certificate functions (CCF). Simulation experiments on a double pendulum empirically demonstrate the advantages of our methods.

Recently, a tensor-on-tensor (ToT) regression model has been proposed to generalize tensor recovery, encompassing scenarios like scalar-on-tensor regression and tensor-on-vector regression. However, the exponential growth in tensor complexity poses challenges for storage and computation in ToT regression. To overcome this hurdle, tensor decompositions have been introduced, with the tensor train (TT)-based ToT model proving efficient in practice due to reduced memory requirements, enhanced computational efficiency, and decreased sampling complexity. Despite these practical benefits, a disparity exists between theoretical analysis and real-world performance. In this paper, we delve into the theoretical and algorithmic aspects of the TT-based ToT regression model. Assuming the regression operator satisfies the restricted isometry property (RIP), we conduct an error analysis for the solution to a constrained least-squares optimization problem. This analysis includes upper error bound and minimax lower bound, revealing that such error bounds polynomially depend on the order $N+M$. To efficiently find solutions meeting such error bounds, we propose two optimization algorithms: the iterative hard thresholding (IHT) algorithm (employing gradient descent with TT-singular value decomposition (TT-SVD)) and the factorization approach using the Riemannian gradient descent (RGD) algorithm. When RIP is satisfied, spectral initialization facilitates proper initialization, and we establish the linear convergence rate of both IHT and RGD.

The Fisher Information Matrix (FIM) provides a way for quantifying the information content of an observable random variable concerning unknown parameters within a model that characterizes the variable. When parameters in a model are directly linked to individual features, the diagonal elements of the FIM can signify the relative importance of each feature. However, in scenarios where feature interactions may exist, a comprehensive exploration of the full FIM is necessary rather than focusing solely on its diagonal elements. This paper presents an end-to-end black box system identification approach that integrates the FIM into the training process to gain insights into dynamic importance and overall model structure. A decision module is added to the first layer of the network to determine the relevance scores using the entire FIM as input. The forward propagation is then performed on element-wise multiplication of inputs and relevance scores. Simulation results demonstrate that the proposed methodology effectively captures various types of interactions between dynamics, outperforming existing methods limited to polynomial interactions. Moreover, the effectiveness of this novel approach is confirmed through its application in identifying a real-world industrial system, specifically the PH neutralization process.

Interpretability is crucial for machine learning algorithms in high-stakes medical applications. However, high-performing neural networks typically cannot explain their predictions. Post-hoc explanation methods provide a way to understand neural networks but have been shown to suffer from conceptual problems. Moreover, current research largely focuses on providing local explanations for individual samples rather than global explanations for the model itself. In this paper, we propose Attri-Net, an inherently interpretable model for multi-label classification that provides local and global explanations. Attri-Net first counterfactually generates class-specific attribution maps to highlight the disease evidence, then performs classification with logistic regression classifiers based solely on the attribution maps. Local explanations for each prediction can be obtained by interpreting the attribution maps weighted by the classifiers' weights. Global explanation of whole model can be obtained by jointly considering learned average representations of the attribution maps for each class (called the class centers) and the weights of the linear classifiers. To ensure the model is ``right for the right reason", we further introduce a mechanism to guide the model's explanations to align with human knowledge. Our comprehensive evaluations show that Attri-Net can generate high-quality explanations consistent with clinical knowledge while not sacrificing classification performance.

Synthetic tabular data generation becomes crucial when real data is limited, expensive to collect, or simply cannot be used due to privacy concerns. However, producing good quality synthetic data is challenging. Several probabilistic, statistical, generative adversarial networks (GANs), and variational auto-encoder (VAEs) based approaches have been presented for synthetic tabular data generation. Once generated, evaluating the quality of the synthetic data is quite challenging. Some of the traditional metrics have been used in the literature but there is lack of a common, robust, and single metric. This makes it difficult to properly compare the effectiveness of different synthetic tabular data generation methods. In this paper we propose a new universal metric, TabSynDex, for robust evaluation of synthetic data. The proposed metric assesses the similarity of synthetic data with real data through different component scores which evaluate the characteristics that are desirable for ``high quality'' synthetic data. Being a single score metric and having an implicit bound, TabSynDex can also be used to observe and evaluate the training of neural network based approaches. This would help in obtaining insights that was not possible earlier. We present several baseline models for comparative analysis of the proposed evaluation metric with existing generative models. We also give a comparative analysis between TabSynDex and existing synthetic tabular data evaluation metrics. This shows the effectiveness and universality of our metric over the existing metrics. Source Code: \url{https://github.com/vikram2000b/tabsyndex}

This letter studies a distribution-free, finite-sample data perturbation (DP) method, the Residual-Permuted Sums (RPS), which is an alternative of the Sign-Perturbed Sums (SPS) algorithm, to construct confidence regions. While SPS assumes independent (but potentially time-varying) noise terms which are symmetric about zero, RPS gets rid of the symmetricity assumption, but assumes i.i.d. noises. The main idea is that RPS permutes the residuals instead of perturbing their signs. This letter introduces RPS in a flexible way, which allows various design-choices. RPS has exact finite sample coverage probabilities and we provide the first proof that these permutation-based confidence regions are uniformly strongly consistent under general assumptions. This means that the RPS regions almost surely shrink around the true parameters as the sample size increases. The ellipsoidal outer-approximation (EOA) of SPS is also extended to RPS, and the effectiveness of RPS is validated by numerical experiments, as well.

The paper investigates the escalating concerns surrounding the surge in diabetes cases, exacerbated by the COVID-19 pandemic, and the subsequent strain on medical resources. The research aims to construct a predictive model quantifying factors influencing inpatient hospital stay durations for diabetes patients, offering insights to hospital administrators for improved patient management strategies. The literature review highlights the increasing prevalence of diabetes, emphasizing the need for continued attention and analysis of urban-rural disparities in healthcare access. International studies underscore the financial implications and healthcare burden associated with diabetes-related hospitalizations and complications, emphasizing the significance of effective management strategies. The methodology involves a quantitative approach, utilizing a dataset comprising 10,000 observations of diabetic inpatient encounters in U.S. hospitals from 1999 to 2008. Predictive modeling techniques, particularly Generalized Linear Models (GLM), are employed to develop a model predicting hospital stay durations based on patient demographics, admission types, medical history, and treatment regimen. The results highlight the influence of age, medical history, and treatment regimen on hospital stay durations for diabetes patients. Despite model limitations, such as heteroscedasticity and deviations from normality in residual analysis, the findings offer valuable insights for hospital administrators in patient management. The paper concludes with recommendations for future research to address model limitations and explore the implications of predictive models on healthcare management strategies, ensuring equitable patient care and resource allocation.

We study gradient descent (GD) dynamics on logistic regression problems with large, constant step sizes. For linearly-separable data, it is known that GD converges to the minimizer with arbitrarily large step sizes, a property which no longer holds when the problem is not separable. In fact, the behaviour can be much more complex -- a sequence of period-doubling bifurcations begins at the critical step size $2/\lambda$, where $\lambda$ is the largest eigenvalue of the Hessian at the solution. Using a smaller-than-critical step size guarantees convergence if initialized nearby the solution: but does this suffice globally? In one dimension, we show that a step size less than $1/\lambda$ suffices for global convergence. However, for all step sizes between $1/\lambda$ and the critical step size $2/\lambda$, one can construct a dataset such that GD converges to a stable cycle. In higher dimensions, this is actually possible even for step sizes less than $1/\lambda$. Our results show that although local convergence is guaranteed for all step sizes less than the critical step size, global convergence is not, and GD may instead converge to a cycle depending on the initialization.

A typical process of learning to play a piece on a piano consists of a progression through a series of practice units that focus on individual dimensions of the skill, the so-called practice modes. Practice modes in learning to play music comprise a particularly large set of possibilities, such as hand coordination, posture, articulation, ability to read a music score, correct timing or pitch, etc. Self-guided practice is known to be suboptimal, and a model that schedules optimal practice to maximize a learner's progress still does not exist. Because we each learn differently and there are many choices for possible piano practice tasks and methods, the set of practice modes should be dynamically adapted to the human learner, a process typically guided by a teacher. However, having a human teacher guide individual practice is not always feasible since it is time-consuming, expensive, and often unavailable. In this work, we present a modeling framework to guide the human learner through the learning process by choosing the practice modes generated by a policy model. To this end, we present a computational architecture building on a Gaussian process that incorporates 1) the learner state, 2) a policy that selects a suitable practice mode, 3) performance evaluation, and 4) expert knowledge. The proposed policy model is trained to approximate the expert-learner interaction during a practice session. In our future work, we will test different Bayesian optimization techniques, e.g., different acquisition functions, and evaluate their effect on the learning progress.

In the field of machine learning, regression problems are pivotal due to their ability to predict continuous outcomes. Traditional error metrics like mean squared error, mean absolute error, and coefficient of determination measure model accuracy. The model accuracy is the consequence of the selected model and the features, which blurs the analysis of contribution. Predictability, in the other hand, focus on the predictable level of a target variable given a set of features. This study introduces conditional entropy estimators to assess predictability in regression problems, bridging this gap. We enhance and develop reliable conditional entropy estimators, particularly the KNIFE-P estimator and LMC-P estimator, which offer under- and over-estimation, providing a practical framework for predictability analysis. Extensive experiments on synthesized and real-world datasets demonstrate the robustness and utility of these estimators. Additionally, we extend the analysis to the coefficient of determination \(R^2 \), enhancing the interpretability of predictability. The results highlight the effectiveness of KNIFE-P and LMC-P in capturing the achievable performance and limitations of feature sets, providing valuable tools in the development of regression models. These indicators offer a robust framework for assessing the predictability for regression problems.