We consider the problem of mixed sparse linear regression with two components, where two real $k$-sparse signals $\beta_1, \beta_2$ are to be recovered from $n$ unlabelled noisy linear measurements. The sparsity is allowed to be sublinear in the dimension, and additive noise is assumed to be independent Gaussian with variance $\sigma^2$. Prior work has shown that the problem suffers from a $\frac{k}{SNR^2}$-to-$\frac{k^2}{SNR^2}$ statistical-to-computational gap, resembling other computationally challenging high-dimensional inference problems such as Sparse PCA and Robust Sparse Mean Estimation; here $SNR$ is the signal-to-noise ratio. We establish the existence of a more extensive computational barrier for this problem through the method of low-degree polynomials, but show that the problem is computationally hard only in a very narrow symmetric parameter regime. We identify a smooth information-computation tradeoff between the sample complexity $n$ and runtime for any randomized algorithm in this hard regime. Via a simple reduction, this provides novel rigorous evidence for the existence of a computational barrier to solving exact support recovery in sparse phase retrieval with sample complexity $n = \tilde{o}(k^2)$. Our second contribution is to analyze a simple thresholding algorithm which, outside of the narrow regime where the problem is hard, solves the associated mixed regression detection problem in $O(np)$ time with square-root the number of samples and matches the sample complexity required for (non-mixed) sparse linear regression; this allows the recovery problem to be subsequently solved by state-of-the-art techniques from the dense case. As a special case of our results, we show that this simple algorithm is order-optimal among a large family of algorithms in solving exact signed support recovery in sparse linear regression.

Complex diseases are caused by a multitude of factors that may differ between patients even within the same diagnostic category. A few underlying root causes may nevertheless initiate the development of disease within each patient. We therefore focus on identifying patient-specific root causes of disease, which we equate to the sample-specific predictivity of the exogenous error terms in a structural equation model. We generalize from the linear setting to the heteroscedastic noise model where $Y = m(X) + \varepsilon\sigma(X)$ with non-linear functions $m(X)$ and $\sigma(X)$ representing the conditional mean and mean absolute deviation, respectively. This model preserves identifiability but introduces non-trivial challenges that require a customized algorithm called Generalized Root Causal Inference (GRCI) to extract the error terms correctly. GRCI recovers patient-specific root causes more accurately than existing alternatives.

The paper uses structured machine learning regressions for nowcasting with panel data consisting of series sampled at different frequencies. Motivated by the problem of predicting corporate earnings for a large cross-section of firms with macroeconomic, financial, and news time series sampled at different frequencies, we focus on the sparse-group LASSO regularization which can take advantage of the mixed frequency time series panel data structures. Our empirical results show the superior performance of our machine learning panel data regression models over analysts' predictions, forecast combinations, firm-specific time series regression models, and standard machine learning methods.

Large language models (LLMs) have brought significant and transformative changes in human society. These models have demonstrated remarkable capabilities in natural language understanding and generation, leading to various advancements and impacts across several domains. We consider the in-context learning under two formulation for attention related regression in this work. Given matrices $A_1 \in \mathbb{R}^{n \times d}$, and $A_2 \in \mathbb{R}^{n \times d}$ and $B \in \mathbb{R}^{n \times n}$, the purpose is to solve some certain optimization problems: Normalized version $\min_{X} \| D(X)^{-1} \exp(A_1 X A_2^\top) - B \|_F^2$ and Rescaled version $\| \exp(A_1 X A_2^\top) - D(X) \cdot B \|_F^2$. Here $D(X) := \mathrm{diag}( \exp(A_1 X A_2^\top) {\bf 1}_n )$. Our regression problem shares similarities with previous studies on softmax-related regression. Prior research has extensively investigated regression techniques related to softmax regression: Normalized version $\| \langle \exp(Ax) , {\bf 1}_n \rangle^{-1} \exp(Ax) - b \|_2^2$ and Resscaled version $\| \exp(Ax) - \langle \exp(Ax), {\bf 1}_n \rangle b \|_2^2 $ In contrast to previous approaches, we adopt a vectorization technique to address the regression problem in matrix formulation. This approach expands the dimension from $d$ to $d^2$, resembling the formulation of the regression problem mentioned earlier. Upon completing the lipschitz analysis of our regression function, we have derived our main result concerning in-context learning.

Learning processes by exploiting restricted domain knowledge is an important task across a plethora of scientific areas, with more and more hybrid methods combining data-driven and model-based approaches. However, while such hybrid methods have been tested in various scientific applications, they have been mostly tested on dynamical systems, with only limited study about the influence of each model component on global performance and parameter identification. In this work, we assess the performance of hybrid modeling against traditional machine learning methods on standard regression problems. We compare, on both synthetic and real regression problems, several approaches for training such hybrid models. We focus on hybrid methods that additively combine a parametric physical term with a machine learning term and investigate model-agnostic training procedures. We also introduce a new hybrid approach based on partial dependence functions. Experiments are carried out with different types of machine learning models, including tree-based models and artificial neural networks.

High-precision modeling of systems is one of the main areas of industrial data analysis. Models of systems, their digital twins, are used to predict their behavior under various conditions. We have developed several models of a storage system using machine learning-based generative models. The system consists of several components: hard disk drive (HDD) and solid-state drive (SSD) storage pools with different RAID schemes and cache. Each storage component is represented by a probabilistic model that describes the probability distribution of the component performance in terms of IOPS and latency, depending on their configuration and external data load parameters. The results of the experiments demonstrate the errors of 4-10 % for IOPS and 3-16 % for latency predictions depending on the components and models of the system. The predictions show up to 0.99 Pearson correlation with Little's law, which can be used for unsupervised reliability checks of the models. In addition, we present novel data sets that can be used for benchmarking regression algorithms, conditional generative models, and uncertainty estimation methods in machine learning.

We use angular clustering of luminous red galaxies from the Dark Energy Spectroscopic Instrument (DESI) imaging surveys to constrain the local primordial non-Gaussianity parameter fNL. Our sample comprises over 12 million targets, covering 14,000 square degrees of the sky, with redshifts in the range 0.2< z < 1.35. We identify Galactic extinction, survey depth, and astronomical seeing as the primary sources of systematic error, and employ linear regression and artificial neural networks to alleviate non-cosmological excess clustering on large scales. Our methods are tested against log-normal simulations with and without fNL and systematics, showing superior performance of the neural network treatment in reducing remaining systematics. Assuming the universality relation, we find fNL $= 47^{+14(+29)}_{-11(-22)}$ at 68\%(95\%) confidence. With a more aggressive treatment, including regression against the full set of imaging maps, our maximum likelihood value shifts slightly to fNL$ \sim 50$ and the uncertainty on fNL increases due to the removal of large-scale clustering information. We apply a series of robustness tests (e.g., cuts on imaging, declination, or scales used) that show consistency in the obtained constraints. Despite extensive efforts to mitigate systematics, our measurements indicate fNL > 0 with a 99.9 percent confidence level. This outcome raises concerns as it could be attributed to unforeseen systematics, including calibration errors or uncertainties associated with low-\ell systematics in the extinction template. Alternatively, it could suggest a scale-dependent fNL model--causing significant non-Gaussianity around large-scale structure while leaving cosmic microwave background scales unaffected. Our results encourage further studies of fNL with DESI spectroscopic samples, where the inclusion of 3D clustering modes should help separate imaging systematics.

The multivariate adaptive regression spline (MARS) is one of the popular estimation methods for nonparametric multivariate regressions. However, as MARS is based on marginal splines, to incorporate interactions of covariates, products of the marginal splines must be used, which leads to an unmanageable number of basis functions when the order of interaction is high and results in low estimation efficiency. In this paper, we improve the performance of MARS by using linear combinations of the covariates which achieve sufficient dimension reduction. The special basis functions of MARS facilitate calculation of gradients of the regression function, and estimation of the linear combinations is obtained via eigen-analysis of the outer-product of the gradients. Under some technical conditions, the asymptotic theory is established for the proposed estimation method. Numerical studies including both simulation and empirical applications show its effectiveness in dimension reduction and improvement over MARS and other commonly-used nonparametric methods in regression estimation and prediction.

Mean decrease in impurity (MDI) is a popular feature importance measure for random forests (RFs). We show that the MDI for a feature $X_k$ in each tree in an RF is equivalent to the unnormalized $R^2$ value in a linear regression of the response on the collection of decision stumps that split on $X_k$. We use this interpretation to propose a flexible feature importance framework called MDI+. Specifically, MDI+ generalizes MDI by allowing the analyst to replace the linear regression model and $R^2$ metric with regularized generalized linear models (GLMs) and metrics better suited for the given data structure. Moreover, MDI+ incorporates additional features to mitigate known biases of decision trees against additive or smooth models. We further provide guidance on how practitioners can choose an appropriate GLM and metric based upon the Predictability, Computability, Stability framework for veridical data science. Extensive data-inspired simulations show that MDI+ significantly outperforms popular feature importance measures in identifying signal features. We also apply MDI+ to two real-world case studies on drug response prediction and breast cancer subtype classification. We show that MDI+ extracts well-established predictive genes with significantly greater stability compared to existing feature importance measures. All code and models are released in a full-fledged python package on Github.

During the past decade, metal additive manufacturing (MAM) has experienced significant developments and gained much attention due to its ability to fabricate complex parts, manufacture products with functionally graded materials, minimize waste, and enable low-cost customization. Despite these advantages, predicting the impact of processing parameters on the characteristics of an MAM printed clad is challenging due to the complex nature of MAM processes. Machine learning (ML) techniques can help connect the physics underlying the process and processing parameters to the clad characteristics. In this study, we introduce a hybrid approach which involves utilizing the data provided by a calibrated multi-physics computational fluid dynamic (CFD) model and experimental research for preparing the essential big dataset, and then uses a comprehensive framework consisting of various ML models to predict and understand clad characteristics. We first compile an extensive dataset by fusing experimental data into the data generated using the developed CFD model for this study. This dataset comprises critical clad characteristics, including geometrical features such as width, height, and depth, labels identifying clad quality, and processing parameters. Second, we use two sets of processing parameters for training the ML models: machine setting parameters and physics-aware parameters, along with versatile ML models and reliable evaluation metrics to create a comprehensive and scalable learning framework for predicting clad geometry and quality. This framework can serve as a basis for clad characteristics control and process optimization. The framework resolves many challenges of conventional modeling methods in MAM by solving t the issue of data scarcity using a hybrid approach and introducing an efficient, accurate, and scalable platform for clad characteristics prediction and optimization.