Lasso

Noise2Noise (N2N) trains denoisers on pairs of independently corrupted observations, eliminating clean references. We stress-test two natural conjectures about why the L1 loss outperforms L2 here. First, the hypothesis that the L1 loss confers robustness via parameter sparsity confuses the loss with Lasso regularization: an explicit Lasso penalty produces the predicted sparsity yet fails to reproduce L1's cross-noise behavior, while L1- and L2-trained weight distributions are indistinguishable. Second, the population optima of the two losses coincide exactly for symmetric signal posteriors and nearly so for concentrated ones. Measured differences are therefore dominated by optimization dynamics (bounded-influence gradients), which we probe with gradient statistics and contaminated-target training. On Kodak24 with five synthetic noise families, the L1 loss holds a statistically significant edge over L2, below 1 dB PSNR, holding across three seeds on 13 of the 14 noise columns. On real camera noise the loss is not the decisive variable in distribution: on official SIDD validation blocks, synthetic-Gaussian-trained N2N models gain only 0.8 to 3.7 dB over the noisy input regardless of loss, while retraining on SIDD's own noisy pairs, never reading ground truth, gains 9.4 to 11.0 dB, far ahead of BM3D. All metrics are on raw network outputs, and the study makes no leaderboard claim. The training pair distribution, not the loss, carries the inductive bias. That design rule applies wherever clean references are unobtainable, from microscopy to industrial inspection sensors.

To avoid missing important variables and their connections in networks, more and more variables are included in network analysis. Here we show that in a setting with many more parameters than observations (high-dimensional) it is possible to get a conservative (i.e., low false positive rate) estimate of the neighbourhood for each node (which connections are in the network). A neighbourhood is often estimated with a linear model, and this leads to two interesting cases: (i) If the true model is linear, then neighbourhood selection work reasonably well, and (ii) if the true model is nonlinear, then neighbourhood selection requires a penalty for the high dimensions. Here we show the impact of the ridge parameter on the mean squared error, and how this leads to low test variance and hence to neighbourhoods with large numbers of edges. We connect these insights with results from machine learning, where the so-called double descent (when more parameters are included than observations, the mean squared error goes down a second time) has put the traditional view on model selection upside down. Essentially, for adequate neighbourhood selection in models with a large number of parameters, the volume of the model space needs to be included in the penalty. Most neighbourhood selection methods (e.g., Lasso, AIC, BIC) lead to spurious edges (high false positive rate), but we prove that in the high-dimensional setting, minimum description length leads to correct neighbourhood selection or smaller (low false positive rates) in both cases when either the model is correctly or incorrectly assumed linear

Selective inference (SI) provides statistically valid $p$-values for hypotheses selected by applying an algorithm to the data, correcting for the bias that arises when the same data are used both to select and to test a hypothesis. Developing an SI procedure for a new algorithm, however, has required an expert to derive, and then implement, the selection event, i.e., the conditions under which the hypothesis is selected. Repeating this specialized effort for every new algorithm is why exact SI has so far been available for only a narrow class. We propose AutoSI, a framework that removes this barrier in two ways. First, AutoSI constructs the selection event automatically from the algorithm's individual operations, so the user only writes the algorithm as ordinary NumPy-like code and derives nothing by hand. Second, AutoSI broadens the class of selection events SI can handle: existing exact methods are limited to selection events characterized by linear or quadratic inequalities in the data, whereas AutoSI covers any algorithm expressible through rational functions of the data (ratios of polynomials). We prove that the $p$-values computed by AutoSI are exactly valid in finite samples. We demonstrate AutoSI on three feature-selection methods, each written in a few dozen lines of code. One of these methods, the lasso with its tuning parameter selected by cross-validated $R^2$, cannot be handled within existing exact SI frameworks and is made possible by AutoSI. Experiments on synthetic and real datasets show that the resulting $p$-values control the type I error rate (i.e., the false positive rate) at the nominal level while retaining high power.

On-device training of deep neural networks is fundamentally constrained by the computational and memory costs of large-scale datasets. Coreset selection offers a practical solution by retaining only a compact subset of real training samples. However, existing gradient-based methods commonly rely on gradients computed at a single model snapshot and employ greedy or pursuit-based selection procedures, limiting their ability to capture evolving optimization dynamics and handle strongly correlated samples. We propose GLOBE (Gradient Local-Balanced Extraction), a trajectory-aligned coreset selection framework that formulates sample selection as a globally optimized sparse weighting problem. GLOBE represents each sample by a gradient trajectory constructed across multiple training checkpoints, thereby capturing its influence throughout different stages of optimization. To preserve the training behavior of the full dataset, we introduce a multi-order matching objective that jointly aligns the first-order mean and projected uncentered second-order moments of gradient trajectories. GLOBE further combines Group LASSO, Elastic Net regularization, and nonnegative budget constraints to induce group- and sample-level sparsity while stabilizing the weights of correlated trajectories. Finally, class-balanced Top-K selection maintains adequate category coverage under limited sampling budgets. Experiments across six benchmarks and five evaluation architectures demonstrate that GLOBE consistently outperforms existing coreset selection methods in downstream test accuracy, particularly at low retention ratios. These results highlight the effectiveness of combining dynamic gradient information, multi-order distribution matching, and structured sparsity for data-efficient learning.

We describe Causal-TS, an open-source Python library for causal discovery in high-dimensional and nonstationary multivariate time series. Causal-TS provides four specialized algorithms-CDNOTS, CDNOTS+, CEDAR, and GRACE-along with wrappers for GES, Granger, LASSO-VAR, and LGES, all sharing a unified conditional independence (CI) test layer with GPU acceleration via PyTorch. A regime discovery pipeline detects structural breaks via pluggable changepoint detectors and runs discovery per regime with regime-specific parameters. A command-line interface, synthetic data generators, and optional DoWhy integration provide an end-to-end pipeline from raw time series to causal effect estimates. The library is pip-installable, tested on Python 3.10--3.12, and available at https://github.com/bloomberg/causal-ts.

Discovering the direct causes and effects of a target variable from observational data is a fundamental problem in causal discovery, with broad applications in domains such as gene regulatory analysis and biomedical research. Existing causal discovery methods either learn a global causal structure, which incurs substantial computational cost, or assume the absence of latent variables and selection bias, assumptions that are often violated in real-world settings. Motivated by these challenges, we study local causal structure learning in the presence of latent variables and selection bias. Specifically, we first characterize a local region that enables target-specific causal discovery without recovering the entire global structure. We then establish a theoretical bridge between causal information learned from the observed distribution induced on this local region and the corresponding information in the global causal structure. Building on these foundations, we propose LoCaLS, a local causal structure learning algorithm that is sound and complete under standard assumptions and identifies the same direct causes and effects of a target variable as those identifiable by global causal discovery methods, while allowing for latent variables and selection bias. Extensive experiments on random and real-world structures demonstrate that the proposed method consistently achieves higher structural accuracy than existing local methods while requiring substantially less computational effort than state-of-the-art global methods. Furthermore, applications to two real-world gene expression datasets reveal biologically plausible target-specific causal structures, demonstrating its practical applicability in large-scale biological data analysis.

We present msPCA: an open-source R package for sparse principal component analysis with multiple components. It implements an alternating maximization algorithm to generate a set of sparse loading vectors that collectively explain a large fraction of the variance in a dataset, while remaining non-redundant. The algorithm supports two definitions of non-redundancy: either orthogonality of the loading vectors or zero pairwise correlation between principal components (PCs). In the reported benchmarks, msPCA solves sparse PCA problems with thousands of features, achieving competitive runtimes while producing sparse components with controlled feasibility violations and a high fraction of variance explained.

Sparse feature selection is critical for high-dimensional machine learning, yet traditional $\ell_1$-regularized methods are often brittle under observational noise and spurious correlations, leading to unstable feature supports and degraded generalization. Although adversarial training has been widely used to improve model robustness, its interaction with hierarchical sparse feature selection remains underexplored. In this work, we propose Adversarial LassoNet (AdLNet), a stability-driven sparse feature selection framework that integrates input-space adversarial perturbations with the hierarchical sparsity mechanism of LassoNet. We derive a tractable first-order adversarial approximation under local smoothness assumptions and provide an NTK-inspired spectral analysis to characterize how perturbation-driven training can reduce gradient concentration. Experiments on high-dimensional SERS data, six public benchmark datasets, and ColoredMNIST show that AdLNet maintains competitive sparse-selection performance while improving out-of-distribution robustness by 4.4% and feature support reproducibility by 6.3% under nearly matched support sparsity on ColoredMNIST. On the high-dimensional lung cancer screening dataset, AdLNet achieves a 5.3% test accuracy gain and a 6.0% AUC improvement over vanilla LassoNet. Code and dataset are available at https://github.com/719573/Adversarial-LassoNet.

The exact computation of the Normalized Maximum Likelihood (NML) codelength for regular non-smooth estimators (e.g., Lasso) has been historically limited by the cubic scaling walls of manifold-constrained projection and volume integration. At each step of the geometric Propose-and-Project Metropolis--Hastings (PPMH) sampler, evaluating the projection operator requires inverting an $(N+k) \times (N+k)$ generalized KKT matrix, while calculating the volume factor requires the determinant of an $(N-k) \times (N-k)$ Gram matrix. This paper presents an exact, mathematically equivalent formulation that bypasses both bottlenecks by utilizing the block Schur complement and Sylvester's determinant identity. We prove that the computational complexity of both operations collapses from $\mathcal{O}(N^3)$ to $\mathcal{O}(k^3 + N^2 k)$ per step. We generalize this reduction to Sparse Support Vector Machines (SVMs), Elastic Net, and Group Lasso. Finally, we provide a rigorous numerical stability analysis and evaluate the sampler's efficiency using the Effective Sample Size (ESS) per second. Our empirical benchmarks on high-dimensional datasets confirm a constant speedup exceeding $14{,}100\times$ while maintaining double-precision numerical equivalence, rendering exact non-smooth NML estimation highly tractable for large-scale statistical inference.

A prominent research direction in mechanistic interpretability is learning sparse circuits over LLM components to reveal how they jointly produce model behavior. However, raw neurons are polysemantic, making learned circuits hard to interpret. Sparse autoencoder (SAE) features alleviate this, but their high dimensionality makes existing intervention-based circuit learning methods computationally prohibitive. We propose CircuitLasso, a scalable circuit-learning approach based on sparse linear regression. CircuitLasso recovers circuits whose structural accuracy matches that of state-of-the-art intervention-based methods on the benchmark data, at a fraction of the computational cost. For interpretability, CircuitLasso efficiently uncovers relationships among SAE features, showing how human-interpretable semantic features propagate through the model and influence its predictions. Finally, we validate the utility of our learned circuits by leveraging their insights to achieve comparable performance at substantially lower cost on a domain-generalization task.

We study online estimation for high-dimensional generalized linear models with streaming data. First, for the non-distributed setting, we propose a gradient-enhanced surrogate loss that approximates the cumulative loss using only historical summaries, which modifies and improves upon the existing renewable estimation approach for the same model in the high-dimensional setting, and removes the batch-number constraint in previous studies. We then extend the method to distributed streaming data under the master-client architecture, where batches are partitioned across sites and only summaries (gradient vectors) are exchanged. Instead of directing applying the popular method of Jordan et al. (2019) to the surrogate quadratic loss, our adjusted approach does not require the clients to compute the full surrogate loss. We derive non-asymptotic error bounds under the high-dimensional scaling, without the stringent constraint on the number of batches in the previous studies. Simulation results under linear and logistic models, together with a real-data application, show improved accuracy over existing renewable estimators.

Retrieval algorithms are used to estimate atmospheric concentrations of greenhouse gases (GHGs), such as carbon dioxide (CO2) and methane (CH4), by solving inverse problems from high-spectral-resolution satellite radiance measurements. However, these algorithms are computationally expensive, which makes real-time estimation at scale difficult. Machine-learning models have therefore been proposed as fast emulators of retrieval algorithms. Most existing studies, however, evaluate them only on test data from the same period as the training data. We study the stability over time of such emulators using data from the Greenhouse Gases Observing SATellite (GOSAT). We show that prediction accuracy generally deteriorates when the test period moves away from the training period. We also show that including time as an input feature substantially improves XCH4 prediction for Lasso and neural-network models. Among the methods considered, a simple Lasso model performs as well as or better than more complex methods such as neural networks, and yields more stable predictions over time. We further validate the results using the Total Carbon Column Observing Network (TCCON), a ground-based observation network. On the TCCON-matched dataset, the time-augmented Lasso achieves errors against TCCON that are comparable to the disagreement between GOSAT and TCCON for both XCO2 and XCH4.

We study the generalization of ridge-regularized nonlinear least-squares models via on-average algorithmic stability, deriving error bounds for local minimizers in terms of a data-dependent effective dimension that reflects the geometry of the gradient model at the trained parameters, through the empirical Jacobian Gram matrix and a residual-curvature term. In the linear case, where the curvature term vanishes, this recovers the classical effective dimension of the Jacobian kernel covariance, but evaluated at the trained model rather than at initialization as is typical in neural tangent kernel analyses. We further bound this effective dimension via covering complexity of the gradient features, leading to guarantees that depend on learned geometry rather than parameter count. In particular, for manifold-supported data and piecewise Lipschitz Jacobians, the bounds scale with intrinsic dimension, while for one-hidden-layer ReLU networks, the mechanism can be made explicit through counts of activation-stable regions. Experiments on synthetic manifolds, clustered distributions, and benchmark datasets illustrate trained-Jacobian compression, the tightness of the residual-curvature linearization, and agreement between the stability bound and observed generalization gaps. A key feature of our bounds is the simplicity of their derivation, which follows from first principles using the Brascamp-Lieb inequality under strongly log-concave noise.

Large language models (LLMs) offer a scalable mechanism to elicit domain-informed prior information for high-dimensional variable selection. However, existing methods such as LLM-Lasso are sensitive to weight quality, with performance degrading substantially when LLM-generated weights are inaccurate. To address this challenge, we first introduce a framework for quantifying the quality of LLM-generated weights, enabling rigorous evaluation of LLM-informed methods across varying weight regimes. We then propose the LLM Sparsity Prior (LSP), which integrates LLM-generated weights into the prior inclusion probabilities of Spike-and-Slab and Spike-and-Slab Lasso models via two interpretable hyperparameters governing global sparsity and weight concentration. Hierarchical hyperpriors on these parameters allow the model to dynamically discount uninformative or misleading weights, improving robustness without sacrificing gains when weights are accurate. Finally, we develop principled prompt engineering strategies and validate the method on a private medical dataset studying Acute Kidney Injury. LSP improves prediction accuracy and identifies clinically relevant features missed by the baselines, with robustness to prompt variation and particular effectiveness in low-data regimes.

In complex multivariate systems, interactions among variables are defined by dependency structures, often encoded as directed acyclic graphs ($\text{DAGs}$). However, dependency structures can vary across subjects, and ignoring this structural heterogeneity introduces bias and obscures subpopulation-specific dependencies. To address this, we propose Directed Acyclic Graph-based Dependency Clustering via Alternating Direction Method of Multipliers (DAG-DC-ADMM), a unified framework built upon Structural Equation Modeling (SEM) that jointly learns cluster assignments and cluster-specific dependency structures. We encode acyclicity via a smooth constraint and integrate a groupwise truncated Lasso fusion penalty (gTLP) to cluster subjects based on their structural similarity. This yields a nonconvex optimization problem that incorporates sparsity, acyclicity, and structural consensus constraints. We address the nonconvexity by using the augmented Lagrangian method and solve it with an adapted version of the Alternating Direction Method of Multipliers (ADMM) for difference-of-convex programs. For certain graph structures, such as upper triangular adjacency matrices, our algorithm is guaranteed to converge to a Karush-Kuhn-Tucker (KKT) point. Experiments demonstrate that our method recovers cluster-specific causal dependency structures with a high true positive rate and a low false discovery rate. This capability enables the robust discovery of heterogeneous dependencies across subjects where the subpopulation label is unknown.

We propose a subgraph importance estimation method for pretrained Graph Neural Networks (GNNs) on graph-level tasks, formulated as a linear Group Lasso regression problem in the embedding space. Our method effectively leverages prior domain knowledge of graph substructures, while remaining independent of the specific form of the output layer or readout function used in the GNN architecture, and it does not require access to ground-truth target labels. Experiments on real-world graph datasets demonstrate that our method consistently outperforms existing baselines in subgraph importance estimation. Furthermore, we extend our method to identify important nodes within the graph.

Data-driven identification of partial differential equations (PDEs) relies on sparse regression over a candidate library of differential operators, where larger libraries inflate false positives under observation noise and smaller libraries risk missing true terms. We introduce Equivariant Operator Discovery (EqOD), a fully automatic method combining two library reduction mechanisms. When Galilean invariance is detected from trajectory data via a weak-form structural test, EqOD uses the symmetry-reduced library, eliminating terms that our Galilean exclusion result proves to be absent from the governing equation. Otherwise, it applies randomized LASSO stability selection guided by classical false-positive bounds. A residual-based fallback prevents degradation below the full-library baseline. On 8 PDEs at 4 noise levels, EqOD attains $F_1 = 1.000 \pm 0.000$ on Heat at $20\%$ noise, where WF-LASSO obtains $0.475 \pm 0.181$, official PySINDy 2.0 obtains $0.000$, and the WSINDy reimplementation obtains $0.789$. Under the strict criterion that the mean F1 difference exceeds the larger of the two standard deviations, EqOD wins 7 of 32 cells. WF-LASSO wins none, and the remaining 25 cells are ties. Across all 32 cells, EqOD outperforms PySINDy 2.0.0 in 23 of 32 cells, and all 5 PySINDy wins occur on reaction PDEs. External validation on WeakIdent and PINN-SR datasets gives $F_1 = 1.000$ on all 5 clean benchmarks. NLS, 2D, coupled-system, and cylinder-wake extensions are reported. The Galilean library reduction is proved under explicit autonomy and library assumptions. The stability-selection step is motivated by classical false-positive bounds, while formal guarantees for correlated PDE design matrices remain open.

We study sparse recovery when observations come from mixed-quality sources: a small collection of high-quality measurements with small noise variance and a larger collection of lower-quality measurements with higher variance. For this heterogeneous-noise setting, we establish sample-size conditions for information-theoretic and algorithmic recovery. On the information-theoretic side, we show that it is sufficient for $(n_1, n_2)$ to satisfy a linear trade-off defining the Price of Quality: the number of low-quality samples needed to replace one high-quality sample. In the agnostic setting, where the decoder is completely agnostic to the quality of the data, it is uniformly bounded, and in particular one high-quality sample is never worth more than two low-quality samples for this sufficient condition to hold. In the informed setting, where the decoder is informed of per-sample variances, the price of quality can grow arbitrarily large. On the algorithmic side, we analyze the LASSO in the agnostic setting and show that the recovery threshold matches the homogeneous-noise case and only depends on the average noise level, revealing a striking robustness of computational recovery to data heterogeneity. Together, these results give the first conditions for sparse recovery with mixed-quality data and expose a fundamental difference between how the information-theoretic and algorithmic thresholds adapt to changes in data quality.

Background: Radiogenomics allows identification of radiological biomarkers for genomic phenotypes. In glioblastoma, these biomarkers could potentially complement patient stratification strategies. We aim to develop and analytically validate radiological biomarkers that capture immune cell signatures within IDH-wildtype glioblastoma microenvironment using radiogenomic analysis. Methods: This was a retrospective multicenter study using curated open-access anonymized imaging and genomic data from TCGA-GBM, CPTAC, IvyGAP, REMBRANDT and CGGA datasets. Imaging data consisted of MRI-based radiomic features extracted from necrotic core, enhancing and edema regions of deep learning-based auto-segmented tumors. Radiomic feature selections were performed using nested cross-validated LASSO. Support vector machine and ensemble models were trained using seventeen immune and cell-specific score labels extracted from deconvoluted transcriptomic data using pan-cancer and glioblastoma immune signature matrices as reference standards. Seventeen classifier models trained in three cross-cohort strategies were validated on three held-out datasets assessing stability and generalizability. Results: One-hundred-and-seventy-six patients were included in the study. The immune-related radiomic signatures obtained after feature selection were shape, first order and higher order radiomic features. Models predicting macrophage subtype immune signature showed stable mean performance on balanced accuracy (0.67) and precision (0.89) metrics for three independent holdout datasets with ensemble model outperforming support vector machine model. Conclusion: Radiogenomic models non-invasively predicted the macrophage subtype M0 immune signature in IDH-wildtype glioblastoma. These biomarkers have the potential to stratify patients for immunotherapy within prospective glioblastoma clinical trials.

Benign overfitting is well-characterized in $\ell_2$ geometries, but its behavior under the $\ell_1$ implicit bias of greedy ensembles remains challenging. The analytical barrier stems from the non-linear coupling of coordinate selection thresholds, which invalidates standard spectral resolvent tools. To isolate this algorithmic bias, we characterize the high-dimensional risk of continuous-time $\ell_2$-Boosting over $p$ features and $n$ samples. By coupling the Convex Gaussian Minimax Theorem with delicate asymptotic expansions of double-sided truncated Gaussian moments, we analytically resolve the non-smooth $\ell_1$ interpolant. Under an isotropic pure-noise model, we prove that benign overfitting fails at the linear rate: greedy selection localizes noise into sparse active sets, and the excess variance decays at a logarithmic rate $Θ(σ^2/\log(p/n))$ for noise variance $σ^2$. We remark that while this localization mechanism should persist in the presence of signals, the exact signal-noise decomposition remains an open problem. For spiked-isotropic designs with $k^*$ head eigenvalues and $r_2 = p - k^*$ tail dimensions, the risk converges to zero when $r_{2} \gg n$, but only at a logarithmic rate $Θ(σ^2/\log(r_2/n))$, which is slower than the linear decay observed in $\ell_2$ geometries. To avoid this slow convergence, we analyze the non-smooth subdifferential dynamics of the boosting flow. This yields a tuning-free early stopping rule that, under a bounded $\ell_1$-path condition, recovers the Lasso basic inequality and attains the minimax-optimal empirical prediction rate for $\ell_1$-bounded signals.

We study high-dimensional LASSO under differential privacy via objective perturbation with heterogeneous covariate scales. In practical scenarios, covariates often exhibit diverse scales; however, standard preprocessing is problematic under privacy constraints, as it consumes additional privacy budget. This heterogeneity induces effective anisotropy in the objective perturbation via the inverse Gram matrix of covariates, which can degrade the stability and accuracy of algorithms. To address this, we propose a Gram-based anisotropic objective perturbation, a ``pre-distortion" strategy that counteracts the distortion from the covariate structure to restore isotropy in the estimation process. Using an Approximate Message Passing (AMP) framework and state evolution analysis, we demonstrate that our proposed perturbation significantly stabilizes convergence and improves both statistical efficiency and privacy performance compared to standard uniform noise injection. Our results provide theoretical insights into designing stable and efficient private estimators without relying on data-dependent preprocessing.

In this paper, we study norm-based regularization methods for neural networks. We compare existing penalization approaches and introduce two regularization strategies that extend classical ridge- and lasso-type penalties to neural network models. The first strategy modifies weight decay by incorporating the covariance structure of the input features into a ridge-type $\ell_2$ penalty, allowing regularization to account for feature dependence. The second combines an $\ell_1$ sparsity penalty with covariance-aware $\ell_2$ regularization, producing neural network weights that are both sparse and structurally informed. Monte Carlo simulations are used to evaluate these methods under different data-generating settings, followed by two real-data applications on building cooling-load prediction and leukemia cell-type classification from high-dimensional gene expression data. Across simulated and real-data examples, the proposed regularizers improve predictive performance on unseen data and provide more effective complexity control than standard norm-based penalties, particularly when features are correlated or high-dimensional.

Variable selection in linear regression models has been a problem since hypothesis testing began. Which variables to include or exclude from a model is not an easy task. Techniques such as Forward, Back ward, Stepwise Regression sequentially add or delete variables from a model. Penalized likelihood methods such as AIC, BIC, etc. seek to choose variables that have a significant contribution to the likelihood. Penalized sum of square methods such as LASSO and Elastic Net have been used to penalize small coefficients to only allow variables with large coefficients in the model. This work introduces an Artificial Intelligence approach to model selection where an ANN is trained to determine the significance of the variables based on OLS estimates. A simulation study shows the accuracy across various sample sizes and variances. Furthermore, a simulation study is conducted to compare the performance of the approach against Forward, Backward, AIC, BIC and LASSO. The approach is illustrated using a dataset from the World Health Organization regarding Life Expectancy. A github link is provided to the pretrained ANN that can handle up to 100 predictor variables, the original WHO dataset and the subset used in this work.

The present study aims to investigate a cluster cleaning algorithm that is both computationally simple and capable of solving the PU classification when the SCAR condition is unsatisfied. A secondary objective of this study is to determine the robustness of the LassoJoint method to perturbations of the SCAR condition. In the first step of our algorithm, we obtain cleaning labels from 2-means clustering. Subsequently, we perform logistic regression on the cleaned data, assigning positive labels from the cleaning algorithm with additional true positive observations. The remaining observations are assigned the negative label. The proposed algorithm is evaluated by comparing 11 real data sets from machine learning repositories and a synthetic set. The findings obtained from this study demonstrate the efficacy of the clustering algorithm in scenarios where the SCAR condition is violated and further underscore the moderate robustness of the LassoJoint algorithm in this context.

The paper considers the computation of L1 regularization paths in a state space setting, which includes L1 regularized Kalman smoothing, linear SVM, LASSO, and more. The paper proposes two new algorithms, which are duals of each other; the first algorithm applies to L1 regularization of independent variables while the second applies to L1 regularization of dependent variables. The heart of the proposed algorithms is parametric Gaussian message passing (i.e., Kalman-type forward-backward recursions) in the pertinent factor graphs. The proposed methods are broadly applicable, they (usually) require only matrix multiplications, and their complexity can be competitive with prior methods in some cases.

We revisit Cox's proportional hazards model to improve variable selection in survival analysis. A square-root transformation of the partial likelihood renders the selection of the regularization parameter pivotal, free of the unknown baseline hazard and censoring mechanism. The resulting criterion borrows from information criteria such as BIC and from penalized regression methods such as the lasso, taking the best of both. On simulated and real data, our method substantially improves upon state-of-the-art approaches used daily in support recovery.

Existing methods for transfer learning struggle to deal with situations where the source datasets are limited and not guaranteed to be well-aligned with the target dataset. A typical strategy is to use the empirical loss minimizer on the source data as a prior mean for the target parameters. Our key conceptual contribution is to use a risk minimizer conditional on source parameters instead. This allows us to construct a single joint prior distribution for all parameters from the source datasets as well as the target dataset. As a consequence, we benefit from full Bayesian uncertainty quantification and can perform model averaging via Gibbs sampling over indicator variables governing the inclusion of each source dataset. We show how a particular instantiation of our prior leads to a Bayesian Lasso in a transformed coordinate system and discuss computational techniques to scale our approach to moderately sized datasets. We discuss connections between the Maximum a Posteriori estimate associated with our approach and the recently proposed Trans-Lasso method and demonstrate that the MAP estimator MSE-dominates the Trans-Lasso in the normal means setting when there is no regularization on the source datasets. Finally, we perform numerical experiments finding that full Bayesian inference provides superior predictive performance relative to Trans-Lasso on a genetics application, especially when the source data are limited.

Explainable artificial intelligence approaches accelerate drug discovery by improving molecular representation learning, identifying key molecular structures, and rationalizing drug property prediction. However, developing end-to-end explainable models for target-specific structure-activity relationship modeling remains challenging because compound-protein interaction data are often limited for individual targets, and small changes in chemical substituents or local structural motifs can cause large differences in molecular properties. Therefore, effectively leveraging structural and property information to identify key moieties associated with compound-protein affinity is essential. We propose a graph neural network (GNN) framework that uses property and structural information from activity-cliff molecule pairs targeting specific proteins to predict compound-protein affinity, measured by half-maximal inhibitory concentration (IC50), and explain property differences. To improve explainability, we trained GNNs with structure-aware loss functions using group lasso and sparse group lasso regularization, which prune and highlight molecular subgraphs relevant to activity differences. We applied this framework to activity-cliff data from molecules targeting six tyrosine-protein kinases across the Src, Abl, and Tec families, as well as anaplastic lymphoma kinase. Integrating common- and uncommon-node information with sparse group lasso improved target-specific molecular property prediction, producing lower root mean square errors and higher Pearson correlation coefficients. Regularization also enhanced GNN feature attribution by improving graph-level global direction scores and atom-level coloring accuracy. These results support more interpretable drug discovery pipelines, particularly for identifying critical molecular substructures during lead optimization.

Given a high-dimensional covariate matrix and a response vector, ridge-regularized sparse linear regression selects a subset of features that explains the relationship between covariates and the response in an interpretable manner. To choose hyperparameters that control the sparsity level and amount of regularization, practitioners commonly use k-fold cross-validation. However, cross-validation substantially increases the computational cost of sparse regression as it requires solving many mixed-integer optimization problems (MIOs) for each hyperparameter combination. To address this computational burden, we derive computationally tractable relaxations of the k-fold cross-validation loss, facilitating hyperparameter selection while solving $50$--$80\%$ fewer MIOs in practice. Our computational results demonstrate, across eleven real-world UCI datasets, that exact MIO-based cross-validation can be competitive with mature software packages such as glmnet and L0Learn.

Mixtures of experts (MoE) are conditional mixture models in which both the mixing proportions and the component densities depend on the predictors, and are widely used for regression, classification and model-based clustering of heterogeneous data. Fitting MoE by maximum likelihood becomes unstable, and sometimes infeasible, when the predictors are numerous or correlated. We propose a regularized maximum likelihood framework for simultaneous parameter estimation and feature selection in MoE whose experts belong to the generalized linear model family, covering Gaussian, Poisson and multinomial responses within a single formulation. Sparsity is induced in both the gating network and the experts through $\ell_1$ penalties, and the penalized log-likelihood is maximized by a proximal Newton-EM algorithm whose M-step reduces to weighted Lasso problems with closed-form coordinate-ascent updates. Unlike existing penalized MoE procedures, the algorithm requires neither a local quadratic approximation of the penalty nor any matrix inversion, it returns exactly sparse estimates without thresholding, and a proximal Newton-type variant guarantees a monotone increase of the penalized objective at every iteration. On simulated data and five real data sets, the method recovers the actual sparsity support and delivers prediction and clustering accuracy that is competitive with, and often better than, state-of-the-art regularized MoE. The source codes of our developed algorithms and their documentation are publicly available on Github at https://github.com/nv-thin/GLM-RMoE.

We study learning to learn through the lens of hyperparameter tuning. We propose the Langevin Gradient Descent Algorithm (LGD), which approximates the mean of the posterior distribution defined by the loss function and regularizer of a regression task with convex objective. For classification tasks, the LGD algorithm estimates the posterior probabilities of each class on the test set. We prove the existence of an optimal hyperparameter configuration for which the LGD algorithm achieves the Bayes' optimal solution for squared loss on regression tasks, and for which LGD closely approximates the posterior probabilities for well-specified classification tasks. Subsequently, we study generalization guarantees on meta learning optimal hyperparameters for the LGD algorithm from a given set of tasks in the data-driven setting. For a number of parameters $d$ and hyperparameter dimension $h$, we show a pseudo-dimension bound of $O(dh)$, up to logarithmic terms under mild assumptions on LGD. This matches the dependence of the bounds on number of parameters obtained in prior work for linear regression using the elastic net, which only allows for $h=2$ hyperparameters, and extends their bounds to regression on convex loss. Compared to bounds on regularized logistic regression that allow for only $h=1$ hyperparameter, our bounds improve greatly on the dependence on samples per task at the cost of worse dependence on the number of parameters by accounting for hardware-aware procedures. Finally, we show empirical evidence of the success of LGD and the meta learning procedure for few-shot learning on linear and logistic regression using synthetically created datasets.