Least absolute shrinkage and selection operator (Lasso), a popular method for high-dimensional regression, is now used widely for estimating high-dimensional time series models such as the vector autoregression (VAR). Selecting its tuning parameter efficiently and accurately remains a challenge, despite the abundance of available methods for doing so. We propose $\mathsf{autotune}$, a strategy for Lasso to automatically tune itself by optimizing a penalized Gaussian log-likelihood alternately over regression coefficients and noise standard deviation. Using extensive simulation experiments on regression and VAR models, we show that $\mathsf{autotune}$ is faster, and provides better generalization and model selection than established alternatives in low signal-to-noise regimes. In the process, $\mathsf{autotune}$ provides a new estimator of noise standard deviation that can be used for high-dimensional inference, and a new visual diagnostic procedure for checking the sparsity assumption on regression coefficients. Finally, we demonstrate the utility of $\mathsf{autotune}$ on a real-world financial data set. An R package based on C++ is also made publicly available on Github.

Softmax attention is a central component of transformer architectures, yet its nonlinear structure poses significant challenges for theoretical analysis. We develop a unified, measure-based framework for studying single-layer softmax attention under both finite and infinite prompts. For i.i.d. Gaussian inputs, we lean on the fact that the softmax operator converges in the infinite-prompt limit to a linear operator acting on the underlying input-token measure. Building on this insight, we establish non-asymptotic concentration bounds for the output and gradient of softmax attention, quantifying how rapidly the finite-prompt model approaches its infinite-prompt counterpart, and prove that this concentration remains stable along the entire training trajectory in general in-context learning settings with sub-Gaussian tokens. In the case of in-context linear regression, we use the tractable infinite-prompt dynamics to analyze training at finite prompt length. Our results allow optimization analyses developed for linear attention to transfer directly to softmax attention when prompts are sufficiently long, showing that large-prompt softmax attention inherits the analytical structure of its linear counterpart. This, in turn, provides a principled and broadly applicable toolkit for studying the training dynamics and statistical behavior of softmax attention layers in large prompt regimes.

Data integration is essential across diverse domains, from historical records to biomedical research, facilitating joint statistical inference. A crucial initial step in this process involves merging multiple data sources based on matching individual records, often in the absence of unique identifiers. When the datasets are networks, this problem is typically addressed through graph matching methodologies. For such cases, auxiliary features or covariates associated with nodes or edges can be instrumental in achieving improved accuracy. However, most existing graph matching techniques do not incorporate this information, limiting their performance against non-identifiable and erroneous matches. To overcome these limitations, we propose two novel covariate-assisted seeded graph matching methods, where a partial alignment for a set of nodes, called seeds, is known. The first one solves a quadratic assignment problem (QAP) over the whole graph, while the second one only leverages the local neighborhood structure of seed nodes for computational scalability. Both methods are grounded in a conditional modeling framework, where elements of one graph's adjacency matrix are modeled using a generalized linear model (GLM), given the other graph and the available covariates. We establish theoretical guarantees for model estimation error and exact recovery of the solution of the QAP. The effectiveness of our methods is demonstrated through numerical experiments and in an application to matching the statistics academic genealogy and the collaboration networks. By leveraging additional covariates, we achieve improved alignment accuracy. Our work highlights the power of integrating covariate information in the classical graph matching setup, offering a practical and improved framework for combining network data with wide-ranging applications.

Clustering is a fundamental unsupervised learning task for uncovering patterns in data. While Gaussian Blurring Mean Shift (GBMS) has proven effective for identifying arbitrarily shaped clusters in Euclidean space, it struggles with datasets exhibiting hierarchical or tree-like structures. In this work, we introduce HypeGBMS, a novel extension of GBMS to hyperbolic space. Our method replaces Euclidean computations with hyperbolic distances and employs Möbius-weighted means to ensure that all updates remain consistent with the geometry of the space. HypeGBMS effectively captures latent hierarchies while retaining the density-seeking behavior of GBMS. We provide theoretical insights into convergence and computational complexity, along with empirical results that demonstrate improved clustering quality in hierarchical datasets. This work bridges classical mean-shift clustering and hyperbolic representation learning, offering a principled approach to density-based clustering in curved spaces. Extensive experimental evaluations on $11$ real-world datasets demonstrate that HypeGBMS significantly outperforms conventional mean-shift clustering methods in non-Euclidean settings, underscoring its robustness and effectiveness.

We identify test prediction variance (TPV) -- the first-order sensitivity of model outputs to parameter perturbations around a trained solution -- as a unifying quantity that links several classical observations about generalization in deep networks. TPV is a fully label-free object whose trace form separates the geometry of the trained model from the specific perturbation mechanism, allowing a broad family of parameter perturbations like SGD noise, label noise, finite-precision noise, and other post-training perturbations to be analyzed under a single framework. Theoretically, we show that TPV estimated on the training set converges to its test-set value in the overparameterized limit, providing the first result that prediction variance under local parameter perturbations can be inferred from training inputs alone. Empirically, TPV exhibits a striking stability across datasets and architectures -- including extremely narrow networks -- and correlates well with clean test loss. Finally, we demonstrate that modeling pruning as a TPV perturbation yields a simple label-free importance measure that performs competitively with state-of-the-art pruning methods, illustrating the practical utility of TPV. Code available at github.com/devansharpit/TPV.

We study outlier (a.k.a., anomaly) detection for single-pass non-stationary streaming data. In the well-studied offline or batch outlier detection problem, traditional methods such as kernel One-Class SVM (OCSVM) are both computationally heavy and prone to large false-negative (Type II) errors under non-stationarity. To remedy this, we introduce SONAR, an efficient SGD-based OCSVM solver with strongly convex regularization. We show novel theoretical guarantees on the Type I/II errors of SONAR, superior to those known for OCSVM, and further prove that SONAR ensures favorable lifelong learning guarantees under benign distribution shifts. In the more challenging problem of adversarial non-stationary data, we show that SONAR can be used within an ensemble method and equipped with changepoint detection to achieve adaptive guarantees, ensuring small Type I/II errors on each phase of data. We validate our theoretical findings on synthetic and real-world datasets.

We present an approach to denoising spatial transcriptomics images that is particularly effective for uncovering cell identities in the regime of ultra-low sequencing depths, and also allows for interpolation of gene expression. The method -- Spatial Transcriptomics via Adaptive Regularization and Kernels (STARK) -- augments kernel ridge regression with an incrementally adaptive graph Laplacian regularizer. In each iteration, we (1) perform kernel ridge regression with a fixed graph to update the image, and (2) update the graph based on the new image. The kernel ridge regression step involves reducing the infinite dimensional problem on a space of images to finite dimensions via a modified representer theorem. Starting with a purely spatial graph, and updating it as we improve our image makes the graph more robust to noise in low sequencing depth regimes. We show that the aforementioned approach optimizes a block-convex objective through an alternating minimization scheme wherein the sub-problems have closed form expressions that are easily computed. This perspective allows us to prove convergence of the iterates to a stationary point of this non-convex objective. Statistically, such stationary points converge to the ground truth with rate $\mathcal{O}(R^{-1/2})$ where $R$ is the number of reads. In numerical experiments on real spatial transcriptomics data, the denoising performance of STARK, evaluated in terms of label transfer accuracy, shows consistent improvement over the competing methods tested.

Verifying uniform conditions over continuous spaces through random sampling is fundamental in machine learning and control theory, yet classical coverage analyses often yield conservative bounds, particularly at small failure probabilities. We study uniform random sampling on the $d$-dimensional unit hypercube and analyze the number of uncovered subcubes after discretization. By applying a concentration inequality to the uncovered-count statistic, we derive a sample complexity bound with a logarithmic dependence on the failure probability ($δ$), i.e., $M =O( \tilde{C}\ln(\frac{2\tilde{C}}δ))$, which contrasts sharply with the classical linear $1/δ$ dependence. Under standard Lipschitz and uniformity assumptions, we present a self-contained derivation and compare our result with classical coupon-collector rates. Numerical studies across dimensions, precision levels, and confidence targets indicate that our bound tracks practical coverage requirements more tightly and scales favorably as $δ\to 0$. Our findings offer a sharper theoretical tool for algorithms that rely on grid-based coverage guarantees, enabling more efficient sampling, especially in high-confidence regimes.

Transport-based methods have emerged as a leading paradigm for building generative models from large, clean datasets. However, in many scientific and engineering domains, clean data are often unavailable: instead, we only observe measurements corrupted through a noisy, ill-conditioned channel. A generative model for the original data thus requires solving an inverse problem at the level of distributions. In this work, we introduce a novel approach to this task based on Stochastic Interpolants: we iteratively update a transport map between corrupted and clean data samples using only access to the corrupted dataset as well as black box access to the corruption channel. Under appropriate conditions, this iterative procedure converges towards a self-consistent transport map that effectively inverts the corruption channel, thus enabling a generative model for the clean data. We refer to the resulting method as the self-consistent stochastic interpolant (SCSI). It (i) is computationally efficient compared to variational alternatives, (ii) highly flexible, handling arbitrary nonlinear forward models with only black-box access, and (iii) enjoys theoretical guarantees. We demonstrate superior performance on inverse problems in natural image processing and scientific reconstruction, and establish convergence guarantees of the scheme under appropriate assumptions.

We report the discovery that binary encoding allows neural networks to extrapolate periodic functions beyond their training bounds. We introduce Normalized Base-2 Encoding (NB2E) as a method for encoding continuous numerical values and demonstrate that, using this input encoding, vanilla multi-layer perceptrons (MLP) successfully extrapolate diverse periodic signals without prior knowledge of their functional form. Internal activation analysis reveals that NB2E induces bit-phase representations, enabling MLPs to learn and extrapolate signal structure independently of position.