Data in many scientific domains are often collected under multiple distinct states (e.g., different clinical interventions), wherein latent processes (e.g., internal biological factors) can create complex variability between individual trials both within single states and between states. A promising approach for addressing this complexity is uncovering fundamental representational units within the data, i.e., functional Building Blocks (BBs), that can adjust their temporal activity and component structure across trials to capture the diverse spectrum of cross-trial variability. However, existing methods for understanding such multi-dimensional data often rely on tensor factorization under assumptions that may not align with the characteristics of real-world data, and struggle to accommodate trials of different durations, missing samples, and varied sampling rates. Here, we present a framework for Similarity-driven Building Block Inference using Graphs across States (SiBBlInGS). SiBBlInGS employs a robust graph-based dictionary learning approach for BB discovery that considers shared temporal activity, inter- and intra-state relationships, non-orthogonal components, and variations in session counts and duration across states, while remaining resilient to noise, random initializations, and missing samples. Additionally, it enables the identification of state-specific vs. state-invariant BBs and allows for cross-state controlled variations in BB structure and per-trial temporal variability. We demonstrate SiBBlInGS on synthetic and several real-world examples to highlight its ability to provide insights into the underlying mechanisms of complex phenomena across fields.

Data augmentation is critical to the empirical success of modern self-supervised representation learning, such as contrastive learning and masked language modeling. However, a theoretical understanding of the exact role of augmentation remains limited. Recent work has built the connection between self-supervised learning and the approximation of the top eigenspace of a graph Laplacian operator, suggesting that learning a linear probe atop such representation can be connected to RKHS regression. Building on this insight, this work delves into a statistical analysis of augmentation-based pretraining. Starting from the isometry property, a geometric characterization of the target function given by the augmentation, we disentangle the effects of the model and the augmentation, and prove two generalization bounds that are free of model complexity. Our first bound works for an arbitrary encoder, where the prediction error is decomposed as the sum of an estimation error incurred by fitting a linear probe with RKHS regression, and an approximation error entailed by RKHS approximation. Our second bound specifically addresses the case where the encoder is near-optimal, that is it approximates the top-d eigenspace of the RKHS induced by the augmentation. A key ingredient in our analysis is the augmentation complexity, which we use to quantitatively compare different augmentations and analyze their impact on downstream performance. It is widely acknowledged that better data augmentation techniques have been a major driving force in many recent breakthroughs in self-supervised representation learning. For example, contrastive learning (Chen et al., 2020) with aggressive random cropping and strong color distortion greatly improves the performance of vision tasks, and masked prediction with random masking and replacement is among the state-of-the-art representation learning methods for both natural language processing (Devlin et al., 2019) and vision (He et al., 2022). Due to their extraordinary empirical successes, developing a rigorous theoretical understanding of augmentation-based representation learning is an important open problem, and is crucial for inventing new, better and principled augmentation techniques. A key advance was recently made by HaoChen et al. (2021), who analyzed a variant of contrastive representation learning, termed spectral contrastive learning, by connecting it to learning eigenfunctions of a Laplacian operator over a population augmentation graph. This has been followed up by multiple works which extended these results to more general contrastive learning approaches (Saunshi et al., 2022; Johnson et al., 2023; Cabannes et al., 2023).

$K$-means clustering is a widely used machine learning method for identifying patterns in large datasets. Semidefinite programming (SDP) relaxations have recently been proposed for solving the $K$-means optimization problem that enjoy strong statistical optimality guarantees, but the prohibitive cost of implementing an SDP solver renders these guarantees inaccessible to practical datasets. By contrast, nonnegative matrix factorization (NMF) is a simple clustering algorithm that is widely used by machine learning practitioners, but without a solid statistical underpinning nor rigorous guarantees. In this paper, we describe an NMF-like algorithm that works by solving a nonnegative low-rank restriction of the SDP relaxed $K$-means formulation using a nonconvex Burer--Monteiro factorization approach. The resulting algorithm is just as simple and scalable as state-of-the-art NMF algorithms, while also enjoying the same strong statistical optimality guarantees as the SDP. In our experiments, we observe that our algorithm achieves substantially smaller mis-clustering errors compared to the existing state-of-the-art.

Recently, significant progress has been made in understanding the generalization of neural networks (NNs) trained by gradient descent (GD) using the algorithmic stability approach. However, most of the existing research has focused on one-hidden-layer NNs and has not addressed the impact of different network scaling parameters. In this paper, we greatly extend the previous work \cite{lei2022stability,richards2021stability} by conducting a comprehensive stability and generalization analysis of GD for multi-layer NNs. For two-layer NNs, our results are established under general network scaling parameters, relaxing previous conditions. In the case of three-layer NNs, our technical contribution lies in demonstrating its nearly co-coercive property by utilizing a novel induction strategy that thoroughly explores the effects of over-parameterization. As a direct application of our general findings, we derive the excess risk rate of $O(1/\sqrt{n})$ for GD algorithms in both two-layer and three-layer NNs. This sheds light on sufficient or necessary conditions for under-parameterized and over-parameterized NNs trained by GD to attain the desired risk rate of $O(1/\sqrt{n})$. Moreover, we demonstrate that as the scaling parameter increases or the network complexity decreases, less over-parameterization is required for GD to achieve the desired error rates. Additionally, under a low-noise condition, we obtain a fast risk rate of $O(1/n)$ for GD in both two-layer and three-layer NNs.

Lung cancer is a leading cause of cancer mortality globally, highlighting the importance of understanding its mortality risks to design effective patient-centered therapies. The National Lung Screening Trial (NLST) employed computed tomography texture analysis, which provides objective measurements of texture patterns on CT scans, to quantify the mortality risks of lung cancer patients. Partially linear Cox models have gained popularity for survival analysis by dissecting the hazard function into parametric and nonparametric components, allowing for the effective incorporation of both well-established risk factors (such as age and clinical variables) and emerging risk factors (e.g., image features) within a unified framework. However, when the dimension of parametric components exceeds the sample size, the task of model fitting becomes formidable, while nonparametric modeling grapples with the curse of dimensionality. We propose a novel Penalized Deep Partially Linear Cox Model (Penalized DPLC), which incorporates the SCAD penalty to select important texture features and employs a deep neural network to estimate the nonparametric component of the model. We prove the convergence and asymptotic properties of the estimator and compare it to other methods through extensive simulation studies, evaluating its performance in risk prediction and feature selection. The proposed method is applied to the NLST study dataset to uncover the effects of key clinical and imaging risk factors on patients' survival. Our findings provide valuable insights into the relationship between these factors and survival outcomes.

Extracting low-dimensional summary statistics from large datasets is essential for efficient (likelihood-free) inference. We characterise different classes of summaries and demonstrate their importance for correctly analysing dimensionality reduction algorithms. We propose obtaining summaries by minimising the expected posterior entropy (EPE) under the prior predictive distribution of the model. Many existing methods are equivalent to or are special or limiting cases of minimising the EPE. We develop a method to obtain high-fidelity summaries that minimise the EPE; we apply it to benchmark and real-world examples. We both offer a unifying perspective for obtaining informative summaries and provide concrete recommendations for practitioners.

Uncertainty is prevalent in engineering design, statistical learning, and decision making broadly. Due to inherent risk-averseness and ambiguity about assumptions, it is common to address uncertainty by formulating and solving conservative optimization models expressed using measures of risk and related concepts. We survey the rapid development of risk measures over the last quarter century. From their beginning in financial engineering, we recount the spread to nearly all areas of engineering and applied mathematics. Solidly rooted in convex analysis, risk measures furnish a general framework for handling uncertainty with significant computational and theoretical advantages. We describe the key facts, list several concrete algorithms, and provide an extensive list of references for further reading. The survey recalls connections with utility theory and distributionally robust optimization, points to emerging applications areas such as fair machine learning, and defines measures of reliability.

The spatial prediction (interpolation) problem arises in various fields of science and engineering that study spatially distributed variables. In the case of scattered data, filling gaps facilitates understanding of the spatial features, visualization of the observed process, and it is also necessary to obtain fully populated grids of spatially dependent parameters used in partial differential equations. Spatial prediction is highly relevant to many disciplines, such as environmental mapping, risk assessment(Christakos, 2012) and environmental health studies (Christakos and Hristopulos, 2013), subsurface hydrology (Kitanidis, 1997; Rubin, 2003), mining (Goovaerts, 1997), and oil reserves estimation (Hohn, 1988; Hamzehpour and Sahimi, 2006). In addition, remote sensing images often include gaps with missing data (e.g., clouds, snow, heavy precipitation, ground vegetation coverage, etc.) that need to be filled (Rossi et al, 1994). Spatial prediction is also useful in image analysis (Winkler, 2003; Gui and Wei, 2004) and signal processing (Unser and Blu, 2005; Ramani and Unser, 2006) including medical applications (Parrott et al, 1993; Cao and Worsley, 2001).

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

Contrastive deep graph clustering (CDGC) utilizes contrastive learning to group nodes into different clusters. Better augmentation techniques benefit the quality of the contrastive samples, thus being one of key factors to improve performance. However, the augmentation samples in existing methods are always predefined by human experiences, and agnostic from the downstream task clustering, thus leading to high human resource costs and poor performance. To this end, we propose an Attribute Graph Clustering method via Learnable Augmentation (\textbf{AGCLA}), which introduces learnable augmentors for high-quality and suitable augmented samples for CDGC. Specifically, we design two learnable augmentors for attribute and structure information, respectively. Besides, two refinement matrices, including the high-confidence pseudo-label matrix and the cross-view sample similarity matrix, are generated to improve the reliability of the learned affinity matrix. During the training procedure, we notice that there exist differences between the optimization goals for training learnable augmentors and contrastive learning networks. In other words, we should both guarantee the consistency of the embeddings as well as the diversity of the augmented samples. Thus, an adversarial learning mechanism is designed in our method. Moreover, a two-stage training strategy is leveraged for the high-confidence refinement matrices. Extensive experimental results demonstrate the effectiveness of AGCLA on six benchmark datasets.