Texture plays a vital role in enhancing visual richness in both real photographs and computer-generated imagery. However, the process of editing textures often involves laborious and repetitive manual adjustments of textons, which are the small, recurring local patterns that define textures. In this work, we introduce a fully unsupervised approach for representing textures using a compositional neural model that captures individual textons. We represent each texton as a 2D Gaussian function whose spatial support approximates its shape, and an associated feature that encodes its detailed appearance. By modeling a texture as a discrete composition of Gaussian textons, the representation offers both expressiveness and ease of editing. Textures can be edited by modifying the compositional Gaussians within the latent space, and new textures can be efficiently synthesized by feeding the modified Gaussians through a generator network in a feed-forward manner. This approach enables a wide range of applications, including transferring appearance from an image texture to another image, diversifying textures, texture interpolation, revealing/modifying texture variations, edit propagation, texture animation, and direct texton manipulation. The proposed approach contributes to advancing texture analysis, modeling, and editing techniques, and opens up new possibilities for creating visually appealing images with controllable textures.

Similarly to many other inference problems in planted models, we are interested in understanding the fundamental information-theoretical limits as well as the computational hardness of graph alignment. First, we study the Gaussian setting, when the graphs are complete and the signal lies on correlated Gaussian edges weights. We prove that the exact recovery task exhibits a sharp information-theoretic threshold (and characterize it), and study a simple and natural spectral method for recovery, EIG1, which consists in aligning the leading eigenvectors of the adjacency matrices of the two graphs. While most of the recent work on the subject was dedicated to recovering the hidden signal in dense graphs, we next explore graph alignment in the sparse regime, where the mean degree of the nodes are constant, not scaling with the graph size. In this particularly challenging setting, for sparse Erdős-Rényi graphs, only a fraction of the nodes can be correctly matched by any algorithm.

Machine learning (ML)-based solutions are rapidly changing the landscape of many fields, including structural engineering. Despite their promising performance, these approaches are usually only demonstrated as proof-of-concept in structural engineering, and are rarely deployed for real-world applications. This paper aims to illustrate the challenges of developing ML models suitable for deployment through two illustrative examples. Among various pitfalls, the presented discussion focuses on model overfitting and underspecification, training data representativeness, variable omission bias, and cross-validation.

The purpose of this paper is twofold. First, we propose a novel algorithm for estimating parameters in one-dimensional Gaussian mixture models (GMMs). The algorithm takes advantage of the Hankel structure inherent in the Fourier data obtained from independent and identically distributed (i.i.d) samples of the mixture. For GMMs with a unified variance, a singular value ratio functional using the Fourier data is introduced and used to resolve the variance and component number simultaneously. The consistency of the estimator is derived. Compared to classic algorithms such as the method of moments and the maximum likelihood method, the proposed algorithm does not require prior knowledge of the number of Gaussian components or good initial guesses. Numerical experiments demonstrate its superior performance in estimation accuracy and computational cost. Second, we reveal that there exists a fundamental limit to the problem of estimating the number of Gaussian components or model order in the mixture model if the number of i.i.d samples is finite. For the case of a single variance, we show that the model order can be successfully estimated only if the minimum separation distance between the component means exceeds a certain threshold value and can fail if below. We derive a lower bound for this threshold value, referred to as the computational resolution limit, in terms of the number of i.i.d samples, the variance, and the number of Gaussian components. Numerical experiments confirm this phase transition phenomenon in estimating the model order. Moreover, we demonstrate that our algorithm achieves better scores in likelihood, AIC, and BIC when compared to the EM algorithm.

The generalization ability of kernel interpolation in large dimensions (i.e., $n \asymp d^{\gamma}$ for some $\gamma>0$) might be one of the most interesting problems in the recent renaissance of kernel regression, since it may help us understand the 'benign overfitting phenomenon' reported in the neural networks literature. Focusing on the inner product kernel on the sphere, we fully characterized the exact order of both the variance and bias of large-dimensional kernel interpolation under various source conditions $s\geq 0$. Consequently, we obtained the $(s,\gamma)$-phase diagram of large-dimensional kernel interpolation, i.e., we determined the regions in $(s,\gamma)$-plane where the kernel interpolation is minimax optimal, sub-optimal and inconsistent.

In the transfer learning paradigm models learn useful representations (or features) during a data-rich pretraining stage, and then use the pretrained representation to improve model performance on data-scarce downstream tasks. In this work, we explore transfer learning with the goal of optimizing downstream performance. We introduce a simple linear model that takes as input an arbitrary pretrained feature transform. We derive exact asymptotics of the downstream risk and its fine-grained bias-variance decomposition. Our finding suggests that using the ground-truth featurization can result in "double-divergence" of the asymptotic risk, indicating that it is not necessarily optimal for downstream performance. We then identify the optimal pretrained representation by minimizing the asymptotic downstream risk averaged over an ensemble of downstream tasks. Our analysis reveals the relative importance of learning the task-relevant features and structures in the data covariates and characterizes how each contributes to controlling the downstream risk from a bias-variance perspective. Moreover, we uncover a phase transition phenomenon where the optimal pretrained representation transitions from hard to soft selection of relevant features and discuss its connection to principal component regression.

The $k$-parity problem is a classical problem in computational complexity and algorithmic theory, serving as a key benchmark for understanding computational classes. In this paper, we solve the $k$-parity problem with stochastic gradient descent (SGD) on two-layer fully-connected neural networks. We demonstrate that SGD can efficiently solve the $k$-sparse parity problem on a $d$-dimensional hypercube ($k\le O(\sqrt{d})$) with a sample complexity of $\tilde{O}(d^{k-1})$ using $2^{\Theta(k)}$ neurons, thus matching the established $\Omega(d^{k})$ lower bounds of Statistical Query (SQ) models. Our theoretical analysis begins by constructing a good neural network capable of correctly solving the $k$-parity problem. We then demonstrate how a trained neural network with SGD can effectively approximate this good network, solving the $k$-parity problem with small statistical errors. Our theoretical results and findings are supported by empirical evidence, showcasing the efficiency and efficacy of our approach.

We study minimax optimization problems defined over infinite-dimensional function classes. In particular, we restrict the functions to the class of overparameterized two-layer neural networks and study (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural network. As an initial step, we consider the minimax optimization problem stemming from estimating a functional equation defined by conditional expectations via adversarial estimation, where the objective function is quadratic in the functional space. For this problem, we establish convergence under the mean-field regime by considering the continuous-time and infinite-width limit of the optimization dynamics. Under this regime, gradient descent-ascent corresponds to a Wasserstein gradient flow over the space of probability measures defined over the space of neural network parameters. We prove that the Wasserstein gradient flow converges globally to a stationary point of the minimax objective at a $\mathcal{O}(T^{-1} + \alpha^{-1} ) $ sublinear rate, and additionally finds the solution to the functional equation when the regularizer of the minimax objective is strongly convex. Here $T$ denotes the time and $\alpha$ is a scaling parameter of the neural network. In terms of representation learning, our results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $\mathcal{O}(\alpha^{-1})$, measured in terms of the Wasserstein distance. Finally, we apply our general results to concrete examples including policy evaluation, nonparametric instrumental variable regression, and asset pricing.

We propose a novel method (floZ), based on normalizing flows, for estimating the Bayesian evidence (and its numerical uncertainty) from a set of samples drawn from the unnormalized posterior distribution. We validate it on distributions whose evidence is known analytically, up to 15 parameter space dimensions, and compare with two state-of-the-art techniques for estimating the evidence: nested sampling (which computes the evidence as its main target) and a k-nearest-neighbors technique that produces evidence estimates from posterior samples. Provided representative samples from the target posterior are available, our method is more robust to posterior distributions with sharp features, especially in higher dimensions. It has wide applicability, e.g., to estimate the evidence from variational inference, Markov-chain Monte Carlo samples, or any other method that delivers samples from the unnormalized posterior density.

Modern compression methods can summarize a target distribution $\mathbb{P}$ more succinctly than i.i.d. sampling but require access to a low-bias input sequence like a Markov chain converging quickly to $\mathbb{P}$. We introduce a new suite of compression methods suitable for compression with biased input sequences. Given $n$ points targeting the wrong distribution and quadratic time, Stein Kernel Thinning (SKT) returns $\sqrt{n}$ equal-weighted points with $\widetilde{O}(n^{-1/2})$ maximum mean discrepancy (MMD) to $\mathbb {P}$. For larger-scale compression tasks, Low-rank SKT achieves the same feat in sub-quadratic time using an adaptive low-rank debiasing procedure that may be of independent interest. For downstream tasks that support simplex or constant-preserving weights, Stein Recombination and Stein Cholesky achieve even greater parsimony, matching the guarantees of SKT with as few as $\operatorname{poly-log}(n)$ weighted points. Underlying these advances are new guarantees for the quality of simplex-weighted coresets, the spectral decay of kernel matrices, and the covering numbers of Stein kernel Hilbert spaces. In our experiments, our techniques provide succinct and accurate posterior summaries while overcoming biases due to burn-in, approximate Markov chain Monte Carlo, and tempering.