In statistics on manifolds, the notion of the mean of a probability distribution becomes more involved than in a linear space. Several location statistics have been proposed, which reduce to the ordinary mean in Euclidean space. A relatively new family of contenders in this field are Diffusion Means, which are a one parameter family of location statistics modeled as initial points of isotropic diffusion with the diffusion time as parameter. It is natural to consider limit cases of the diffusion time parameter and it turns out that for short times the diffusion mean set approaches the intrinsic mean set. For long diffusion times, the limit is less obvious but for spheres of arbitrary dimension the diffusion mean set has been shown to converge to the extrinsic mean set. Here, we extend this result to the real projective spaces in their unique smooth isometric embedding into a linear space. We conjecture that the long time limit is always given by the extrinsic mean in the isometric embedding for connected compact symmetric spaces with unique isometric embedding.
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Although diffusion models have achieved remarkable success in the field of image generation, their latent space remains under-explored. Current methods for identifying semantics within latent space often rely on external supervision, such as textual information and segmentation masks. In this paper, we propose a method to identify semantic attributes in the latent space of pre-trained diffusion models without any further training. By projecting the Jacobian of the targeted semantic region into a low-dimensional subspace which is orthogonal to the non-masked regions, our approach facilitates precise semantic discovery and control over local masked areas, eliminating the need for annotations. We conducted extensive experiments across multiple datasets and various architectures of diffusion models, achieving state-of-the-art performance. In particular, for some specific face attributes, the performance of our proposed method even surpasses that of supervised approaches, demonstrating its superior ability in editing local image properties.
While deep generative models (DGMs) have gained popularity, their susceptibility to biases and other inefficiencies that lead to undesirable outcomes remains an issue. With their growing complexity, there is a critical need for early detection of issues to achieve desired results and optimize resources. Hence, we introduce a progressive analysis framework to monitor the training process of DGMs. Our method utilizes dimensionality reduction techniques to facilitate the inspection of latent representations, the generated and real distributions, and their evolution across training iterations. This monitoring allows us to pause and fix the training method if the representations or distributions progress undesirably. This approach allows for the analysis of a models' training dynamics and the timely identification of biases and failures, minimizing computational loads. We demonstrate how our method supports identifying and mitigating biases early in training a Generative Adversarial Network (GAN) and improving the quality of the generated data distribution.
Understanding the dependence structure between response variables is an important component in the analysis of correlated multivariate data. This article focuses on modeling dependence structures in multivariate binary data, motivated by a study aiming to understand how patterns in different U.S. senators' votes are determined by similarities (or lack thereof) in their attributes, e.g., political parties and social network profiles. To address such a research question, we propose a new Ising similarity regression model which regresses pairwise interaction coefficients in the Ising model against a set of similarity measures available/constructed from covariates. Model selection approaches are further developed through regularizing the pseudo-likelihood function with an adaptive lasso penalty to enable the selection of relevant similarity measures. We establish estimation and selection consistency of the proposed estimator under a general setting where the number of similarity measures and responses tend to infinity. Simulation study demonstrates the strong finite sample performance of the proposed estimator, particularly compared with several existing Ising model estimators in estimating the matrix of pairwise interaction coefficients. Applying the Ising similarity regression model to a dataset of roll call voting records of 100 U.S. senators, we are able to quantify how similarities in senators' parties, businessman occupations and social network profiles drive their voting associations.
Many high-dimensional data sets suffer from hidden confounding which affects both the predictors and the response of interest. In such situations, standard regression methods or algorithms lead to biased estimates. This paper substantially extends previous work on spectral deconfounding for high-dimensional linear models to the nonlinear setting and with this, establishes a proof of concept that spectral deconfounding is valid for general nonlinear models. Concretely, we propose an algorithm to estimate high-dimensional sparse additive models in the presence of hidden dense confounding: arguably, this is a simple yet practically useful nonlinear scope. We prove consistency and convergence rates for our method and evaluate it on synthetic data and a genetic data set.
The sensitivity of machine learning algorithms to outliers, particularly in high-dimensional spaces, necessitates the development of robust methods. Within the framework of $\epsilon$-contamination model, where the adversary can inspect and replace up to $\epsilon$ fraction of the samples, a fundamental open question is determining the optimal rates for robust stochastic convex optimization (robust SCO), provided the samples under $\epsilon$-contamination. We develop novel algorithms that achieve minimax-optimal excess risk (up to logarithmic factors) under the $\epsilon$-contamination model. Our approach advances beyonds existing algorithms, which are not only suboptimal but also constrained by stringent requirements, including Lipschitzness and smoothness conditions on sample functions.Our algorithms achieve optimal rates while removing these restrictive assumptions, and notably, remain effective for nonsmooth but Lipschitz population risks.
This paper addresses the problem of estimating the positions of points from distance measurements corrupted by sparse outliers. Specifically, we consider a setting with two types of nodes: anchor nodes, for which exact distances to each other are known, and target nodes, for which complete but corrupted distance measurements to the anchors are available. To tackle this problem, we propose a novel algorithm powered by Nystr\"om method and robust principal component analysis. Our method is computationally efficient as it processes only a localized subset of the distance matrix and does not require distance measurements between target nodes. Empirical evaluations on synthetic datasets, designed to mimic sensor localization, and on molecular experiments, demonstrate that our algorithm achieves accurate recovery with a modest number of anchors, even in the presence of high levels of sparse outliers.
A challenge in high-dimensional inverse problems is developing iterative solvers to find the accurate solution of regularized optimization problems with low computational cost. An important example is computed tomography (CT) where both image and data sizes are large and therefore the forward model is costly to evaluate. Since several years algorithms from stochastic optimization are used for tomographic image reconstruction with great success by subsampling the data. Here we propose a novel way how stochastic optimization can be used to speed up image reconstruction by means of image domain sketching such that at each iteration an image of different resolution is being used. Hence, we coin this algorithm ImaSk. By considering an associated saddle-point problem, we can formulate ImaSk as a gradient-based algorithm where the gradient is approximated in the same spirit as the stochastic average gradient am\'elior\'e (SAGA) and uses at each iteration one of these multiresolution operators at random. We prove that ImaSk is linearly converging for linear forward models with strongly convex regularization functions. Numerical simulations on CT show that ImaSk is effective and increasing the number of multiresolution operators reduces the computational time to reach the modeled solution.
In structured additive distributional regression, the conditional distribution of the response variables given the covariate information and the vector of model parameters is modelled using a P-parametric probability density function where each parameter is modelled through a linear predictor and a bijective response function that maps the domain of the predictor into the domain of the parameter. We present a method to perform inference in structured additive distributional regression using stochastic variational inference. We propose two strategies for constructing a multivariate Gaussian variational distribution to estimate the posterior distribution of the regression coefficients. The first strategy leverages covariate information and hyperparameters to learn both the location vector and the precision matrix. The second strategy tackles the complexity challenges of the first by initially assuming independence among all smooth terms and then introducing correlations through an additional set of variational parameters. Furthermore, we present two approaches for estimating the smoothing parameters. The first treats them as free parameters and provides point estimates, while the second accounts for uncertainty by applying a variational approximation to the posterior distribution. Our model was benchmarked against state-of-the-art competitors in logistic and gamma regression simulation studies. Finally, we validated our approach by comparing its posterior estimates to those obtained using Markov Chain Monte Carlo on a dataset of patents from the biotechnology/pharmaceutics and semiconductor/computer sectors.
We consider the problem of explaining the predictions of graph neural networks (GNNs), which otherwise are considered as black boxes. Existing methods invariably focus on explaining the importance of graph nodes or edges but ignore the substructures of graphs, which are more intuitive and human-intelligible. In this work, we propose a novel method, known as SubgraphX, to explain GNNs by identifying important subgraphs. Given a trained GNN model and an input graph, our SubgraphX explains its predictions by efficiently exploring different subgraphs with Monte Carlo tree search. To make the tree search more effective, we propose to use Shapley values as a measure of subgraph importance, which can also capture the interactions among different subgraphs. To expedite computations, we propose efficient approximation schemes to compute Shapley values for graph data. Our work represents the first attempt to explain GNNs via identifying subgraphs explicitly and directly. Experimental results show that our SubgraphX achieves significantly improved explanations, while keeping computations at a reasonable level.
Residual networks (ResNets) have displayed impressive results in pattern recognition and, recently, have garnered considerable theoretical interest due to a perceived link with neural ordinary differential equations (neural ODEs). This link relies on the convergence of network weights to a smooth function as the number of layers increases. We investigate the properties of weights trained by stochastic gradient descent and their scaling with network depth through detailed numerical experiments. We observe the existence of scaling regimes markedly different from those assumed in neural ODE literature. Depending on certain features of the network architecture, such as the smoothness of the activation function, one may obtain an alternative ODE limit, a stochastic differential equation or neither of these. These findings cast doubts on the validity of the neural ODE model as an adequate asymptotic description of deep ResNets and point to an alternative class of differential equations as a better description of the deep network limit.
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