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The minimum covariance determinant (MCD) estimator is a popular method for robustly estimating the mean and covariance of multivariate data. We extend the MCD to the setting where the observations are matrices rather than vectors and introduce the matrix minimum covariance determinant (MMCD) estimators for robust parameter estimation. These estimators hold equivariance properties, achieve a high breakdown point, and are consistent under elliptical matrix-variate distributions. We have also developed an efficient algorithm with convergence guarantees to compute the MMCD estimators. Using the MMCD estimators, we can compute robust Mahalanobis distances that can be used for outlier detection. Those distances can be decomposed into outlyingness contributions from each cell, row, or column of a matrix-variate observation using Shapley values, a concept for outlier explanation recently introduced in the multivariate setting. Simulations and examples reveal the excellent properties and usefulness of the robust estimators.

Graph neural networks (GNNs) are the predominant architectures for a variety of learning tasks on graphs. We present a new angle on the expressive power of GNNs by studying how the predictions of a GNN probabilistic classifier evolve as we apply it on larger graphs drawn from some random graph model. We show that the output converges to a constant function, which upper-bounds what these classifiers can express uniformly. This convergence phenomenon applies to a very wide class of GNNs, including state of the art models, with aggregates including mean and the attention-based mechanism of graph transformers. Our results apply to a broad class of random graph models, including the (sparse) Erd\H{o}s-R\'enyi model and the stochastic block model. We empirically validate these findings, observing that the convergence phenomenon already manifests itself on graphs of relatively modest size.

We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.

Under interference, the potential outcomes of a unit depend on treatments assigned to other units. A network interference structure is typically assumed to be given and accurate. In this paper, we study the problems resulting from misspecifying these networks. First, we derive bounds on the bias arising from estimating causal effects under a misspecified network. We show that the maximal possible bias depends on the divergence between the assumed network and the true one with respect to the induced exposure probabilities. Then, we propose a novel estimator that leverages multiple networks simultaneously and is unbiased if one of the networks is correct, thus providing robustness to network specification. Additionally, we develop a probabilistic bias analysis that quantifies the impact of a postulated misspecification mechanism on the causal estimates. We illustrate key issues in simulations and demonstrate the utility of the proposed methods in a social network field experiment and a cluster-randomized trial with suspected cross-clusters contamination.

Deep Ensembles (DEs) demonstrate improved accuracy, calibration and robustness to perturbations over single neural networks partly due to their functional diversity. Particle-based variational inference (ParVI) methods enhance diversity by formalizing a repulsion term based on a network similarity kernel. However, weight-space repulsion is inefficient due to over-parameterization, while direct function-space repulsion has been found to produce little improvement over DEs. To sidestep these difficulties, we propose First-order Repulsive Deep Ensemble (FoRDE), an ensemble learning method based on ParVI, which performs repulsion in the space of first-order input gradients. As input gradients uniquely characterize a function up to translation and are much smaller in dimension than the weights, this method guarantees that ensemble members are functionally different. Intuitively, diversifying the input gradients encourages each network to learn different features, which is expected to improve the robustness of an ensemble. Experiments on image classification datasets and transfer learning tasks show that FoRDE significantly outperforms the gold-standard DEs and other ensemble methods in accuracy and calibration under covariate shift due to input perturbations.

Current deep neural networks (DNNs) are overparameterized and use most of their neuronal connections during inference for each task. The human brain, however, developed specialized regions for different tasks and performs inference with a small fraction of its neuronal connections. We propose an iterative pruning strategy introducing a simple importance-score metric that deactivates unimportant connections, tackling overparameterization in DNNs and modulating the firing patterns. The aim is to find the smallest number of connections that is still capable of solving a given task with comparable accuracy, i.e. a simpler subnetwork. We achieve comparable performance for LeNet architectures on MNIST, and significantly higher parameter compression than state-of-the-art algorithms for VGG and ResNet architectures on CIFAR-10/100 and Tiny-ImageNet. Our approach also performs well for the two different optimizers considered -- Adam and SGD. The algorithm is not designed to minimize FLOPs when considering current hardware and software implementations, although it performs reasonably when compared to the state of the art.

It has been shown that deep neural networks of a large enough width are universal approximators but they are not if the width is too small. There were several attempts to characterize the minimum width $w_{\min}$ enabling the universal approximation property; however, only a few of them found the exact values. In this work, we show that the minimum width for $L^p$ approximation of $L^p$ functions from $[0,1]^{d_x}$ to $\mathbb R^{d_y}$ is exactly $\max\{d_x,d_y,2\}$ if an activation function is ReLU-Like (e.g., ReLU, GELU, Softplus). Compared to the known result for ReLU networks, $w_{\min}=\max\{d_x+1,d_y\}$ when the domain is $\smash{\mathbb R^{d_x}}$, our result first shows that approximation on a compact domain requires smaller width than on $\smash{\mathbb R^{d_x}}$. We next prove a lower bound on $w_{\min}$ for uniform approximation using general activation functions including ReLU: $w_{\min}\ge d_y+1$ if $d_x<d_y\le2d_x$. Together with our first result, this shows a dichotomy between $L^p$ and uniform approximations for general activation functions and input/output dimensions.

We consider problems where agents in a network seek a common quantity, measured independently and periodically by each agent through a local time-varying process. Numerous solvers addressing such problems have been developed in the past, featuring various adaptations of the local processing and the consensus step. However, existing solvers still lack support for advanced techniques, such as superiorization and over-the-air function computation (OTA-C). To address this limitation, we introduce a comprehensive framework for the analysis of distributed algorithms by characterizing them using the quasi-Fej\'er type algorithms and an extensive communication model. Under weak assumptions, we prove almost sure convergence of the algorithm to a common estimate for all agents. Moreover, we develop a specific class of algorithms within this framework to tackle distributed optimization problems with time-varying objectives, and, assuming that a time-invariant solution exists, prove its convergence to a solution. We also present a novel OTA-C protocol for consensus step in large decentralized networks, reducing communication overhead and enhancing network autonomy as compared to the existing protocols. The effectiveness of the algorithm, featuring superiorization and OTA-C, is demonstrated in a real-world application of distributed supervised learning over time-varying wireless networks, highlighting its low-latency and energy-efficiency compared to standard approaches.

Approximation capability of reservoir systems whose reservoir is a recurrent neural network (RNN) is discussed. In our problem setting, a reservoir system approximates a set of functions just by adjusting its linear readout while the reservoir is fixed. We will show what we call uniform strong universality of a family of RNN reservoir systems for a certain class of functions to be approximated. This means that, for any positive number, we can construct a sufficiently large RNN reservoir system whose approximation error for each function in the class of functions to be approximated is bounded from above by the positive number. Such RNN reservoir systems are constructed via parallel concatenation of RNN reservoirs.

Recent advances in 3D fully convolutional networks (FCN) have made it feasible to produce dense voxel-wise predictions of volumetric images. In this work, we show that a multi-class 3D FCN trained on manually labeled CT scans of several anatomical structures (ranging from the large organs to thin vessels) can achieve competitive segmentation results, while avoiding the need for handcrafting features or training class-specific models. To this end, we propose a two-stage, coarse-to-fine approach that will first use a 3D FCN to roughly define a candidate region, which will then be used as input to a second 3D FCN. This reduces the number of voxels the second FCN has to classify to ~10% and allows it to focus on more detailed segmentation of the organs and vessels. We utilize training and validation sets consisting of 331 clinical CT images and test our models on a completely unseen data collection acquired at a different hospital that includes 150 CT scans, targeting three anatomical organs (liver, spleen, and pancreas). In challenging organs such as the pancreas, our cascaded approach improves the mean Dice score from 68.5 to 82.2%, achieving the highest reported average score on this dataset. We compare with a 2D FCN method on a separate dataset of 240 CT scans with 18 classes and achieve a significantly higher performance in small organs and vessels. Furthermore, we explore fine-tuning our models to different datasets. Our experiments illustrate the promise and robustness of current 3D FCN based semantic segmentation of medical images, achieving state-of-the-art results. Our code and trained models are available for download: //github.com/holgerroth/3Dunet_abdomen_cascade.