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The univariate generalized extreme value (GEV) distribution is the most commonly used tool for analyzing the properties of rare events. The ever greater utilization of Bayesian methods for extreme value analysis warrants detailed theoretical investigation, which has thus far been underdeveloped. Even the most basic asymptotic results are difficult to obtain because the GEV fails to satisfy standard regularity conditions. Here, we prove that the posterior distribution of the GEV parameter vector, given $n$ independent and identically distributed samples, converges in distribution to a trivariate normal distribution. The proof necessitates analyzing integrals of the GEV likelihood function over the entire parameter space, which requires considerable care because the support of the GEV density depends on the parameters in complicated ways.

The unprecedented growth in DNN model complexity, size, and amount of training data has led to a commensurate increase in demand for computing and a search for minimal encoding. Recent research advocates Hybrid Block Floating Point (HBFP) to minimize silicon provisioning in accelerators by converting the majority of arithmetic operations in training to 8-bit fixed point. In this paper, we perform a full-scale exploration of the HBFP design space using mathematical tools to study the interplay among various parameters and identify opportunities for even smaller encodings across layers and epochs. Based on our findings, we propose Accuracy Boosters, an epoch-driven mixed-mantissa HBFP technique that uses 6-bit mantissas only in the last epoch and first/last layers, and 4-bit mantissas for $99.7\%$ of all other arithmetic operations in training. Using analytic models, we show Accuracy Boosters enable increasing arithmetic density for an HBFP training accelerator by up to $21.3\times$ compared to FP32 and up to $4.4\times$ compared to another SOTA format Bfloat16, while preserving or outperforming FP32 accuracy.

In this paper, practically computable low-order approximations of potentially high-dimensional differential equations driven by geometric rough paths are proposed and investigated. In particular, equations are studied that cover the linear setting, but we allow for a certain type of dissipative nonlinearity in the drift as well. In a first step, a linear subspace is found that contains the solution space of the underlying rough differential equation (RDE). This subspace is associated to covariances of linear Ito-stochastic differential equations which is shown exploiting a Gronwall lemma for matrix differential equations. Orthogonal projections onto the identified subspace lead to a first exact reduced order system. Secondly, a linear map of the RDE solution (quantity of interest) is analyzed in terms of redundant information meaning that state variables are found that do not contribute to the quantity of interest. Once more, a link to Ito-stochastic differential equations is used. Removing such unnecessary information from the RDE provides a further dimension reduction without causing an error. Finally, we discretize a linear parabolic rough partial differential equation in space. The resulting large-order RDE is subsequently tackled with the exact reduction techniques studied in this paper. We illustrate the enormous complexity reduction potential in the corresponding numerical experiments.

This paper investigates an intelligent reflecting surface (IRS) enabled multiuser integrated sensing and communications (ISAC) system, which consists of one multi-antenna base station (BS), one IRS, multiple single-antenna communication users (CUs), and one target at the non-line-of-sight (NLoS) region of the BS. The IRS is deployed to not only assist the communication from the BS to the CUs, but also enable the BS's NLoS target sensing based on the echo signals from the BS-IRS-target-IRS-BS link. We consider two types of targets, namely the extended and point targets, for which the BS aims to estimate the complete target response matrix and the target direction-of-arrival (DoA) with respect to the IRS, respectively. To provide full degrees of freedom for sensing, we consider that the BS sends dedicated sensing signals in addition to the communication signals. Accordingly, we model two types of CU receivers, namely Type-I and Type-II CU receivers, which do not have and have the capability of canceling the interference from the sensing signals, respectively. Under each setup, we jointly optimize the transmit beamforming at the BS and the reflective beamforming at the IRS to minimize the Cram\'er-Rao bound (CRB) for target estimation, subject to the minimum signal-to-interference-plus-noise ratio (SINR) constraints at the CUs and the maximum transmit power constraint at the BS. We present efficient algorithms to solve the highly non-convex SINR-constrained CRB minimization problems, by using the techniques of alternating optimization, semi-definite relaxation, and successive convex approximation. Numerical results show that the proposed design achieves lower estimation CRB than other benchmark schemes, and the sensing signal interference cancellation at Type-II CU receivers is beneficial when the number of CUs is greater than one.

In the problem of aggregation, the aim is to combine a given class of base predictors to achieve predictions nearly as accurate as the best one. In this flexible framework, no assumption is made on the structure of the class or the nature of the target. Aggregation has been studied in both sequential and statistical contexts. Despite some important differences between the two problems, the classical results in both cases feature the same global complexity measure. In this paper, we revisit and tighten classical results in the theory of aggregation in the statistical setting by replacing the global complexity with a smaller, local one. Some of our proofs build on the PAC-Bayes localization technique introduced by Catoni. Among other results, we prove localized versions of the classical bound for the exponential weights estimator due to Leung and Barron and deviation-optimal bounds for the Q-aggregation estimator. These bounds improve over the results of Dai, Rigollet and Zhang for fixed design regression and the results of Lecu\'e and Rigollet for random design regression.

We consider the problem of approximating a $d \times d$ covariance matrix $M$ with a rank-$k$ matrix under $(\varepsilon,\delta)$-differential privacy. We present and analyze a complex variant of the Gaussian mechanism and show that the Frobenius norm of the difference between the matrix output by this mechanism and the best rank-$k$ approximation to $M$ is bounded by roughly $\tilde{O}(\sqrt{kd})$, whenever there is an appropriately large gap between the $k$'th and the $k+1$'th eigenvalues of $M$. This improves on previous work that requires that the gap between every pair of top-$k$ eigenvalues of $M$ is at least $\sqrt{d}$ for a similar bound. Our analysis leverages the fact that the eigenvalues of complex matrix Brownian motion repel more than in the real case, and uses Dyson's stochastic differential equations governing the evolution of its eigenvalues to show that the eigenvalues of the matrix $M$ perturbed by complex Gaussian noise have large gaps with high probability. Our results contribute to the analysis of low-rank approximations under average-case perturbations and to an understanding of eigenvalue gaps for random matrices, which may be of independent interest.

Quantization is commonly used in Deep Neural Networks (DNNs) to reduce the storage and computational complexity by decreasing the arithmetical precision of activations and weights, a.k.a. tensors. Efficient hardware architectures employ linear quantization to enable the deployment of recent DNNs onto embedded systems and mobile devices. However, linear uniform quantization cannot usually reduce the numerical precision to less than 8 bits without sacrificing high performance in terms of model accuracy. The performance loss is due to the fact that tensors do not follow uniform distributions. In this paper, we show that a significant amount of tensors fit into an exponential distribution. Then, we propose DNA-TEQ to exponentially quantize DNN tensors with an adaptive scheme that achieves the best trade-off between numerical precision and accuracy loss. The experimental results show that DNA-TEQ provides a much lower quantization bit-width compared to previous proposals, resulting in an average compression ratio of 40% over the linear INT8 baseline, with negligible accuracy loss and without retraining the DNNs. Besides, DNA-TEQ leads the way in performing dot-product operations in the exponential domain, which saves 66% of energy consumption on average for a set of widely used DNNs.

In distributed computing, slower nodes (stragglers) usually become a bottleneck. Gradient Coding (GC), introduced by Tandon et al., is an efficient technique that uses principles of error-correcting codes to distribute gradient computation in the presence of stragglers. In this paper, we consider the distributed computation of a sequence of gradients $\{g(1),g(2),\ldots,g(J)\}$, where processing of each gradient $g(t)$ starts in round-$t$ and finishes by round-$(t+T)$. Here $T\geq 0$ denotes a delay parameter. For the GC scheme, coding is only across computing nodes and this results in a solution where $T=0$. On the other hand, having $T>0$ allows for designing schemes which exploit the temporal dimension as well. In this work, we propose two schemes that demonstrate improved performance compared to GC. Our first scheme combines GC with selective repetition of previously unfinished tasks and achieves improved straggler mitigation. In our second scheme, which constitutes our main contribution, we apply GC to a subset of the tasks and repetition for the remainder of the tasks. We then multiplex these two classes of tasks across workers and rounds in an adaptive manner, based on past straggler patterns. Using theoretical analysis, we demonstrate that our second scheme achieves significant reduction in the computational load. In our experiments, we study a practical setting of concurrently training multiple neural networks over an AWS Lambda cluster involving 256 worker nodes, where our framework naturally applies. We demonstrate that the latter scheme can yield a 16\% improvement in runtime over the baseline GC scheme, in the presence of naturally occurring, non-simulated stragglers.

Phase noise (PN) is a major disturbance in MIMO systems, where the contribution of different oscillators at the transmitter and the receiver side may degrade the overall performance and offset the gains offered by MIMO techniques. This is even more crucial in the case of massive MIMO, since the number of PN sources may increase considerably. In this work, we propose an iterative receiver based on the application of the expectation-maximization algorithm. We consider a massive MIMO framework with a general association of oscillators to antennas, and include other channel disturbances like imperfect channel state information and Rician block fading. At each receiver iteration, given the information on the transmitted symbols, steepest descent is used to estimate the PN samples, with an optimized adaptive step size and a threshold-based stopping rule. The results obtained for several test cases show how the bit error rate and mean square error can benefit from the proposed phase-detection algorithm, even to the point of reaching the same performance as in the case where no PN is present{\color{black}, offering better results than a state-of-the-art alternative}. Further analysis of the results allow to draw some useful trade-offs respecting final performance and consumption of resources.

In this paper we study a non-local Cahn-Hilliard equation with singular single-well potential and degenerate mobility. This results as a particular case of a more general model derived for a binary, saturated, closed and incompressible mixture, composed by a tumor phase and a healthy phase, evolving in a bounded domain. The general system couples a Darcy-type evolution for the average velocity field with a convective reaction-diffusion type evolution for the nutrient concentration and a non-local convective Cahn-Hilliard equation for the tumor phase. The main mathematical difficulties are related to the proof of the separation property for the tumor phase in the Cahn-Hilliard equation: up to our knowledge, such problem is indeed open in the literature. For this reason, in the present contribution we restrict the analytical study to the Cahn-Hilliard equation only. For the non-local Cahn- Hilliard equation with singular single-well potential and degenerate mobility, we study the existence and uniqueness of weak solutions for spatial dimensions $d\leq 3$. After showing existence, we prove the strict separation property in three spatial dimensions, implying the same property also for lower spatial dimensions, which opens the way to the proof of uniqueness of solutions. Finally, we propose a well posed and gradient stable continuous finite element approximation of the model for $d\leq 3$, which preserves the physical properties of the continuos solution and which is computationally efficient, and we show simulation results in two spatial dimensions which prove the consistency of the proposed scheme and which describe the phase ordering dynamics associated to the system.