We initiate the study of Boolean function analysis on high-dimensional expanders. We give a random-walk based definition of high-dimensional expansion, which coincides with the earlier definition in terms of two-sided link expanders. Using this definition, we describe an analog of the Fourier expansion and the Fourier levels of the Boolean hypercube for simplicial complexes. Our analog is a decomposition into approximate eigenspaces of random walks associated with the simplicial complexes. Our random-walk definition and the decomposition have the additional advantage that they extend to the more general setting of posets, encompassing both high-dimensional expanders and the Grassmann poset, which appears in recent work on the unique games conjecture. We then use this decomposition to extend the Friedgut-Kalai-Naor theorem to high-dimensional expanders. Our results demonstrate that a constant-degree high-dimensional expander can sometimes serve as a sparse model for the Boolean slice or hypercube, and quite possibly additional results from Boolean function analysis can be carried over to this sparse model. Therefore, this model can be viewed as a derandomization of the Boolean slice, containing only $|X(k-1)|=O(n)$ points in contrast to $\binom{n}{k}$ points in the $(k)$-slice (which consists of all $n$-bit strings with exactly $k$ ones).
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Fully-strict fork-join parallelism is a powerful model for shared-memory programming due to its optimal time scaling and strong bounds on memory scaling. The latter is rarely achieved due to the difficulty of implementing continuation stealing in traditional High Performance Computing (HPC) languages -- where it is often impossible without modifying the compiler or resorting to non-portable techniques. We demonstrate how stackless coroutines (a new feature in C++20) can enable fully-portable continuation stealing and present libfork a lock-free fine-grained parallelism library, combining coroutines with user-space, geometric segmented-stacks. We show our approach is able to achieve optimal time/memory scaling, both theoretically and empirically, across a variety of benchmarks. Compared to openMP (libomp), libfork is on average 7.2x faster and consumes 10x less memory. Similarly, compared to Intel's TBB, libfork is on average 2.7x faster and consumes 6.2x less memory. Additionally, we introduce non-uniform memory access (NUMA) optimizations for schedulers that demonstrate performance matching busy-waiting schedulers.
We describe a `discretize-then-relax' strategy to globally minimize integral functionals over functions $u$ in a Sobolev space subject to Dirichlet boundary conditions. The strategy applies whenever the integral functional depends polynomially on $u$ and its derivatives, even if it is nonconvex. The `discretize' step uses a bounded finite element scheme to approximate the integral minimization problem with a convergent hierarchy of polynomial optimization problems over a compact feasible set, indexed by the decreasing size $h$ of the finite element mesh. The `relax' step employs sparse moment-sum-of-squares relaxations to approximate each polynomial optimization problem with a hierarchy of convex semidefinite programs, indexed by an increasing relaxation order $\omega$. We prove that, as $\omega\to\infty$ and $h\to 0$, solutions of such semidefinite programs provide approximate minimizers that converge in a suitable sense (including in certain $L^p$ norms) to the global minimizer of the original integral functional if it is unique. We also report computational experiments showing that our numerical strategy works well even when technical conditions required by our theoretical analysis are not satisfied.
Latitude on the choice of initialisation is a shared feature between one-step extended state-space and multi-step methods. The paper focuses on lattice Boltzmann schemes, which can be interpreted as examples of both previous categories of numerical schemes. We propose a modified equation analysis of the initialisation schemes for lattice Boltzmann methods, determined by the choice of initial data. These modified equations provide guidelines to devise and analyze the initialisation in terms of order of consistency with respect to the target Cauchy problem and time smoothness of the numerical solution. In detail, the larger the number of matched terms between modified equations for initialisation and bulk methods, the smoother the obtained numerical solution. This is particularly manifest for numerical dissipation. Starting from the constraints to achieve time smoothness, which can quickly become prohibitive for they have to take the parasitic modes into consideration, we explain how the distinct lack of observability for certain lattice Boltzmann schemes -- seen as dynamical systems on a commutative ring -- can yield rather simple conditions and be easily studied as far as their initialisation is concerned. This comes from the reduced number of initialisation schemes at the fully discrete level. These theoretical results are successfully assessed on several lattice Boltzmann methods.
We introduce a fine-grained framework for uncertainty quantification of predictive models under distributional shifts. This framework distinguishes the shift in covariate distributions from that in the conditional relationship between the outcome ($Y$) and the covariates ($X$). We propose to reweight the training samples to adjust for an identifiable covariate shift while protecting against worst-case conditional distribution shift bounded in an $f$-divergence ball. Based on ideas from conformal inference and distributionally robust learning, we present an algorithm that outputs (approximately) valid and efficient prediction intervals in the presence of distributional shifts. As a use case, we apply the framework to sensitivity analysis of individual treatment effects with hidden confounding. The proposed methods are evaluated in simulation studies and three real data applications, demonstrating superior robustness and efficiency compared with existing benchmarks.
This work proposes the extended functional tensor train (EFTT) format for compressing and working with multivariate functions on tensor product domains. Our compression algorithm combines tensorized Chebyshev interpolation with a low-rank approximation algorithm that is entirely based on function evaluations. Compared to existing methods based on the functional tensor train format, the adaptivity of our approach often results in reducing the required storage, sometimes considerably, while achieving the same accuracy. In particular, we reduce the number of function evaluations required to achieve a prescribed accuracy by up to over 96% compared to the algorithm from [Gorodetsky, Karaman and Marzouk, Comput. Methods Appl. Mech. Eng., 347 (2019)].
Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a collection of data points on a Riemannian manifold while matching a prescribed derivative at each point. We propose a novel procedure relying on the general concept of retractions to solve this problem on a large class of manifolds, including those for which computing the Riemannian exponential or logarithmic maps is not straightforward, such as the manifold of fixed-rank matrices. We analyze the well-posedness of the method by introducing and showing the existence of retraction-convex sets, a generalization of geodesically convex sets. We extend to the manifold setting a classical result on the asymptotic interpolation error of Hermite interpolation. We finally illustrate these results and the effectiveness of the method with numerical experiments on the manifold of fixed-rank matrices and the Stiefel manifold of matrices with orthonormal columns.
Modern Out-of-Order (OoO) CPUs are complex systems with many components interleaved in non-trivial ways. Pinpointing performance bottlenecks and understanding the underlying causes of program performance issues are critical tasks to make the most of hardware resources. We provide an in-depth overview of performance bottlenecks in recent OoO microarchitectures and describe the difficulties of detecting them. Techniques that measure resources utilization can offer a good understanding of a program's execution, but, due to the constraints inherent to Performance Monitoring Units (PMU) of CPUs, do not provide the relevant metrics for each use case. Another approach is to rely on a performance model to simulate the CPU behavior. Such a model makes it possible to implement any new microarchitecture-related metric. Within this framework, we advocate for implementing modeled resources as parameters that can be varied at will to reveal performance bottlenecks. This allows a generalization of bottleneck analysis that we call sensitivity analysis. We present Gus, a novel performance analysis tool that combines the advantages of sensitivity analysis and dynamic binary instrumentation within a resource-centric CPU model. We evaluate the impact of sensitivity on bottleneck analysis over a set of high-performance computing kernels.
We prove explicit uniform two-sided bounds for the phase functions of Bessel functions and of their derivatives. As a consequence, we obtain new enclosures for the zeros of Bessel functions and their derivatives in terms of inverse values of some elementary functions. These bounds are valid, with a few exceptions, for all zeros and all Bessel functions with non-negative indices. We provide numerical evidence showing that our bounds either improve or closely match the best previously known ones.
Digital credentials represent a cornerstone of digital identity on the Internet. To achieve privacy, certain functionalities in credentials should be implemented. One is selective disclosure, which allows users to disclose only the claims or attributes they want. This paper presents a novel approach to selective disclosure that combines Merkle hash trees and Boneh-Lynn-Shacham (BLS) signatures. Combining these approaches, we achieve selective disclosure of claims in a single credential and creation of a verifiable presentation containing selectively disclosed claims from multiple credentials signed by different parties. Besides selective disclosure, we enable issuing credentials signed by multiple issuers using this approach.
Living organisms interact with their surroundings in a closed-loop fashion, where sensory inputs dictate the initiation and termination of behaviours. Even simple animals are able to develop and execute complex plans, which has not yet been replicated in robotics using pure closed-loop input control. We propose a solution to this problem by defining a set of discrete and temporary closed-loop controllers, called "tasks", each representing a closed-loop behaviour. We further introduce a supervisory module which has an innate understanding of physics and causality, through which it can simulate the execution of task sequences over time and store the results in a model of the environment. On the basis of this model, plans can be made by chaining temporary closed-loop controllers. The proposed framework was implemented for a real robot and tested in two scenarios as proof of concept.
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