Under-approximations of reachable sets and tubes have been receiving growing research attention due to their important roles in control synthesis and verification. Available under-approximation methods applicable to continuous-time linear systems typically assume the ability to compute transition matrices and their integrals exactly, which is not feasible in general, and/or suffer from high computational costs. In this note, we attempt to overcome these drawbacks for a class of linear time-invariant (LTI) systems, where we propose a novel method to under-approximate finite-time forward reachable sets and tubes, utilizing approximations of the matrix exponential and its integral. In particular, we consider the class of continuous-time LTI systems with an identity input matrix and initial and input values belonging to full dimensional sets that are affine transformations of closed unit balls. The proposed method yields computationally efficient under-approximations of reachable sets and tubes, when implemented using zonotopes, with first-order convergence guarantees in the sense of the Hausdorff distance. To illustrate its performance, we implement our approach in three numerical examples, where linear systems of dimensions ranging between 2 and 200 are considered.
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This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with structured quadrature points (QMC), which provide stable $L_2$ reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a RKHS of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the $d$-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings.
In this work we make progress in understanding the relationship between learning models with access to entangled, separable and statistical measurements in the quantum statistical query (QSQ) model. To this end, we show the following results. $\textbf{Entangled versus separable measurements.}$ The goal here is to learn an unknown $f$ from the concept class $C\subseteq \{f:\{0,1\}^n\rightarrow [k]\}$ given copies of $\frac{1}{\sqrt{2^n}}\sum_x \vert x,f(x)\rangle$. We show that, if $T$ copies suffice to learn $f$ using entangled measurements, then $O(nT^2)$ copies suffice to learn $f$ using just separable measurements. $\textbf{Entangled versus statistical measurements}$ The goal here is to learn a function $f \in C$ given access to separable measurements and statistical measurements. We exhibit a class $C$ that gives an exponential separation between QSQ learning and quantum learning with entangled measurements (even in the presence of noise). This proves the "quantum analogue" of the seminal result of Blum et al. [BKW'03]. that separates classical SQ and PAC learning with classification noise. $\textbf{QSQ lower bounds for learning states.}$ We introduce a quantum statistical query dimension (QSD), which we use to give lower bounds on the QSQ learning. With this we prove superpolynomial QSQ lower bounds for testing purity, shadow tomography, Abelian hidden subgroup problem, degree-$2$ functions, planted bi-clique states and output states of Clifford circuits of depth $\textsf{polylog}(n)$. $\textbf{Further applications.}$ We give and $\textit{unconditional}$ separation between weak and strong error mitigation and prove lower bounds for learning distributions in the QSQ model. Prior works by Quek et al. [QFK+'22], Hinsche et al. [HIN+'22], and Nietner et al. [NIS+'23] proved the analogous results $\textit{assuming}$ diagonal measurements and our work removes this assumption.
The European Union's Artificial Intelligence (AI) Act is set to be a landmark legal instrument for regulating AI technology. While stakeholders have primarily focused on the governance of fixed purpose AI applications (also known as narrow AI), more attention is required to understand the nature of highly and broadly capable systems. As of the beginning of 2023, several definitions for General Purpose AI Systems (GPAIS) exist in relation to the AI Act, attempting to distinguish between systems with and without a fixed purpose. In this article, we operationalise these differences through the concept of "distinct tasks" and examine four approaches (quantity, performance, adaptability, and emergence) to determine whether an AI system should be classified as a GPAIS. We suggest that EU stakeholders use the four approaches as a starting point to discriminate between fixed-purpose and GPAIS.
The discrete $\alpha$-neighbor $p$-center problem (d-$\alpha$-$p$CP) is an emerging variant of the classical $p$-center problem which recently got attention in literature. In this problem, we are given a discrete set of points and we need to locate $p$ facilities on these points in such a way that the maximum distance between each point where no facility is located and its $\alpha$-closest facility is minimized. The only existing algorithms in literature for solving the d-$\alpha$-$p$CP are approximation algorithms and two recently proposed heuristics. In this work, we present two integer programming formulations for the d-$\alpha$-$p$CP, together with lifting of inequalities, valid inequalities, inequalities that do not change the optimal objective function value and variable fixing procedures. We provide theoretical results on the strength of the formulations and convergence results for the lower bounds obtained after applying the lifting procedures or the variable fixing procedures in an iterative fashion. Based on our formulations and theoretical results, we develop branch-and-cut (B&C) algorithms, which are further enhanced with a starting heuristic and a primal heuristic. We evaluate the effectiveness of our B&C algorithms using instances from literature. Our algorithms are able to solve 116 out of 194 instances from literature to proven optimality, with a runtime of under a minute for most of them. By doing so, we also provide improved solution values for 116 instances.
We define the complexity of a continuous-time linear system to be the minimum number of bits required to describe its forward increments to a desired level of fidelity, and compute this quantity using the rate distortion function of a Gaussian source of uncertainty in those increments. The complexity of a linear system has relevance in control-communications contexts requiring local and dynamic decision-making based on sampled data representations. We relate this notion of complexity to the design of attention-varying controllers, and demonstrate a novel methodology for constructing source codes via the endpoint maps of so-called emulating systems, with potential for non-parametric, data-based simulation and analysis of unknown dynamical systems.
CRYSTALS-Dilithium is a lattice-based signature scheme to be standardized by NIST as the primary post-quantum signature algorithm. In this work, we make a thorough study of optimizing the implementations of Dilithium by utilizing the Advanced Vector Extension (AVX) instructions, specifically AVX2 and the latest AVX512. We first present an improved parallel small polynomial multiplication with tailored early evaluation (PSPM-TEE) to further speed up the signing procedure, which results in a speedup of 5\%-6\% compared with the original PSPM Dilithium implementation. We then present a tailored reduction method that is simpler and faster than Montgomery reduction. Our optimized AVX2 implementation exhibits a speedup of 3\%-8\% compared with the state-of-the-art of Dilithium AVX2 software. Finally, for the first time, we propose a fully and highly vectorized implementation of Dilithium using AVX-512. This is achieved by carefully vectorizing most of Dilithium functions with the AVX512 instructions in order to improve efficiency both for time and for space simultaneously. With all the optimization efforts, our AVX-512 implementation improves the performance by 37.3\%/50.7\%/39.7\% in key generation, 34.1\%/37.1\%/42.7\% in signing, and 38.1\%/38.7\%/40.7\% in verification for the parameter sets of Dilithium2/3/5 respectively. To the best of our knowledge, our AVX512 implementation has the best performance for Dilithium on the Intel x64 CPU platform to date.
Addressing the challenge of scaling-up epidemiological inference to complex and heterogeneous models, we introduce Poisson Approximate Likelihood (PAL) methods. In contrast to the popular ODE approach to compartmental modelling, in which a large population limit is used to motivate a deterministic model, PALs are derived from approximate filtering equations for finite-population, stochastic compartmental models, and the large population limit drives consistency of maximum PAL estimators. Our theoretical results appear to be the first likelihood-based parameter estimation consistency results which apply to a broad class of partially observed stochastic compartmental models and address the large population limit. PALs are simple to implement, involving only elementary arithmetic operations and no tuning parameters, and fast to evaluate, requiring no simulation from the model and having computational cost independent of population size. Through examples we demonstrate how PALs can be used to: fit an age-structured model of influenza, taking advantage of automatic differentiation in Stan; compare over-dispersion mechanisms in a model of rotavirus by embedding PALs within sequential Monte Carlo; and evaluate the role of unit-specific parameters in a meta-population model of measles.
Stein Variational Gradient Descent (SVGD) is a nonparametric particle-based deterministic sampling algorithm. Despite its wide usage, understanding the theoretical properties of SVGD has remained a challenging problem. For sampling from a Gaussian target, the SVGD dynamics with a bilinear kernel will remain Gaussian as long as the initializer is Gaussian. Inspired by this fact, we undertake a detailed theoretical study of the Gaussian-SVGD, i.e., SVGD projected to the family of Gaussian distributions via the bilinear kernel, or equivalently Gaussian variational inference (GVI) with SVGD. We present a complete picture by considering both the mean-field PDE and discrete particle systems. When the target is strongly log-concave, the mean-field Gaussian-SVGD dynamics is proven to converge linearly to the Gaussian distribution closest to the target in KL divergence. In the finite-particle setting, there is both uniform in time convergence to the mean-field limit and linear convergence in time to the equilibrium if the target is Gaussian. In the general case, we propose a density-based and a particle-based implementation of the Gaussian-SVGD, and show that several recent algorithms for GVI, proposed from different perspectives, emerge as special cases of our unified framework. Interestingly, one of the new particle-based instance from this framework empirically outperforms existing approaches. Our results make concrete contributions towards obtaining a deeper understanding of both SVGD and GVI.
How do statistical dependencies in measurement noise influence high-dimensional inference? To answer this, we study the paradigmatic spiked matrix model of principal components analysis (PCA), where a rank-one matrix is corrupted by additive noise. We go beyond the usual independence assumption on the noise entries, by drawing the noise from a low-order polynomial orthogonal matrix ensemble. The resulting noise correlations make the setting relevant for applications but analytically challenging. We provide the first characterization of the Bayes-optimal limits of inference in this model. If the spike is rotation-invariant, we show that standard spectral PCA is optimal. However, for more general priors, both PCA and the existing approximate message passing algorithm (AMP) fall short of achieving the information-theoretic limits, which we compute using the replica method from statistical mechanics. We thus propose a novel AMP, inspired by the theory of Adaptive Thouless-Anderson-Palmer equations, which saturates the theoretical limit. This AMP comes with a rigorous state evolution analysis tracking its performance. Although we focus on specific noise distributions, our methodology can be generalized to a wide class of trace matrix ensembles at the cost of more involved expressions. Finally, despite the seemingly strong assumption of rotation-invariant noise, our theory empirically predicts algorithmic performance on real data, pointing at remarkable universality properties.
Recently, the use of deep equilibrium methods has emerged as a new approach for solving imaging and other ill-posed inverse problems. While learned components may be a key factor in the good performance of these methods in practice, a theoretical justification from a regularization point of view is still lacking. In this paper, we address this issue by providing stability and convergence results for the class of equilibrium methods. In addition, we derive convergence rates and stability estimates in the symmetric Bregman distance. We strengthen our results for regularization operators with contractive residuals. Furthermore, we use the presented analysis to gain insight into the practical behavior of these methods, including a lower bound on the performance of the regularized solutions. In addition, we show that the convergence analysis leads to the design of a new type of loss function which has several advantages over previous ones. Numerical simulations are used to support our findings.
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