1,918 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Semantic Security with Infinite Dimensional Quantum Eavesdropping Channel

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    We propose a new proof method for direct coding theorems for wiretap channels where the eavesdropper has access to a quantum version of the transmitted signal on an infinite-dimensional Hilbert space and the legitimate parties communicate through a classical channel or a classical input, quantum output (cq) channel. The transmitter input can be subject to an additive cost constraint, which specializes to the case of an average energy constraint. This method yields errors that decay exponentially with increasing block lengths. Moreover, it provides a guarantee of a quantum version of semantic security, which is an established concept in classical cryptography and physical layer security. Therefore, it complements existing works which either do not prove the exponential error decay or use weaker notions of security. The main part of this proof method is a direct coding result on channel resolvability which states that there is only a doubly exponentially small probability that a standard random codebook does not solve the channel resolvability problem for the cq channel. Semantic security has strong operational implications meaning essentially that the eavesdropper cannot use its quantum observation to gather any meaningful information about the transmitted signal. We also discuss the connections between semantic security and various other established notions of secrecy

    Random block coordinate methods for inconsistent convex optimisation problems

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    We develop a novel randomised block coordinate primal-dual algorithm for a class of non-smooth ill-posed convex programs. Lying in the midway between the celebrated Chambolle-Pock primal-dual algorithm and Tseng's accelerated proximal gradient method, we establish global convergence of the last iterate as well optimal O(1/k)O(1/k) and O(1/k2)O(1/k^{2}) complexity rates in the convex and strongly convex case, respectively, kk being the iteration count. Motivated by the increased complexity in the control of distribution level electric power systems, we test the performance of our method on a second-order cone relaxation of an AC-OPF problem. Distributed control is achieved via the distributed locational marginal prices (DLMPs), which are obtained \revise{as} dual variables in our optimisation framework.Comment: Changed title and revised manuscrip

    Advances and Applications of DSmT for Information Fusion. Collected Works, Volume 5

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    This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered. First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of combination with degree of intersection, coarsening techniques, interval calculus for PCR thanks to set inversion via interval analysis (SIVIA), rough set classifiers, canonical decomposition of dichotomous belief functions, fast PCR fusion, fast inter-criteria analysis with PCR, and improved PCR5 and PCR6 rules preserving the (quasi-)neutrality of (quasi-)vacuous belief assignment in the fusion of sources of evidence with their Matlab codes. Because more applications of DSmT have emerged in the past years since the apparition of the fourth book of DSmT in 2015, the second part of this volume is about selected applications of DSmT mainly in building change detection, object recognition, quality of data association in tracking, perception in robotics, risk assessment for torrent protection and multi-criteria decision-making, multi-modal image fusion, coarsening techniques, recommender system, levee characterization and assessment, human heading perception, trust assessment, robotics, biometrics, failure detection, GPS systems, inter-criteria analysis, group decision, human activity recognition, storm prediction, data association for autonomous vehicles, identification of maritime vessels, fusion of support vector machines (SVM), Silx-Furtif RUST code library for information fusion including PCR rules, and network for ship classification. Finally, the third part presents interesting contributions related to belief functions in general published or presented along the years since 2015. These contributions are related with decision-making under uncertainty, belief approximations, probability transformations, new distances between belief functions, non-classical multi-criteria decision-making problems with belief functions, generalization of Bayes theorem, image processing, data association, entropy and cross-entropy measures, fuzzy evidence numbers, negator of belief mass, human activity recognition, information fusion for breast cancer therapy, imbalanced data classification, and hybrid techniques mixing deep learning with belief functions as well

    Structured Semidefinite Programming for Recovering Structured Preconditioners

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    We develop a general framework for finding approximately-optimal preconditioners for solving linear systems. Leveraging this framework we obtain improved runtimes for fundamental preconditioning and linear system solving problems including the following. We give an algorithm which, given positive definite KRd×d\mathbf{K} \in \mathbb{R}^{d \times d} with nnz(K)\mathrm{nnz}(\mathbf{K}) nonzero entries, computes an ϵ\epsilon-optimal diagonal preconditioner in time O~(nnz(K)poly(κ,ϵ1))\widetilde{O}(\mathrm{nnz}(\mathbf{K}) \cdot \mathrm{poly}(\kappa^\star,\epsilon^{-1})), where κ\kappa^\star is the optimal condition number of the rescaled matrix. We give an algorithm which, given MRd×d\mathbf{M} \in \mathbb{R}^{d \times d} that is either the pseudoinverse of a graph Laplacian matrix or a constant spectral approximation of one, solves linear systems in M\mathbf{M} in O~(d2)\widetilde{O}(d^2) time. Our diagonal preconditioning results improve state-of-the-art runtimes of Ω(d3.5)\Omega(d^{3.5}) attained by general-purpose semidefinite programming, and our solvers improve state-of-the-art runtimes of Ω(dω)\Omega(d^{\omega}) where ω>2.3\omega > 2.3 is the current matrix multiplication constant. We attain our results via new algorithms for a class of semidefinite programs (SDPs) we call matrix-dictionary approximation SDPs, which we leverage to solve an associated problem we call matrix-dictionary recovery.Comment: Merge of arXiv:1812.06295 and arXiv:2008.0172

    A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces

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    We consider gradient flow/gradient descent and heavy ball/accelerated gradient descent optimization for convex objective functions. In the gradient flow case, we prove the following: 1. If ff does not have a minimizer, the convergence f(xt)infff(x_t)\to \inf f can be arbitrarily slow. 2. If ff does have a minimizer, the excess energy f(xt)infff(x_t) - \inf f is integrable/summable in time. In particular, f(xt)inff=o(1/t)f(x_t) - \inf f = o(1/t) as tt\to\infty. 3. In Hilbert spaces, this is optimal: f(xt)infff(x_t) - \inf f can decay to 00 as slowly as any given function which is monotone decreasing and integrable at \infty, even for a fixed quadratic objective. 4. In finite dimension (or more generally, for all gradient flow curves of finite length), this is not optimal: We prove that there are convex monotone decreasing integrable functions g(t)g(t) which decrease to zero slower than f(xt)infff(x_t)-\inf f for the gradient flow of any convex function on Rd\mathbb R^d. For instance, we show that any gradient flow xtx_t of a convex function ff in finite dimension satisfies lim inft(tlog2(t){f(xt)inff})=0\liminf_{t\to\infty} \big(t\cdot \log^2(t)\cdot \big\{f(x_t) -\inf f\big\}\big)=0. This improves on the commonly reported O(1/t)O(1/t) rate and provides a sharp characterization of the energy decay law. We also note that it is impossible to establish a rate O(1/(tϕ(t))O(1/(t\phi(t)) for any function ϕ\phi which satisfies limtϕ(t)=\lim_{t\to\infty}\phi(t) = \infty, even asymptotically. Similar results are obtained in related settings for (1) discrete time gradient descent, (2) stochastic gradient descent with multiplicative noise and (3) the heavy ball ODE. In the case of stochastic gradient descent, the summability of E[f(xn)inff]\mathbb E[f(x_n) - \inf f] is used to prove that f(xn)infff(x_n)\to \inf f almost surely - an improvement on the convergence almost surely up to a subsequence which follows from the O(1/n)O(1/n) decay estimate

    On the latent dimension of deep autoencoders for reduced order modeling of PDEs parametrized by random fields

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    Deep Learning is having a remarkable impact on the design of Reduced Order Models (ROMs) for Partial Differential Equations (PDEs), where it is exploited as a powerful tool for tackling complex problems for which classical methods might fail. In this respect, deep autoencoders play a fundamental role, as they provide an extremely flexible tool for reducing the dimensionality of a given problem by leveraging on the nonlinear capabilities of neural networks. Indeed, starting from this paradigm, several successful approaches have already been developed, which are here referred to as Deep Learning-based ROMs (DL-ROMs). Nevertheless, when it comes to stochastic problems parameterized by random fields, the current understanding of DL-ROMs is mostly based on empirical evidence: in fact, their theoretical analysis is currently limited to the case of PDEs depending on a finite number of (deterministic) parameters. The purpose of this work is to extend the existing literature by providing some theoretical insights about the use of DL-ROMs in the presence of stochasticity generated by random fields. In particular, we derive explicit error bounds that can guide domain practitioners when choosing the latent dimension of deep autoencoders. We evaluate the practical usefulness of our theory by means of numerical experiments, showing how our analysis can significantly impact the performance of DL-ROMs
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