Set-membership Problem Research Articles

Operator Splitting for a Homogeneous Embedding of the Linear Complementarity Problem

We present a first-order quadratic cone programming algorithm that can scale to very large problem sizes and produce modest accuracy solutions quickly. Our algorithm returns primal-dual optimal solutions when available or certificates of infeasibility otherwise. It is derived by applying Douglas--Rachford splitting to a homogeneous embedding of the linear complementarity problem, which is a general set membership problem that includes quadratic cone programs (QCPs) as a special case. Each iteration of our procedure requires projecting onto a convex cone and solving a linear system with a fixed coefficient matrix. If a sequence of related problems are solved, then the procedure can easily be warm-started and make use of factorization caching of the linear system. We demonstrate on a range of public and synthetic datasets that for feasible problems our approach tends to be somewhat faster than applying operator splitting directly to the QCP, and in cases of infeasibility our approach can be significantly faster than alternative approaches based on diverging iterates. The algorithm we describe has been implemented in C and is available open-source in the solver SCS v3.0.

Set-membership filtering for a class of nonlinear complex networks with mixed time-delays and incomplete measurements

In this paper, the set-membership filtering problem is discussed for a class of discrete time-varying nonlinear complex networks with mixed time-delays and incomplete measurements. Both the process noise and the measurement noise are unknown but bounded. The nonlinear function is assumed to satisfy the sector bounded condition and the incomplete measurement is model by a diagonal matrix. The main purpose of this paper is to design a set-membership filtering method in order to provide the certain regions, which include the real states. Accordingly, the filtering errors at each time can be limited to the ellipsoid sets for the considered nonlinear delayed complex networks in the simultaneous presence of unknown but bounded noise, mixed time-delays and incomplete measurements. Moreover, a convex optimization method is presented to obtain the minimum ellipsoid domain. In addition, the filter gain matrix can be obtained by solving a series of linear matrix inequalities. Finally, a simulation example is used to illustrate the effectiveness of the proposed set-membership filtering method.

High-dimensional vector semantics

In this paper we explore the “vector semantics” problem from the perspective of “almost orthogonal” property of high-dimensional random vectors. We show that this intriguing property can be used to “memorize” random vectors by simply adding them, and we provide an efficient probabilistic solution to the set membership problem. Also, we discuss several applications to word context vector embeddings, document sentences similarity, and spam filtering.

Satisfiability-based Set Membership Filters

Introduced here is a novel application of Satisfiability (SAT) to the set membership problem with specific focus on efficiently testing whether large sets contain a given element. Such tests can be greatly enhanced via the use of filters, probabilistic algorithms that can quickly decide whether or not a given element is in a given set. This article proposes SAT filters (i.e., filters based on SAT) and their use in the set membership problem. Both the theoretical advantages of SAT filters and experimental results show that this technique yields significant performance improvements over previous techniques. Specifically, a SAT filter is a filter construction that is simple yet efficient in terms of both query time and filter size; i.e., SAT filters asymptotically achieve the information-theoretic limit while providing fast querying. As well, this is the first application that makes use of the random k-SAT phase transition results and may drive research into efficient solvers for this and similar applications.

Set-membership filtering for polytopic uncertain discrete-time systems

In this paper, a set-membership filtering problem is considered for systems with polytopic uncertainty. A recursive algorithm for calculating an ellipsoid which always contains the state is developed. In the prediction step, a predicted state ellipsoid is determined; in the update step, a state estimation ellipsoid is computed by combining the predicted state ellipsoid and the set of states compatible with the measurement equation. A smallest possible estimate set is calculated recursively by solving the semi-definite programming problems. Hence, the proposed set-membership filter relies on a two-step prediction–correction structure, which is similar to the Kalman filter. Simulation results are provided to demonstrate the effectiveness of the proposed method.

A property of quantum relative entropy with an application to privacy in quantum communication

We prove the following information-theoretic property about quantum states. Substate theorem: Let ρ and σ be quantum states in the same Hilbert space with relative entropy S (ρ ‖ σ) ≔ Tr ρ (log ρ - log σ) = c . Then for all ϵ > 0, there is a state ρ′ such that the trace distance ‖ρ′ - ρ‖ tr : Tr √(ρ′ - ρ) 2 ≤ ϵ, and ρ′/2 O ( c /ϵ 2 ) ≤ σ. It states that if the relative entropy of ρ and σ is small, then there is a state ρ′ close to ρ, i.e. with small trace distance ‖ρ′ - ρ‖ tr , that when scaled down by a factor 2 O ( c ) ‘sits inside’, or becomes a ‘substate’ of, σ. This result has several applications in quantum communication complexity and cryptography. Using the substate theorem, we derive a privacy trade-off for the set membership problem in the two-party quantum communication model. Here Alice is given a subset A ⊆ [ n ], Bob an input i ∈ [ n ], and they need to determine if i ∈ A . Privacy trade-off for set membership: In any two-party quantum communication protocol for the set membership problem, if Bob reveals only k bits of information about his input, then Alice must reveal at least n /2 O( k ) bits of information about her input. We also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem.

From Set Membership to Group Membership: A Separation of Concerns

We revisit the well-known group membership problem and show how it can be considered a special case of a simple problem, the set membership problem. In the set membership problem, processes maintain a set whose elements are drawn from an arbitrary universe: They can request the addition or removal of elements to/from that set, and they agree on the current value of the set. Group membership corresponds to the special case where the elements of the set happen to be processes. We exploit this new way of looking at group membership to give a simple and succinct specification of this problem and to outline a simple implementation approach based on the state machine paradigm. This treatment of group membership separates several issues that are often mixed in existing specifications and/or implementations of group membership. We believe that this separation of concerns greatly simplifies the understanding of this problem.

The Quantum Complexity of Set Membership

Abstract. We study the quantum complexity of the static set membership problem: given a subset S (|S| ≤ n ) of a universe of size m ( >> n ), store it as a table, T: {0,1} r --> {0,1} , of bits so that queries of the form ``Is x in S ?'' can be answered. The goal is to use a small table and yet answer queries using few bit probes. This problem was considered recently by Buhrman et al. [BMRV], who showed lower and upper bounds for this problem in the classical deterministic and randomized models. In this paper we formulate this problem in the ``quantum bit probe model''. We assume that access to the table T is provided by means of a black box (oracle) unitary transform O T that takes the basis state | y,b > to the basis state | y,b \oplus T(y) > . The query algorithm is allowed to apply O T on any superposition of basis states. We show tradeoff results between space (defined as 2 r ) and number of probes (oracle calls) in this model. Our results show that the lower bounds shown in [BMRV] for the classical model also hold (with minor differences) in the quantum bit probe model. These bounds almost match the classical upper bounds. Our lower bounds are proved using linear algebraic arguments.

The fuzzy set membership problem using the hierarchy decision method

Abstract The membership values of the elements of a fuzzy set are central to any discussion of fuzzy set theory. This paper investigates the soundness of using the Analytic Hierarchy Decision Model of T.L. Saaty for the purpose of specifying the fuzzy set membership values. Other papers have posed the question and have shown experimentally through other methods that no specific evaluation technique within the framework of the model always obtains the best calculable membership values. This work defines a different metric to measure a best calculable membership vector and argues that the input data is sufficiently restrictive that the numerical precision being sought simply cannot be obtained. Thus the Analytic Hierarchy Decision process remains a valuable first stage heuristic, but it can not be expected to report fine grain differences in membership values.