Multilinear Research Articles

The zonoid algebra, generalized mixed volumes, and random determinants

We show that every multilinear map between Euclidean spaces induces a unique, continuous, Minkowski multilinear map of the corresponding real cones of zonoids. Applied to the wedge product of the exterior algebra of a Euclidean space, this yields a multiplication of zonoids, defining the structure of a commutative, associative, and partially ordered ring, which we call the zonoid algebra. This framework gives a new perspective on classical objects in convex geometry, and it allows to introduce new functionals on zonoids, in particular generalizing the notion of mixed volume. We also analyze a similar construction based on the complex wedge product, which leads to the new notion of mixed J–volume. These ideas connect to the theory of random determinants.

Effects of Similarity Score Functions in Attention Mechanisms on the Performance of Neural Question Answering Systems

Attention mechanisms have been incorporated into many neural network-based natural language processing (NLP) models. They enhance the ability of these models to learn and reason with long input texts. A critical part of such mechanisms is the computation of attention similarity scores between two elements of the texts using a similarity score function. Given that these models have different architectures, it is difficult to comparatively evaluate the effectiveness of different similarity score functions. In this paper, we proposed a baseline model that captures the common components of recurrent neural network-based Question Answering (QA) systems found in the literature. By isolating the attention function, this baseline model allows us to study the effects of different similarity score functions on the performance of such systems. Experimental results show that a trilinear function produced the best results among the commonly used functions. Based on these insights, a new T-trilinear similarity function is proposed which achieved the higher predictive EM and F1 scores than these existing functions. A heatmap visualization of the attention score matrix explains why this T-trilinear function is effective.

Multilinear target-based decision analysis with hybrid-information targets and performance levels

For several classes of decisions, the use of target-based decision analysis (TBDA) is more appropriate than utility analysis. Recent literature on TBDA mainly focuses on different expression forms of targets and performance levels, and on different types of aggregation operators in terms of multi-attribute preference functions. However, different expression forms are usually introduced separately in literature, which cannot fully support the inherent complexity of problems along with different perception and knowledge levels of decision makers during assessing targets and performance levels. Furthermore, although there are attractive features of multilinear target-based preference functions (MTPFs), its applications are seldom considered because of the complexity of identifying their coefficients. In order to solve these two issues, an integrated approach is proposed for multilinear hybrid-information target-based decision analysis, which can deal with diverse forms of targets and performance levels, and identify the coefficients of MTPFs. First, given targets and performance levels for each attribute being expressed in multiple forms simultaneously, a generalized procedure is proposed to transform different forms into probability distributions, and to measure probability of target achievement for each attribute. Second, a novel procedure to identify the coefficients of MTPFs is proposed. This procedure is based on the multilinear model and 2-additive fuzzy measures, which is based on the equivalence between multilinear model based on fuzzy measures and MTPFs. The approach is applied to a case study involving customer competitive evaluation of smart thermometer patches to demonstrate its feasibility and advantages.

An IoT Privacy-Oriented selective disclosure credential system

Abstract Personal credentials, such as passports and drivers’ licenses, can be implemented electronically using multi-show protocols. In this paper, we introduce an IoT Privacy-Oriented selective disclosure credential system, i.e. based on bilinear pairings and multilinear maps. The proposed system consists of three protocols, which allow users to be in control of their personal credentials. The Credentials Authority (CA) verifies and attests to the users credentials. Once the CA signs these credentials, the users cannot modify any of them. Moreover, the users can mask these credentials in every showing process to protect their identity from being revealed through a collusion between the CA and the verifiers. The proposed system maintains unlinkability between the issuing and showing protocols. Furthermore, it achieves unlinkability in the showing protocol such that the verifier cannot distinguish a user in two different sessions of the showing protocol. The proposed system is novel and practical in terms of introducing a new multi-show credential system that supports selective disclosure (Some credentials can be disclosed and others kept secret during the showing protocol.) The proposed system is the first that utilizes multilinear maps in the identification protocol. Making use of bilinear pairings and multilinear maps are suitable for IoT devices that have limited capabilities in terms of power consumption, key storage, and computing power. The security analysis of the proposed system is discussed using Burrows–Abadi–Needham (BAN) logic.

Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces revisited

Continuous orthosymmetric multilinear maps and homogeneous polynomials on Riesz spaces revisited

Global solution to the Cauchy problem of fractional drift diffusion system with power-law nonlinearity

<abstract><p>In this paper, we consider the global existence, regularizing decay rate and asymptotic behavior of mild solutions to Cauchy problem of fractional drift diffusion system with power-law nonlinearity. Using the properties of fractional heat semigroup and the classical estimates of fractional heat kernel, we first prove the global-in-time existence and uniqueness of the mild solutions in the frame of mixed time-space Besov space with multi-linear continuous mappings. Then, we show the asymptotic behavior and regularizing-decay rate estimates of the solution to equations with power-law nonlinearity by the method of multi-linear operator and the classical Hardy-Littlewood-Sobolev inequality.</p></abstract>

Generalized MBI Algorithm for Designing Sequence Set and Mismatched Filter Bank With Ambiguity Function Constraints

A sequence set with ambiguity function (AF) specifications is frequently required in multi-transmit active sensing systems which exploit waveform diversity. This paper formulates a new model to jointly design sequence set and mismatched filter bank with AF requirements, which is a generalization of the auto-AF and cross-AF adopted in the matched filter scheme to attain lower AF sidelobe levels with an increased degree-of-freedom. The aforementioned designs result in nonconvex and nonlinear high-order polynomial (HOP) optimization problems with HOP constraints. Although the maximum block improvement (MBI) method has exhibited the powerful HOP optimization ability to design a short sequence with slow-time AF, it cannot tackle HOP constraints and involves high-complexity tensor operations. To address these issues, we develop a generalized MBI method for the HOP constrained optimization formulations. In addition, the proposed algorithm significantly reduces the computational complexity via designing an equivalent polynomial function for the original multi-linear tensor function. Numerical results demonstrate the excellent performance of our design solutions.

АЛГОРИТМ КОНСЕРВАТИВНОГО СГЛАЖИВАНИЯ РАЗРЫВНЫХ ВОЛН НАПРЯЖЕНИЙ В МКЭ

Three-dimensional geometrically and physically nonlinear problems of nonstationary deformation of structures are considered. The defining system of equations is formulated in Lagrange variables. The equation of motion is derived from the balance of virtual work capacities. The elastic-plastic deformation of structural materials is described by the relations of the flow theory with isotropic hardening. The solution of the problem is based on the moment scheme of the finite element method. The discretization of the problem by spatial variables is carried out by eight nodal finite elements with multilinear functions of the approximation form of the displacement velocity. Time integration is performed according to an explicit finite-difference scheme of the “cross” type, which does not have the monotonicity property. Due to the dispersion in the vicinity of the gap, it generates non-physical oscillations, which significantly limits the scope of its applicability. In this paper, to suppress high-frequency oscillations, it is proposed to use an algorithm for conservative smoothing of a numerical solution with a space-time monotony analyzer. Based on it, software modules have been developed for the “Dynamics-3” computing complex. Verification of the developed technique and its software implementation was carried out by solving a one-dimensional problem in a three-dimensional formulation of the compression wave passing through a fixed elastic layer. For comparison, the results of numerical solution of the problem according to the “cross” scheme without smoothing, with linear viscosity, with conservative smoothing using spatial and spatiotemporal monotonicity analyzers are obtained. The problem of penetration of an elastic cylinder into a round steel plate is solved in a three-dimensional formulation. It is shown that the developed technique not only suppresses high-frequency oscillations, but also prevents the occurrence of zero-energy modes. So, in the second problem, without using the procedure of conservative smoothing of the numerical solution, the final elements of the plate in the collision zone are significantly distorted, which leads to an early interruption of the calculation.

Cryptanalysis of FRS obfuscation based on the CLT13 multilinear map

The authors present a classical polynomial-time attack against the branching program obfuscator of Fernando–Rasmussen–Sahai (for short FRS, Asiacrypt’17) (with one zerotest parameter), which is robust against all known classical cryptanalyses on obfuscators when instantiated with the CLT13 multilinear map. The first step is to recover a plaintext modulus of the CLT13 multilinear map. To achieve the goal, the Coron and Notarnicola (Asiacrypt’19) algorithm is applied. However, because of parameter issues, the algorithm cannot be used directly. In order to detour the issue, the authors convert an FRS obfuscator into a new programme containing a small message space. Through the conversion, the authors obtain two zerotest parameters and encodings of zero except for two non-zero slots. Then, they are used to mitigate parameter constraints of the message space recovering algorithm. Then, a cryptanalysis of the FRS obfuscation based on the recovered message space is proposed. The authors show that there exist two functionally equivalent programmes such that their obfuscated programmes are computationally distinguishable. Thus, the FRS scheme does not satisfy the desired security without any additional constraints.

A Novel Hierarchical Key Assignment Scheme for Data Access Control in IoT

Hierarchical key assignment scheme is an efficient cryptographic method for hierarchical access control, in which the encryption keys of lower classes can be derived by the higher classes. Such a property is an effective way to ensure the access control security of Internet of Things data markets. However, many researchers on this field cannot avoid potential single point of failure in key distribution, and some key assignment schemes are insecure against collusive attack or sibling attack or collaborative attack. In this paper, we propose a hierarchical key assignment scheme based on multilinear map to solve the multigroup access control in Internet of Things data markets. Compared with previous hierarchical key assignment schemes, our scheme can avoid potential single point of failure in key distribution. Also the central authority of our scheme (corresponding to the data owner in IoT data markets) does not need to assign the corresponding encryption keys to each user directly, and users in each class can obtain the encryption key via only a one-round key agreement protocol. We then show that our scheme satisfies the security of key indistinguishability under decisional multilinear Diffie-Hellman assumption. Finally, comparisons show the efficiency of our scheme and indicates that our proposed scheme can not only resist the potential attacks, but also guarantee the forward and backward security.

An Efficient Digital Realization of Retinal Light Adaptation in Cone Photoreceptors

In recent years, hardware modeling for various parts of the body’s sensitive organs, including the brain and nervous system, heart and eyes, has been considered for the treatment of diseases and rehabilitation, as well as for moving towards the construction of artificial prostheses. The retina is a thin layer that is the innermost layer of the human eye. In this paper, low-cost hardware implementation for retinal cone cells is performed. Existing mathematical models for implementing the behavior of these cells include a series of nonlinear functions that, if implemented directly, would require a large amount of hardware and, in addition, would not have the desired speed. The proposed model uses multi-linear functions to approximate the nonlinear terms and eliminate the multiplication expressions. The simulation results show that the proposed model tracks the behavior of the original model with high precision. There is also a good match between the main model and the proposed model in terms of dynamic behaviors. The results of hardware implementation using the virtex5 XC5VLX20T (2FF323) reconfigurable board (FPGA) show that the proposed model is fully valid and has a lower hardware volume as well as a 4 times higher frequency, and 22% less power consumption than the original model.

Nonlinear–nonquadratic optimal and inverse optimal control for discrete‐time stochastic dynamical systems

AbstractIn this article, we investigate the role of Lyapunov functions in evaluating nonlinear–nonquadratic cost functionals for Itô‐type nonlinear stochastic difference equations. Specifically, it is shown that the cost functional can be evaluated in closed‐form as long as the cost functional is related in a specific way to an underlying Lyapunov function that guarantees asymptotic stability in probability. This result is then used to analyze discrete‐time linear as well as nonlinear stochastic dynamical systems with polynomial and multilinear cost functionals. Furthermore, a stochastic optimal control framework is developed by exploiting connections between stochastic Lyapunov theory and stochastic Bellman theory. In particular, we show that asymptotic and geometric stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady state form of the stochastic Bellman equation, and hence, guaranteeing both stochastic stability and optimality.

A closer look at the multilinear cryptography using nilpotent groups

In Kahrobaei et al. [Multilinear cryptography using nilpotent groups, Proceedings of Elementary Theory of Groups and Group Rings, and Related Topics conference. Conference held at Fairfield University and at the Graduate Center, CUNY, New York, NY, USA, November 1–2, 2018, De Gruyter, 2020, pp. 127–133] we generalized the definition of a multilinear map to arbitrary groups and introduced two multiparty key-exchange protocols using nilpotent groups. In this paper we have a closer look at the protocols and will address some incorrect cryptanalysis which has been proposed in Roman'kov [Discrete logarithm for nilpotent groups and cryptanalysis of polylinear cryptographic system, Prikl. Diskretn. Mat. Suppl. (12), (2019), pp. 154–160].

A Radon–Nikodým theorem for local completely positive invariant multilinear maps

In this article, we introduce local completely positive k-linear maps between locally -algebras and obtain Stinespring type representation by adopting the notion of ‘invariance’ defined by J. Heo for k-linear maps between -algebras. Also, we supply the minimality condition to make certain that minimal representation is unique up to unitary equivalence. As a consequence, we prove Radon–Nikodým theorem for unbounded operator-valued local completely positive invariant k-linear maps. The obtained Radon–Nikodým derivative is a positive contraction on some Hilbert space with several reducing subspaces.

A New Variant of the Winternitz One Time Signature Based on Graded Encoding Schemes

Digital signature schemes are used to guarantee for non-repudiation and authenticity of any kind of data like documents, messages or software. The Winternitz one-time signature (WOTS) scheme, which can be described using a certain number of so-called “function chains”, plays an important role in the design of both stateless and stateful many-time signature schemes. The main idea of WOTS scheme is the use of a limited number of function chains, all of which begin at some random values. This work introduces WOTS-GES, a new WOTS type signature scheme in which the need for computing all of the intermediate values of the chains is eliminated. More precisely, to compute each algorithm of the proposed scheme, we only need to calculate one intermediate value. This significantly reduces the number of required operations needed to calculate the algorithms of WOTS-GES. To achieve this results, we have used the concept of “leveled” multilinear maps which is alsoreferred to as graded encoding schemes. We expect these results to increase the efficiency of Winternitz based digital signature schemes.

Novel correlations for prediction of SARA contents of vacuum residues

Vacuum residues (VRs) are among the heaviest petroleum fractions, and their proper characterization is of vital importance for a better understanding of their structure. Saturate, aromatic, resin, and asphaltene (SARA) analysis is widely used due to the unique information that can provide about chemical nature of petroleum fractions. However, the analysis is time-consuming and expensive, which may restrict its application. In this study, a new set of correlations has been developed for facile prediction of VRs’ SARA values. The correlations have been obtained through multi-linear (MLR) regression based on density, viscosity, and Conradson carbon residue (CCR) variables. The saturate and asphaltene contents are predicted by linear regression while the aromatic and resin contents are predicted by linear regression of asphaltene to aromatic (Asph/Aro) and asphaltene to resin (Asph/Res) ratios, respectively, and hence, is considered non-linear regression. The predicted SARA values are in good agreement with the measured values. The model is validated by a testing set of data from various worldwide VRs. The average absolute relative errors (AARE) of presented correlations for saturate, aromatic, resin, and asphaltene predictions are 0.1134, 0.0695, 0.2019, and 0.1736, respectively. Root mean squared errors (RMSE) of saturate, aromatic, resin, and asphaltene prediction are 3.9883, 3.9502, 2.2722, and 1.5288, respectively.

Weak and Strong Type Estimates for the Multilinear Littlewood–Paley Operators

Let \(S_{\alpha }\) be the multilinear square function defined on the cone with aperture \(\alpha \ge 1\). In this paper, we investigate several kinds of weighted norm inequalities for \(S_{\alpha }\). We first obtain a sharp weighted estimate in terms of aperture \(\alpha \) and \(\vec {w} \in A_{\vec {p}}\). By means of some pointwise estimates, we also establish two-weight inequalities including bump and entropy bump estimates, and Fefferman–Stein inequalities with arbitrary weights. Beyond that, we consider the mixed weak type estimates corresponding Sawyer’s conjecture, for which a Coifman–Fefferman inequality with the precise \(A_{\infty }\) norm is proved. Finally, we present the local decay estimates using the extrapolation techniques and dyadic analysis respectively. All the conclusions aforementioned hold for the Littlewood–Paley \(g^*_{\lambda }\) function. Some results are new even in the linear case.

Unsupervised sleep staging system based on domain adaptation

Currently, most deep-learning-based sleep staging system relies heavily on a large number of labeled physiological signals. However, sleep-related data, such as polysommography (PSG), are often manually labeled by one or more than one professional experts with much effort. Meanwhile, due to physiological differences that existed among different subjects, how to boost the performance of trained models on an unseen dataset is still an open issue. One potential solution to this issue is to borrow knowledge from a labeled dataset to train an unlabeled or few labeled dataset by way of unsupervised or semi-unsupervised domain adaptation. To overcome the problem of insufficient labeled data for training robust sleep staging systems, this study aims to investigate the training of an unlabeled target sleep dataset from a labeled source sleep dataset in a deep learning framework, which integrates a conditional and collaborative adversarial domain adaptation module. To facilitate the network to learn domain-invariant features, a domain classifier is deployed for each feature extraction block at different scale. The input to the domain classifier at different level is the multilinear mapping of the sleep stage prediction vector and the corresponding feature vector at this level. It is assumed that the feedback of the class information provided by the network into the domain classifier can be beneficial to help the network to reduce the feature distribution distance between different domains. Experiments on public Sleep-EDF dataset demonstrate the effectiveness of the proposed approach. Compared to other domain adaptation approaches, the proposed approaches can provide better sleep staging performance in different model transferring tasks.

Some estimates for the commutators of multilinear maximal function on Morrey-type space

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

Carbon Isotope Fractionation in Noelaerhabdaceae Algae in Culture and a Critical Evaluation of the Alkenone Paleobarometer

AbstractThe carbon isotope fractionation in algal organic matter (εp), including the long‐chain alkenones produced by the coccolithophorid family Noelaerhabdaceae, is used to reconstruct past atmospheric CO2 levels. The conventional proxy linearly relates εp to changes in cellular carbon demand relative to diffusive CO2 supply, with larger εp values occurring at lower carbon demand relative to supply (i.e., abundant CO2). However, the response of Gephyrocapsa oceanica, one of the dominant alkenone producers of the last few million years, has not been studied closely. Here, we subject G. oceanica to various CO2 levels by increasing pCO2 in the culture headspace, as opposed to increasing dissolved inorganic carbon (DIC) and alkalinity concentrations at constant pH. We note no substantial change in physiology, but observe an increase in εp as carbon demand relative to supply decreases, consistent with DIC manipulations. We compile existing Noelaerhabdaceae εp data and show that the diffusive model poorly describes the data. A meta‐analysis of individual treatments (unique combinations of lab, strain, and light conditions) shows that the slope of the εp response depends on the light conditions and range of carbon demand relative to CO2 supply in the treatment, which is incompatible with the diffusive model. We model εp as a multilinear function of key physiological and environmental variables and find that both photoperiod duration and light intensity are critical parameters, in addition to CO2 and cell size. While alkenone carbon isotope ratios indeed record CO2 information, irradiance and other factors are also necessary to properly describe alkenone εp.