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Fifth generation (5G) cellular standards are set to utilize millimeter wave (mmWave) frequencies, which enable data speeds greater than 10 Gbps and sub-centimeter localization accuracy. These capabilities rely on accurate estimates of the channel parameters, which we define as the angle of arrival, angle of departure, and path distance for each path between the transmitter and receiver. Estimating the channel parameters in a computationally efficient manner poses a challenge because it requires estimation of parameters from a high-dimensional measurement - particularly for multi-carrier systems since each subcarrier must be estimated separately. Additionally, channel parameter estimation must be able to handle hybrid beamforming, which uses a combination of digital and analog beamforming to reduce the number of required analog to digital converters. This paper introduces a channel parameter estimation technique based on the multilinear singular value decomposition (MSVD), a Tucker form tensor analog of the singular value decomposition, for massive multiple input multiple output (MIMO) multi-carrier systems with hybrid beamforming. The MSVD tensor estimation approach is more computationally efficient than methods such as the canonical polyadic decomposition (CPD) and the Tucker form of the MSVD enables paths to be extracted based on signal energy. The algorithms performance is compared to the CPD method and shown to closely match the Cramer-Rao bound (CRB) of channel parameter estimates through simulations. Additionally, limitations of channel parameter estimation and communication waveform effects are studied.

The following functional equation is solved: $$f\left( {{x_1} + z} \right) \cdots f\left( {{x_2} + z} \right)f\left( {{x_1} + \cdots + {x_{s - 1}} - z} \right) = {\phi _1}\left( x \right){\psi _1}\left( z \right) + \cdots + {\phi _m}\left( x \right){\psi _m}\left( z \right),$$ where x =(x1,…,xs−1), for the unknowns $$f,{\psi _j}:\mathbb{C} \to \mathbb{C}$$ and $${\phi _j}:{\mathbb{C}^{s - 1}} \to \mathbb{C}$$ for s ≥ 3 and m ≤ 4s − 5.

This paper is devoted to the computation of low multilinear rank approximations of tensors. Combining the stretegy of power scheme, random projection, and singular value decomposition, we derive a ...

This paper considers the existence and uniqueness of positive solutions to a class of nonlinear systems of equations, which has wide applications in fluid mechanics and electrical circuit theory. In this paper, we define a new class of functions, called M-type function, and then, we show some good properties of M-type function as well as the relationship between this kind of functions and other kinds of functions. In particular, by fully exploiting the properties of M-type function, we present that under a reasonable condition, the nonlinear system of equations with an M-type function, which can be seen as a generalization of the multilinear system with a nonsingular M-tensor, and a positive right-hand side vector has a unique positive solution. Meanwhile, for this class of problems, we give a preliminary and simple algorithm framework. What is more, under an additional assumption, we illustrate the feasibility of using Newton step when improving the original iteration algorithm to obtain better convergence. Numerical experiments show the stability and efficiency of the proposed algorithms.

A systematic read on evaluating the machining characteristics of wire cut electrical discharge machining (WEDM) using analysis of variance (ANOVA) and multilinear regression model based multi objective optimization is provided during this analysis article. The current work explores, the surface roughness of copper metal matrix composite (MMCs) is minimized by optimizing the process parameters like spark on time, spark off time, peak current and wire feed. Taguchi’s L9 orthogonal array has been used to conduct the experiments for two samples having a different composition to measure the surface roughness. The order of significance of parameters on surface roughness (Ra) for sample 1 and 2 MMCs were found using ANOVA and a multilinear regression model (MLRM) was developed to predict the Ra value. The new contribution in the present work is that, the coefficients of MLRM were optimized using artificial immune system algorithm with the objective of minimizing the mean absolute percentage error. Finally, the optimum process parameters were obtained to minimize the surface roughness and reported that the reduced value of Ra for 2.5% and 5% WC Copper MMCs were 0.9 μm and 1.1 μm, respectively later which were established by confirmation experiments.

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

This paper presents the modeling of large elasto-plastic deformations in bi-material engineering components by employing the conventional finite element method (FEM). The updated Lagrangian Approach has been used to take into account the large deformations occurring in the domain. The Von-Mises yield criterion with isotropic hardening has been used to model the elasto-plastic behaviour of the material. Bilinear and multilinear plasticity models have been used to model the plastic behaviour of the material. Finally, several numerical problems are solved using FEM to demonstrate the applicability of this method in the modeling and simulation of large deformation phenomenon in bi-material components.

In the recent period, significant progress has been achieved towards reconstructing the 3D face model from face image. With the support of the render engines and sufficient data, the reconstruction results are fine in detail. Nevertheless, the research on the 3D face reconstruction with texture from a single unrestricted face image is imperfect. The rebuild process lacks essential structure and texture information in the profile and the craniofacial region. To address this problem, we present a method of creating a 3D full face-and-head model with photorealistic texture from a single “in-the-wild” face image in this paper. To this end, we introduce a pipeline to integrate the highly-detailed face model into the basic model. Specifically, the basic model was built by multilinear optimization, and the highly-detailed face model which represents the facial features generated by constrained illumination distribution. Additionally, to infer the invisible region texture information corresponding to the input face image, we design an effective architecture with the generative adversarial network (GAN) for panoramic UV texture generation. The final results after UV texture mapping were visualized in the experiment, which demonstrates that the model faithfully recovers the photorealistic details in arbitrary perspective. Furthermore, compared to the state-of-the-art facial modeling techniques and existing commercial solutions, our method takes less input and performs better in surface detail.

Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R with the extended centroid C and f(x1,…,xn) be a non-central multilinear polynomial over C. Suppose that F a...

The Internet of Things (IoT) has become a research hotspot in recent years. With the increase of smart devices which are connected in IoT, the privacy of IoT has become an important problem. Multi-recipient public key encryption (MRPKE) system plays an important role in protecting the privacy of those smart devices in IoT. At present, the existing ID-MRPKE schemes can only achieve the security in the random oracle model. To enhance the security level, a new ID-MRPKE is constructed by using the programmable hash function from multilinear maps. The security of the novel scheme can be proven in the standard model, instead of the random oracle. Furthermore, based on the k-level Multilinear-maps Decisional Deffie-Hellman (MDDH) assumption, we prove that the proposed scheme has the indistinguishability under the selective multi-identity attack and chosen plaintext attack (IND-sMID-CPA).

Stream gauge-based information is the foundation for many hydrological applications in a river basin including the aquatic-habitat conservation. A simple two-parameter model for routing streamflow ...

We present a method of memory footprint reduction for FFT-based, electromagnetic (EM) volume integral equation (VIE) formulations. The arising Green's function tensors have low multilinear rank, which allows Tucker decomposition to be employed for their compression, thereby greatly reducing the required memory storage for numerical simulations. Consequently, the compressed components are able to fit inside a graphical processing unit (GPU) on which highly parallelized computations can vastly accelerate the iterative solution of the arising linear system. In addition, the element-wise products throughout the iterative solver's process require additional flops, thus, we provide a variety of novel and efficient methods that maintain the linear complexity of the classic element-wise product with an additional multiplicative small constant. We demonstrate the utility of our approach via its application to VIE simulations for the Magnetic Resonance Imaging (MRI) of a human head. For these simulations we report an order of magnitude acceleration over standard techniques., Comment: 11 pages, 10 figures, 5 tables, 2 algorithms, journal

In this paper, we obtain the boundedness of the multilinear fractional integral operators and their commutators on central Morrey spaces with variable exponent.

In recent years, radar has been employed as a fall detector because of its effective sensing capabilities and penetration through walls. In this paper, we introduce a multilinear subspace human activity recognition scheme that exploits the three radar signal variables: slow-time, fast-time, and Doppler frequency. The proposed approach attempts to find the optimum subspaces that minimize the reconstruction error for different modes of the radar data cube. A comprehensive analysis of the optimization considerations is performed, such as initialization, number of projections, and convergence of the algorithms. Finally, a boosting scheme is proposed combining the unsupervised multilinear principal component analysis (PCA) with the supervised methods of linear discriminant analysis and shallow neural networks. Experimental results based on real radar data obtained from multiple subjects, different locations, and aspect angles (0 $^{\circ }$ , 30 $^{\circ }$ , 45 $^{\circ }$ , 60 $^{\circ }$ , and 90 $^{\circ }$ ) demonstrate that the proposed algorithm yields the highest overall classification accuracy among spectrogram-based methods including predefined physical features, one- and two-dimensional PCA and convolutional neural networks.

We obtain the formula for the inverse of polynomial maps of of the form where M(x) is a homogeneous of degree m nilpotent matrix, all of whose powers are exact; and we obtain recursion relations for the scalars cmk . Such maps have recently been considered by Connell and Zweibel, but our derivations is based on our earlier result that F −1(a) where x(t z a) is the unique solution, with initial condition x(0z a)=z, of the Wazewski differential equation dx/dt=Ft (x)−1 a=a−M(x)a, with vector parameter a. Basic to our method is the multilinear matrix function B(x y,…z) uniquely determined by M(x). We give a new proof that all powers of M are exact provided only that both M and M 2 are exact.

Let R be a non commutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C. Let d and $$\delta $$ be two derivations of R and S be the set of evaluations of a multilinear polynomial $$f(x_1,\ldots ,x_n)$$ over C which is not central valued. Let $$p,q\in R$$ . We prove the followings.

In this paper, we consider the {\it tensor absolute value equations} (TAVEs), which is a newly introduced problem in the context of multilinear systems. Although the system of TAVEs is an interesting generalization of matrix {\it absolute value equations} (AVEs), the well-developed theory and algorithms for AVEs are not directly applicable to TAVEs due to the nonlinearity (or multilinearity) of the problem under consideration. Therefore, we first study the solutions existence of some classes of TAVEs with the help of degree theory, in addition to showing, by fixed point theory, that the system of TAVEs has at least one solution under some checkable conditions. Then, we give a bound of solutions of TAVEs for some special cases. To find a solution to TAVEs, we employ the generalized Newton method and report some preliminary results., Comment: 26 pages and 4 tables

Low-rank tensor reconstruction has attracted a great deal of research interest in signal processing, image processing and machine learning. To deal with outliers that are now ubiquitous in many modern applications, a representative approach is to decompose and reconstruct the data into a low-rank component and a sparse outlier component. As surrogate functions of the matrix rank and the sparsity that are non-convex and discontinuous, the nuclear norm and the $\ell _1$ norm are often exploited for recovering higher order tensors and removing outliers, respectively. However, the nuclear norm and the $\ell _1$ norm are loose approximations of the matrix rank and the sparsity, respectively. Furthermore, the tensor nuclear norm is not guaranteed to be the tightest convex envelope of a multilinear rank. In this paper, we develop a new method for robust tensor reconstruction. By capitalizing on the empirical Bayes method, whose performance has been shown to act as state-of-the-art to the performance of standard sparse and low-rank matrix recovery, we formulate the objective function of the empirical Bayes method for robust tensor reconstruction, and then adjust the objective function to facilitate algorithm derivation, while keeping desirable attributes including concavity and symmetry of the objective function of the empirical Bayes method. We demonstrate the superior performance of the proposed algorithm in comparison with state-of-the-art alternatives by conducting experiments on both synthetic data and real data.

In recent studies, tensor ring decomposition (TRD) has become a promising model for tensor completion. However, TRD suffers from the rank selection problem due to the undetermined multilinear rank. For tensor decomposition with missing entries, the sub-optimal rank selection of traditional methods leads to the overfitting/underfitting problem. In this paper, we first explore the latent space of the TRD and theoretically prove the relationship between the TR-rank and the rank of the tensor unfoldings. Then, we propose two tensor completion models by imposing the different low-rank regularizations on the TR-factors, by which the TR-rank of the underlying tensor is minimized and the low-rank structures of the underlying tensor are exploited. By employing the alternating direction method of multipliers scheme, our algorithms obtain the TR factors and the underlying tensor simultaneously. In experiments of tensor completion tasks, our algorithms show robustness to rank selection and high computation efficiency, in comparison to traditional low-rank approximation algorithms.

Constrained verifiable random functions (VRFs) were introduced by Fuchsbauer. In a constrained VRF, one can drive a constrained key skS from the master secret key sk, where S is a subset of the domain. Using the constrained key skS, one can compute function values at points which are not in the set S. The security of constrained VRFs requires that the VRFs’ output should be indistinguishable from a random value in the range. They showed how to construct constrained VRFs for the bit-fixing class and the circuit constrained class based on multilinear maps. Their construction can only achieve selective security where an attacker must declare which point he will attack at the beginning of experiment. In this work, we propose a novel construction for constrained verifiable random function from bilinear maps and prove that it satisfies a new security definition which is stronger than the selective security. We call it semiadaptive security where the attacker is allowed to make the evaluation queries before it outputs the challenge point. It can immediately get that if a scheme satisfied semiadaptive security, and it must satisfy selective security.

In asymmetric group key agreement (ASGKA) protocols, a group of users establish a common encryption key which is publicly accessible and compute pairwise different decryption keys. It is left as an open problem to design an ASGKA protocol with traitor traceability in Eurocrypt 2009. A one-round dynamic authenticated ASGKA protocol with public traitor traceability is proposed in this study. It provides a black-box tracing algorithm. Ind-CPA security with key compromise impersonation resilience (KCIR) and forward secrecy of ASGKA protocols is formally defined. The proposed protocol is proved to be Ind-CPA secure with KCIR and forward secrecy under D k-HDHE assumption. It is also proved that the proposed protocol resists collusion attack. In Setup algorithm and Join algorithm, one communication round is required. In Leave algorithm, no message is required to be transmitted. The proposed protocol adopts O ( log ⁡ N ) -way asymmetric multilinear map to make the size of public key and the size of ciphertext both achieve O ( log ⁡ N ) , where N is the number of potential group members. This is the first ASGKA protocol with public traitor traceability which is more efficient than trivial construction of ASGKA protocols.

We investigate the multilinear variants of the quantities which measure the noncompactness of multilinear operators taking values in Banach spaces with the uniform approximation property. We show applications to multilinear variant of the Hilbert and Riesz transform on rearrangement invariant spaces. We derive lower estimates of the essential norm of these transforms. Along the way we obtain also as a by-product the lower estimates of the measure noncompactness of these transforms. As a consequence we conclude that the Hilbert and Riesz transforms are not compact multilinear transforms.

Multi-index fusion has demonstrated impressive performances in retrieval task by integrating different visual representations in a unified framework. However, previous works mainly consider propagating similarities via neighbor structure, ignoring the high order information among different visual representations. In this paper, we propose a new multi-index fusion scheme for image retrieval. By formulating this procedure as a multilinear based optimization problem, the complementary information hidden in different indexes can be explored more thoroughly. Specially, we first build our multiple indexes from various visual representations. Then a so-called index-specific functional matrix, which aims to propagate similarities, is introduced for updating the original index. The functional matrices are then optimized in a unified tensor space to achieve a refinement, such that the relevant images can be pushed more closer. The optimization problem can be efficiently solved by the augmented Lagrangian method with theoretical convergence guarantee. Unlike the traditional multi-index fusion scheme, our approach embeds the multi-index subspace structure into the new indexes with sparse constraint, thus it has little additional memory consumption in online query stage. Experimental evaluation on three benchmark datasets reveals that the proposed approach achieves the state-of-the-art performance, i.e., N-score 3.94 on UKBench, mAP 94.1\% on Holiday and 62.39\% on Market-1501., 12 pages

Although correlation filter (CF)-based trackers have shown promising results in addressing problematic challenges of visual tracking, common holistic-wise CF-based trackers mostly drift away from the target object when undergoing partial occlusion. On the other hand, part-based models provide a prosperous basis for handling occlusion problem, due to preserving local structure of the target object. Employing local-global appearance models of the object, we propose a robust tracking algorithm based on the weighted cumulative fusion of CF-based part regressors. Indeed, we dynamically learn importance weights of each part via a multilinear ridge regression optimization model aiming at enhancing discrimination power of our tracker. To alleviate tracking drift caused by the object size changes, we further present an accurate method that jointly estimates object scale and aspect ratio by analyzing relative deformation cost of importance pair-wise parts. Also, to reduce the computational complexity, we introduce a feature sharing strategy for all constituent parts. Extensive experiments on OTB-2013, OTB-50, OTB-100, and VOT2016 datasets demonstrate that our tracker not only impressively enhances the performance of target-wise KCF tracker as its baseline but also performs favorably against state-of-the-art trackers in terms of qualitative and quantitative measures while running about 30 fps using Matlab on 3.2 GHz core-i5.

This article investigates the level of influence that strong motion duration may have on the inelastic demand of reinforced concrete structures. Sets of short-duration spectrally equivalent records are generated using as target the response spectrum of an actual long-duration record. The sets of short-duration records are applied to carefully calibrated numerical models of the structures along with the target long-duration records. The input motions are applied in an incremental dynamic analysis fashion, so that the duration effect at different levels of inelastic demand can be investigated. It was found that long-duration records tend to impose larger inelastic demands. However, such influence is difficult to quantify, as it was found to depend on the dynamic properties of the structure, the strength, and stiffness degrading characteristics, the approach used to generate the numerical model and the seismic scenario (target spectrum). While for some scenarios, the dominance of the long record was evident; in other scenarios, the set of short records clearly imposed larger demands than the long record. The detrimental effect of large strong motion durations was mainly observed in relatively rigid structures and poorly detailed flexible structures. The modeling approach was found to play an important role in the perceived effect of duration, with the lumped plasticity multilinear hysteretic models suggesting that the demands from the long records can be up to twice the inferred from distributed plasticity fiber models.

We focus on two classes of problems in graph mininghere: (1) finding trees and (2) anomaly detection using networkscan statistics in complex networks. These are fundamentalproblems in a broad class of applications. Most of the parallelalgorithms for such problems are either based on heuristics,which do not scale very well, or use techniques like colorcoding, which have a high memory overhead. In this paper, wedevelop a novel approach for parallelizing both these classesof problems, using an algebraic representation of subgraphsas monomials—this methodology involves detecting multilinearterms in multivariate polynomials. Our algorithms show goodscaling over a large regime, and they run on networks with closeto half a billion edges. The resulting parallel algorithm for treesis able to scale to subgraphs of size18, which has not beenshown before, and it significantly outperforms the best priorcolor coding based method (FASCIA) by more than two ordersof magnitude. Our algorithm for network scan statistics is thefirst such parallelization, and it is able to handle a broad class ofscan statistics functions (both parametric and non-parametric),with the same approach.

A phenomenology-based virtual metrology (VM) for monitoring SiO2 etching depth was proposed by Park (2015). It achieved high prediction accuracy by introducing newly developed plasma information (PI) variables as designated inputs, called PI-VM. The PI variables represent the state of the plasma, the sheath, and the target during the process. We investigate how a PI variable can help to improve prediction accuracy of VM and how it plays a special role in the statistical selection. We choose only PIEEDF among the three PI variables to focus on the investigation. The PIEEDF is determined from the ratio of line-intensities of optical emission spectroscopy. We apply Pearson's correlation filter (PCF), principal component analysis (PCA), and stepwise variable selection (SVS) as statistical selection methods on the variables set including PIEEDF or not. Multilinear regression is used to model the VM. This study reveals that PIEEDF variable is a good variable in terms of independence from other input variables and explanatory power for an output variable. Especially, VM using SVS method applied to variable sets including PIEEDF achieves the highest accuracy, comparable to Park's PI-VM. This study shows that PIEEDF variable is particularly useful for monitoring of the fine variations in semiconductor manufacturing process and it also extends the utilization of OES sensor data.

Stem volume is a key attribute of Eucalyptus forest plantations upon which decision-making is based at diverse levels of planning. Quantifying volume through remote sensing can support a proper management of forests. Because of limitations on spaceborne optical and synthetic aperture radar sensors, this study integrated both types of datasets assembled using support vector regression (SVR) to retrieve the stand volume of Eucalyptus plantations. We assessed different combinations of sensors and a minimum number of plots to develop an SVR model. Finally, the best SVR performance was compared with other analytical methods already tested and in the literature: multilinear regression, artificial neural networks (ANN), and random forest (RF). Here, we introduce a test for comparative analysis of the performance of different methods. We found that SVR accurately predicted stem volume of Brazilian fast-growing Eucalyptus forest plantations. Gaussian radial basis was the most suitable kernel function. Integrating the optical and L-band backscatter data increased the predictive accuracy compared to a single sensor model. Combining NIR-band data from ALOS AVNIR-2 and backscatter of L-band horizontal emitted and vertical received (HV) electric fields from ALOS PALSAR produced the most accurate SVR model (with an R2 of 0.926 and root mean square error of 11.007 m3/ha). The number of field plots sufficient for model development with non-redundant explanatory variables was 77. Under this condition, SVR performed similarly to ANN and outperformed the multiple linear regression and random forest methods.

Given a symmetric non-degenerate bilinear form b on a vector space g , G. Pinczon and R. Ushirobira have defined a bracket { , } on the space of multilinear skewsymmetric forms on g . With this bracket, the quadratic Lie algebra structure equation on ( g , b ) becomes simply { Ω , Ω } = 0 . Following the same program, we characterize the quadratic hom-right symmetric structures on ( g , b ) by the same equation { Ω , Ω } = 0 , on the space of ‘bi-symmetric’ forms. This characterization extends to quadratic hom-right symmetric algebras up to homotopy and allows us to describe the corresponding cohomology.

Enhancing the spatial resolution of hyperspectral (HS) images by fusing with higher spatial resolution multispectral (MS) data is of significance for applications. However, due to the narrow bandwidth, HS images (HSIs) are vulnerable to various types of noise, such as Gaussian noise and stripes, which can severely affect the fusion performance. This paper focuses on antinoise HS and MS image fusion to enhance the spatial details and suppress the noise. By analysis of the intrinsic structure and noise properties, we formulate this problem as the minimization of an objective function. Under the optimization framework, small multilinear ranks in tensor are first used to identify the intrinsic structures of the clean HSI part. Then, considering the high spectral correlation, it is assumed that any bands can be represented by the combination of certain adjacent bands. The difference between one band and its corresponding combination can be used to preserve the spatio-spectral consistency and characterize the distribution of sparse noise (such as stripe noise), based on the variational properties along two directions. The alternating direction method of multipliers (ADMM) is applied to solve and accelerate the model optimization. Experiments with both simulated- and real-data demonstrate the effectiveness of the proposed model and its robustness to the noise, in terms of both qualitative and quantitative perspectives.

Significance The white matter architecture of brain networks promotes synchrony among neuronal populations, giving rise to richly patterned functional networks. Relating structure and function is a fundamental question for systems neuroscience, but the nature of the relationship is unknown. Here we examine the possibility that structure–function relationships are not uniform in the brain. We find that structure and function are closely aligned in unimodal cortex (primary sensory and motor regions), but diverge in transmodal cortex (default mode and salience networks). The divergence between structure and function closely follows representational and cytoarchitectonic hierarchies, reflecting a macroscale gradient. Our findings suggest structure and function are not uniformly related, but gradually decouple in parallel to this macroscale gradient., The white matter architecture of the brain imparts a distinct signature on neuronal coactivation patterns. Interregional projections promote synchrony among distant neuronal populations, giving rise to richly patterned functional networks. A variety of statistical, communication, and biophysical models have been proposed to study the relationship between brain structure and function, but the link is not yet known. In the present report we seek to relate the structural and functional connection profiles of individual brain areas. We apply a simple multilinear model that incorporates information about spatial proximity, routing, and diffusion between brain regions to predict their functional connectivity. We find that structure–function relationships vary markedly across the neocortex. Structure and function correspond closely in unimodal, primary sensory, and motor regions, but diverge in transmodal cortex, particularly the default mode and salience networks. The divergence between structure and function systematically follows functional and cytoarchitectonic hierarchies. Altogether, the present results demonstrate that structural and functional networks do not align uniformly across the brain, but gradually uncouple in higher-order polysensory areas.

The aim of this paper is to give necessary and sufficient conditions for the boundedness of a general class of multilinear Hausdorff operators that acts on the product of some weighted function spa...

In this paper, we investigate the weighted multilinear boundedness properties of the maximal higher order Calder\'on commutator for the dimensions larger than two. We establish all weighted multilinear estimates on the product of the $L^p(\mathbb{R}^d,w)$ space, including some peculiar endpoint estimates of the higher dimensional Calder\'on commutator., Comment: 36 pages, Canadian Journal of Mathematics, to appear. arXiv admin note: text overlap with arXiv:1712.09020

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space $V$ given together with a linear map $\Delta : V \to V \otimes V$. In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on $V$, and our main result concerns a certain transformation $\Delta^{*}$ on such families of multilinear functionals., Comment: Version 2: Minor revision, added references

Low-rank or sparse tensor recovery finds many applications in computer vision and machine learning. The recently proposed regularized multilinear regression and selection (Remurs) model assumes the true tensor to be simultaneously low-Tucker-rank and sparse, and has been successfully applied in fMRI analysis. However, the statistical performance of Remurs-like models is still lacking. To address this problem, a minimization problem based on a newly defined tensor nuclear-l1-norm is proposed, to recover a simultaneously low-Tucker-rank and sparse tensor from its degraded observations. Then, an M-ADMM-based algorithm is developed to efficiently solve the problem. Further, the statistical performance is analyzed by establishing a deterministic upper bound on the estimation error for general noise. Also, under Gaussian noise, non-asymptotic upper bounds for two specific settings, i.e., noisy tensor decomposition and random Gaussian design, are given. Experiments on synthetic datasets demonstrate that the proposed theorems can precisely predict the scaling behavior of the estimation error.

The estimation of seismic demand that connects the ground motion intensity measure and the damage measure of structures is one of the most important components in the performance-based design. The random nature of earthquake ground motion with great uncertainties in structural properties has posed challenges before this estimation. In this study, the suitability of different scalar and vector intensity measures of ground motion is quantified by using information theory and relative entropy concepts in the representation of ground motion uncertainty. The multilinear demand models for a better estimation of the seismic demand of structures are developed for near-fault pulse-like and ordinary ground motions, separately. The results indicate that the measures based on the spectral acceleration, velocity, and vector values are more appropriate intensity measures in the low, moderate, and high levels of damage for pulse-like records, respectively. For ordinary records, the use of vector IMs is not recommended due to their complexities and slight superiorities to scalar IMs. If the evaluations require a simple IM in a single-linear demand model for all the damage, the measures of VSI and SIH would be recommended.

How to implement an effective factorization for nonrigid structure from motion (NRSFM) has attracted much attention in recent years. A straightforward factorization scheme is to multilinearly solve NRSFM in an alternating manner, where each of the unknown variables in NRSFM is updated by fixing the others at each iteration. However, recent works show that most existing multilinear factorization (MLF) methods achieve poorer performances than some state-of-the-art sequential factorization methods. In this paper, we reinvestigate the MLF scheme for improving factorization accuracy, and first propose an MLF method with the only low-rank prior for NRSFM in the presence of missing data. Then, for further improving the performances of such MLF methods, a latent "smoothness" characteristic on unknown 3-D deformable shapes is investigated, which is independent of temporal relations among deformable shapes. Accordingly, a latent-smoothness prior for solving NRSFM is derived from the latent smoothness characteristic, and it is able to effectively recover 3-D deformable shapes from unordered data, which is hard for the traditional temporal-smoothness prior to handle. Finally, a regularized factorization method is proposed by integrating MLF with the explored latent-smoothness prior for further pursuing better performances. Extensive experimental results show the effectiveness of our methods in comparison to eight existing multilinear/sequential methods.

We consider global optimization of nonconvex problems whose factorable reformulations contain a collection of multilinear equations of the form $$z_e = \prod _{v \in e} {z_v}$$, $$e \in E$$, where E denotes a set of subsets of cardinality at least two of a ground set. Important special cases include multilinear and polynomial optimization problems. The multilinear polytope is the convex hull of the set of binary points z satisfying the system of multilinear equations given above. Recently Del Pia and Khajavirad introduced running intersection inequalities, a family of facet-defining inequalities for the multilinear polytope. In this paper we address the separation problem for this class of inequalities. We first prove that separating flower inequalities, a subclass of running intersection inequalities, is NP-hard. Subsequently, for multilinear polytopes of fixed degree, we devise an efficient polynomial-time algorithm for separating running intersection inequalities and embed the proposed cutting-plane generation scheme at every node of the branch-and-reduce global solver BARON. To evaluate the effectiveness of the proposed method we consider two test sets: randomly generated multilinear and polynomial optimization problems of degree three and four, and computer vision instances from an image restoration problem Results show that running intersection cuts significantly improve the performance of BARON and lead to an average CPU time reduction of 50% for the random test set and of 63% for the image restoration test set.

International audience; A general bilinear optimal control problem subject to an infinitedimensional state equation is considered. Polynomial approximations of the associated value function are derived around the steady state by repeated forma ldifferentiation of the Hamilton–Jacobi-Bellman equation. The terms of the approximations are described by multilinear forms, which can be obtained as solutions to generalized Lyapunov equations with recursively defined right-handsides.They form the basis for defining a sub-optimal feedback law.The approximation properties of this feedback law are investigated. An application to the optimal control of a Fokker-Planck equation is also provided.

In this paper, a 4D tensor model is firstly constructed to explore efficient structural information and correlations from multi-modal data (both 2D and 3D face data). As the dimensionality of the generated 4D tensor is high, a tensor dimensionality reduction technique is in need. Since many real-world high-order data often reside in a low dimensional subspace, Tucker decomposition as a powerful technique is utilized to capture multilinear low-rank structure and to extract useful information from the generated 4D tensor data. Our goal is to use Tucker decomposition to obtain a set of core tensors with smaller sizes and factor matrices which are projected into the 4D tensor data for classification prediction. To characterize the involved similarities of the 4D tensor, the low-rank and sparse representation is built in terms of the low-rank structure of factor matrices and the sparsity of the core tensor in the Tucker decomposition of the generated 4D tensor. A tensor completion (TC) framework is embedded to recover the missing information in the 4D tensor modeling process. Thus, a novel tensor dimensionality reduction approach for 2D+3D facial expression recognition via low-rank tensor completion (FERLrTC) is proposed to solve the factor matrices in a majorization–minimization manner by using a rank reduction strategy. Numerical experiments are conducted with a full implementation on the BU-3DFE and Bosphorus databases and synthetic data to illustrate the effectiveness of the proposed approach.

We consider the initial value problem for the Dirac–Klein–Gordon equations in two space dimensions. Global regularity for C ∞ data was proved by Grunrock and Pecher. Here we consider analytic data, proving that if the initial radius of analyticity is σ 0 > 0 , then for later times t > 0 the radius of analyticity obeys a lower bound σ ( t ) ≥ σ 0 exp ⁡ ( − A t ) . This provides information about the possible dynamics of the complex singularities of the holomorphic extension of the solution at time t. The proof relies on an analytic version of Bourgain's Fourier restriction norm method, multilinear space–time estimates of null form type and an approximate conservation of charge.

In this paper, the boundedness for some multilinear operators related to certain integral operators on Lp spaces with variable exponent are obtained by using a sharp estimate of the multili...

We present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones. In particular, we solve a problem about the boundedness of the commutators of the bilinear Hilbert transform with functions in BMO., Simons Foundation, grants No. 246024 and No.160427, Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa” Programme for Centres of Excellence in R&D (SEV-2015-0554), European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ ERC agreement no. 615112 HAPDEGMT, NSF grant DMS 1069015

Let R be a prime ring of characteristic different from 2, U be its Utumi quotient ring with extended centroid C and $$f(x_1,\ldots ,x_n)$$ be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations on R such that $$F^2(f(r))f(r)-G(f(r)^2)=0$$ for all $$r=(r_1,\ldots ,r_n)\in R^{n}$$. Then one of the following holds

The first author is supported by NSF Grant DMS-1362425 and research funds from the Dean of the College of Arts & Sciences, the University of Alabama. The second author is supported by the Simons Foundation. We establish the boundedness of the multilinear Calderon{Zygmund operators from a product of weighted Hardy spaces into a weighted Hardy or Lebesgue space. Our results generalize to the weighted setting results obtained by Grafakos and Kalton [18] and recent work by the third author, Grafakos, Nakamura, and Sawano [20]. As part of our proof we provide a finite atomic decomposition theorem for weighted Hardy spaces, which is interesting in its own right. As a consequence of our weighted results, we prove the corresponding estimates on variable Hardy spaces. Our main tool is a multilinear extrapolation theorem that generalizes a result of the first author and Naibo [10].

Summary In this article, using the Mizar system [5], [2], the isomorphisms from the space of multilinear operators are discussed. In the first chapter, two isomorphisms are formalized. The former isomorphism shows the correspondence between the space of multilinear operators and the space of bilinear operators. The latter shows the correspondence between the space of multilinear operators and the space of the composition of linear operators. In the last chapter, the above isomorphisms are extended to isometric mappings between the normed spaces. We referred to [6], [11], [9], [3], [10] in this formalization.