Simultaneously Structured Models with Application to Sparse and Low-rank Matrices

The topic of recovery of a structured model given a small number of linear observations has been well-studied in recent years. Examples include recovering sparse or group-sparse vectors, low-rank matrices, and the sum of sparse and low-rank matrices, among others. In various applications in signal processing and machine learning, the model of interest is known to be structured in several ways at the same time, for example, a matrix that is simultaneously sparse and low-rank. Often norms that promote each individual structure are known, and allow for recovery using an order-wise optimal number of measurements (e.g., $\ell_1$ norm for sparsity, nuclear norm for matrix rank). Hence, it is reasonable to minimize a combination of such norms. We show that, surprisingly, if we use multi-objective optimization with these norms, then we can do no better, order-wise, than an algorithm that exploits only one of the present structures. This result suggests that to fully exploit the multiple structures, we need an entirely new convex relaxation, i.e. not one that is a function of the convex relaxations used for each structure. We then specialize our results to the case of sparse and low-rank matrices. We show that a nonconvex formulation of the problem can recover the model from very few measurements, which is on the order of the degrees of freedom of the matrix, whereas the convex problem obtained from a combination of the $\ell_1$ and nuclear norms requires many more measurements. This proves an order-wise gap between the performance of the convex and nonconvex recovery problems in this case. Our framework applies to arbitrary structure-inducing norms as well as to a wide range of measurement ensembles. This allows us to give performance bounds for problems such as sparse phase retrieval and low-rank tensor completion.Comment: 38 pages, 9 figure

Comments on the paper "COINCIDENCE THEOREMS FOR SOME MULTIVALUED MAPPINGS" by B. E. RHOADES, S. L. SINGH AND CHITRA KULSHRESTHA

The aim of this note is to point out an error in the proof of Theorem 1 in the paper entitled \u201cCoincidence theorems for some multivalued mappings\u201d by B. E. Rhoades, S. L. Singh and Chitra Kulshrestha [Internat. J. Math. & Math. Sci., 7 (1984), 429-434], and to indicate a way to repair it

A Perturbed Cauchy Viscoelastic Problem in an Exterior Domain

A Cauchy viscoelastic problem perturbed by an inverse-square potential, and posed in an exterior domain of R^N, is considered under a Dirichlet boundary condition. Using nonlinear capacity estimates specifically adapted to the non-local nature of the problem, the potential function and the boundary condition, we establish sufficient conditions for the nonexistence of weak solutions

On the critical behavior for inhomogeneous wave inequalities with Hardy potential in an exterior domain

We study the wave inequality with a Hardy potential (Eqation Presented), where Ω is the exterior of the unit ball in ℝN, N ≥ 2, p > 1, and λ ≥ - (N-2/2)2, under the inhomogeneous boundary condition 'Equation Presented', where α, β ≥ 0 and (α, β) ≠ (0, 0). Namely, we show that there exists a critical exponent pc(N, λ) ∈ (1, ∞] for which, if 1 < p < pc(N, λ), the above problem admits no global weak solution for any w ∈ L1 (∂Ω) with ∫∂Ω w(x) dσ > 0, while if p > pc(N, λ), the problem admits global solutions for some w > 0. To the best of our knowledge, the study of the critical behavior for wave inequalities with a Hardy potential in an exterior domain was not considered in previous works. Some open questions are also mentioned in this paper

Higher order evolution inequalities involving Leray-Hardy potential singular on the boundary

We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray-Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of solutions

Nonexistence results for higher order fractional differential inequalities with nonlinearities involving Caputo fractional derivative

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function

Characterization of an embedded RF-MEMS switch

An RF-MEMS capacitive switch for mm-wave integrated circuits, embedded in the BEOL of 0.25μm BiCMOS process, has been characterized. First, a mechanical model based on Finite-Element-Method (FEM) was developed by taking the residual stress of the thin film membrane into account. The pull-in voltage and the capacitance values obtained with the mechanical model agree very well with the measured values. Moreover, S-parameters were extracted using Electromagnetic (EM) solver. The data observed in this way also agree well with the experimental ones measured up to 110GHz. The developed RF model was applied to a transmit/receive (T/R) antenna switch design. The results proved the feasibility of using the FEM model in circuit simulations for the development of RF-MEMS switch embedded, single-chip multi-band RF ICs

Reroute Prediction Service

The cost of delays was estimated as 33 billion US dollars only in 2019 for the US National Airspace System, a peak value following a growth trend in past years. Aiming to address this huge inefficiency, we designed and developed a novel Data Analytics and Machine Learning system, which aims at reducing delays by proactively supporting re-routing decisions. Given a time interval up to a few days in the future, the system predicts if a reroute advisory for a certain Air Route Traffic Control Center or for a certain advisory identifier will be issued, which may impact the pertinent routes. To deliver such predictions, the system uses historical reroute data, collected from the System Wide Information Management (SWIM) data services provided by the FAA, and weather data, provided by the US National Centers for Environmental Prediction (NCEP). The data is huge in volume, and has many items streamed at high velocity, uncorrelated and noisy. The system continuously processes the incoming raw data and makes it available for the next step where an interim data store is created and adaptively maintained for efficient query processing. The resulting data is fed into an array of ML algorithms, which compete for higher accuracy. The best performing algorithm is used in the final prediction, generating the final results. Mean accuracy values higher than 90% were obtained in our experiments with this system. Our algorithm divides the area of interest in units of aggregation and uses temporal series of the aggregate measures of weather forecast parameters in each geographical unit, in order to detect correlations with reroutes and where they will most likely occur. Aiming at practical application, the system is formed by a number of microservices, which are deployed in the cloud, making the system distributed, scalable and highly available.Comment: Submitted to the 2023 IEEE/AIAA Digital Aviation Systems Conference (DASC

Shrinkage and mechanical performance of geopolymeric mortars based on calcined Tunisian clay

Infrastructure rehabilitation represents a multitrillion dollar opportunity for the construction industry. Since the majority of the existent infrastructures are Portland cement concrete based this means that concrete infrastructure rehabilitation is a hot issue to be dealt with. Geopolymers are novel inorganic binders with high potential to replace Portland cement based ones. Geopolymerization is a complex chemical process evolving various aluminosilicate oxides with silicates under highly alkaline conditions, yielding polymeric units, similar to those of an aluminosilicate glass. So far very few studies in the geopolymer field have addressed the rehabilitation of deteriorated concrete structures. This paper discloses some results of an investigation concerning the development geopolymeric repair mortars based on a calcined Tunisian clay. The results show that Tunisian calcined clay based mortars have hydration products with typical geopolymeric phases. Results also show that the geopolymeric mortar shows a high unrestrained shrinkage behavior and that its modulus of elasticity is below the threshold required for this repair mortars