Batch Greenkhorn Algorithm for Entropic-Regularized Multimarginal Optimal Transport: Linear Rate of Convergence and Iteration Complexity

In this work we propose a batch multimarginal version of the Greenkhorn algorithm for the entropic-regularized optimal transport problem. This framework is general enough to cover, as particular cases, existing Sinkhorn and Greenkhorn algorithms for the bi-marginal setting, and greedy MultiSinkhorn for the general multimarginal case. We provide a comprehensive convergence analysis based on the properties of the iterative Bregman projections method with greedy control. Linear rate of convergence as well as explicit bounds on the iteration complexity are obtained. When specialized to the above mentioned algorithms, our results give new convergence rates or provide key improvements over the state-of-the-art rates. We present numerical experiments showing that the flexibility of the batch can be exploited to improve performance of Sinkhorn algorithm both in bi-marginal and multimarginal settings

Sharp Spectral Rates for Koopman Operator Learning

Non-linear dynamical systems can be handily described by the associated Koopman operator, whose action evolves every observable of the system forward in time. Learning the Koopman operator and its spectral decomposition from data is enabled by a number of algorithms. In this work we present for the first time non-asymptotic learning bounds for the Koopman eigenvalues and eigenfunctions. We focus on time-reversal-invariant stochastic dynamical systems, including the important example of Langevin dynamics. We analyze two popular estimators: Extended Dynamic Mode Decomposition (EDMD) and Reduced Rank Regression (RRR). Our results critically hinge on novel minimax estimation bounds for the operator norm error, that may be of independent interest. Our spectral learning bounds are driven by the simultaneous control of the operator norm error and a novel metric distortion functional of the estimated eigenfunctions. The bounds indicates that both EDMD and RRR have similar variance, but EDMD suffers from a larger bias which might be detrimental to its learning rate. Our results shed new light on the emergence of spurious eigenvalues, an issue which is well known empirically. Numerical experiments illustrate the implications of the bounds in practice.Comment: 10 pages, 3 figures, 6 appendice

On the intersection of the classes of doubly diagonally dominant matrices and S-strictly diagonally dominant matrices

We denote by H0 the subclass of H-matrices consisting of all the matrices that lay simultaneously on the classes of doubly diagonally dominant (DDD) matrices (A = [aij ] ∈ Cn×n : |aii||ajj | ≥ k =i |aik| k =j |ajk|, i = j) and S-strictly diagonally dominant (S-SDD) matrices. Notice that strictly doubly diagonally dominant matrices (also called Ostrowsky matrices) are a subclass of H0. Strictly diagonally dominant matrices (SDD) are also a subclass of H0. In this paper we analyze some properties of the class H0 = DDD ∩ S-SDD

On Matrix Nearness Problems: Distance to Delocalization

This paper introduces two new matrix nearness problems that are intended to generalize the distance to instability and the distance to stability. They are named the distance to delocalization and the distance to localization due to their applicability in analyzing the robustness of eigenvalues with respect to arbitrary localization sets (domains) in the complex plane. For the open left-half plane or the unit circle, the distance to the nearest unstable/stable matrix is obtained as a special case. Then, following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present a new Newton-type algorithm for the distance to delocalization (D2D) and study its implementations using both an explicit and an implicit computation of the desired singular values. Since our investigations are motivated by several practical applications, we will illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to instability, we will validate our algorithms against the state of the art computational method

Proširenje frekvencijskog opsega za eksperimentalno određivanje disperzione relacija talasa savijanja u štapovima primenom korelacione metode

The dispersion relation is the key for studies of wave propagation. The experimental determination of a dispersion relation by measurements of wave properties in different points in space meets the challenge of spatial aliasing, which is further augmented by numerical instability of calculations. This paper presents and discusses a concept aimed to overcome the spatial aliasing problem in measurements of dispersion relation of mechanical waves propagating through beams. The concept is based on the properties of the dispersion relationship and it may be extended to the case of all other waves with monotonous dispersion relationship.Publishe

A review of the risk and precipitating factors for spontaneous coronary artery dissection

IntroductionSpontaneous coronary artery dissection (SCAD) accounts for 1%–4% of cases of acute coronary syndrome (ACS). SCAD is caused by separation occurring within or between any of the three tunics of the coronary artery wall. This leads to intramural hematoma and/or formation of false lumen in the artery, which leads to ischemic changes or infarction of the myocardium. The incidence of SCAD is higher in women than in men, with a ratio of approximately 9:1. It is estimated that SCAD is responsible for 35% of ACS cases in women under the age of 60. The high frequency is particularly observed during pregnancy and in the peripartum period (first week). Traditional risk factors are rare in patients with SCAD, except for hypertension. Patients diagnosed with SCAD have different combinations of risk factors compared with patients who have atherosclerotic changes in their coronary arteries. We presented the most common so-called “non-traditional” risk factors associated with SCAD patients.Risk factors and precipitating disorders which are associated with SCADIn the literature, there are few diseases frequently associated with SCAD, and they are identified as predisposing factors. The predominant cause is fibromuscular dysplasia, followed by inherited connective tissue disorders, systemic inflammatory diseases, pregnancy, use of sex hormones or steroids, use of cocaine or amphetamines, thyroid disorders, migraine, and tinnitus. In recent years, the genetic predisposition for SCAD is also recognized as a predisposing factor. The precipitating factors are also different in women (emotional stress) compared with those in men (physical stress). Women experiencing SCAD frequently describe symptoms of anxiety and depression. These conditions could increase shear stress on the arterial wall and dissection of the coronary artery wall. Despite the advancement of SCAD, we can find significant differences in the clinical presentation between women and men.ConclusionWhen evaluating patients with chest pain or other ACS symptoms who have a low cardiovascular risk, particularly female patients, it is important to consider the possibility of ACS due to SCAD, particularly in conditions often associated with SCAD. This will increase the recognition of SCAD and the timely treatment of affected patients

Functional Characterization of Rare RAB12 Variants and Their Role in Musician's and Other Dystonias

Mutations in RAB (member of the Ras superfamily) genes are increasingly recognized as cause of a variety of disorders including neurological conditions. While musician's dystonia (MD) and writer's dystonia (WD) are task-specific movement disorders, other dystonias persistently affect postures as in cervical dystonia. Little is known about the underlying etiology. Next-generation sequencing revealed a rare missense variant (c.586A> G; p.Ile196Val) in RAB12 in two of three MD/WD families. Next, we tested 916 additional dystonia patients; 512 Parkinson's disease patients; and 461 healthy controls for RAB12 variants and identified 10 additional carriers of rare missense changes among dystonia patients (1.1%) but only one carrier in non-dystonic individuals (0.1%; p = 0.005). The detected variants among index patients comprised p.Ile196Val (n = 6); p.Ala174Thr (n = 3); p.Gly13Asp; p.Ala148Thr; and p.Arg181Gln in patients with MD; cervical dystonia; or WD. Two relatives of MD patients with WD also carried p.Ile196Val. The two variants identified in MD patients (p.Ile196Val; p.Gly13Asp) were characterized on endogenous levels in patient-derived fibroblasts and in two RAB12-overexpressing cell models. The ability to hydrolyze guanosine triphosphate (GTP), so called GTPase activity, was increased in mutants compared to wildtype. Furthermore, subcellular distribution of RAB12 in mutants was altered in fibroblasts. Soluble Transferrin receptor 1 levels were reduced in the blood of all three tested p.Ile196Val carriers. In conclusion, we demonstrate an enrichment of missense changes among dystonia patients. Functional characterization revealed altered enzyme activity and lysosomal distribution in mutants suggesting a contribution of RAB12 variants to MD and other dystonias