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Reduced-rank approach has been used for decades in robust linear estimation of both deterministic and random vector of parameters in linear model y=Hx+\sqrt{epsilon}n. In practical settings, estimation is frequently performed under…
Decentralized learning (DL) has recently employed local updates to reduce the communication cost for general non-convex optimization problems. Specifically, local updates require each node to perform multiple update steps on the parameters…
The distributional single index model is a semiparametric regression model in which the conditional distribution functions $P(Y \leq y | X = x) = F_0(\theta_0(x), y)$ of a real-valued outcome variable $Y$ depend on $d$-dimensional…
We consider the stochastic variational inequality problem in which the map is expectation-valued in a component-wise sense. Much of the available convergence theory and rate statements for stochastic approximation schemes are limited to…
Design-space dimensionality reduction is essential to mitigate the cost of high-fidelity simulation-based optimization, especially when dealing with high-dimensional geometric parameterizations. Traditional linear techniques, such as…
By means of the concept of balanced estimation of diffusion entropy we evaluate reliable scale-invariance embedded in different sleep stages and stride records. Segments corresponding to Wake, light sleep, REM, and deep sleep stages are…
Recent research has observed that in machine learning optimization, gradient descent (GD) often operates at the edge of stability (EoS) [Cohen, et al., 2021], where the stepsizes are set to be large, resulting in non-monotonic losses…
We study the convergence rate for the last iterate of stochastic gradient descent (SGD) and stochastic heavy ball (SHB) in the parametric setting when the objective function $F$ is globally convex or non-convex whose gradient is…
In this work, we derive a result of exponential stability for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a…
Magnetic Resonance Elastography (MRE) has become an essential tool in assessing the mechanical properties of soft tissues in-vivo, prompting significant progress in new inversion algorithms. This creates a need for a benchmarking framework…
When calculating properties of periodic systems at the thermodynamic limit (TDL), the dominant source of finite size error (FSE) arises from the long-range Coulomb interaction, and can manifest as a slowly converging quadrature error when…
Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although…
With the emergence of precision medicine, estimating optimal individualized decision rules (IDRs) has attracted tremendous attention in many scientific areas. Most existing literature has focused on finding optimal IDRs that can maximize…
We study the dynamics of a continuous-time model of the Stochastic Gradient Descent (SGD) for the least-square problem. Indeed, pursuing the work of Li et al. (2019), we analyze Stochastic Differential Equations (SDEs) that model SGD either…
Fractional-order stochastic gradient descent (FOSGD) leverages fractional exponents to capture long-memory effects in optimization. However, its utility is often limited by the difficulty of tuning and stabilizing these exponents. We…
Some continuous optimization methods can be connected to ordinary differential equations (ODEs) by taking continuous limits, and their convergence rates can be explained by the ODEs. However, since such ODEs can achieve any convergence rate…
In this paper, both semidiscrete and completely discrete finite volume element methods (FVEMs) are analyzed for approximating solutions of a class of linear hyperbolic integro- differential equations in a two-dimensional convex polygonal…
Reliability theory is used to assess the sensitivity of a passive flexion and active flexion of the human lower leg Finite Element (FE) models with Total Knee Replacement (TKR) to the variability in the input parameters of the respective FE…
Designing and analyzing optimization methods via continuous-time models expressed as ordinary differential equations (ODEs) is a promising approach for its intuitiveness and simplicity. A key concern, however, is that the convergence rates…
In the first part of the paper, we study reflected backward stochastic differential equations (RBSDEs) with lower obstacle which is assumed to be right upper-semicontinuous but not necessarily right-continuous. We prove existence and…