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This paper introduces a novel algorithmic framework for a deep neural network (DNN), which in a mathematically rigorous manner, allows us to incorporate history (or memory) into the network -- it ensures all layers are connected to one…
Unsigned Distance Functions (UDFs) can be used to represent non-watertight surfaces in a deep learning framework. However, UDFs tend to be brittle and difficult to learn, in part because the surface is located exactly where the UDF is…
We propose a new approach, called as functional deep neural network (FDNN), for classifying multi-dimensional functional data. Specifically, a deep neural network is trained based on the principle components of the training data which shall…
Many functional datasets are observed sparsely and irregularly. Ordering such data is challenging because only limited information is available from each observation, while the underlying trajectories remain infinite-dimensional. This paper…
Fine-Grained Domain Generalization (FGDG) presents greater challenges than conventional domain generalization due to the subtle inter-class differences and relatively pronounced intra-class variations inherent in fine-grained recognition…
Over the last decade, it has been demonstrated that many systems in science and engineering can be modeled more accurately by fractional-order than integer-order derivatives, and many methods are developed to solve the problem of fractional…
In complex physical systems, conventional differential equations often fall short in capturing non-local and memory effects, as they are limited to local dynamics and integer-order interactions. This study introduces a stepwise data-driven…
Motivated by the application of Lyapunov methods to partial differential equations (PDEs), we study functional inequalities of the form $f(I_1(u),\ldots,I_k(u))\geq 0$ where $f$ is a polynomial, $u$ is any function satisfying prescribed…
Set function learning has emerged as a crucial area in machine learning, addressing the challenge of modeling functions that take sets as inputs. Unlike traditional machine learning that involves fixed-size input vectors where the order of…
Data depth functions are a generalization of one-dimensional order statistics and medians to real spaces of dimension greater than one; in particular, a data depth function quantifies the centrality of a point with respect to a data set or…
Dimensionality Reduction is a commonly used element in a machine learning pipeline that helps to extract important features from high-dimensional data. In this work, we explore an alternative federated learning system that enables…
During the past two decades there has been a lot of interest in developing statistical depth notions that generalize the univariate concept of ranking to multivariate data. The notion of depth has also been extended to regression models and…
The modulating functions method has been used for the identification of linear and nonlinear systems. In this paper, we generalize this method to the on-line identification of fractional order systems based on the Riemann-Liouville…
Over the past few years, the federated learning ($\texttt{FL}$) community has witnessed a proliferation of new $\texttt{FL}$ algorithms. However, our understating of the theory of $\texttt{FL}$ is still fragmented, and a thorough, formal…
We consider the concept of rank as a measure of the vertical levels and positions of elements of partially ordered sets (posets). We are motivated by the need for algorithmic measures on large, real-world hierarchically-structured data…
A uniform gradient for functions u which satisfy a system of N second-order partial differential inequalities is given in this paper. Some structure conditions are given for the coefficients of the matrices of second-order terms and of…
In this work, we study the classical distributed optimization problem over digraphs, where the objective function is a sum of smooth local functions. Inspired by the implicit tracking mechanism proposed in our earlier work, we develop a…
We study the statistical complexity of estimating partition functions given sample access to a proposal distribution and an unnormalized density ratio for a target distribution. While partition function estimation is a classical problem,…
Constrained low-rank matrix approximations have been known for decades as powerful linear dimensionality reduction techniques to be able to extract the information contained in large data sets in a relevant way. However, such low-rank…
As machine learning becomes increasingly central to molecular design, it is vital to ensure the reliability of learnable protein-ligand scoring functions on novel protein targets. While many scoring functions perform well on standard…