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We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem…

The max-min fair (MMF) multicasting problem is known to be NP-hard. In this work, we analytically derive the optimal solution to this NP-hard problem and establish the equivalence between rate balancing and the optimal MMF multicasting…

This article introduces a novel, geometric approach for multi-manifold clustering (MMC), i.e. for clustering a collection of potentially intersecting, d-dimensional manifolds into the individual manifold components. We first compute a…

Deep Neural Networks (DNNs) approaches for the Optimal Power Flow (OPF) problem received considerable attention recently. A key challenge of these approaches lies in ensuring the feasibility of the predicted solutions to physical system…

We study the problem of finding a minimum homology basis, that is, a lightest set of cycles that generates the $1$-dimensional homology classes with $\mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been…

Fractures form the main pathways for flow in the subsurface within low-permeability rock. For this reason, accurately predicting flow and transport in fractured systems is vital for improving the performance of subsurface applications.…

Deep neural networks (DNNs) have achieved significant success in a variety of real world applications, i.e., image classification. However, tons of parameters in the networks restrict the efficiency of neural networks due to the large model…

Continuous normalizing flows (CNFs) can model data distributions with expressive infinite-length architectures. But this modeling involves computationally expensive process of solving an ordinary differential equation (ODE) during maximum…

Optimization-based filtering smoothes an image by minimizing a fidelity function and simultaneously preserves edges by exploiting a sparse norm penalty over gradients. It has obtained promising performance in practical problems, such as…

We introduce a high-performance cost-effective network topology called Slim Fly that approaches the theoretically optimal network diameter. Slim Fly is based on graphs that approximate the solution to the degree-diameter problem. We analyze…

In this paper we present algorithms for a number of problems in geometric pattern matching where the input consist of a collections of segments in the plane. Our work consists of two main parts. In the first, we address problems and…

We develop DeepOPF as a Deep Neural Network (DNN) approach for solving security-constrained direct current optimal power flow (SC-DCOPF) problems, which are critical for reliable and cost-effective power system operation.DeepOPF is inspired…

Heterogeneous graph neural networks (GNNs) achieve strong performance on node classification tasks in a semi-supervised learning setting. However, as in the simpler homogeneous GNN case, message-passing-based heterogeneous GNNs may struggle…

Decentralized optimization is critical for solving large-scale machine learning problems over distributed networks, where multiple nodes collaborate through local communication. In practice, the variances of stochastic gradient estimators…

Two fundamental objects in knot theory are the minimal genus surface and the least area surface bounded by a knot in a 3-dimensional manifold. When the knot is embedded in a general 3-manifold, the problems of finding these surfaces were…

We analyze the stability of synchronized state for coupled nearly identical dynamical systems on networks by deriving an approximate Master Stability Function (MSF). Using this MSF we treat the problem of designing a network having the best…

Symmetry in finite model finding problems of many-sorted first-order logic (MSFOL) can be exploited to reduce the number of interpretations considered during search, thereby improving solver performance. In this thesis, we situate symmetry…

Smoothing splines are twice differentiable by construction, so they cannot capture potential discontinuities in the underlying signal. In this work, we consider a special case of the weak rod model of Blake and Zisserman (1987) that allows…

We propose a deep neural network (DNN) based least distance (LD) estimator (DNN-LD) for a multivariate regression problem, addressing the limitations of the conventional methods. Due to the flexibility of a DNN structure, both linear and…

Multi-region segmentation algorithms often have the onus of incorporating complex anatomical knowledge representing spatial or geometric relationships between objects, and general-purpose methods of addressing this knowledge in an…