Related papers: A visualisation for conveying the dynamics of iter…
A common algorithm for the computation of eigenvalues of real symmetric tridiagonal matrices is the iteration of certain special maps $F_\sigma$ called shifted $QR$ steps. Such maps preserve spectrum and a natural common domain is ${\cal…
In machine learning and statistical data analysis, we often run into objective function that is a summation: the number of terms in the summation possibly is equal to the sample size, which can be enormous. In such a setting, the stochastic…
Understanding dynamics in complex systems is challenging because there are many degrees of freedom, and those that are most important for describing events of interest are often not obvious. The leading eigenfunctions of the transition…
Self-supervised representation learning is a fundamental problem in computer vision with many useful applications (e.g., image search, instance level recognition, copy detection). In this paper we present a new contrastive self-supervised…
Detecting and visualizing what are the most relevant changes in an evolving network is an open challenge in several domains. We present a fast algorithm that filters subsets of the strongest nodes and edges representing an evolving weighted…
Imitation learning (IL) has seen remarkable progress, yet field deployment of IL-powered robots remains hindered by the challenge of out-of-distribution (OOD) scenarios. Fine-tuning pre-trained policies with end-user demonstrations…
We analyze the convergence rate of various momentum-based optimization algorithms from a dynamical systems point of view. Our analysis exploits fundamental topological properties, such as the continuous dependence of iterates on their…
For large-scale eigenvalue problems requiring many mutually orthogonal eigenvectors, traditional numerical methods suffer substantial computational and communication costs with limited parallel scalability, primarily due to explicit…
In our earlier work [Fareed et al., Comput. Math. Appl. 75 (2018), no. 6, 1942-1960], we developed an incremental approach to compute the proper orthogonal decomposition (POD) of PDE simulation data. Specifically, we developed an…
In this paper, I present VQ-DRAW, an algorithm for learning compact discrete representations of data. VQ-DRAW leverages a vector quantization effect to adapt the sequential generation scheme of DRAW to discrete latent variables. I show that…
Iterative procedures for parameter estimation based on stochastic gradient descent allow the estimation to scale to massive data sets. However, in both theory and practice, they suffer from numerical instability. Moreover, they are…
Randomized iterative algorithms have attracted much attention in recent years because they can approximately solve large-scale linear systems of equations without accessing the entire coefficient matrix. In this paper, we propose two novel…
Interactive exploration of large, multidimensional datasets plays a very important role in various scientific fields. It makes it possible not only to identify important structural features and forms, such as clusters of vertices and their…
The generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large-scale problems is challenging. Motivated by applications in hyper-differential…
We develop a fixed-point iterative algorithm that computes the matrix projection with respect to the Bures distance on the set of positive definite matrices that are invariant under some symmetry. We prove that the fixed-point iteration…
Some fast algorithms for computing the eigenvalues of a block companion matrix $A = U + XY^H$, where $U\in \mathbb C^{n\times n}$ is unitary block circulant and $X, Y \in\mathbb{C}^{n \times k}$, have recently appeared in the literature.…
At the core of self-supervised learning for vision is the idea of learning invariant or equivariant representations with respect to a set of data transformations. This approach, however, introduces strong inductive biases, which can render…
We present a greedy algorithm for computing selected eigenpairs of a large sparse matrix $H$ that can exploit localization features of the eigenvector. When the eigenvector to be computed is localized, meaning only a small number of its…
Differential geometric approaches to the analysis and processing of data in the form of symmetric positive definite (SPD) matrices have had notable successful applications to numerous fields including computer vision, medical imaging, and…
Features in machine learning problems are often time-varying and may be related to outputs in an algebraic or dynamical manner. The dynamic nature of these machine learning problems renders current higher order accelerated gradient descent…