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Point-source transient events (PSTEs) - optical events that are both extremely fast and extremely small - pose several challenges to an imaging system. Due to their speed, accurately characterizing such events often requires detectors with very high frame rates. Due to their size, accurately detecting such events requires maintaining coverage over an extended field-of-view, often through the use of imaging focal plane arrays (FPA) with a global shutter readout. Traditional imaging systems that meet these requirements are costly in terms of price, size, weight, power consumption, and data bandwidth, and there is a need for cheaper solutions with adequate temporal and spatial coverage. To address these issues, we develop a novel compressed sensing algorithm adapted to the rolling shutter readout of an imaging system. This approach enables reconstruction of a PSTE signature at the sampling rate of the rolling shutter, offering a 1-2 order of magnitude temporal speedup and a proportional reduction in data bandwidth. We present empirical results demonstrating accurate recovery of PSTEs using measurements that are spatially undersampled by a factor of 25, and our simulations show that, relative to other compressed sensing algorithms, our algorithm is both faster and yields higher quality reconstructions. We also present theoretical results characterizing our algorithm and corroborating simulations. The potential impact of our work includes the development of much faster, cheaper sensor solutions for PSTE detection and characterization., Comment: 20 pages, 11 figures

We analyze a random projection method for adjacency matrices, studying its utility in representing sparse graphs. We show that these random projections retain the functionality of their underlying adjacency matrices while having extra properties that make them attractive as dynamic graph representations. In particular, they can represent graphs of different sizes and vertex sets in the same space, allowing for the aggregation and manipulation of graphs in a unified manner. We also provide results on how the size of the projections need to scale in order to preserve accurate graph operations, showing that the size of the projections can scale linearly with the number of vertices while accurately retaining first-order graph information. We conclude by characterizing our random projection as a distance-preserving map of adjacency matrices analogous to the usual Johnson-Lindenstrauss map., Comment: 21 pages

Following up on a previous analysis of graph embeddings, we generalize and expand some results to the general setting of vector symbolic architectures (VSA) and hyperdimensional computing (HDC). Importantly, we explore the mathematical relationship between superposition, orthogonality, and tensor product. We establish the tensor product representation as the central representation, with a suite of unique properties. These include it being the most general and expressive representation, as well as being the most compressed representation that has errorrless unbinding and detection., Comment: 18 pages

In this paper, we interpret disentanglement as the discovery of local charts of the data manifold and trace how this definition naturally leads to an equivalent condition for disentanglement: commutativity between factors of variation. We study the impact of this manifold framework to two classes of problems: learning matrix exponential operators and compressing data-generating models. In each problem, the manifold perspective yields interesting results about the feasibility and fruitful approaches their solutions. We also link our manifold framework to two other common disentanglement paradigms: group theoretic and probabilistic approaches to disentanglement. In each case, we show how these frameworks can be merged with our manifold perspective. Importantly, we recover commutativity as a central property in both alternative frameworks, further highlighting its importance in disentanglement., Comment: 26 pages

We propose a dynamic graph representation method, showcasing its rich representational capacity and establishing some of its theoretical properties. Our representation falls under the bind-and-sum approach in hyperdimensional computing (HDC), and we show that the tensor product is the most general binding operation that respects the superposition principle employed in HDC. We also establish some precise results characterizing the behavior of our method, including a memory vs. size analysis of how our representation's size must scale with the number of edges in order to retain accurate graph operations. True to its HDC roots, we also compare our graph representation to another typical HDC representation, the Hadamard-Rademacher scheme, showing that these two graph representations have the same memory-capacity scaling. We establish a link to adjacency matrices, showing that our method is a pseudo-orthogonal generalization of adjacency matrices. In light of this, we briefly discuss its applications toward a dynamic compressed representation of large sparse graphs., Comment: 59 pages, 2 tables. arxiv admin note: substantial text overlap with arXiv:2208.08769

THIS PAPER IS NOW DEFUNCT: Check out "Graph Embeddings via Tensor Products and Approximately Orthonormal Codes", where it has been combined into one paper., Comment: 23 Pages, 2 Figures

Discrete spatial patterns and their continuous transformations are two important regularities contained in natural signals. Lie groups and representation theory are mathematical tools that have been used in previous works to model continuous image transformations. On the other hand, sparse coding is an important tool for learning dictionaries of patterns in natural signals. In this paper, we combine these ideas in a Bayesian generative model that learns to disentangle spatial patterns and their continuous transformations in a completely unsupervised manner. Images are modeled as a sparse superposition of shape components followed by a transformation that is parameterized by n continuous variables. The shape components and transformations are not predefined, but are instead adapted to learn the symmetries in the data, with the constraint that the transformations form a representation of an n-dimensional torus. Training the model on a dataset consisting of controlled geometric transformations of specific MNIST digits shows that it can recover these transformations along with the digits. Training on the full MNIST dataset shows that it can learn both the basic digit shapes and the natural transformations such as shearing and stretching that are contained in this data.