Search for: All records

Hyperbolic space is particularly useful for embedding data with hierarchical structure; however, representing hyperbolic space with ordinary floating-point numbers greatly affects the performance due to its \emph{ineluctable} numerical errors. Simply increasing the precision of floats fails to solve the problem and incurs a high computation cost for simulating greater-than-double-precision floats on hardware such as GPUs, which does not support them. In this paper, we propose a simple, feasible-on-GPUs, and easy-to-understand solution for numerically accurate learning on hyperbolic space. We do this with a new approach to represent hyperbolic space using multi-component floating-point (MCF) in the Poincar{\'e} upper-half space model. Theoretically and experimentally we show our model has small numerical error, and on embedding tasks across various datasets, models represented by multi-component floating-points gain more capacity and run significantly faster on GPUs than prior work. 

Abstract We show that if a sequence of normalized polynomials gives rise to a positive basis of the skein algebra of a surface, then it is sandwiched between the two types of Chebyshev polynomials. For the closed torus, we show that the normalized sequence of Chebyshev polynomials of type one $(\hat{T}_n)$ is the only one that gives a positive basis. 

Point-of-care COVID-19 assays that are more sensitive than the current RT-PCR (reverse transcription polymerase chain reaction) gold standard assay are needed to improve disease control efforts. We describe the development of a portable, ultrasensitive saliva-based COVID-19 assay with a 15-min sample-to-answer time that does not require RNA isolation or laboratory equipment. This assay uses CRISPR-Cas12a activity to enhance viral amplicon signal, which is stimulated by the laser diode of a smartphone-based fluorescence microscope device. This device robustly quantified viral load over a broad linear range (1 to 10 5 copies/μl) and exhibited a limit of detection (0.38 copies/μl) below that of the RT-PCR reference assay. CRISPR-read SARS-CoV-2 (severe acute respiratory syndrome coronavirus 2) RNA levels were similar in patient saliva and nasal swabs, and viral loads measured by RT-PCR and the smartphone-read CRISPR assay demonstrated good correlation, supporting the potential use of this portable assay for saliva-based point-of-care COVID-19 diagnosis. 

Graph Convolutional Networks (GCNs) and their variants have experienced significant attention and have become the de facto methods for learning graph representations. GCNs derive inspiration primarily from recent deep learning approaches, and as a result, may inherit unnecessary complexity and redundant computation. In this paper, we reduce this excess complexity through successively removing nonlinearities and collapsing weight matrices between consecutive layers. We theoretically analyze the resulting linear model and show that it corresponds to a fixed low-pass filter followed by a linear classifier. Notably, our experimental evaluation demonstrates that these simplifications do not negatively impact accuracy in many downstream applications. Moreover, the resulting model scales to larger datasets, is naturally interpretable, and yields up to two orders of magnitude speedup over FastGCN.