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We study fast algorithms for computing basic properties of an n x n positive semidefinite kernel matrix K corresponding to n points x_1,...,x_n in R^d. In particular, we consider the estimating the sum of kernel matrix entries, along with its top eigenvalue and eigenvector. These are some of the most basic problems defined over kernel matrices. We show that the sum of matrix entries can be estimated up to a multiplicative factor of 1+epsilon in time sublinear in n and linear in d for many popular kernel functions, including the Gaussian, exponential, and rational quadratic kernels. For these kernels, we also show that the top eigenvalue (and a witnessing approximate eigenvector) can be approximated to a multiplicative factor of 1+epsilon in time subquadratic in n and linear in d. Our algorithms represent significant advances in the best known runtimes for these problems. They leverage the positive definiteness of the kernel matrix, along with a recent line of work on efficient kernel density estimation.

We initiate the study of biologicallyinspired spiking neural networks from the perspective of streaming algorithms. Like computers, human brains face memory limitations, which pose a significant obstacle when processing large scale and dynamically changing data. In computer science, these challenges are captured by the wellknown streaming model, which can be traced back to Munro and Paterson `78 and has had significant impact in theory and beyond. In the classical streaming setting, one must compute a function f of a stream of updates š® = {uā,ā¦,u_m}, given restricted singlepass access to the stream. The primary complexity measure is the space used by the algorithm. In contrast to the large body of work on streaming algorithms, relatively little is known about the computational aspects of data processing in spiking neural networks. In this work, we seek to connect these two models, leveraging techniques developed for streaming algorithms to better understand neural computation. Our primary goal is to design networks for various computational tasks using as few auxiliary (noninput or output) neurons as possible. The number of auxiliary neurons can be thought of as the "space" required by the network. Previous algorithmic work in spiking neural networks has many similarities with streaming algorithms.more »