We study algorithms for approximating pairwise similarity matrices that arise in natural language processing. Generally, computing a matrix n data points requires Omega(n^2) computations. This quadratic scaling is significant bottleneck, especially when similarities are computed via expensive functions, e.g., transformer models. Approximation methods reduce this complexity, often by using small...

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

We consider the case of ranking a very large set of labels, items, or documents, which is common to information retrieval, recommendation, and large-scale annotation tasks. We present a general approach for converting an algorithm which has linear time in the size of the set to a sublinear one via label partitioning. Our method consists of learning an input partition and a label assignment to e...

Proof Idea For any two specific inputs there exists an algorithm that is adapted to these inputs. For example, if we take the two inputs x = (1, 2, ..., n) as a “yes” example, and y = (1, 1, 2, 2, ..., n, n, n + 1, n + 2, ..., n(1 − )) as a “no” example, then an algorithm that checks the second coordinate can distinguish between them. Therefore, we look on distributions over inputs. We need two...

We study the problem of estimating the best k term Fourier representation for a given frequency-sparse signal (i.e., vector) A of length N k. More explicitly, we investigate how to deterministically identify k of the largest magnitude frequencies of Â, and estimate their coefficients, in polynomial(k, log N) time. Randomized sublinear time algorithms which have a small (controllable) probabilit...

The central aim of most fields of data analysis and experimental scientific investigation is to draw valid conclusions from a given data set. But when past inferences guide future inquiries into the same dataset, reaching valid inferences becomes significantly more difficult. In addition to avoiding the overfitting that can result from adaptive analysis, a data analyst often wants to use as lit...

We prove that the cycle–convergence of the restarted GMRES applied to a system of linear equations with a normal coefficient matrix is sublinear.

Given a set of n points in <d, a family of shapes S and a number of clusters k, the projective clustering problem is to find a collection of k shapes in S such that the maximum distance from a point to its nearest shape is minimized. Some special cases of the problem include the k-line center problem where the goal is to cover the points with minimum radius hypercylinders and the k-hyperplane c...