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In a geometric $k$-clustering problem the goal is to partition a set of points in $\mathbb{R}^d$ into $k$ subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering…
We present a classical algorithm that, for any 3D geometrically-local, polylogarithmic-depth quantum circuit $C$ acting on $n$ qubits, and any bit string $x\in\{0,1\}^n$, can compute the quantity $|< x |C|0^{\otimes n}>|^2$ to within any…
Clustering high-dimensional datasets is hard because interpoint distances become less informative in high-dimensional spaces. We present a clustering algorithm that performs nonlinear dimensionality reduction and clustering jointly. The…
We study approximation of embeddings between finite dimensional L_p spaces in the quantum model of computation. For the quantum query complexity of this problem matching (up to logarithmic factors) upper and lower bounds are obtained. The…
We initiate the rigorous study of classification in semimetric spaces, which are point sets with a distance function that is non-negative and symmetric, but need not satisfy the triangle inequality. For metric spaces, the doubling dimension…
We study the optimal design problems where the goal is to choose a set of linear measurements to obtain the most accurate estimate of an unknown vector in $d$ dimensions. We study the $A$-optimal design variant where the objective is to…
First, we study geometric variants of the standard set cover motivated by assignment of directional antenna and shipping with deadlines, providing the first known polynomial-time exact solutions. Next, we consider the following general…
Clustering multi-dimensional points is a fundamental task in many fields, and density-based clustering supports many applications as it can discover clusters of arbitrary shapes. This paper addresses the problem of Density-Peaks Clustering…
A data structure is presented for point location in connected planar subdivisions when the distribution of queries is known in advance. The data structure has an expected query time that is within a constant factor of optimal. More…
Dimension reduction is often the first step in statistical modeling or prediction of multivariate spatial data. However, most existing dimension reduction techniques do not account for the spatial correlation between observations and do not…
Finding a maximum or minimum is a fundamental building block in many mathematical models. Compared with classical algorithms, Durr, Hoyer's quantum algorithm (DHA) achieves quadratic speed. However, its key step, the quantum exponential…
Variational quantum algorithms are the centerpiece of modern quantum programming. These algorithms involve training parameterized quantum circuits using a classical co-processor, an approach adapted partly from classical machine learning.…
Quantum Interior Point Methods (QIPMs) have been attracting significant interests recently due to their potential of solving optimization problems substantially faster than state-of-the-art conventional algorithms. In general, QIPMs use…
This paper studies the statistical query (SQ) complexity of estimating $d$-dimensional submanifolds in $\mathbb{R}^n$. We propose a purely geometric algorithm called Manifold Propagation, that reduces the problem to three natural geometric…
We investigate algorithms with predictions in computational geometry, specifically focusing on the basic problem of computing 2D Delaunay triangulations. Given a set $P$ of $n$ points in the plane and a triangulation $G$ that serves as a…
The task of approximating a function of d variables from its evaluations at a given number of points is ubiquitous in numerical analysis and engineering applications. When d is large, this task is challenged by the so-called curse of…
Recent progress in deep learning has been driven by increasingly larger models. However, their computational and energy demands have grown proportionally, creating significant barriers to their deployment and to a wider adoption of deep…
Many quantum algorithms can be analyzed in a query model to compute Boolean functions where input is given by a black box. As in the classical version of decision trees, different kinds of quantum query algorithms are possible: exact,…
We consider a general task called partial Wasserstein covering with the goal of providing information on what patterns are not being taken into account in a dataset (e.g., dataset used during development) compared with another dataset(e.g.,…
In this paper, we provide a comprehensive rigorous modeling for multidimensional spaces with hierarchically structured dimensions in several layers of abstractions and data cubes that live in such spaces. We model cube queries and their…