Luis Rademacher

Matrix approximation and projective clustering via volume sampling

Frieze et al. [17] proved that a small sample of rows of a given matrix A contains a low-rank approximation D that minimizes ||A - D||F to within small additive error, and the sampling can be done efficiently using just two passes over the matrix [12]. In this paper, we generalize this result in two ways. First, we prove that the additive error drops exponentially by iterating the sampling in an adaptive manner. Using this result, we give a pass-efficient algorithm for computing low-rank approximation with reduced additive error. Our second result is that using a natural distribution on subsets of rows (called volume sampling), there exists a subset of k rows whose span contains a factor (k + 1) relative approximation and a subset of k + k(k + 1)/e rows whose span contains a 1+e relative approximation. The existence of such a small certificate for multiplicative low-rank approximation leads to a PTAS for the following projective clustering problem: Given a set of points P in Rd, and integers k, j, find a set of j subspaces F 1 , . . ., F j , each of dimension at most k, that minimize Σ p∈P min i d(p, F i )2.

Efficient Volume Sampling for Row/Column Subset Selection

We give efficient algorithms for volume sampling, i.e., for picking

Matrix Approximation and Projective Clustering via Volume Sampling

k

Approximating the centroid is hard

-subsets of the rows of any given matrix with probabilities proportional to the squared volumes of the simplices defined by them and the origin (or the squared volumes of the parallelepipeds defined by these subsets of rows). %In other words, we can efficiently sample

Expanders via Random Spanning Trees

k

Testing geometric convexity

-subsets of

On the monotonicity of the expected volume of a random simplex

[m]

Dispersion of Mass and the Complexity of Randomized Geometric Algorithms

with probabilities proportional to the corresponding

Lower Bounds for the Average and Smoothed Number of Pareto-Optima

k

A SIMPLICIAL POLYTOPE THAT MAXIMIZES THE ISOTROPIC CONSTANT MUST BE A SIMPLEX

by