James Saunderson

Spectral clustering with inconsistent advice

Clustering with advice (often known as constrained clustering) has been a recent focus of the data mining community. Success has been achieved incorporating advice into the k-means and spectral clustering frameworks. Although the theory community has explored inconsistent advice, it has not yet been incorporated into spectral clustering. Extending work of De Bie and Cristianini, we set out a framework for finding minimum normalised cuts, subject to inconsistent advice.

Diagonal and Low-Rank Matrix Decompositions, Correlation Matrices, and Ellipsoid Fitting

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix

SEMIDEFINITE DESCRIPTIONS OF THE CONVEX HULL OF ROTATION MATRICES

X

Equivariant Semidefinite Lifts and Sum-of-Squares Hierarchies

formed as the sum of an unknown diagonal matrix and an unknown low-rank positive semidefinite matrix, decompose

A Local-Search 2-Approximation for 2-Correlation-Clustering

X

Sparse sum-of-squares certificates on finite abelian groups

into these constituents. The second problem we consider is to determine the facial structure of the set of correlation matrices, a convex set also known as the elliptope. This convex body, and particularly its facial structure, plays a role in applications from combinatorial optimization to mathematical finance. The third problem is a basic geometric question: given points

Semidefinite relaxations for optimization problems over rotation matrices

v_1,v_2,\ldots,v_n\in \mathbb{R}^k

Sparse sums of squares on finite abelian groups and improved semidefinite lifts

(where

Polynomial-sized semidefinite representations of derivative relaxations of spectrahedral cones

n > k

Equivariant Semidefinite Lifts of Regular Polygons

) determine whether there is a centered ellipsoid passing exactly through all the points. We show that in a precise sense these three problems are equivalent. Furthermore we establish a simple sufficient condition on a su...