# Mathematics Colloquia and Seminars

Several challenging problem in clustering, partitioning and imaging have traditionally been solved using the spectral technique". These problems include the normalized cut problem, the graph expander ratio problem, the Cheeger constant problem and the conductance problem. These problems share several common features: all seek a bipartition of a set of elements; the problems are formulated as a form of ratio cut; the formulation as discrete optimization is equivalent to a quadratic ratio, sometimes referred to as the isoperimetric or Raleigh ratio, on discrete variables and a single sum constraint which we call the balance or orthogonality constraint; when the discrete nature of the variables is disregarded, the continuous relaxation is solved by the spectral method. Indeed the spectral relaxation technique is a dominant method providing an approximate solution to these problems.