The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard, because it contains vector cardinality minimization as a special case. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to solving the norm minimization relaxations, and illustrate our results with numerical examples.

Abstract. We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix. 1.

Abstract. We present new algorithms for computing the values of the Schur sλ(x1,x2,...,xn)andJackJ α λ (x1,x2,...,xn) functions in floating point arithmetic. These algorithms deliver guaranteed high relative accuracy for positive data (xi,α>0) and run in time that is only linear in n. 1.

Let {(Zi,Wi) : i = 1,...,n} be uniformly distributed in [0,1] d × G(k,d), where G(k,d) denotes the space of k-dimensional linear subspaces of R d. For a differentiable function f: [0,1] k → [0,1] d, we say that f interpolates (z,w) ∈ [0,1] d × G(k,d) if there exists x ∈ [0,1] k such that f(x) = z and ⃗ f(x) = w, where ⃗ f(x) denotes the tangent space at x defined by f. For a smoothness class F of Hölder type, we obtain probability bounds on the maximum number of points a function f ∈ F interpolates. 1

Abstract. We present explicit formulas for the distributions of the extreme eigenvalues of the β–Jacobi random matrix ensemble in terms of the hypergeometric function of a matrix argument. For β =1, 2, 4, these formulas specialize to the well-known real, complex, and quaternion Jacobi ensembles, respectively.

Abstract. Spectral Clustering has reached a wide level of diffusion among unsupervised learning applications. Despite its practical success we believe that for a correct usage one has to face a difficult problem: given a target number of classes K the optimal K-dimensional subspace is not necessarily spanned by the first K eigenvectors of the graph Normalized Laplacian. The contribution of this paper is twofold. First, we show a bound for choosing a correct number of eigenvectors. Second, we propose a randomized spectral algorithm able to find a clustering solution. We show the efficacy of the algorithm with experiments on real world graphs. Our proposal is a scheme that naturally extends the current usage of Spectral Clustering. 1