Given a set of n points in ℜ d, a family of shapes S and a number of clusters k, the projective clustering problem is to find a collection of k shapes in S such that the maximum distance from a point to its nearest shape is minimized. Some special cases of the problem include the k-line center problem where the goal is to cover the points with minimum radius hypercylinders and the k-hyperplane center problem where the goal is to cover the points with minimum width slabs. In practice, projective clustering algorithms are often used as a dimension reduction technique to enable more effective data representation for indexing and data mining purposes on massively large datasets (See, for example, [8, 9]). In typical applications the number of points n is extremely large, the dimensionality d is large, the data possesses some outliers, while the number of clusters, k is small. Consequently, the emphasis of this paper will be on the running times of the algorithms. We present for the first time sublinear time randomized algorithms for the k-line and hyperplane center problems, where the running times of our algorithms are independent of n. Related Work Both the k-line and k-hyperplane center problems are computationally difficult to solve in any exact or single-criteria approximation sense. Megiddo and Tamir [7] show that it is NP-hard to decide whether a set of points in the plane can be covered by k lines, i.e., cylinders with

Abstract. We consider problems on data sets where each data point has uncertainty described by an individual probability distribution. We develop several frameworks and algorithms for calculating statistics on these uncertain data sets. Our examples focus on geometric shape fitting problems. We prove approximation guarantees for the algorithms with respect to the full probability distributions. We then empirically demonstrate that our algorithms are simple and practical, solving for a constant hidden by asymptotic analysis so that a user can reliably trade speed and size for accuracy. 1

We consider the problem of approximating a set P of n points in R d by a collection of j-dimensional flats, and extensions thereof, under the standard median / mean / center measures, in which we wish to minimize, respectively, the sum of the distances from each point of P to its nearest flat, the sum of the squares of these distances, or the maximal such distance. Such problems cannot be approximated unless P=NP but do allow bi-criteria approximations where one allows some leeway in both the number of flats and the quality of the objective function. We give a very simple bi-criteria approximation algorithm, which produces at most α(k, j, n) = log n · (jk log log n) O(j) flats, which exceeds the optimal objective value for any k j-dimensional flats by a factor of no more than β(j) = 2 O(j). Given this bi-criteria approximation, we can use it to reduce the approximation factor arbitrarily, at the cost of increasing the number of flats. Our algorithm has many advantages over previous work, in that it is much more widely applicable (wider set of objective functions and classes of clusters) and much more efficient — reducing the running time bound from O(n poly(k,j) ) to dn · (jk) O(j). Our algorithm is randomized and successful with probability 1/2 (easily boosted to probabilities arbitrarily close to 1). ∗ Supported by the German-Israel Foundation for Scientific Research and Development.

We consider parameterized convex optimization problems over the unit simplex, that depend on one parameter. We provide a simple and efficient scheme for maintaining an ε-approximate solution (and a corresponding ε-coreset) along the entire parameter path. We prove correctness and parameterized optimization problem are for example regularization paths of support vector machines, multiple kernel learning, and minimum enclosing balls of moving points. 1

“And so ended Svejk’s Budejovice anabasis. It is certain that if Svejk had been granted liberty of movement he would have got to Budejovice on his own. However much the authorities may boast that it was they who brought Svejk to his place of duty, this is nothing but a mistake. With Svejk energy and irresistible desire to fight, the authorities action was like throwing a spanner into the works.” – – The good soldier Svejk, Jaroslav Hasek 23.1 Covering problems, expansion and shell sets Consider a set P of n points in IR d, that we are interested in covering by the best shape in a family of shapes F. For example, F might be the set of all balls in IR d, and we are looking for the minimum enclosing ball of P. A ε-coreset S ⊆ P would guarantee that any ball that covers S will cover the whole point set if we expand it by (1 + ε). However, sometimes, computing the coreset is computationally expensive, the coreset does not exist at all, or its size is prohibitively large. It is still natural to look for a small subset S of the points, such that finding the optimal solution for S generates (after appropriate expansion) an approximate solution to the original problem. Definition 23.1.1 (Shell sets) Given a set P of points (or geometric objects) in IR d, and F be a family of shapes in IR d. Let f: F → IR be a target optimization function, and assume that there is a natural expansion operation defined over F. Namely, given a set r ∈ F, one can compute a set (1 + ε)r which is the expansion of r by a factor of 1 + ε. In particular, we would require that f ((1 + ε)r) ≤ (1 + ε) f (r). Let f opt(P) = minr∈F,P⊆r f (r) be the shape in F that bests fits P. Furthermore, assume that f opt(·) is a monotone function, that is for A ⊆ B ⊆ P we have f opt(A) ≤ f opt(B). A subset S ⊆ P is a ε-shell set for P, if SlowAlg on a set B that contains S, if the range r returned by SlowAlg(S) covers S, (1 + ε)r covers P, and f (r) ≤ (1 + ε) f opt(S). Namely, the range (1 + ε)r is an (1 + ε)-approximation to the optimal range of F covering P.

Ignored exact dependency on ε. (I don’t care, you shouldn’t care, nobody should care.) Show ideas, cares less for fastest known results For more exact info, see survey (AHV04a). Tried to give insight into ideas instead of being exhaustive. Meta claims would be proven by example. Coresets – p.1/127Part I

“And so ended Svejk’s Budejovice anabasis. It is certain that if Svejk had been granted liberty of movement he would have got to Budejovice on his own. However much the authorities may boast that it was they who brought Svejk to his place of duty, this is nothing but a mistake. With Svejk energy and irresistible desire to fight, the authorities action was like throwing a spanner into the works.” – – The good soldier Svejk, Jaroslav Hasek 24.1 Covering problems, expansion and shell sets Consider a set P of n points in IR d, that we are interested in covering by the best shape in a family of shapes F. For example, F might be the set of all balls in IR d, and we are looking for the minimum enclosing ball of P. A ε-coreset S ⊆ P would guarantee that any ball that covers S will cover the whole point set if we expand it by (1 + ε). However, sometimes, computing the coreset is computationally expensive, the coreset does not exist at all, or its size is prohibitively large. It is still natural to look for a small subset S of the points, such that finding the optimal solution for S generates (after appropriate expansion) an approximate solution to the original problem. Definition 24.1.1 (Shell sets) Given a set P of points (or geometric objects) in IR d, and F be a

“And so ended Svejk’s Budejovice anabasis. It is certain that if Svejk had been granted liberty of movement he would have got to Budejovice on his own. However much the authorities may boast that it was they who brought Svejk to his place of duty, this is nothing but a mistake. With Svejk energy and irresistible desire to fight, the authorities action was like throwing a spanner into the works.” – – The good soldier Svejk, Jaroslav Hasek 24.1 Covering problems, expansion and shell sets Consider a set P of n points in IR d, that we are interested in covering by the best shape in a family of shapes F. For example, F might be the set of all balls in IR d, and we are looking for the minimum enclosing ball of P. A ε-coreset S ⊆ P would guarantee that any ball that covers S will cover the whole point set if we expand it by (1 + ε). However, sometimes, computing the coreset is computationally expensive, the coreset does not exist at all, or its size is prohibitively large. It is still natural to look for a small subset S of the points, such that finding the optimal solution for S generates (after appropriate expansion) an approximate solution to the original problem. Definition 24.1.1 (Shell sets) Given a set P of points (or geometric objects) in IR d, and F be a

Abstract not found

Frieze, Kannan, and Vempala (JACM 2004) proved that a small sample of rows of a given matrix A spans the rows of a low-rank approximation D that minimizes �A−D�F within a small additive error, and the sampling can be done efficiently using just two passes over the matrix. 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 (adaptive sampling). Using this result, we give a pass-efficient algorithm for computing a low-rank approximation with reduced additive error. Our second result is that there exist k rows of A whose span contains the rows of a multiplicative (k + 1)-approximation to the best rank-k matrix; moreover, this subset can be found by sampling k-subsets of rows from a natural distribution (volume sampling). Combining volume sampling with adaptive sampling yields the existence of a set of k + k(k + 1)/ε rows whose span contains the rows of a multiplicative (1 + ε)-approximation. This leads to a PTAS for the following NP-hard