In this paper we look at the classical set-maxima problem. We give a new result using a sparse lattice of subsets. We then extend this to a geometric setting. Let @math be a set of @math points in a plane and @math be a set of convex regions. Let a key value be assigned to each point. The problems is to determine for each convex region of @math the point having the largest key. We give a parameterized result for the comparison complexity.
@cite_2 gave the first general algorithm. Their rank-sequence algorithm'' determines a rank sequence @math according to the application domain. Specifically a rank sequence is an ordered sequence of @math ranks @math . The corresponding partition of @math is computed. Each @math is reduced to just those elements in on block of the partition. When the elements are points and the sets are hyperplanes that form a projective space, it can be computed with linear comparisons, for a suitable rank sequence. However, the rank-sequence algorithm is no better than the trivial algorithm above in the worst case. It was shown by @cite_1 that for some collection of subsets there are no good rank sequence for which the number of comparisons made by the algorithm is linear.
@cite_1 @cite_2
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The probability that a user will click a search result depends both on its relevance and its position on the results page. The position based model explains this behavior by ascribing to every item an attraction probability, and to every position an examination probability. To be clicked, a result must be both attractive and examined. The probabilities of an item-position pair being clicked thus form the entries of a rank- @math matrix. We propose the learning problem of a Bernoulli rank- @math bandit where at each step, the learning agent chooses a pair of row and column arms, and receives the product of their Bernoulli-distributed values as a reward. This is a special case of the stochastic rank- @math bandit problem considered in recent work that proposed an elimination based algorithm Rank1Elim, and showed that Rank1Elim's regret scales linearly with the number of rows and columns on "benign" instances. These are the instances where the minimum of the average row and column rewards @math is bounded away from zero. The issue with Rank1Elim is that it fails to be competitive with straightforward bandit strategies as @math . In this paper we propose Rank1ElimKL which simply replaces the (crude) confidence intervals of Rank1Elim with confidence intervals based on Kullback-Leibler (KL) divergences, and with the help of a novel result concerning the scaling of KL divergences we prove that with this change, our algorithm will be competitive no matter the value of @math . Experiments with synthetic data confirm that on benign instances the performance of Rank1ElimKL is significantly better than that of even Rank1Elim, while experiments with models derived from real data confirm that the improvements are significant across the board, regardless of whether the data is benign or not. We study random submatrices of a large matrix A. We show how to approximately compute A from its random submatrix of the smallest possible size O(rlog r) with a small error in the spectral norm, where r e VAV2F VAV22 is the numerical rank of A. The numerical rank is always bounded by, and is a stable relaxation of, the rank of A. This yields an asymptotically optimal guarantee in an algorithm for computing low-rank approximations of A. We also prove asymptotically optimal estimates on the spectral norm and the cut-norm of random submatrices of A. The result for the cut-norm yields a slight improvement on the best-known sample complexity for an approximation algorithm for MAX-2CSP problems. We use methods of Probability in Banach spaces, in particular the law of large numbers for operator-valued random variables. Given a matrix A e ℝ m ×n of rank r, and an integer k < r, the top k singular vectors provide the best rank-k approximation to A. When the columns of A have specific meaning, it is desirable to find (provably) "good" approximations to A k which use only a small number of columns in A. Proposed solutions to this problem have thus far focused on randomized algorithms. Our main result is a simple greedy deterministic algorithm with guarantees on the performance and the number of columns chosen. Specifically, our greedy algorithm chooses c columns from A with @math such that @math where C gr is the matrix composed of the c columns, @math is the pseudo-inverse of C gr ( @math is the best reconstruction of A from C gr), and ¼(A) is a measure of the coherence in the normalized columns of A. The running time of the algorithm is O(SVD(A k) + mnc) where SVD(A k) is the running time complexity of computing the first k singular vectors of A. To the best of our knowledge, this is the first deterministic algorithm with performance guarantees on the number of columns and a (1 + e) approximation ratio in Frobenius norm. The algorithm is quite simple and intuitive and is obtained by combining a generalization of the well known sparse approximation problem from information theory with an existence result on the possibility of sparse approximation. Tightening the analysis along either of these two dimensions would yield improved results. A simple partitioning algorithm for merging two disjoint linearly ordered sets is given, and an upper bound on the average number of comparisons required is established. The upper bound is @math , where n is the number of elements in the larger of the two sets, m the number of the smaller, and @math . An immediate corollary is that any sorting problem can be done with an average number of comparisons within @math of the information theoretic bound using repeated merges; it does not matter what the merging order used is. Although the provable bound is @math over the lower bound, computations indicate that the algorithm will asymptotically make only @math more comparisons than the lower bound. The algorithm is compared with the Hwang-Lin algorithm, and modifications to improve average efficiency of this well known algorithm are given.