2559830
Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm
As a sub-procedure our algorithm executes the Berlekamp/Massey algorithm on a sequence of large integers or polynomials. We give a fraction-free version of the Berlekamp/Massey algorithm, which does not require rational numbers or functions and GCD operations on the arising numerators and denominators. The relationship between the solution of Toeplitz systems, Padé approximations, and the Euclidean algorithm is classical. Fraction-free versions [3] can be obtained from the subresultant PRS algorithm [2]. Dornstetter [6] gives an interpretation of the Berlekamp/Massey algorithm as a partial extended Euclidean algorithm. We map the subresultant PRS algorithm onto Dornstetter's formulation. We note that the Berlekamp/Massey algorithm is more efficient than the classical extended Euclidean algorithm.
86127
Symbolic-numeric sparse interpolation of multivariate polynomials
We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating-point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or/Tiwari sparse interpolation algorithm and the classical Prony's method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony's method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications.
7994736
Algorithms for computing sparsest shifts of polynomials in power, Chebyshev, and Pochhammer bases
We give a new class of algorithms for computing sparsest shifts of a given polynomial. Our algorithms are based on the early termination version of sparse interpolation algorithms: for a symbolic set of interpolation points, a sparsest shift must be a root of the first possible zero discrepancy that can be used as the early termination test. Through reformulating as multivariate shifts in a designated set, our algorithms can compute the sparsest shifts that simultaneously minimize the terms of a given set of polynomials. Our algorithms can also be applied to the Pochhammer and Chebyshev bases for the polynomials, and potentially to other bases as well. For a given univariate polynomial, we give a lower bound for the optimal sparsity. The efficiency of our algorithms can be further improved by imposing such a bound and pruning the highest degree terms.
7026830
A fully distributed spatial index for wireless data broadcast
To support location-based services in wireless data broadcast systems, a distributed spatial index (called DSI) is proposed in this paper. DSI is highly efficient because it has a linear yet fully distributed structure that naturally facilitates multiple replications of the index by sharing links in different search trees. Search algorithms for point queries, window queries, and kNN queries, based on DSI are presented. Empirical evaluation of DSI are conducted. Result shows that DSI significantly out-performs R-tree and Hilbert curve index, two state-of-the-art spatial indexing techniques for wireless data broadcast.
1489685
Optimal channel allocation for data dissemination in mobile computing environments
We discuss the wireless channel allocation problem for data dissemination in mobile computing systems. Methods for accessing data through broadcast and on-demand channels are described. We provide analytical models and cost formulae for the exclusive broadcast channels and the exclusive on-demand channels and propose a dynamic channel allocation algorithm for optimizing system performance. Our performance evaluation shows that dynamic channel allocation significantly improves system performance and the channel allocation algorithm gives us the optimal solution for various system parameter settings.
3113906
A Theoretical Analysis of Query Selection for Collaborative Filtering
AbstractWe consider the problem of determining which of a set of experts has tastes most similar to a given user by asking the user questions about his likes and dislikes. We describe a simple algorithm for generating queries for a theoretical model of this problem. We show that the algorithm requires at most opt(F)(ln(|F|/opt(F)) + 1) + 1 queries to find the correct expert, where opt(F) is the optimal worst-case bound on the number of queries for learning arbitrary elements of the set of experts F. The algorithm runs in time polynomial in |F| and |X| (where X is the domain) and we prove that no polynomial-time algorithm can have a significantly better bound on the number of queries unless all problems in NP have nO(log log n) time algorithms. We also study a more general case where the user ratings come from a finite set Y and there is an integer-valued loss function ℓ on Y that is used to measure the distance between the ratings. Assuming that the loss function is a metric and that there is an expert within a distance η from the user, we give a polynomial-time algorithm that is guaranteed to find such an expert after at most 2opt(F, η) ln 
$$\tfrac{{\left| F \right|}}{{1 + \deg (F,\eta )}}$$
 + 2(η + 1)(1 + deg(F, η)) queries, where deg(F, η) is the largest number of experts in F that are within a distance 2η of any f ∈ F.
7965599
q-gram matching using tree models
q-gram matching is used for approximate substring matching problems in a wide range of application areas, including intrusion detection. In this paper, we present a tree-based model to perform fast linear time q-gram matching. All q-grams present in the text are stored in a tree structure similar to trie. We use a tree redundancy pruning algorithm to reduce the size of the tree without losing any information. We also use suffix links for fast q-gram search during query matching. We compare our work with the Rabin-Karp-based hash-table technique, commonly used for multiple q-gram search. We present results of experiments on system call sequence data used for intrusion detection.
9023584
Text classification with kernels on the multinomial manifold
Support Vector Machines (SVMs) have been very successful in text classification. However, the intrinsic geometric structure of text data has been ignored by standard kernels commonly used in SVMs. It is natural to assume that the documents are on the multinomial manifold, which is the simplex of multinomial models furnished with the Riemannian structure induced by the Fisher information metric. We prove that the Negative Geodesic Distance (NGD) on the multinomial manifold is conditionally positive definite (cpd), thus can be used as a kernel in SVMs. Experiments show the NGD kernel on the multinomial manifold to be effective for text classification, significantly outperforming standard kernels on the ambient Euclidean space.