Early termination in Ben-Or/Tiwari sparse interpolation and a hybrid of Zippel's algorithm

title: Symbolic-numeric sparse interpolation of multivariate polynomials
abstract: 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.

title: Algorithms for computing sparsest shifts of polynomials in power, Chebyshev, and Pochhammer bases
abstract: 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.

title: Algorithms for computing the sparsest shifts of polynomials via the Berlekamp/Massey algorithm
abstract: 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  can be obtained from the subresultant PRS algorithm . Dornstetter  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.

title: Early termination in sparse interpolation algorithms
abstract: A probabilistic strategy, early termination, enables different interpolation algorithms to adapt to the degree or the number of terms in the target polynomial when neither is supplied in the input. In addition to dense algorithms, we implement this strategy in sparse interpolation algorithms. Based on early termination, racing algorithms execute simultaneously dense and sparse algorithms. The racing algorithms can be embedded as the univariate interpolation substep within Zippel's multivariate method. In addition, we experimentally verify some heuristics of early termination, which make use of thresholds and post-verification.

title: Tiling and adaptive image compression
abstract: We investigate the task of compressing an image by using different probability models for compressing different regions of the image. In this task, using a larger number of regions would result in better compression, but would also require more bits for describing the regions and the probability models used in the regions. We discuss using quadtree methods for performing the compression. We introduce a class of probability models for images, the k-rectangular tilings of an image, that is formed by partitioning the image into k rectangular regions and generating the coefficients within each region by using a probability model selected from a finite class of N probability models. For an image of size n/spl times/n, we give a sequential probability assignment algorithm that codes the image with a code length which is within O(k log(Nn/k) of the code length produced by the best probability model in the class. The algorithm has a computational complexity of O(Nn/sup 3/). An interesting subclass of the class of k-rectangular tilings is the class of tilings using rectangles whose widths are powers of two. This class is far more flexible than quadtrees and yet has a sequential probability assignment algorithm that produces a code length that is within O(k log(Nn/k) of the best model in the class with a computational complexity of O(Nn/sup 2/logn) (similar to the computational complexity of sequential probability assignment using quadtrees). We also consider progressive transmission of the coefficients of the image.

title: A Theoretical Analysis of Query Selection for Collaborative Filtering
abstract: We 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 and fast algorithm for a theoretical model of this problem with a provable approximation guarantee, and prove that solving the problem exactly is NP-Hard.

title: Statistical Causality Analysis of INFOSEC Alert Data
abstract: With the increasingly widespread deployment of security mechanisms, such as firewalls, intrusion detection systems (IDSs), antiviras software and authentication services, the problem of alert analysis has become very important. The large amount of alerts can overwhelm security administrators and prevent them from adequately understanding and analyzing the security state of the network, and initiating appropriate response in a timely fashion. Recently, several approaches for alert correlation and attack scenario analysis have been proposed. However, these approaches all have limited capabilities in detecting new attack scenarios. In this paper, we study the problem of security alert correlation with an emphasis on attack scenario analysis. In our framework, we use clustering techniques to process low-level alert data into high-level aggregated alerts, and conduct causal analysis based on statistical tests to discover new relationships among attacks. Our statistical causality approach complements other approaches that use hard-coded prior knowledge for pattern matching. We perform a series of experiments to validate our method using DARPA’s Grand Challenge Problem (GCP) datasets, the 2000 DARPA Intrusion Detection Scenario datasets, and the DEF CON 9 datasets. The results show that our approach can discover new patterns of attack relationships when the alerts of attacks are statistically correlated.

title: MPEG-4 compatible faces from orthogonal photos
abstract: MPEG-4 is scheduled to become an international standard in March 1999. The paper demonstrates an experiment for a virtual cloning method and animation system, which is compatible with the MPEG-4 standard facial object specification. Our method uses orthogonal photos (front and side view) as input and reconstructs the 3D facial model. The method is based on extracting MPEG-4 face definition parameters (FDP) from photos, which initializes a custom face in a more capable interface, and deforming a generic model. Texture mapping is employed using an image composed of the two orthogonal images, which is done completely automatically. A reconstructed head can be animated immediately inside our animation system, which is adapted to the MPEG-4 standard specification of face animation parameters (FAP). The result is integrated into our virtual human director (VHD) system.

title: Genetic Algorithmic Topology Control for Two-Tiered Wireless Sensor Networks
abstract: This paper proposes an optimized topology control scheme (GA-OTC) based on a genetic algorithm for clustering-based hierarchical wireless sensor networks (WSNs). By using a genetic algorithm, we can obtain optimal solutions to multiple objective functions according to the two criteria of both balanced energy consumption and minimized total energy consumption of cluster heads while the entire WSNs field is covered by clusters. Through performance evaluation studies, we show that GA-OTCachieves desirable properties.