Learning mixtures of structured distributions over discrete domains

We study the performance of anonymous posted-price selling mechanisms for a standard Bayesian auction setting, where n bidders have i.i.d. valuations for a single item. We show that for the natural class of Monotone Hazard Rate (MHR) distributions, offering the same, take-it-or-leave-it price to all bidders can achieve an (asymptotically) optimal revenue. In particular, the approximation ratio is shown to be 1+ O (ln ln n / ln n ), matched by a tight lower bound for the case of exponential distributions. This improves upon the previously best-known upper bound of e /( e −1)≈ 1.58 for the slightly more general class of regular distributions. In the worst case (over n ), we still show a global upper bound of 1.35. We give a simple, closed-form description of our prices, which, interestingly enough, relies only on minimal knowledge of the prior distribution, namely, just the expectation of its second-highest order statistic. Furthermore, we extend our techniques to handle the more general class of λ-regular distributions that interpolate between MHR (λ =0) and regular (λ =1). Our anonymous pricing rule now results in an asymptotic approximation ratio that ranges smoothly, with respect to λ, from 1 (MHR distributions) to e /( e −1) (regular distributions). Finally, we explicitly give a class of continuous distributions that provide matching lower bounds, for every λ.

Consider a seller who seeks to provide service to a collection of interested parties, subject to feasibility constraints on which parties may be simultaneously served. Assuming that a distribution is known on the value of each party for service---arguably a strong assumption---Myerson's seminal work provides revenue optimizing auctions [12]. We show instead that, for very general feasibility constraints, only knowledge of the median of each party's value distribution, or any other quantile of these distributions, or approximations thereof, suffice for designing simple auctions that simultaneously approximate both the optimal revenue and the optimal welfare. Our results apply to all downward-closed feasibility constraints under the assumption that the underlying, unknown value distributions are monotone hazard rate, and to all matroid feasibility constraints under the weaker assumption of regularity of the underlying distributions. Our results jointly generalize the single-item results obtained by Azar and Micali [2] on parametric auctions, and Daskalakis and Pierrakos [6] for simultaneously approximately optimal and efficient auctions.

Algebraic coding theory is one of the areas that routinely gives rise to computational problems involving various structured matrices, such as Hankel, Vandermonde, Cauchy matrices, and certain generalizations thereof. Their structure has often been used to derive efficient algorithms; however, the use of the structure was pattern-specific, without applying a unified technique. In contrast, in several other areas, where structured matrices are also widely encountered, the concept of displacement rank was found to be useful to derive efficient algorithms in a unified manner (i.e., not depending on a particular pattern of structure). The latter technique allows one to compress, in a unified way, different types of n × n structured matrices to only αn parameters. This typically leads to computational savings (in many applications the number α, called the displacement rank, is a small fixed constant).In this paper we demonstrate the power of the displacement structure approach by deriving in a unified way efficient algorithms for a number of decoding problems. We accelerate the Sudan's list decoding algorithm for Reed-Solomon codes, its generalization to algebraic-geometric codes by Shokrollahi and Wasserman, and the improvement of Guruswami and Sudan in the case of Reed-Solomon codes. In particular, we notice that matrices that occur in the context of list decoding have low displacement rank, and use this fact to derive algorithms that use O(n2 l) and O(n7/3 l) operations over the base field for list decoding of Reed-Solomon codes and algebraic-geometric codes from certain plane curves, respectively. Here l denotes the length of the list; assuming that l is constant, this gives algorithms of running time O(n2) and O(n7/3), which is the same as the running time of conventional decoding algorithms. We also present efficient parallel algorithms for the above tasks.To the best of our knowledge this is the first application of the concept of displacement rank to the unified derivation of several decoding algorithms; the technique can be useful in finding efficient and fast methods for solving other decoding problems.

A complete and efficient algorithm is proposed for searching form-closure grasps of n hard fingers on the surface of a three-dimensional object represented by discrete points. Both frictional and frictionless cases are considered. This algorithm starts to search a form-closure grasp from a randomly selected grasp using an efficient local search procedure until encountering a local minimum. The local search procedure employs the powerful ray-shooting technique to search in the direction of reducing the distance between the convex hull corresponding to the grasp and the origin of the wrench space. When the distance reaches a local minimum in the local search procedure, the algorithm decomposes the problem into a few subproblems in subsets of the points according to the existence conditions of form-closure grasps. A search tree whose root represents the original problem is employed to perform the searching process. The subproblems are represented as children of the root node and the same procedure is recursively applied to the children. It is proved that the search tree generates O(KlnK/n) nodes in case a from-closure grasp exists, where K is the number of the local minimum points of the distance in the grasp space and n is the number of fingers. Compared to the exhaustive search, this algorithm is more efficient, and, compared to other heuristic algorithms, the proposed algorithm is complete in the discrete domain. The efficiency of this algorithm is demonstrated by numerical examples.

This article presents applicable bounds for distributions with monotone log-odds rate. The bounds are based on results obtained by Wang et al. (2003) and Zimmer et al. (1998). Due to the complexity of some of the equations, numerical analysis and computer algorithms were necessary to calculate some of the results of these bounds. Five different types of bounds are tabled in this research. Numerical examples of their application and comparisons to the results obtained by Barlow and Marshall (1965) of bounds on distributions with monotone hazard rate are also provided. The first type of tables is for lower bounds of increasing log-odds rate distributions with knowledge of two values of the reliability of the distribution. The second and third types of tables are for upper bounds under the same conditions. The fourth type of tables is for lower bounds of increasing log-odds rate distributions with knowledge of the mean and the median of the distribution. Finally, the fifth type of tables is for lower bounds of increasing log-odds rate distributions with knowledge of the mean and the variance of the distribution.

SUMMARY The generalized Pareto distribution as a model for survival data is considered. The three parameter model is general enough so as to include increasing, decreasing and constant failure rate distributions. Maximum likelihood estimation for the progressively censored case is discussed at length and conditions for existence and uniqueness of estimates are given. Examples are presented to illustrate its appropriateness for a variety of situations. We consider a population of items whose times to failure from a well-defined origin are being observed. Time to failure is assumed to be a random variable T having survivor function S(t) = pr (T > t), density f(t), and hazard rate h(t) = -d log S(t)/dt. Although the most widely used model for h(t) is constant hazard, giving rise to the exponential distribution, this may not be a realistic assumption when modelling chronic disease populations. Also, when the origin corresponds to an event which may dramatically influence survival, such as surgery, the objection to constant hazard may be especially strong. In situations where constant hazard is not a reasonable assumption, monotone hazard usually is. The Weibull and most other parametric families used to model survival data have monotone hazard rates. Another family of monotone hazard rates which forms the basis of this paper is given by

The Riemannian geometry of covariance matrices has been essential to several successful applications, in computer vision, biomedical signal and image processing, and radar data processing. For these applications, an important ongoing challenge is to develop Riemannian-geometric tools which are adapted to structured covariance matrices. This paper proposes to meet this challenge by introducing a new class of probability distributions, Gaussian distributions of structured covariance matrices . These are Riemannian analogs of Gaussian distributions, which only sample from covariance matrices having a preassigned structure, such as complex, Toeplitz, or block-Toeplitz. The usefulness of these distributions stems from three features: 1) they are completely tractable, analytically, or numerically, when dealing with large covariance matrices; 2) they provide a statistical foundation to the concept of structured Riemannian barycentre ( i.e. , Frechet or geometric mean); and 3) they lead to efficient statistical learning algorithms, which realise, among others, density estimation and classification of structured covariance matrices. This paper starts from the observation that several spaces of structured covariance matrices, considered from a geometric point of view, are Riemannian symmetric spaces. Accordingly, it develops an original theory of Gaussian distributions on Riemannian symmetric spaces, of their statistical inference, and of their relationship to the concept of Riemannian barycentre. Then, it uses this original theory to give a detailed description of Gaussian distributions of three kinds of structured covariance matrices, complex, Toeplitz, and block-Toeplitz. Finally, it describes algorithms for density estimation and classification of structured covariance matrices, based on Gaussian distribution mixture models.

The main problem in Computational Real Algebraic Geometry is the development of efficient Quantifier Elimination algorithms. It is well known (Davenport and Heintz 1988) that the general problem of quantifier elimination cannot be solved in polynomial time. Therefore the only way to attack this problem is to consider specific cases where efficient algorithms can be applied. By efficient we do not mean “polynomial time”. Instead we are looking for algorithms, methods and criteria that can be used to solve specific quantifier elimination problems involving low degree polynomials in a reasonable amount of time. This strategy has already been investigated by Hong (1993d) for formulas with quadratic polynomial constraints and by Heintz, Roy, and Solerno (1993) for specific inputs for which the general Quantifier Elimination algorithm presented in (Heintz et al. 1990) is efficient.

A general and efficient remeshing algorithm is presented for the discretization of arbitrary planar domains into triangular elements in consistency with the given node spacing function.The contour lines of the node spacing function at suitable calculated levels provide the natural lines of division of the problem domain into subregions, where finite element meshes of different element sizes are generated using the available general‐purpose mesh generators.1, 2Examples of remeshing for various node spacing functions are given to illustrate that high‐quality gradation meshes can be generated automatically without any user's intervention by this simple contour line method.

Online rated datasets have become a source for large-scale population studies for analysts and a means for end-users to achieve routine tasks such as finding a book club. Existing systems however only provide limited insights into the opinions of different segments of the rater population. In this paper, we develop a framework for finding and exploring population segments and their opinions. We propose rating maps, a collection of (population segment, rating distribution) pairs, where a segment, e.g., {18-29 year old males in CA} has a rating distribution in the form of a histogram that aggregates its ratings for a set of items (e.g., movies starring Russel Crowe). We formalize the problem of building rating maps dynamically given desired input distributions. Our problem raises two challenges: (i) the choice of an appropriate measure for comparing rating distributions, and (ii) the design of efficient algorithms to find segments. We show that the Earth Mover's Distance (EMD) is well-adapted to comparing rating distributions and prove that finding segments whose rating distribution is close to input ones is NP-complete. We propose an efficient algorithm for building Partition Decision Trees and heuristics for combining the resulting partitions to further improve their quality. Our experiments on real and synthetic datasets validate the utility of rating maps for both analysts and end-users.

Quick general reduction algorithms for inconsistent decision tables

As the scale and dimensionality of data continue to grow in many applications of data analytics (e.g., bioinformatics, finance, computer vision, medical informatics), it becomes critical to develop efficient and effective algorithms to solve numerous machine learning and data mining problems. This tutorial will focus on simple yet practically effective techniques and algorithms for big data analytics. In the first part, we plan to present the state-of-the-art large-scale optimization algorithms, including various stochastic gradient descent methods, stochastic coordinate descent methods and distributed optimization algorithms, for solving various machine learning problems. In the second part, we will focus on randomized approximation algorithms for learning from large-scale data. We will discuss i) randomized algorithms for low-rank matrix approximation; ii) approximation techniques for solving kernel learning problems; iii) randomized reduction methods for addressing the high-dimensional challenge. Along with the description of algorithms, we will also present some empirical results to facilitate understanding of different algorithms and comparison between them.

In Systems Biology, system identification, which infers regulatory network in genetic system and metabolic pathways using experimentally observed time-course data, is one of the hottest issues. The efficient numerical optimization algorithm to estimate more than 100 real-coded parameters should be developed for this purpose. New real-coded genetic algorithm (RCGA), the combination of AREX (adaptive real-coded ensemble crossover) with JGG (just generation gap), have applied to the inference of genetic interactions involving more than 100 parameters related to the interactions with using experimentally observed time-course data. Compared with conventional RCGA, the combination of UNDX (unimodal normal distribution crossover) with MGG (minimal generation gap), new algorithm has shown the superiority with improving early convergence in the first stage of search and suppressing evolutionary stagnation in the last stage of search.

In this paper, we propose using smooth robust loss functions to formulate robust low-rank optimization problem in the presence of outliers. The objective of the problem is to recover a low-rank data matrix from noisy entries. Our main contributions are i) providing two smooth robust loss functions to handle respectively two different types of outliers, i.e., the universal outliers with unknown statistical distribution and the sparse spike-like outliers; ii) an efficient algorithm doing parallel minimization instead of alternating update. Numerical results show that the proposed algorithm obtains a better solution at a faster convergence rate than the state-of-art algorithms.

The New <scp>A</scp>merican Guidelines on Hypertension: A Favorable Opinion