Sparse Solutions to Nonnegative Linear Systems and Applications
نویسندگان
چکیده
We give an efficient algorithm for finding sparse approximate solutions to linear systems of equations with nonnegative coefficients. Unlike most known results for sparse recovery, we do not require any assumption on the matrix other than non-negativity. Our algorithm is combinatorial in nature, inspired by techniques for the set cover problem, as well as the multiplicative weight update method. We then present a natural application to learning mixture models in the PAC framework. For learning a mixture of k axisaligned Gaussians in d dimensions, we give an algorithm that outputs a mixture of O(k/�) Gaussians that is �-close in statistical distance to the true distribution, without any separation assumptions. The time and sample complexity is roughly O(kd/�). This is polynomial when d is constant – precisely the regime in which known methods fail to identify the components efficiently. Given that non-negativity is a natural assumption, we believe that our result may find use in other settings in which we wish to approximately explain data using a small number of a (large) candidate set of components.
منابع مشابه
NEW MODELS AND ALGORITHMS FOR SOLUTIONS OF SINGLE-SIGNED FULLY FUZZY LR LINEAR SYSTEMS
We present a model and propose an approach to compute an approximate solution of Fully Fuzzy Linear System $(FFLS)$ of equations in which all the components of the coefficient matrix are either nonnegative or nonpositive. First, in discussing an $FFLS$ with a nonnegative coefficient matrix, we consider an equivalent $FFLS$ by using an appropriate permutation to simplify fuzzy multiplications. T...
متن کاملEquivalence and Strong Equivalence between Sparsest and Least `1-Norm Nonnegative Solutions of Linear Systems and Their Application
Many practical problems can be formulated as `0-minimization problems with nonnegativity constraints, which seek the sparsest nonnegative solutions to underdetermined linear systems. Recent study indicates that `1-minimization is efficient for solving some classes of `0-minimization problems. From a mathematical point of view, however, the understanding of the relationship between `0and `1-mini...
متن کاملModeling Nonnegative Data with Clumping at Zero: A Survey
Applications in which data take nonnegative values but have a substantial proportion of values at zero occur in many disciplines. The modeling of such “clumped-at-zero” or “zero-inflated” data is challenging. We survey models that have been proposed. We consider cases in which the response for the non-zero observations is continuous and in which it is discrete. For the continuous and then the d...
متن کاملFully Fuzzy Linear Systems
As can be seen from the definition of extended operations on fuzzy numbers, subtraction and division of fuzzy numbers are not the inverse operations to addition and multiplication . Hence, to solve the fuzzy equations or a fuzzy system of linear equations analytically, we must use methods without using inverse operators. In this paper, a novel method to find the solutions in which 0 is not ...
متن کاملLinear Program Relaxation of Sparse Nonnegative Recovery in Compressive Sensing Microarrays
Compressive sensing microarrays (CSM) are DNA-based sensors that operate using group testing and compressive sensing principles. Mathematically, one can cast the CSM as sparse nonnegative recovery (SNR) which is to find the sparsest solutions subjected to an underdetermined system of linear equations and nonnegative restriction. In this paper, we discuss the l₁ relaxation of the SNR. By definin...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- CoRR
دوره abs/1501.01689 شماره
صفحات -
تاریخ انتشار 2015