Eigenvalues of a matrix in the streaming model
نویسندگان
چکیده
We study the question of estimating the eigenvalues of a matrix in the streaming model, addressing a question posed in [Mut05]. We show that the eigenvalue “heavy hitters” of a matrix can be computed in a single pass. In particular, we show that the φ-heavy hitters (in the `1 or `2 norms) can be estimated in space proportional to 1/φ. Such a dependence on φ is optimal. We also show how the same techniques may give an estimate of the residual error tail of a rank-k approximation of the matrix (in the Frobenius norm), in space proportional to k. All our algorithms are linear and hence can support arbitrary updates to the matrix in the stream. In fact, what we show can be seen as a form of a bi-linear dimensionality reduction: if we multiply an input matrix with projection matrices on both sides, the resulting matrix preserves the top eigenvalues and the residual Frobenius norm.
منابع مشابه
Online Streaming Feature Selection Using Geometric Series of the Adjacency Matrix of Features
Feature Selection (FS) is an important pre-processing step in machine learning and data mining. All the traditional feature selection methods assume that the entire feature space is available from the beginning. However, online streaming features (OSF) are an integral part of many real-world applications. In OSF, the number of training examples is fixed while the number of features grows with t...
متن کاملLocalization of Eigenvalues in Small Specified Regions of Complex Plane by State Feedback Matrix
This paper is concerned with the problem of designing discrete-time control systems with closed-loop eigenvalues in a prescribed region of stability. First, we obtain a state feedback matrix which assigns all the eigenvalues to zero, and then by elementary similarity operations we find a state feedback which assigns the eigenvalues inside a circle with center and radius. This new algorithm ca...
متن کاملSome remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs
Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...
متن کاملSeidel Signless Laplacian Energy of Graphs
Let $S(G)$ be the Seidel matrix of a graph $G$ of order $n$ and let $D_S(G)=diag(n-1-2d_1, n-1-2d_2,ldots, n-1-2d_n)$ be the diagonal matrix with $d_i$ denoting the degree of a vertex $v_i$ in $G$. The Seidel Laplacian matrix of $G$ is defined as $SL(G)=D_S(G)-S(G)$ and the Seidel signless Laplacian matrix as $SL^+(G)=D_S(G)+S(G)$. The Seidel signless Laplacian energy $E_{SL^+...
متن کاملModelling and Scheduling Lot Streaming Flexible Flow Lines
Although lot streaming scheduling is an active research field, lot streaming flexible flow lines problems have received far less attention than classical flow shops. This paper deals with scheduling jobs in lot streaming flexible flow line problems. The paper mathematically formulates the problem by a mixed integer linear programming model. This model solves small instances to optimality. Moreo...
متن کاملOn the Remarkable Formula for Spectral Distance of Block Southeast Submatrix
This paper presents a remarkable formula for spectral distance of a given block normal matrix $G_{D_0} = begin{pmatrix} A & B \ C & D_0 end{pmatrix} $ to set of block normal matrix $G_{D}$ (as same as $G_{D_0}$ except block $D$ which is replaced by block $D_0$), in which $A in mathbb{C}^{ntimes n}$ is invertible, $ B in mathbb{C}^{ntimes m}, C in mathbb{C}^{mti...
متن کامل