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Implementing regularization implicitly via approximate eigenvector computation [article]

Michael W. Mahoney, Lorenzo Orecchia
2011 arXiv   pre-print
regularization implicitly.  ...  In each case, we provide a precise characterization of the manner in which the approximation method can be viewed as implicitly computing the exact solution to a regularized problem.  ...  lazy random walk-for computing an approximation to the smallest nontrivial eigenvector of a graph Laplacian, and we show that these approximation procedures may be viewed as implicitly solving a regularized  ... 
arXiv:1010.0703v2 fatcat:bzaxcrxqjncvhmienazxrdnola

Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis [article]

Michael W. Mahoney
2012 arXiv   pre-print
By using several case studies, I will illustrate, both theoretically and empirically, the nonobvious fact that approximate computation, in and of itself, can implicitly lead to statistical regularization  ...  Although it is nearly completely absent from computer science, which historically has taken the input data as given and modeled algorithms discretely, regularization in one form or another is central to  ...  That is, approximate computation, in and of itself, can implicitly lead to statistical regularization.  ... 
arXiv:1203.0786v1 fatcat:xm27jkmhivcqtcrvpcatcigoza

Approximate computation and implicit regularization for very large-scale data analysis

Michael W. Mahoney
2012 Proceedings of the 31st symposium on Principles of Database Systems - PODS '12  
By using several case studies, I will illustrate, both theoretically and empirically, the nonobvious fact that approximate computation, in and of itself, can implicitly lead to statistical regularization  ...  Although it is nearly completely absent from computer science, which historically has taken the input data as given and modeled algorithms discretely, regularization in one form or another is central to  ...  via approximate computation in three somewhat different ways.  ... 
doi:10.1145/2213556.2213579 dblp:conf/pods/Mahoney12 fatcat:orrhrufykval3jltt7cks5z5jq

Approximate algebraic methods for curves and surfaces and their applications

Bert Jüttler, Pavel Chalmovianský, Mohamed Shalaby, Elmar Wurm
2005 Proceedings of the 21st spring conference on Computer graphics - SCCG '05  
We report on approximate techniques for conversion between the implicit and the parametric representation of curves and surfaces, i.e., implicitization and parameterization.  ...  In addition, we discuss several applications of these techniques, such as detection of self-intersections, raytracing, footpoint computation and parameterization of scattered data for parametric curve  ...  Acknowledgments This research was supported by the Austrian Science Fund (FWF) through subproject 15 of the Special Research Area (SFB) F013 "Numerical and Symbolic Scientific Computing" at Linz, and by  ... 
doi:10.1145/1090122.1090124 dblp:conf/sccg/JuttlerCSW05 fatcat:fh5552ulavfzplhhym467a27fy

An implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric eigenproblems

V. Mehrmann, C. Schröder, V. Simoncini
2012 Linear Algebra and its Applications  
It computes a few eigenvalues and eigenvectors of the matrix pencil close to a given target point.  ...  A new implicitly-restarted Krylov subspace method for real symmetric/skew-symmetric generalized eigenvalue problems is presented.  ...  One more implementation aspect that needs further discussion is the eigenvector extraction.  ... 
doi:10.1016/j.laa.2009.11.009 fatcat:y2o7anqhozdjjbn2r3xkmcnm3q

Adaptive Randomized Dimension Reduction on Massive Data [article]

Gregory Darnell and Stoyan Georgiev and Sayan Mukherjee and Barbara E Engelhardt
2015 arXiv   pre-print
A key observation in this paper is that randomization serves a dual role, improving both computational and statistical performance by implicitly regularizing the covariance matrix estimate of the random  ...  One approach to implementing scalable algorithms is to compress data into a low dimensional latent space using dimension reduction methods.  ...  Under certain settings, we find that the LMM using ARSVD outperforms current state-of-the-art approaches by implicitly performing regularization on the covariance matrix.  ... 
arXiv:1504.03183v1 fatcat:yz6lrheik5ccpgn5kldx4dhqdy

Computing charge densities with partially reorthogonalized Lanczos

Constantine Bekas, Yousef Saad, Murilo L. Tiago, James R. Chelikowsky
2005 Computer Physics Communications  
This is achievable by exploiting more memory and BLAS3 (dense) computations while avoiding the frequent updates of eigenvectors inherent to all restarted Lanczos methods. *  ...  This paper considers the problem of computing charge densities in a density functional theory (DFT) framework.  ...  In order to compute the charge density ρ(r) via (1), eigenvectors are normally required. However, it is also possible to compute ρ(r) without explicitly resorting to using eigenvectors.  ... 
doi:10.1016/j.cpc.2005.05.005 fatcat:2nq7blzrojbtxj42nxyaz5xo6y

A mathematical biography of Danny C. Sorensen

Peter Benner, Mark Embree, Richard B. Lehoucq, C.T. Kelley
2012 Linear Algebra and its Applications  
Finally, the paper "Large-scale Tikhonov regularization via reduction by orthogonal projection" by Lampe, Reichel, and Voss presents a sequential Krylov projection method to compute an approximate solution  ...  While the computed eigenvalues were quite satisfactory, the orthogonality of computed eigenvectors posed a further challenge, to which Dan and others devoted attention [30, 53] .  ... 
doi:10.1016/j.laa.2012.01.031 fatcat:hm3b7fqamvcxjphqletg7ze7va

Preconditioned spectral clustering for stochastic block partition streaming graph challenge (Preliminary version at arXiv.)

David Zhuzhunashvili, Andrew Knyazev
2017 2017 IEEE High Performance Extreme Computing Conference (HPEC)  
For streaming graph partitioning, LOBPCG is initiated with approximate eigenvectors of the graph Laplacian already computed for the previous graph, in many cases reducing 2-3 times the number of required  ...  , in 10-stage streaming comparison with the base code for the 5K graph, the quality of our clusters is similar or better starting at stage 4 (7) for emerging edging (snowballing) streaming, while the computations  ...  Streaming graphs are treated via warm-starts of LOBPCG, where the approximate eigenvectors already computed for the previous graph in the stream serve as high-quality initial approximations in LOBPCG for  ... 
doi:10.1109/hpec.2017.8091045 dblp:conf/hpec/ZhuzhunashviliK17 fatcat:gn5gsuldengchbn4k264ne4drm

Semi-supervised Eigenvectors for Large-scale Locally-biased Learning [article]

Toke J. Hansen, Michael W. Mahoney
2013 arXiv   pre-print
We show that these semi-supervised eigenvectors can be computed quickly as the solution to a system of linear equations; and we also describe several variants of our basic method that have improved scaling  ...  machine learning algorithms that use global eigenvectors of the graph Laplacian.  ...  Several things should be noted about our implementation of our main algorithm. First, as we will discuss in more detail below, we compute the projection matrix F F T only implicitly.  ... 
arXiv:1304.7528v1 fatcat:riosluvi2zh3de7qts2yw23dvy

Nonlinear multiclass discriminant analysis

Junshui Ma, J.L. Sancho-Gomez, S.C. Ahalt
2003 IEEE Signal Processing Letters  
Moreover, we propose a method to determine the value of the regularization parameter , based on this derivation. Index Terms-Discriminant analysis, feature extraction, kernel method.  ...  In fact, we can further reduce the computational complexity by replacing in (23) with one of its approximations. For example, we can replace with the one-norm of in practice.  ...  Section IV deals with this implementation issue. IV.  ... 
doi:10.1109/lsp.2003.813680 fatcat:4kx7go2ysfdkroti2nhylcf2au

Spectral coarsening of geometric operators

Hsueh-Ti Derek Liu, Alec Jacobson, Maks Ovsjanikov
2019 ACM Transactions on Graphics  
for sharing implementations and results; Zih-Yin Chen for early discussions.  ...  Fig. 2 . 2 Our coarsening directly preserves eigenvectors so eigenvalues are also implicitly preserved: eigenvalue plot ofFig. 1.  ...  Using Eq. 8, the functional map C N, M can be computed by solving a simple least squares problem, via a single linear solve.  ... 
doi:10.1145/3306346.3322953 fatcat:qyfygyrzn5gsdd4zshtytcwt7q

Voronoi-based Variational Reconstruction of Unoriented Point Sets [article]

Pierre Alliez, David Cohen-Steiner, Yiying Tong, Mathieu Desbrun
2007 Symposium on geometry processing : [proceedings]. Symposium on Geometry Processing  
An implicit function is then computed by solving a generalized eigenvalue problem such that its gradient is most aligned with the principal axes of the tensor field, providing a best-fitting isosurface  ...  Second, an implicit function is computed via a generalized eigenvalue problem (Section 3) so as to make its gradient best fit the normal directions.  ...  The final mesh is extracted via isocontouring of a scalar function computed through optimization. Figure 2 : 2 Figure 2: Our reconstruction procedure at a glance.  ... 
doi:10.2312/sgp/sgp07/039-048 fatcat:zdvddaqoevc43jvzhcum72huvm

Computation of Large Invariant Subspaces Using Polynomial Filtered Lanczos Iterations with Applications in Density Functional Theory

C. Bekas, E. Kokiopoulou, Yousef Saad
2008 SIAM Journal on Matrix Analysis and Applications  
This technique employs a well-selected low pass filter polynomial, obtained via a conjugate residual-type algorithm in polynomial space.  ...  Experiments are reported to illustrate the efficiency of the proposed scheme compared to state-of-the-art implicitly restarted techniques.  ...  Yang for useful discussions concerning implicitly restarted methods.  ... 
doi:10.1137/060675435 fatcat:etjvdesl3rg53fzqpakfx3vgle

Algorithm 873:

Marielba Rojas, Sandra A. Santos, Danny C. Sorensen
2008 ACM Transactions on Mathematical Software  
LSTRS relies on matrix-vector products only and has low and fixed storage requirements, features that make it suitable for large-scale computations.  ...  A MATLAB 6.0 implementation of the LSTRS method is presented. LSTRS was described in Rojas et al. [2000] . LSTRS is designed for large-scale quadratic problems with one norm constraint.  ...  We also thank Bill Hager and Henry Wolkowicz for providing MATLAB implementations of their methods so that comparisons with LSTRS could be performed. M.  ... 
doi:10.1145/1326548.1326553 fatcat:sdoaitlyareytca7fehqho4qpi
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