Low-rank approximation of a matrix by means of random sampling has been consistently efficient in its empirical studies by many scientists who applied it with various sparse and structured multipliers, but adequate formal support for this empirical phenomenon has been missing so far. Our novel insight into the subject leads to such an elusive formal support and promises significant acceleration of the known algorithms for some fundamental problems of matrix computations and data mining and analysis. Our formal results and our numerical tests are in good accordance with each other. We also outline extensions of low-rank approximation algorithms and of our progress to the Least Squares Regression, the Fast Multipole Method, and the Conjugate Gradient algorithms.
Part (ii) of our Theorem is implied by [Theorem 10.8] HMT11 , but our specific supporting estimates are more compact, cover the case of any @math (whereas @cite_36 assumes that @math ), and we deduce them by using a shorter proof (see Remark ). Our approach, our results listed in Section , some of our techniques, e.g., expan -sion compres -sion in Section , and even the concept of factor-Gaussian matrices are new, and so are our families of multipliers and policies of their generation, combination, and application in Sections and as well. Moreover, our progress can be extended to a variety of important matrix computations. In Section (Conclusions) we outline such novel extensions to the highly popular and much studied computations for Least Squares Regression, the Fast Multipole Method and the Conjugate Gradient Algorithms. Hereafter we use the acronyms LSR" for Least Squares Regression", FMM" for Fast Multipole Method", and CG " for Conjugate Gradient". The extensions provide new insights and new opportunities and should motivate further effort and further progress.
@cite_36
2051142108 1769664844 2170546552 2014344607
We describe new algorithms of the locally optimal block preconditioned conjugate gradient (LOBPCG) method for symmetric eigenvalue problems, based on a local optimization of a three-term recurrence, and suggest several other new methods. To be able to compare numerically different methods in the class, with different preconditioners, we propose a common system of model tests, using random preconditioners and initial guesses. As the "ideal" control algorithm, we advocate the standard preconditioned conjugate gradient method for finding an eigenvector as an element of the null-space of the corresponding homogeneous system of linear equations under the assumption that the eigenvalue is known. We recommend that every new preconditioned eigensolver be compared with this "ideal" algorithm on our model test problems in terms of the speed of convergence, costs of every iteration, and memory requirements. We provide such comparison for our LOBPCG method. Numerical results establish that our algorithm is practically as efficient as the ideal'' algorithm when the same preconditioner is used in both methods. We also show numerically that the LOBPCG method provides approximations to first eigenpairs of about the same quality as those by the much more expensive global optimization method on the same generalized block Krylov subspace. We propose a new version of block Davidson's method as a generalization of the LOBPCG method. Finally, direct numerical comparisons with the Jacobi--Davidson method show that our method is more robust and converges almost two times faster. We consider the problem of learning a structured multi-task regression, where the output consists of multiple responses that are related by a graph and the correlated response variables are dependent on the common inputs in a sparse but synergistic manner. Previous methods such as l1 l2-regularized multi-task regression assume that all of the output variables are equally related to the inputs, although in many real-world problems, outputs are related in a complex manner. In this paper, we propose graph-guided fused lasso (GFlasso) for structured multi-task regression that exploits the graph structure over the output variables. We introduce a novel penalty function based on fusion penalty to encourage highly correlated outputs to share a common set of relevant inputs. In addition, we propose a simple yet efficient proximal-gradient method for optimizing GFlasso that can also be applied to any optimization problems with a convex smooth loss and the general class of fusion penalty defined on arbitrary graph structures. By exploiting the structure of the non-smooth ''fusion penalty'', our method achieves a faster convergence rate than the standard first-order method, sub-gradient method, and is significantly more scalable than the widely adopted second-order cone-programming and quadratic-programming formulations. In addition, we provide an analysis of the consistency property of the GFlasso model. Experimental results not only demonstrate the superiority of GFlasso over the standard lasso but also show the efficiency and scalability of our proximal-gradient method. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. Abstract Conjugate gradient (CG) methods to solve sparse systems of linear equations play an important role in numerical methods for solving discretized partial differential equations. The large size and the condition of many technical or physical applications in this area result in the need for efficient parallelization and preconditioning techniques of the CG method, in particular on massively parallel machines. Here, the data distribution and the communication scheme for the sparse matrix operations of the preconditioned CG are based on the analysis of the indices of the non-zero elements. Polynomial preconditioning is shown to reduce global synchronizations considerably, and a fully local incomplete Cholesky preconditioner is presented. On a PARAGON XP S 10 with 138 processors, the developed parallel methods outperform diagonally scaled CG markedly with respect to both scaling behavior and execution time for many matrices from real finite element applications.