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TENSOR RANK AND THE ILLPOSEDNESS OF THE BEST LOWRANK APPROXIMATION PROBLEM
"... There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, te ..."
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Cited by 75 (10 self)
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There has been continued interest in seeking a theorem describing optimal lowrank approximations to tensors of order 3 or higher, that parallels the Eckart–Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rankr approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Brègman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank3 tensor has an optimal rank2 approximation. The notable exceptions to this misbehavior are rank1 tensors and order2 tensors (i.e. matrices). In a more positive spirit, we propose a natural way of overcoming the illposedness of the lowrank approximation problem, by using weak solutions when true solutions do not exist. For this to work, it is necessary to characterize the set of weak solutions, and we do this in the case of rank 2, order 3 (in arbitrary dimensions). In our work we emphasize the importance of closely studying concrete lowdimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2 × 2 × 2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; in other words, can be determined by a system of polynomial inequalities. We study some of these polynomials in cases of interest to us; in particular we make extensive use of the hyperdeterminant ∆ on R 2×2×2.
Algorithms for Arbitrary Precision Floating Point Arithmetic
 Proceedings of the 10th Symposium on Computer Arithmetic
, 1991
"... We present techniques which may be used to perform computations of very high accuracy using only straightforward floating point arithmetic operations of limited precision, and we prove the validity of these techniques under very general hypotheses satisfied by most implementations of floating point ..."
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Cited by 71 (1 self)
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We present techniques which may be used to perform computations of very high accuracy using only straightforward floating point arithmetic operations of limited precision, and we prove the validity of these techniques under very general hypotheses satisfied by most implementations of floating point arithmetic. To illustrate the application of these techniques, we present an algorithm which computes the intersection of a line and a line segment. The algorithm is guaranteed to correctly decide whether an intersection exists and, if so, to produce the coordinates of the intersection point accurate to full precision. Moreover, the algorithm is usually quite efficient; only in a few cases does guaranteed accuracy necessitate an expensive computation. 1. Introduction "How accurate is a computed result if each intermediate quantity is computed using floating point arithmetic of a given precision?" The casual reader of Wilkinson's famous treatise [21] and similar roundoff error analyses might...
A deterministic strongly polynomial algorithm for matrix scaling and approximate permanents
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A method computing multiple roots of inexact polynomials
 In Sendra [29
, 2003
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Pseudozeros of Polynomials and Pseudospectra of Companion Matrices
, 1994
"... this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The fflpseudozero set Z ffl (p) is the set of zeros of all polynomials p obtained by coefficientwise perturbations of p of size ..."
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Cited by 46 (2 self)
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this paper we take a geometric view of the conditioning of these two problems and of the stability of algorithms for polynomial zerofinding. The fflpseudozero set Z ffl (p) is the set of zeros of all polynomials p obtained by coefficientwise perturbations of p of size
On Properties of Floating Point Arithmetics: Numerical Stability and the Cost of Accurate Computations
, 1992
"... Floating point arithmetics generally possess many regularity properties in addition to those that are typically used in roundoff error analyses; these properties can be exploited to produce computations that are more accurate and cost effective than many programmers might think possible. Furthermore ..."
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Cited by 32 (0 self)
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Floating point arithmetics generally possess many regularity properties in addition to those that are typically used in roundoff error analyses; these properties can be exploited to produce computations that are more accurate and cost effective than many programmers might think possible. Furthermore, many of these properties are quite simple to state and to comprehend, but few programmers seem to be aware of them (or at least willing to rely on them). This dissertation presents some of these properties and explores their consequences for computability, accuracy, cost, and portability. For example, we consider several algorithms for summing a sequence of numbers and show that under very general hypotheses, we can compute a sum to full working precision at only somewhat greater cost than a simple accumulation, which can often produce a sum with no significant figures at all. This example, as well as others we present, can be generalized further by substituting still more complex algorith...
Stability of Methods for Matrix Inversion
, 1992
"... Inversion of a triangular matrix can be accomplished in several ways. The standard methods are characterised by the loop ordering, whether matrixvector multiplication, solution of a triangular system, or a rank1 update is done inside the outer loop, and whether the method is blocked or unblocked. ..."
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Cited by 27 (11 self)
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Inversion of a triangular matrix can be accomplished in several ways. The standard methods are characterised by the loop ordering, whether matrixvector multiplication, solution of a triangular system, or a rank1 update is done inside the outer loop, and whether the method is blocked or unblocked. The numerical stability properties of these methods are investigated. It is shown that unblocked methods satisfy pleasing bounds on the left or right residual. However, for one of the block methods it is necessary to convert a matrix multiplication into the solution of a multiple righthand side triangular system in order to have an acceptable residual bound. The inversion of a full matrix given a factorization PA = LU is also considered, including the special cases of symmetric inde nite and symmetric positive de nite matrices. Three popular methods are shown to possess satisfactory residual bounds, subject to a certain requirement on the implementation, and an attractive new method is described. This work was motivated by the question of what inversion methods should be used in LAPACK.
Collinearity and Least Squares Regression
 Statistical Science
, 1987
"... this paper we introduce certain numbers, called collinearity indices, which are useful in detecting near collinearities in regression problems. The coefficients enter adversely into formulas concerning significance testing and the effects of errors in the regression variables. Thus they provide simp ..."
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Cited by 19 (2 self)
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this paper we introduce certain numbers, called collinearity indices, which are useful in detecting near collinearities in regression problems. The coefficients enter adversely into formulas concerning significance testing and the effects of errors in the regression variables. Thus they provide simple regression diagnostics, suitable for incorporation in regression packages. Keywords and phrases: collinearity, illconditioning, linear regression, errors in the variables, regression diagnostics. 1 Introduction
A Test of a Computer's FloatingPoint Arithmetic Unit
, 1981
"... This paper describes a test of a computer's floatingpoint arithmetic unit. The test has two goals. The first goal deals with the needs of users of computers, and the second goal deals with manufacturers of computers. The first and major goal is to determine if the machine supports a particular ..."
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Cited by 13 (2 self)
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This paper describes a test of a computer's floatingpoint arithmetic unit. The test has two goals. The first goal deals with the needs of users of computers, and the second goal deals with manufacturers of computers. The first and major goal is to determine if the machine supports a particular mathematical model of computer arithmetic. This model was developed as an aid in the design, analysis, implementation and testing of portable, highquality numerical software. If a computer supports the arithmetic model, then software written using the model will perform correctly and to specified accuracy on that machine. The second goal of the test is to check that the basic operations perform as the manufacturer intended. For example, if division ( x x // yy ) is implemented as a composite operation ( x x × (1//yy) ), then the test should detect that fact. Also, the accuracy lost in such a division due to the extra arithmetic operations can tell the manufacturer whether it has been implemente...