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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 71 (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 65 (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
, 1998
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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 36 (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 26 (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...
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 17 (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 Stable Algorithm for Multidimensional Padé Systems and the Inversion of Generalized Sylvester Matrices
, 1994
"... . For k + 1 power series a 0 (z); : : : ; a k (z), we present a new iterative, lookahead algorithm for numerically computing Pad'eHermite systems and simultaneous Pad'e systems along a diagonal of the associated Pad'e tables. The algorithm computes the systems at all those points along the diagona ..."
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Cited by 12 (5 self)
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. For k + 1 power series a 0 (z); : : : ; a k (z), we present a new iterative, lookahead algorithm for numerically computing Pad'eHermite systems and simultaneous Pad'e systems along a diagonal of the associated Pad'e tables. The algorithm computes the systems at all those points along the diagonal at which the associated striped Sylvester and mosaic Sylvester matrices are wellconditioned. It is shown that a good estimate for the condition numbers of these Sylvester matrices at a point is easily determined from the Pad'eHermite system and simultaneous Pad'e system computed at that point. The operation and the stability of the algorithm is controlled by a single parameter ø which serves as a threshold in deciding if the Sylvester matrices at a point are sufficiently wellconditioned. We show that the algorithm is weakly stable, and provide bounds for the error in the computed solutions as a function of ø . Experimental results are given which show that the bounds reflect the actual b...
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 math ..."
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Cited by 12 (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...