Degeneracy in Candecomp/Parafac explained for p × p × 2 arrays of rank p + 1 or higher

The Candecomp/Parafac (CP) model decomposes a three-way array into a prespecified number R of rank-1 arrays and a residual array, in which the sum of squares of the residual array is minimized. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate solutions, in which some components are highly correlated in all three modes and the elements of these components become arbitrarily large. We consider the real-valued CP model in which p × p × 2 arrays of rank p + 1 or higher are decomposed into p rank-1 arrays and a residual array. It is shown that the CP objective function does not have a minimum in these cases, but an infimum. Moreover, any sequence of CP approximations, of which the objective value approaches the infimum, will become degenerate. This result extends Ten Berge, Kiers, & De Leeuw (1988), who consider a particular 2 × 2 × 2 array of rank 3. Request for reprints should be sent to Alwin Stegeman, Heijmans Institute of Psychological Research, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands. The author is obliged to Jos ten Berge and Henk Kiers for helpful comments. Also, the author would like to thank the Associate Editor and the anonymous reviewers for many suggestions on how to improve the contents and the presentation of the paper.     

On the Non-Existence of Optimal Solutions and the Occurrence of “Degeneracy” in the CANDECOMP/PARAFAC Model

The CANDECOMP/PARAFAC (CP) model decomposes a three-way array into a prespecified number of R factors and a residual array by minimizing the sum of squares of the latter. It is well known that an optimal solution for CP need not exist. We show that if an optimal CP solution does not exist, then any sequence of CP factors monotonically decreasing the CP criterion value to its infimum will exhibit the features of a so-called “degeneracy”. That is, the parameter matrices become nearly rank deficient and the Euclidean norm of some factors tends to infinity. We also show that the CP criterion function does attain its infimum if one of the parameter matrices is constrained to be column-wise orthonormal.     

The typical rank of tall three-way arrays

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J. The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.     

Transforming three-way arrays to maximal simplicity

Transforming the core array in Tucker three-way component analysis to simplicity is an intriguing way of revealing structures in between standard Tucker three-way PCA, where the core array is unconstrained, and CANDECOMP/PARAFAC, where the core array has a generalized diagonal form. For certain classes of arrays, transformations to simplicity, that is, transformations that produce a large number of zeros, can be obtained explicitly by solving sets of linear equations. The present paper extends these results. First, a method is offered to simplifyJ ×J × 2 arrays. Next, it is shown that the transformation that simplifies anI ×J ×K array can be used to also simplify the (complementary) arrays of order (JK −I) ×J ×K, of orderI × (IK −J) ×K and of orderI ×J × (IJ −K). Finally, the question of what constitutes the maximal simplicity for arrays (the maximal number of zero elements) will be considered. It is shown that cases of extreme simplicity, considered in the past, are, in fact, cases of maximal simplicity.     

Explicit candecomp/parafac solutions for a contrived 2 × 2 × 2 array of rank three

Kruskal, Harshman and Lundy have contrived a special 2 × 2 × 2 array to examine formal properties of degenerate Candecomp/Parafac solutions. It is shown that for this array the Candecomp/Parafac loss has an infimum of 1. In addition, the array will be used to challenge the tradition of fitting Indscal and related models by means of the Candecomp/Parafac process.     

Kruskal's polynomial for 2×2×2 arrays and a generalization to 2×n×n arrays

A remarkable difference between the concept of rank for matrices and that for three-way arrays has to do with the occurrence of non-maximal rank. The set ofn×n matrices that have a rank less thann has zero volume. Kruskal pointed out that a 2×2×2 array has rank three or less, and that the subsets of those 2×2×2 arrays for which the rank is two or three both have positive volume. These subsets can be distinguished by the roots of a certain polynomial. The present paper generalizes Kruskal's results to 2×n×n arrays. Incidentally, it is shown that twon ×n matrices can be diagonalized simultaneously with positive probability.The author is obliged to Joe Kruskal and Henk Kiers for commenting on an earlier draft, and to Tom Wansbeek for raising stimulating questions.     

Cohen’s Linearly Weighted Kappa is a Weighted Average of 2×2 Kappas

An agreement table with n∈ℕ_≥3 ordered categories can be collapsed into n−1 distinct 2×2 tables by combining adjacent categories. Vanbelle and Albert (Stat. Methodol. 6:157–163, 2009c) showed that the components of Cohen’s weighted kappa with linear weights can be obtained from these n−1 collapsed 2×2 tables. In this paper we consider several consequences of this result. One is that the weighted kappa with linear weights can be interpreted as a weighted arithmetic mean of the kappas corresponding to the 2×2 tables, where the weights are the denominators of the 2×2 kappas. In addition, it is shown that similar results and interpretations hold for linearly weighted kappas for multiple raters.     

On uniqueness in candecomp/parafac

One of the basic issues in the analysis of three-way arrays by CANDECOMP/PARAFAC (CP) has been the question of uniqueness of the decomposition. Kruskal (1977) has proved that uniqueness is guaranteed when the sum of thek-ranks of the three component matrices involved is at least twice the rank of the solution plus 2. Since then, little has been achieved that might further qualify Kruskal's sufficient condition. Attempts to prove that it is also necessary for uniqueness (except for rank 1 or 2) have failed, but counterexamples to necessity have not been detected. The present paper gives a method for generating the class of all solutions (or at least a subset of that class), given a CP solution that satisfies certain conditions. This offers the possibility to examine uniqueness for a great variety of specific CP solutions. It will be shown that Kruskal's condition is necessary and sufficient when the rank of the solution is three, but that uniqueness may hold even if the condition is not satisfied, when the rank is four or higher. The authors are obliged to Henk Kiers for commenting on a previous draft, and to Tom Snijders for suggesting a proof mentioned in the appendix.     

Simplicity and Typical Rank Results for Three-Way Arrays

Matrices can be diagonalized by singular vectors or, when they are symmetric, by eigenvectors. Pairs of square matrices often admit simultaneous diagonalization, and always admit block wise simultaneous diagonalization. Generalizing these possibilities to more than two (non-square) matrices leads to methods of simplifying three-way arrays by nonsingular transformations. Such transformations have direct applications in Tucker PCA for three-way arrays, where transforming the core array to simplicity is allowed without loss of fit. Simplifying arrays also facilitates the study of array rank. The typical rank of a three-way array is the smallest number of rank-one arrays that have the array as their sum, when the array is generated by random sampling from a continuous distribution. In some applications, the core array of Tucker PCA is constrained to have a vast majority of zero elements. Both simplicity and typical rank results can be applied to distinguish constrained Tucker PCA models from tautologies. An update of typical rank results over the real number field is given in the form of two tables.     

Basic visual perceptual processes in children with typical development and cerebral palsy: The processing of surface,length, orientation,and position

The present study aims to assess how the processing of basic visual perceptual (VP) components (length, surface, orientation, and position) develops in typically developing (TD) children (n = 215, 4–14 years old) and adults (n = 20, 20–25 years old), and in children with cerebral palsy (CP) (n = 86, 5–14 years old) using the first four subtests of the Battery for the Evaluation of Visual Perceptual and Spatial processing in children. Experiment 1 showed that these four basic VP processes follow distinct developmental trajectories in typical development. Experiment 2 revealed that children with CP present global and persistent deficits for the processing of basic VP components when compared with TD children matched on chronological age and nonverbal reasoning abilities.     

The Assessment of National Stereotypes: A Methodological Study

Abstract

The effect of instructions on judgments of German students concerning national stereotypes was assessed. Focus of stereotype (typical individual/whole group), nature of judgment (extent/percentage/probability), and nationality were manipulated in a 2 × 3 × 5 design, with repeated measures on the third factor. Instructions had no effect on ratings of content, certainty, or uniformity of stereotypes, but percentage-based stereotypes were significantly less intense (p<.01). All three levels of the second instructions factor were highly correlated with liking for the national groups, but not so clearly with similarity and contact. Overall, the subjects viewed the stereotyping task negatively, especially in the probability condition (p<.025).     

Accuracy and stability of metacognitive monitoring: A new measure

An examination of gamma (?; Goodman & Kruskal, 1954) and the Hamann coefficient (HC; see Schraw, 1995) as a measure of feeling-of-knowing (FOK) accuracy in the prediction of subsequent memory performance shows that neither coefficient provides a pure measure of accuracy in 2 (feeling-of-not-knowing and FOK) × 2 (correct and incorrect recognition) cases. γ underestimated accuracy in the m × 2 case with m being an integer larger than 2, producing low and unstable values of γ. In the present study, FOK predictive accuracy is treated as the extent to which the percentage of correct recognition can be increased by an increase from one FOK rank to another. A statistic computing the relative magnitude of the percentage increase to its maximal increase, designated as C, is able to provide a correct estimate of accuracy. Furthermore, the value of C is stable across different portions of FOK data and is free from restricted-/truncated-range and fineness/ coarseness effects.     

Two-Valued Logics of Intentionality: Temporality,Truth, Modality,and Identity

The essay introduces a non-Diodorean, non-Kantian temporal modal semantics based on part-whole, rather than class, theory. Formalizing Edmund Husserl’s theory of inner time consciousness, §3 uses his protention and retention concepts to define a relation of self-awareness on intentional events. §4 introduces a syntax and two-valued semantics for modal first-order predicate object-languages, defines semantic assignments for variables and predicates, and truth for formulae in terms of the axiomatic version of Edmund Husserl’s dependence ontology (viz. the Calculus [CU] of Urelements) introduced by The Ontology of Intentionality I & II. It then uses the §3 results to define the modalities of truth, and §5 extends the semantics to identity claims. §6 defines and contrasts synthetic a priori truths to analytic a priori truths, and §7 compares Brentano School noetic semantic and Leibnizian possible-world semantic perspectives on modality. The essay argues that the modal logics it defines semantically are two-valued, first-order versions of the type of language which Husserl viewed as the language of any ontology of experience (i.e. of any science), and conceived as the logic of intentionality.     

An efficient algorithm for PARAFAC of three-way data with large numbers of observation units

The CANDECOMP algorithm for the PARAFAC analysis ofn×m×p three-way arrays is adapted to handle arrays in whichn>mp more efficiently. For such arrays, the adapted algorithm needs less memory space to store the data during the iterations, and uses less computation time than the original CANDECOMP algorithm. The size of the arrays that can be handled by the new algorithm is in no way limited by the number of observation units (n) in the data.The authors are obliged to Jos ten Berge for his comments on an earlier version of this paper. The research of Henk A. L. Kiers has been made possible by a fellowship of the Royal Netherlands Academy of Arts and Sciences.     

Matrix differential calculus with applications to simple,hadamard, and kronecker products

Several definitions are in use for the derivative of an m × p matrix function F(X) with respect to its n × q matrix argument X. We argue that only one of these definitions is a viable one, and that to study smooth maps from the space of n × q matrices to the space of m × p matrices it is often more convenient to study the map from nq-space to mp-space. Also, several procedures exist for a calculus of functions of matrices. It is argued that the procedure based on differentials is superior to other methods of differentiation, and leads inter alia to a satisfactory chain rule for matrix functions.     

Mangabeys (<Emphasis Type="Italic">Cercocebus torquatus lunulatus</Emphasis>) solve the reverse contingency task without a modified procedure

Problem solving often relies on generating new responses while inhibiting others, particularly prepotent ones. A paradigm to study inhibitory abilities is the reverse contingency task (Boysen and Berntson in J Exp Psychol Anim Behav Process 21:82–86, 1995), in which two different quantities of food are offered to an individual who receives the array he did not choose. Therefore, mastery of the task demands selecting the smaller quantity to obtain the larger one. Several non-human primates have been tested in the reverse contingency task. To date, only great apes and rhesus monkeys (Macaca mulatta) have succeeded in the original task, with no need of procedural modifications as the large-or-none contingency, correction trials or symbolic stimuli substituting for actual food quantities. Here, four mangabeys were presented with two stimulus arrays of one and four raisins in the context of the reverse contingency task. Three of them learned to perform the task well above chance without a modified procedure. They also reached above-chance performance when presented with two stimulus arrays of zero and four raisins, despite the initial difficulty of choosing a null quantity. After a period of 7–10 months, in which the animals were not tested on any task, all three subjects continued to perform well, even when presented with novel quantity pairs.     

A simplification of a result by zellini on the maximal rank of symmetric three-way arrays

Zellini (1979, Theorem 3.1) has shown how to decompose an arbitrary symmetric matrix of ordern ×n as a linear combination of 1/2n(n+1) fixed rank one matrices, thus constructing an explicit tensor basis for the set of symmetricn ×n matrices. Zellini's decomposition is based on properties of persymmetric matrices. In the present paper, a simplified tensor basis is given, by showing that a symmetric matrix can also be decomposed in terms of 1/2n(n+1) fixed binary matrices of rank one. The decomposition implies that ann ×n ×p array consisting ofp symmetricn ×n slabs has maximal rank 1/2n(n+1). Likewise, an unconstrained INDSCAL (symmetric CANDECOMP/PARAFAC) decomposition of such an array will yield a perfect fit in 1/2n(n+1) dimensions. When the fitting only pertains to the off-diagonal elements of the symmetric matrices, as is the case in a version of PARAFAC where communalities are involved, the maximal number of dimensions can be further reduced to 1/2n(n–1). However, when the saliences in INDSCAL are constrained to be nonnegative, the tensor basis result does not apply. In fact, it is shown that in this case the number of dimensions needed can be as large asp, the number of matrices analyzed.     

Working memory and fine motor skills predict early numeracy performance of children with cerebral palsy

Early numeracy is an important precursor for arithmetic performance, academic proficiency, and work success. Besides their apparent motor difficulties, children with cerebral palsy (CP) often show additional cognitive disturbances. In this study, we examine whether working memory, non-verbal intelligence, linguistic skills, counting and fine motor skills are positively related to the early numeracy performance of 6-year-old children with CP. A total of 56 children (M = 6.0, SD = 0.61, 37 boys) from Dutch special education schools participated in this cross-sectional study. Of the total group, 81% of the children have the spastic type of CP (33% unilateral and 66% bilateral), 9% have been diagnosed as having diskinetic CP, 8% have been diagnosed as having spastic and diskinetic CP and 2% have been diagnosed as having a combination of diskinetic and atactic CP. The children completed standardized tests assessing early numeracy performance, working memory, non-verbal intelligence, sentence understanding and fine motor skills. In addition, an experimental task was administered to examine their basic counting performance. Structural equation modeling showed that working memory and fine motor skills were significantly related to the early numeracy performance of the children (β = .79 and p < .001, β = .41 and p < .001, respectively). Furthermore, counting was a mediating variable between working memory and early numeracy (β = .57, p < .001). Together, these findings highlight the importance of working memory for early numeracy performance in children with CP and they warrant further research into the efficacy of intervention programs aimed at working memory training.     

WoMMBAT: A user interface for hierarchical Bayesian estimation of working memory capacity

The change detection paradigm has become an important tool for researchers studying working memory. Change detection is especially useful for studying visual working memory, because recall paradigms are difficult to employ in the visual modality. Pashler (Perception & Psychophysics, 44, 369–378, 1988) and Cowan (Behavioral and Brain Sciences, 24, 87–114, 2001) suggested formulas for estimating working memory capacity from change detection data. Although these formulas have become widely used, Morey (Journal of Mathematical Psychology, 55, 8–24, 2011) showed that the formulas suffer from a number of issues, including inefficient use of information, bias, volatility, uninterpretable parameter estimates, and violation of ANOVA assumptions. Morey presented a hierarchical Bayesian extension of Pashler’s and Cowan’s basic models that mitigates these issues. Here, we present WoMMBAT (Working Memory Modeling using Bayesian Analysis Techniques) software for fitting Morey’s model to data. WoMMBAT has a graphical user interface, is freely available, and is cross-platform, running on Windows, Linux, and Mac operating systems.     

Ghrelin level negatively predicts quality of life in obese women

A cross-sectional cohort study was conducted to investigate whether ghrelin level in obese women predicts the quality of life (QOL). A total of 307 subjects fulfilled the criteria: (1) age between 20 and 65 years old, (2) body mass index ≥27 kg/m² (3) waist circumference ≥80 cm were enrolled in the study. All subjects were assigned to one of the plasma ghrelin level categories according to the quartiles. The median of age and BMI of the 307 obese women were 45 ± 18 years and 29.9 ± 4.1 kg/m², respectively. The main outcome evaluated is the associations of plasma ghrelin level and QOL, which were evaluated using multiple linear regression analysis. Results of linear trend test show significant statistical difference in plasma lipoproteins (triglyceride, cholesterol, HDL-cholestero and LDL-cholesterol = and levels of obesity-related hormone peptides, including leptin, adiponectin, insulin among quartiles of ghrelin. Multiple liner regression analysis of serum obesity-related hormone peptide level and QOL using stepwise method shows ghrelin concentration was the only predictor of QOL, including PCS-12 level (β = ?0.18, p = 0.001), MCS-12 level (β = ?0.14, p = 0.009), WHOQOL-BREF scores: physical (β = ?0.13, p = 0.03), psychological (β = ?0.16, p = 0.007), social (β = ?0.21, p?= < 0.001), and environmental (β = ?0.22, p?= <0.001), after adjusting other factors for obese female subjects. This study demonstrated that ghrelin concentration is strongly associated with QOL level among obese women. Hence, ghrelin concentration might be a valuable marker to be monitored in obese women.