Degeneracy in Candecomp/Parafac and Indscal Explained For Several Three-Sliced Arrays With A Two-Valued Typical Rank

The Candecomp/Parafac (CP) method decomposes a three-way array into a prespecified number R of rank-1 arrays, by minimizing the sum of squares of the residual array. The practical use of CP is sometimes complicated by the occurrence of so-called degenerate sequences of solutions, in which several rank-1 arrays become highly correlated in all three modes and some elements of the rank-1 arrays become arbitrarily large. We consider the real-valued CP decomposition of all known three-sliced arrays, i.e., of size p×q×3, with a two-valued typical rank. These are the 5×3×3 and 8×4×3 arrays, and the 3×3×4 and 3×3×5 arrays with symmetric 3×3 slices. In the latter two cases, CP is equivalent to the Indscal model. For a typical rank of {m,m+1}, we consider the CP decomposition with R=m of an array of rank m+1. We show that (in most cases) the CP objective function does not have a minimum but an infimum. Moreover, any sequence of feasible CP solutions in which the objective value approaches the infimum will become degenerate. We use the tools developed in Stegeman (2006), who considers p×p×2 arrays, and present a framework of analysis which is of use to the future study of CP degeneracy related to a two-valued typical rank. Moreover, our examples show that CP uniqueness is not necessary for degenerate solutions to occur. The author is supported by the Dutch Organisation for Scientific Research (NWO), VENI grant 451-04-102.     

PH?nomenologische Beschreibung, Essentielles Apriori, Transzendentale Subjektivit?t und Was Dann?

Ohne Zusammenfassung     

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.     

Who Is This God of Whom You Speak? Counseling Students' Concept of God

Many clients bring a belief in God or a Higher Power into counseling sessions. What is less clear, however, is how counselors conceptualize “god,” which may affect how spirituality is integrated in the counseling process. To explore this question, the authors conducted a concept map study of god images among counseling students. Results suggest that students harbor broad and diverse conceptualizations of god, which may affect the counseling process.     

The Cultural Divide and Changing Beliefs about Gender in the United States, 1974–2010

The present paper examines claims of a growing cultural divide in the United States. We analyze social change in beliefs about gender over a period of 36 years (from 1974 to 2010) in the United States using data from the nationally representative General Social Survey (GSS). We find evidence of growing gender egalitarianism until the mid-1990s, with a reversal between 1996 and 2000, and a decline in state differences in beliefs about gender over time in our decomposition analysis and multilevel models. Although we find significant differences in gender beliefs among states in the 1970s based on their voting record on the Equal Rights Amendment and based on patterns of family formation and family life associated with the Second Demographic Transition, these differences among states decreased or disappeared entirely by the early years of the twenty-first century. We highlight the implications of our findings for the ongoing public and academic debate surrounding growing cultural differences among states.     

Assessing the validity of the achievement motive in the presence of random measurement error

The presence of random measurement error in indicators of theoretical constructs biases observed estimates of relations among those constructs. Correcting for this bias is particularly important when random measurement error is substantial, or is substantially different for indicators of distinct constructs included in a theoretical model. Validity assessment in the case of thematic apperceptive measures of the achievement motive (TAT n Achievement) has been vulnerable to interpretive errors because these indicators of the achievement motive are typically much less reliable than indicators of other constructs to which the motive may be related, and no correction has been made for the bias introduced by such differential measurement error. Here we illustrate a causal modeling approach to validity assessment for TAT n Achievement that incorporates explicit true-score measurement models of theoretical constructs. We confirm the hypothesis that the achievement motive construct is positively related to work satisfaction in a representative sample of adult males in the United States, taken in 1976 (N = 413). Evidence for the discriminant validity of story content versus story length, an issue raised in the literature on the TAT, is also presented in this nomological network.