Received: 5 November 2007, Revised: 7 December 2007, Accepted: 14 January 2008, Published online in Wiley InterScience: 2008 GRAM satisfies Kruskal’s condition Nicolaas (Klaas) M. Faber a * It is well known that least-squares (LS) methods uniquely identify the parameters of the CANDECOMP/PARAFAC model if Kruskal’s condition is satisfied. By contrast, a stricter sufficient condition applies to eigenvalue-based methods like the generalized rank annihilation method (GRAM). This discrepancy suggests that LS methods can solve problems for which GRAM must fail. However, GRAM has been specifically introduced for the special case of a three-way array with two frontal slices only (i.e. K ¼ 2). Here, it is shown that the two conditions are equivalent for this special case. Copyright ß 2008 John Wiley & Sons, Ltd. Keywords: CANDECOMP; PARAFAC; generalized rank annihilation method; identifiability; Kruskal’s condition 1. THE MODEL AND IDENTIFIABILITY Consider an I  J  K three-way array X, containing two I  J frontal slices X 1 and X 2 , that is the special case K ¼ 2. When ignoring noise terms to simplify the presentation, the F-component CANDECOMP/PARAFAC model [1,2] for X can be written as X 1 ¼ AD 1 ðCÞB T X 2 ¼ AD 2 ðCÞB T (1) where A is an I  F matrix, B is a J  F matrix and D 1 (C) and D 2 (C) are diagonal matrices (F  F) constructed out of rows 1 and 2 of a 2  F matrix C, respectively. Kruskal [3] has shown that the decomposition in Equation (1) is unique if k A þ k B þ k C  2F þ 2 (2) where k symbolizes the so-called k-rank and the subscript refers to the particular component matrix (A, B or C). The concept of k-rank was termed universal k-column independence by Kruskal [4], but named k-rank (‘Kruskal-rank’) by Harshman and Lundy [5], a term that appears to be generally adopted. The k-rank of a matrix is defined as the largest value of k such that every subset of k columns of the matrix is linearly independent. The k-rank is related, but not equal, to the rank of a matrix, as the k-rank can never exceed the rank. Leurgans et al. [6], among others, have shown that eigenvalue-based methods, such as the popular generalized rank annihilation method (GRAM) [7], uniquely recover the decomposition in Equation (1) if the component matrices A and B are of full rank (F) and every pair of columns of C is linearly independent (i.e. k C  2). It is easily verified that this stricter condition is implied by Kruskal’s condition for the special case that K ¼ 2, but a proof could not be found in the literature. A simple proof follows. Since the rank of a matrix cannot exceed the number of rows and columns, it holds that k A  r A  min(I,F)  F , in which r A is the rank of A, likewise k A  F . It is immediate that Kruskal’s condition can only hold (as an equality) for the special case if k C ¼ 2 and k A ¼ k B ¼ F . Noting that full k-rank implies full rank completes the proof. 2. CONCLUDING REMARKS Least-squares (LS) methods have been shown to outperform variants of GRAM when K > 2 in terms of the quality of the decomposition [8–11]. In addition, these variants are inferior for not satisfying Kruskal’s condition when K > 2. By contrast, GRAM has been shown to compare well with LS methods when K ¼ 2 in terms of the quality of the decomposition [12]. Here, it has been shown that GRAM satisfies Kruskal’s condition when K ¼ 2. It is noted that this special case is of considerable interest to analytical chemistry-related fields, since a single standard addition of all analytes of interest to the calibration sample may already provide sufficient data for solving the analytical problem [7]. Currently, GRAM appears to be controversial, see for example reference [13], partly due to one of the former results [8]. The current observation may help rehabilitating this method. REFERENCES 1. Harshman RA. Foundations of the PARAFAC procedure: Models and conditions for an ‘‘explanatory’’ multi-modal factor analysis. UCLA Working Papers Phonet. 1970; 16: 1–84. 2. Carroll JD, Chang JJ. Analysis of individual differences in multidimen- sional scaling via an n-way generalization of ‘‘Eckart-Young’’ decomposition. Psychometrika 1970; 35: 283–319. 3. Kruskal JB. Three-way arrays: Rank and uniqueness of trilinear decompositions, with applications to arithmetic complexity and statistics. Linear Algebra Appl. 1977; 18: 95–138. 4. Kruskal JB. Rank, decomposition, and uniqueness for 3-way and N-way arrays. In Multiway Data Analysis, Coppi R, Bolasco S (eds). Elsevier: Amsterdam, 1989; 7–18. 5. Harshman RA, Lundy ME. The PARAFAC model for three-way factor analysis and multidimensional scaling. In Research Methods for Multimode Data Analysis, Law HG, Snyder CW Jr, Hattie JA, McDonald RP (eds). Praeger: New York, 1984; 122–215. 6. Leurgans SE, Ross RT, Abel RB. A decomposition for three-way arrays. SIAM J. Matrix Anal. Appl. 1993; 14: 1064–1083. 7. Sa ´ nchez E, Kowalski BR. Generalized rank annihilation factor analysis. Anal. Chem. 1986; 58: 496–499. (www.interscience.wiley.com) DOI: 10.1002/cem.1134 Short Communication * Chemometry Consultancy, Goudenregenstraat 6, 6573 XN Beek-Ubbergen, The Netherlands. E-mail: nmf@chemometry.com a N. M. Faber Chemometry Consultancy, Goudenregenstraat 6, 6573 XN Beek-Ubbergen, The Netherlands J. Chemometrics 2008; 22: 417–418 Copyright ß 2008 John Wiley & Sons, Ltd. 417