Minimum Rank and Minimum Trace of Covariance Matrices

Riccia, Giacomo Della; Shapiro, Alexander

doi:10.1007/bf02293708

Cited by 39 publications

(7 citation statements)

References 6 publications

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“…The relation between minimum-trace factor analysis and minimum-rank factor analysis goes back to Ledermann in [56] (see [22] and [66]). Herein we only refer to two propositions which characterize minimizers for the two problems, Frisch’s and Shapiro’s, respectively.…”

Section: Trace-minimization Heuristicsmentioning

confidence: 99%

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Linear Models Based on Noisy Data and the Frisch Scheme

Ning¹,

Georgiou²,

Tannenbaum³

et al. 2015

SIAM Rev.

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We address the problem of identifying linear relations among variables based on noisy measurements. This is a central question in the search for structure in large data sets. Often a key assumption is that measurement errors in each variable are independent. This basic formulation has its roots in the work of Charles Spearman in 1904 and of Ragnar Frisch in the 1930s. Various topics such as errors-in-variables, factor analysis, and instrumental variables all refer to alternative viewpoints on this problem and on ways to account for the anticipated way that noise enters the data. In the present paper we begin by describing certain fundamental contributions by the founders of the field and provide alternative modern proofs to certain key results. We then go on to consider a modern viewpoint and novel numerical techniques to the problem. The central theme is expressed by the Frisch–Kalman dictum, which calls for identifying a noise contribution that allows a maximal number of simultaneous linear relations among the noise-free variables—a rank minimization problem. In the years since Frisch’s original formulation, there have been several insights, including trace minimization as a convenient heuristic to replace rank minimization. We discuss convex relaxations and theoretical bounds on the rank that, when met, provide guarantees for global optimality. A complementary point of view to this minimum-rank dictum is presented in which models are sought leading to a uniformly optimal quadratic estimation error for the error-free variables. Points of contact between these formalisms are discussed, and alternative regularization schemes are presented.

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Section: Trace-minimization Heuristicsmentioning

confidence: 99%

“…In view of the fundamental nature of this problem and the scarcity of exact solutions, several computational approaches have emerged over the years [68, 69, 22, 8, 79, 9, 59]. A key trend has been to seek numerical decompositions that employ the trace as a surrogate for the rank [28].…”

Section: Introductionmentioning

confidence: 99%

Linear Models Based on Noisy Data and the Frisch Scheme

Ning¹,

Georgiou²,

Tannenbaum³

et al. 2015

SIAM Rev.

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“…Such a D can be obtained e.g., by minimum trace factor analysis (Bentler, 1972; Della Riccia & Shapiro, 1982). …”

Section: Lemmamentioning

confidence: 99%

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Bentler

Yuan

2011

Psychometrika

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Indefinite symmetric matrices that are estimates of positive definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based theory for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk.

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“… 1 A more recent name for the optimization is minimum trace factor analysis or MTFA (e.g., della Riccia & Shapiro, 1982; Shapiro, 1982). …”

mentioning

confidence: 99%

Alpha, Dimension-Free, and Model-Based Internal Consistency Reliability

Bentler¹

2009

Psychometrika

361

283

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As pointed out by Sijtsma (in press), coefficient alpha is inappropriate as a single summary of the internal consistency of a composite score. Better estimators of internal consistency are available. In addition to those mentioned by Sijtsma, an old dimension-free coefficient and structural equation model based coefficients are proposed to improve the routine reporting of psychometric internal consistency. The various ways to measure internal consistency are also shown to be appropriate to binary and polytomous items.Sijtsma (in press) has ably critiqued the unfortunate ascendance of coefficient alpha as the almost universal and sole estimator of the somewhat vague population concept of internal consistency. Among several recommendations, he suggests to report the greatest lower bound, here denoted ρ glb , as a measure of internal consistency reliability, and the explained common variance (ECV) based on minimum rank factor analysis, as a measure of unidimensionality. These are good recommendations, but they are incomplete. We provide additional technical detail on some methods, and also consider several alternative methods.Using a slightly different notation than Sijtsma, we are interested in describing the reliability of a composite X that is a simple sum of p unit-weighted components such as X = X 1 + X 2 +… + X p . An internal consistency reliability coefficient describes the quality of the composite or scale in terms of hypothesized constituents of the components X i . These might represent true and error parts based on classical test theory (X i = T i + E i ), common and unique parts based on common factor analysis (X i = C i + U i ), or the loading of the component on its factor plus residual error (X i = λ i F + E i ). Since U i is the sum of two uncorrelated parts called specificity and error, that is, U i = S i + E i , where S i is specificity, it is a more general decomposition than that of classical test theory and we focus on it. We assume that C i , and U i are uncorrelated, and that the X i are not linearly dependent. Then the covariance structure model ∑ = ∑ C + Ψ follows, where ∑ C is the positive semidefinite (psd) covariance matrix of the common variables and Ψ is the covariance matrix of the unique variables, typically taken as positive definite and diagonal. In general, the C i , are linearly dependent, so that ∑ C can be of low rank and can have such factor analytic (FA) decompositions as ∑ C = ΛΛ′ or ∑ C = ΛΦΛ′, or a structure based on a FA simultaneous equation system such as ∑ C = Λ(I − B) −1 Φ(I − B) −1′ Λ′ (see Bentler, 2007). Here Λis a factor loading matrix,Φ is the covariance matrix of nondependent latent factors, and B is a matrix of coefficients relating latent factors.

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Minimum Rank and Minimum Trace of Covariance Matrices

Abstract: factor analysis, covariance matrices, minimum trace, constrained minimum trace, minimum rank,

Cited by 39 publications

References 6 publications

Linear Models Based on Noisy Data and the Frisch Scheme

Linear Models Based on Noisy Data and the Frisch Scheme

Positive Definiteness via Off-diagonal Scaling of a Symmetric Indefinite Matrix

Alpha, Dimension-Free, and Model-Based Internal Consistency Reliability

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