soft-thresholding, a commonly used operation in modern high-dimensional probability and statistics, is not guaranteed to maintain positive definiteness, even if the original matrix is sparse. This ...result has a deep connection to graphs, and in particular, to the class of trees. We then proceed to fully characterize functions which do preserve positive definiteness. This characterization is in terms of absolutely monotonic functions and turns out to be quite different from the case when the function is also applied to diagonal elements. We conclude by giving bounds on the condition number of a matrix which guarantee that the regularized matrix is positive definite.>
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Entrywise functions preserving the cone of positive semidefinite matrices have been studied by many authors, most notably by Schoenberg Duke Math. J. 9, 1942 and Rudin Duke Math. J. 26, 1959. ...Following their work, it is well-known that entrywise functions preserving Loewner positivity in all dimensions are precisely the absolutely monotonic functions. However, there are strong theoretical and practical motivations to study functions preserving positivity in a fixed dimension 𝑛. Such characterizations for a fixed value of 𝑛 are difficult to obtain, and in fact are only known in the 2 × 2 case. In this paper, using a novel and intuitive approach, we study entrywise functions preserving positivity on distinguished submanifolds inside the cone obtained by imposing rank constraints. These rank constraints are prevalent in applications, and provide a natural way to relax the elusive original problem of preserving positivity in a fixed dimension. In our main result, we characterize entrywise functions mapping 𝑛 × 𝑛 positive semidefinite matrices of rank at most 𝑙 into positive semidefinite matrices of rank at most 𝑘 for 1 ≤ 𝑙 ≤ 𝑛 and 1 ≤ 𝑘 < 𝑛. We also demonstrate how an important necessary condition for preserving positivity by Horn and Loewner Trans. Amer. Math. Soc. 136, 1969 can be significantly generalized by adding rank constraints. Finally, our techniques allow us to obtain an elementary proof of the classical characterization of functions preserving positivity in all dimensions obtained by Schoenberg and Rudin.
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Understanding centennial scale climate variability requires data sets that are accurate, long, continuous and of broad spatial coverage. Since instrumental measurements are generally only available ...after 1850, temperature fields must be reconstructed using paleoclimate archives, known as proxies. Various climate field reconstructions (CFR) methods have been proposed to relate past temperature to such proxy networks. In this work, we propose a new CFR method, called GraphEM, based on Gaussian Markov random fields embedded within an EM algorithm. Gaussian Markov random fields provide a natural and flexible framework for modeling high-dimensional spatial fields. At the same time, they provide the parameter reduction necessary for obtaining precise and well-conditioned estimates of the covariance structure, even in the sample-starved setting common in paleoclimate applications. In this paper, we propose and compare the performance of different methods to estimate the graphical structure of climate fields, and demonstrate how the GraphEM algorithm can be used to reconstruct past climate variations. The performance of GraphEM is compared to the widely used CFR method RegEM with regularization via truncated total least squares, using synthetic data. Our results show that GraphEM can yield significant improvements, with uniform gains over space, and far better risk properties. We demonstrate that the spatial structure of temperature fields can be well estimated by graphs where each neighbor is only connected to a few geographically close neighbors, and that the increase in performance is directly related to recovering the underlying sparsity in the covariance of the spatial field. Our work demonstrates how significant improvements can be made in climate reconstruction methods by better modeling the covariance structure of the climate field.
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Climate field reconstructions(CFRs) enable spatially resolved estimates of past climates, providing important insights about climate variability over the Common Era. In particular, a reconstructed ...“La Niña‐like” pattern during the transition from the Medieval Climate Anomaly (MCA) to the Little Ice Age has been widely tied to medieval droughts in southwest North America. This pattern is now used as a key benchmark for global climate model simulations of the last millennium, which have yet to reproduce it. Here we test the pattern's robustness by using four different CFR methods and two proxy networks. With the older network, we find the reconstructed patterns to be highly method‐dependent, with the La Niña‐like pattern not reproduced by two of the CFR methodologies. With the updated proxy network, a globally uniform MCA emerges with all methods, in agreement with simulations from the Paleoclimate Modelling Intercomparison Project Phase 3 ensemble. Our results caution against drawing dynamical interpretations from a single CFR and affirm the importance of developing CFRs through improved statistical methodology and community‐driven proxy syntheses.
Key Points
Reconstructed spatial patterns of temperature are highly method dependent
CFRs are more consistent with GCM simulations when an updated network is used
Results based on a single CFR should be treated with caution
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Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg Duke Math. J. 9, Rudin ...Duke Math. J. 26, and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph G or a family of graphs Gn
. Our results generalize Schoenberg and Rudin’s results to a modern setting, where functions are often applied entrywise to sparse matrices in order to improve their properties (e.g. better conditioning, graphical models). The only such result known in the literature is for the complete graph K
2. We provide the first such characterization result for a large family of noncomplete graphs. Specifically, we characterize functions preserving Loewner positivity on matrices with zeros according to a tree. These functions are multiplicatively midpoint-convex and superadditive. Leveraging the underlying sparsity in matrices thus admits the use of functions which are not necessarily analytic nor absolutely monotonic. We further show that analytic functions preserving positivity on matrices with zeros according to trees can contain arbitrarily long sequences of negative coefficients, thus obviating the need for absolute monotonicity in a very strong sense. This result leads to the question of exactly when absolute monotonicity is necessary when preserving positivity for an arbitrary class of graphs. We then provide a stronger condition in terms of the numerical range of all symmetric matrices, such that functions satisfying this condition on matrices with zeros according to any family of graphs with unbounded degrees are necessarily absolutely monotonic.
2010 Mathematics Subject Classification. Primary 15B48; Secondary 26E05, 05C50, 26A48.
Key words and phrases. Matrices with structure of zeros, entrywise positive maps, absolutely monotonic functions, multiplicatively convex functions, positive semidefiniteness, Loewner ordering, fractional Schur powers.
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A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's ...work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.
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Let X=(x1,…,xn) be a vector of distinct positive integers. The n×n matrix S=S(X):=(gcd(xi,xj))i,j=1n, where gcd(xi,xj) denotes the greatest common divisor of xi and xj, is called the greatest common ...divisor (GCD) matrix on X. By a surprising result of Beslin and Ligh (1989) 7, all GCD matrices are positive definite. In this paper, we completely characterize the GCD matrices satisfying the stronger property of being totally nonnegative (TN) or totally positive (TP). As we show, a GCD matrix is never TP when n≥3, and is TN if and only if it is TN2, i.e., all its 2×2 minors are nonnegative. We next demonstrate that a GCD matrix is TN2 if and only if the exponents of each prime divisor in the prime factorization of the xi's form a monotonic sequence. Reformulated in the language of kernels, our results characterize the subsets of integers over which the kernel K(x,y)=gcd(x,y) is totally nonnegative. The proofs of our characterizations depend on Gantmacher and Krein's notion of a Green's matrix. We conclude by showing that a GCD matrix is TN if and only if it is a Green's matrix. As a consequence, we obtain explicit formulas for all the minors and for the inverse of totally nonnegative GCD matrices.
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Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently ...in the literature. Following FitzGerald and Horn (1977) 8, it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below and above the critical exponents that are positive, monotone, or convex on the cone of positive semidefinite matrices. We then extend the original problem by fully classifying the positive, monotone, or convex powers in a more general setting where additional rank constraints are imposed on the matrices. We also classify the entrywise powers that are super/sub-additive with respect to the Loewner ordering. Finally, we extend all the previous characterizations to matrices with negative entries. Our analysis consequently allows us to answer a question raised by Bhatia and Elsner (2007) regarding the smallest dimension for which even extensions of the power functions do not preserve Loewner positivity.
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Hirschman and Widder introduced a class of Pólya frequency functions given by linear combinations of one-sided exponential functions. The members of this class are probability densities, and the ...class is closed under convolution but not under pointwise multiplication. We show that, generically, a polynomial function of such a density is a Pólya frequency function only if the polynomial is a homothety, and also identify a subclass for which each positive-integer power is a Pólya frequency function. We further demonstrate connections between the Maclaurin coefficients, the moments of these densities, and the recovery of the density from finitely many moments, via Schur polynomials.
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Positive definite (p.d.) matrices arise naturally in many areas within mathematics and also feature extensively in scientific applications. In modern high-dimensional applications, a common approach ...to finding sparse positive definite matrices is to threshold their small off-diagonal elements. This thresholding, sometimes referred to as hard-thresholding, sets small elements to zero. Thresholding has the attractive property that the resulting matrices are sparse, and are thus easier to interpret and work with. In many applications, it is often required, and thus implicitly assumed, that thresholded matrices retain positive definiteness. In this paper we formally investigate the algebraic properties of p.d. matrices which are thresholded. We demonstrate that for positive definiteness to be preserved, the pattern of elements to be set to zero has to necessarily correspond to a graph which is a union of complete components. This result rigorously demonstrates that, except in special cases, positive definiteness can be easily lost. We then proceed to demonstrate that the class of diagonally dominant matrices is not maximal in terms of retaining positive definiteness when thresholded. Consequently, we derive characterizations of matrices which retain positive definiteness when thresholded with respect to important classes of graphs. In particular, we demonstrate that retaining positive definiteness upon thresholding is governed by complex algebraic conditions.
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