DOCSLIB.ORG

Bayesian Inference of Log Determinants

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bayesian Inference of Log Determinants Bayesian Inference of Log Determinants Jack Fitzsimons1 Kurt Cutajar2 Michael Osborne1 Stephen Roberts1 Maurizio Filippone2 1 Information Engineering, University of Oxford, UK 2 Department of Data Science, EURECOM, France Abstract The standard approach for evaluating the log determinant of a positive definite matrix involves the use of Cholesky The log determinant of a kernel matrix ap- decomposition (Golub & Van Loan, 1996), which is em- pears in a variety of machine learning prob- ployed in various applications of statistical models such lems, ranging from determinantal point pro- as kernel machines. However, the use of Cholesky de- 3 cesses and generalized Markov random fields, composition for general dense matrices requires O(n ) through to the training of Gaussian processes. operations, whilst also entailing memory requirements 2 Exact calculation of this term is often in- of O(n ). In view of this computational bottleneck, var- tractable when the size of the kernel matrix ex- ious models requiring the log determinant for inference ceeds a few thousands. In the spirit of proba- bypass the need to compute it altogether (Anitescu et al., bilistic numerics, we reinterpret the problem of 2012; Stein et al., 2013; Cutajar et al., 2016; Filippone & computing the log determinant as a Bayesian Engler, 2015). inference problem. In particular, we com- Alternatively, several methods exploit sparsity and struc- bine prior knowledge in the form of bounds ture within the matrix itself to accelerate computations. from matrix theory and evidence derived from For example, sparsity in Gaussian Markov random fields stochastic trace estimation to obtain proba- (GMRFs) arises from encoding conditional indepen- bilistic estimates for the log determinant and dence assumptions that are readily available when con- its associated uncertainty within a given com- sidering low-dimensional problems. For such matrices, putational budget. Beyond its novelty and the- the Cholesky decompositions can be computed in fewer oretic appeal, the performance of our proposal than O(n3) operations (Rue & Held, 2005; Rue et al., is competitive with state-of-the-art approaches 2009). Similarly, Kronecker-based linear algebra tech- to approximating the log determinant, while niques may be employed for kernel matrices computed also quantifying the uncertainty due to budget- on regularly spaced inputs (Saatc¸i, 2011). While these constrained evidence. ideas have proven successful for a variety of specific ap- plications, they cannot be extended to the case of general 1 INTRODUCTION dense matrices without assuming special forms or struc- tures for the available data. Developing scalable learning models without compro- To this end, general approximations to the log determi- mising performance is at the forefront of machine learn- nant frequently build upon stochastic trace estimation ing research. The scalability of several learning mod- techniques using iterative methods (Avron & Toledo, els is predominantly hindered by linear algebraic op- 2011). Two of the most widely-used polynomial ap- erations having large computational complexity, among proximations for large-scale matrices are the Taylor and which is the computation of the log determinant of a ma- Chebyshev expansions (Aune et al., 2014; Han et al., trix (Golub & Van Loan, 1996). The latter term features 2015). A more recent approach draws from the possibil- heavily in the machine learning literature, with applica- ity of estimating the trace of functions using stochastic tions including spatial models (Aune et al., 2014; Rue Lanczos quadrature (Ubaru et al., 2016), which has been & Held, 2005), kernel-based models (Davis et al., 2007; shown to outperform polynomial approximations from Rasmussen & Williams, 2006), and Bayesian learn- both a theoretic and empirical perspective. ing (Mackay, 2003). Inspired by recent developments in the field of proba- the logarithm of the matrix, and employ trace estima- bilistic numerics (Hennig et al., 2015), in this work we tion techniques (Hutchinson, 1990) to obtain unbiased propose an alternative approach for calculating the log estimates of these. Chen et al. (2011) propose an itera- determinant of a matrix by expressing this computation tive algorithm to efficiently compute the product of the as a Bayesian quadrature problem. In doing so, we refor- logarithm of a matrix with a vector, which is achieved mulate the problem of computing an intractable quantity by computing a spline approximation to the logarithm as an estimation problem, where the goal is to infer the function. A similar idea using Chebyshev polynomi- correct result using tractable computations that can be als has been developed by Han et al. (2015). Most re- carried out within a given time budget. In particular, we cently, the Lanczos method has been extended to handle model the eigenvalues of a matrix A from noisy obser- stochastic estimates of the trace and obtain probabilistic vations of Tr(Ak) obtained through stochastic trace es- error bounds for the approximation (Ubaru et al., 2016). timation using the Taylor approximation method (Zhang Blocking techniques, such as in Ipsen & Lee (2011) and & Leithead, 2007). Such a model can then be used to Ambikasaran et al. (2016), have also been proposed. make predictions on the infinite series of the Taylor ex- In our work, we similarly strive to use a small number pansion, yielding the estimated value of the log deter- of matrix-vector products for approximating log deter- minant. Aside from permitting a probabilistic approach minants. However, we show that by taking a Bayesian for predicting the log determinant, this approach inher- approach we can combine priors with the evidence gath- ently yields uncertainty estimates for the predicted value, ered from the intermediate results of matrix-vector prod- which in turn serves as an indicator of the quality of our ucts involved in the aforementioned methods to obtain approximation. more accurate results. Most importantly, our proposal Our contributions are as follows. has the considerable advantage that it provides a full dis- tribution on the approximated value. 1. We propose a probabilistic approach for computing the log determinant of a matrix which blends differ- Our proposal allows for the inclusion of explicit bounds ent elements from the literature on estimating log on log determinants to constrain the posterior distribu- determinants under a Bayesian framework. tion over the estimated log determinant (Bai & Golub, 1997). Nystrom¨ approximations can also be used to 2. We demonstrate how bounds on the expected value bound the log determinant, as shown in Bardenet & Tit- of the log determinant improve our estimates by sias (2015). Similarly, Gaussian processes (Rasmussen constraining the probability distribution to lie be- & Williams, 2006) have been formulated directly using tween designated lower and upper bounds. the eigendecomposition of its spectrum, where eigenvec- 3. Through rigorous numerical experiments on syn- tors are approximated using the Nystrom¨ method (Peng thetic and real data, we demonstrate how our & Qi, 2015). There has also been work on estimating method can yield superior approximations to com- the distribution of kernel eigenvalues by analyzing the peting approaches, while also having the additional spectrum of linear operators (Braun, 2006; Wathen & benefit of uncertainty quantification. Zhu, 2015), as well as bounds on the spectra of matrices with particular emphasis on deriving the largest eigen- 4. Finally, in order to demonstrate how this technique value (Wolkowicz & Styan, 1980; Braun, 2006). In this may be useful within a practical scenario, we em- work, we directly consider bounds on the log determi- ploy our method to carry out parameter selection for nants of matrices (Bai & Golub, 1997). a large-scale determinantal point process. To the best of our knowledge, this is the first time that 2 BACKGROUND the approximation of log determinants is viewed as a Bayesian inference problem, with the resulting quantifi- As highlighted in the introduction, several approaches cation of uncertainty being hitherto unexplored thus far. for approximating the log determinant of a matrix rely on stochastic trace estimation for accelerating computa- 1.1 RELATED WORK tions. This comes about as a result of the relationship between the log determinant of a matrix, and the corre- The most widely-used approaches for estimating log sponding trace of the log-matrix, whereby determinants involve extensions of iterative algorithms, such as the conjugate gradient and Lanczos methods, to logDet (A) = Trlog (A). (1) obtain estimates of functions of matrices (Chen et al., 2011; Han et al., 2015) or their trace (Ubaru et al., 2016). Provided the matrix log(A) can be efficiently sampled, The idea is to rewrite log determinants as the trace of this simple identity enables the use of stochastic trace es- 100 determinant of matrices having eigenvalues bounded be- 10-1 tween zero and one. In particular, this approach relies on the following logarithm identity, 10-2 1 10-3 X Ak log (I − A) = − : (3) 10-4 º = 1 º = 30 k k=1 Absolute Error º = 10 º = 40 10-5 º = 20 º = 50 -6 While the infinite summation is not explicitly com- 10 0 1 2 3 4 5 6 10 10 10 10 10 10 10 putable in finite time, this may be approximated by com- Order of Truncation puting a truncated series instead. Furthermore, given that the trace of matrices is additive, we find Figure 1: Expected absolute error of truncated Taylor m k series for stationary ν-continuous
Load more

Recommended publications
  • Lecture Notes in One File
    Phys 6124 zipped! World Wide Quest to Tame Math Methods Predrag Cvitanovic´ January 2, 2020 Overview The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. — Sidney Coleman I am leaving the course notes here, not so much for the notes themselves –they cannot be understood on their own, without the (unrecorded) live lectures– but for the hyperlinks to various source texts you might find useful later on in your research. We change the topics covered year to year, in hope that they reflect better what a graduate student needs to know. This year’s experiment are the two weeks dedicated to data analysis. Let me know whether you would have preferred the more traditional math methods fare, like Bessels and such. If you have taken this course live, you might have noticed a pattern: Every week we start with something obvious that you already know, let mathematics lead us on, and then suddenly end up someplace surprising and highly non-intuitive. And indeed, in the last lecture (that never took place), we turn Coleman on his head, and abandon harmonic potential for the inverted harmonic potential, “spatiotemporal cats” and chaos, arXiv:1912.02940. Contents 1 Linear algebra7 HomeworkHW1 ................................ 7 1.1 Literature ................................ 8 1.2 Matrix-valued functions ......................... 9 1.3 A linear diversion ............................ 12 1.4 Eigenvalues and eigenvectors ...................... 14 1.4.1 Yes, but how do you really do it? . 18 References ................................... 19 exercises 20 2 Eigenvalue problems 23 HomeworkHW2 ................................ 23 2.1 Normal modes .............................. 24 2.2 Stable/unstable manifolds .......................
    [Show full text]
  • Introduction to Quantum Field Theory
    Utrecht Lecture Notes Masters Program Theoretical Physics September 2006 INTRODUCTION TO QUANTUM FIELD THEORY by B. de Wit Institute for Theoretical Physics Utrecht University Contents 1 Introduction 4 2 Path integrals and quantum mechanics 5 3 The classical limit 10 4 Continuous systems 18 5 Field theory 23 6 Correlation functions 36 6.1 Harmonic oscillator correlation functions; operators . 37 6.2 Harmonic oscillator correlation functions; path integrals . 39 6.2.1 Evaluating G0 ............................... 41 6.2.2 The integral over qn ............................ 42 6.2.3 The integrals over q1 and q2 ....................... 43 6.3 Conclusion . 44 7 Euclidean Theory 46 8 Tunneling and instantons 55 8.1 The double-well potential . 57 8.2 The periodic potential . 66 9 Perturbation theory 71 10 More on Feynman diagrams 80 11 Fermionic harmonic oscillator states 87 12 Anticommuting c-numbers 90 13 Phase space with commuting and anticommuting coordinates and quanti- zation 96 14 Path integrals for fermions 108 15 Feynman diagrams for fermions 114 2 16 Regularization and renormalization 117 17 Further reading 127 3 1 Introduction Physical systems that involve an infinite number of degrees of freedom are usually described by some sort of field theory. Almost all systems in nature involve an extremely large number of degrees of freedom. A droplet of water contains of the order of 1026 molecules and while each water molecule can in many applications be described as a point particle, each molecule has itself a complicated structure which reveals itself at molecular length scales. To deal with this large number of degrees of freedom, which for all practical purposes is infinite, we often regard a system as continuous, in spite of the fact that, at certain distance scales, it is discrete.
    [Show full text]
  • Journal of Computational and Applied Mathematics Exponentials of Skew
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 233 (2010) 2867–2875 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Exponentials of skew-symmetric matrices and logarithms of orthogonal matrices João R. Cardoso a,b,∗, F. Silva Leite c,b a Institute of Engineering of Coimbra, Rua Pedro Nunes, 3030-199 Coimbra, Portugal b Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290 Coimbra, Portugal c Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal article info a b s t r a c t Article history: Two widely used methods for computing matrix exponentials and matrix logarithms are, Received 11 July 2008 respectively, the scaling and squaring and the inverse scaling and squaring. Both methods become effective when combined with Padé approximation. This paper deals with the MSC: computation of exponentials of skew-symmetric matrices and logarithms of orthogonal 65Fxx matrices. Our main goal is to improve these two methods by exploiting the special structure Keywords: of skew-symmetric and orthogonal matrices. Geometric features of the matrix exponential Skew-symmetric matrix and logarithm and extensions to the special Euclidean group of rigid motions are also Orthogonal matrix addressed. Matrix exponential ' 2009 Elsevier B.V. All rights reserved. Matrix logarithm Scaling and squaring Inverse scaling and squaring Padé approximants 1. Introduction n×n Given a square matrix X 2 R , the exponential of X is given by the absolute convergent power series 1 X X k eX D : kW kD0 n×n X Reciprocally, given a nonsingular matrix A 2 R , any solution of the matrix equation e D A is called a logarithm of A.
    [Show full text]
  • The Exponential Map for the Group of Similarity Transformations and Applications to Motion Interpolation
    The exponential map for the group of similarity transformations and applications to motion interpolation Spyridon Leonardos1, Christine Allen–Blanchette2 and Jean Gallier3 Abstract— In this paper, we explore the exponential map The problem of motion interpolation has been studied and its inverse, the logarithm map, for the group SIM(n) of extensively both in the robotics and computer graphics n similarity transformations in R which are the composition communities. Since rotations can be represented by quater- of a rotation, a translation and a uniform scaling. We give a formula for the exponential map and we prove that it is nions, the problem of quaternion interpolation has been surjective. We give an explicit formula for the case of n = 3 investigated, an approach initiated by Shoemake [21], [22], and show how to efficiently compute the logarithmic map. As who extended the de Casteljau algorithm to the 3-sphere. an application, we use these algorithms to perform motion Related work was done by Barr et al. [2]. Moreover, Kim interpolation. Given a sequence of similarity transformations, et al. [13], [14] corrected bugs in Shoemake and introduced we compute a sequence of logarithms, then fit a cubic spline 4 that interpolates the logarithms and finally, we compute the various kinds of splines on the 3-sphere in R , using the interpolating curve in SIM(3). exponential map. Motion interpolation and rational motions have been investigated by Juttler¨ [9], [10], Juttler¨ and Wagner I. INTRODUCTION [11], [12], Horsch and Juttler¨ [8], and Roschel¨ [20]. Park and Ravani [18], [19] also investigated Bezier´ curves on The exponential map is ubiquitous in engineering appli- Riemannian manifolds and Lie groups and in particular, the cations ranging from dynamical systems and robotics to group of rotations.
    [Show full text]
  • Linear Algebra Review
    Appendix A Linear Algebra Review In this appendix, we review results from linear algebra that are used in the text. The results quoted here are mostly standard, and the proofs are mostly omitted. For more information, the reader is encouraged to consult such standard linear algebra textbooks as [HK]or[Axl]. Throughout this appendix, we let Mn.C/ denote the space of n n matrices with entries in C: A.1 Eigenvectors and Eigenvalues n For any A 2 Mn.C/; a nonzero vector v in C is called an eigenvector for A if there is some complex number such that Av D v: An eigenvalue for A is a complex number for which there exists a nonzero v 2 Cn with Av D v: Thus, is an eigenvalue for A if the equation Av D v or, equivalently, the equation .A I /v D 0; has a nonzero solution v: This happens precisely when A I fails to be invertible, which is precisely when det.A I / D 0: For any A 2 Mn.C/; the characteristic polynomial p of A is given by A previous version of this book was inadvertently published without the middle initial of the author’s name as “Brian Hall”. For this reason an erratum has been published, correcting the mistake in the previous version and showing the correct name as Brian C. Hall (see DOI http:// dx.doi.org/10.1007/978-3-319-13467-3_14). The version readers currently see is the corrected version. The Publisher would like to apologize for the earlier mistake.
    [Show full text]
  • Arxiv:2004.07192V2 [Quant-Ph] 14 May 2021
    Two-way covert quantum communication in the microwave regime R. Di Candia,1, ∗ H. Yi˘gitler,1 G. S. Paraoanu,2 and R. J¨antti1 1Department of Communications and Networking, Aalto University, Espoo, 02150 Finland 2QTF Centre of Excellence, Department of Applied Physics, Aalto University School of Science, FI-00076 AALTO, Finland Quantum communication addresses the problem of exchanging information across macroscopic distances by employing encryption techniques based on quantum mechanical laws. Here, we ad- vance a new paradigm for secure quantum communication by combining backscattering concepts with covert communication in the microwave regime. Our protocol allows communication between Alice, who uses only discrete phase modulations, and Bob, who has access to cryogenic microwave technology. Using notions of quantum channel discrimination and quantum metrology, we find the ultimate bounds for the receiver performance, proving that quantum correlations can enhance the signal-to-noise ratio by up to 6 dB. These bounds rule out any quantum illumination advantage when the source is strongly amplified, and shows that a relevant gain is possible only in the low photon-number regime. We show how the protocol can be used for covert communication, where the carrier signal is indistinguishable from the thermal noise in the environment. We complement our information-theoretic results with a feasible experimental proposal in a circuit QED platform. This work makes a decisive step toward implementing secure quantum communication concepts in the previously uncharted 1 − 10 GHz frequency range, in the scenario when the disposable power of one party is severely constrained. I. INTRODUCTION range. For instance, QKD security proofs require level of noises which at room temperature are reachable only by frequencies at least in the Terahertz band [8].
    [Show full text]
  • Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices
    Randomized Linear Algebra Approaches to Estimate the Von Neumann Entropy of Density Matrices Eugenia-Maria Kontopoulou Ananth Grama Wojciech Szpankowski Petros Drineas Purdue University Purdue University Purdue University Purdue University Computer Science Computer Science Computer Science Computer Science West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA West Lafayette, IN, USA Email: [email protected] Email: [email protected] Email: [email protected] Email: [email protected] Abstract—The von Neumann entropy, named after John von Motivated by the above discussion, we seek numerical Neumann, is the extension of classical entropy concepts to the algorithms that approximate the von Neumann entropy of large field of quantum mechanics and, from a numerical perspective, density matrices, e.g., symmetric positive definite matrices can be computed simply by computing all the eigenvalues of a 3 density matrix, an operation that could be prohibitively expensive with unit trace, faster than the trivial O(n ) approach. Our for large-scale density matrices. We present and analyze two algorithms build upon recent developments in the field of randomized algorithms to approximate the von Neumann entropy Randomized Numerical Linear Algebra (RandNLA), an in- of density matrices: our algorithms leverage recent developments terdisciplinary research area that exploits randomization as a in the Randomized Numerical Linear Algebra (RandNLA) liter- computational resource to develop improved algorithms for ature, such as randomized trace
    [Show full text]
  • Computing Real Square Roots of a Real Matrix* LINEAR ALGEBRA
    Computing Real Square Roots of a Real Matrix* Nicholas J. Higham Department of Mathematics University of Manchester Manchester Ml3 9PL, England In memory of James H. WiIkinson Submitted by Hans Schneider ABSTRACT Bjiirck and Hammarling [l] describe a fast, stable Schur method for computing a square root X of a matrix A (X2 = A).We present an extension of their method which enables real arithmetic to be used throughout when computing a real square root of a real matrix. For a nonsingular real matrix A conditions are given for the existence of a real square root, and for the existence of a real square root which is a polynomial in A; thenumber of square roots of the latter type is determined. The conditioning of matrix square roots is investigated, and an algorithm is given for the computation of a well-conditioned square root. 1. INTRODUCTION Given a matrix A, a matrix X for which X2 = A is called a square root of A. Several authors have considered the computation of matrix square roots [3, 4, 9, 10, 15, 161. A particularly attractive method which utilizes the Schur decomposition is described by Bjiirck and Hammarling [l]; in general it requires complex arithmetic. Our main purpose is to show how the method can be extended so as to compute a real square root of a real matrix, if one exists, in real arithmetic. The theory behind the existence of matrix square roots is nontrivial, as can be seen by noting that while the n x n identity matrix has infinitely many square roots for n > 2 (any involutary matrix such as a Householder transformation is a square root), a nonsingular Jordan block has precisely two square roots (this is proved in Corollary 1).
    [Show full text]
  • Floquet Theory for Time Scales and Putzer Representations of Matrix Logarithms
    Taylor & Francis Journal of Difference Equations and Applications, Ta@rhFrancir tmup 2003 VOL.9 (1). pp. 77-92 Floquet Theory for Time Scales and Putzer Representations of Matrix Logarithms CALVIN D. AHLBRANDP,* and JERRY RIDENHOURb" aDepartment of Mathematics, University of Missouri, Columbia, MO 6521 1, USA; bDepartment of Mathematics and Statistics, Utah State University, Logan, UT 84322-3900, USA (Received 14 September 2001; Revised 9 October 2001; In final form 9 October 2001) Dedicated to Allan Peterson on the occasion of his 60th birthday. A Floquet theory is presented that generalizes known results for differential equations and for difference equations to the setting of dynamic equations on time scales. Since logarithms of matrices play a key role in Floquet theory, considerable effort is expended in producing case-free exact representations of the principal branch of the matrix logarithm. Such representations were fmt produced by Putzer as representations of matrix exponentials. Some representations depend on knowledge of the eigenvalues while others depend only on the coefficients of the characteristic polynomial. Logarithms of special forms of matrices are also considered. In particular, it is shown that the square of a nonsingular matrix with real entries has a logarithm with real entries. Keywords: Floquet theory; Matrix logarithm; Principal branch; Putzer representation 2000 AMS Subject ClassiJication: 39A10 1. INTRODUCTION The theory of dynamic equations on time scales (or, more generally, measure chains) was introduced in Stefan Hilger's PhD thesis in 1988. The theory presents a structure where, once a result is established for general time scales, special cases include a result for differential equations (obtained by taking the time scale to be the real numbers) and a result for difference equations (obtained by taking the time scale to be the integers).
    [Show full text]
  • Fast Computation of Von Neumann Entropy for Large-Scale Graphs Via Quadratic Approximations
    Linear Algebra and its Applications 585 (2020) 127–146 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Fast computation of von Neumann entropy for large-scale graphs via quadratic approximations Hayoung Choi a,∗, Jinglian He b, Hang Hu b, Yuanming Shi b a Research Institute of Mathematics, Seoul National University, Republic of Korea b School of Information Science and Technology, ShanghaiTech University, China a r t i c l e i n f o a b s t r a c t Article history: The von Neumann graph entropy (VNGE) can be used as Received 21 November 2018 a measure of graph complexity, which can be the measure Accepted 24 September 2019 of information divergence and distance between graphs. Available online 1 October 2019 However, computing VNGE is extensively demanding for a Submitted by M. Benzi large-scale graph. We propose novel quadratic approximations MSC: for fast computing VNGE. Various inequalities for error 05C12 between the quadratic approximations and the exact VNGE 05C50 are found. Our methods reduce the cubic complexity of VNGE 05C80 to linear complexity. Computational simulations on random 05C90 graph models and various real network datasets demonstrate 81P45 superior performance. 81P68 © 2019 Elsevier Inc. All rights reserved. Keywords: Von Neumann entropy Von Neumann graph entropy Graph dissimilarity Density matrix 1. Introduction Graph is one of the most common representations of complex data. It has the sophis- ticated capability of representing and summarizing irregular structural features. Today, * Corresponding author. E-mail addresses: [email protected] (H. Choi), [email protected] (J.
    [Show full text]
  • Logarithms and Square Roots of Real Matrices Existence, Uniqueness, and Applications in Medical Imaging
    Logarithms and Square Roots of Real Matrices Existence, Uniqueness, and Applications in Medical Imaging Jean Gallier Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104, USA [email protected] September 2, 2019 Abstract. The need for computing logarithms or square roots of real matrices arises in a number of applied problems. A significant class of problems comes from medical imag- ing. One of these problems is to interpolate and to perform statistics on data represented by certain kinds of matrices (such as symmetric positive definite matrices in DTI). Another important and difficult problem is the registration of medical images. For both of these prob- lems, the ability to compute logarithms of real matrices turns out to be crucial. However, not all real matrices have a real logarithm and thus, it is important to have sufficient conditions for the existence (and possibly the uniqueness) of a real logarithm for a real matrix. Such conditions (involving the eigenvalues of a matrix) are known, both for the logarithm and the square root. As far as I know, with the exception of Higham's recent book [18], proofs of the results involving these conditions are scattered in the literature and it is not easy to locate them. Moreover, Higham's excellent book assumes a certain level of background in linear algebra that readers interested in applications to medical imaging may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham [18]. In this paper, I present a unified exposition of these results, including a proof of the existence of the Real Jordan Form, and give more direct proofs of some of these results using the Real Jordan Form.
    [Show full text]
  • Bayesian Inference of Log Determinants
    Bayesian Inference of Log Determinants Jack Fitzsimons1 Kurt Cutajar2 Michael Osborne1 Stephen Roberts1 Maurizio Filippone2 1 Information Engineering, University of Oxford, UK 2 Department of Data Science, EURECOM, France Abstract The standard approach for evaluating the log- determinant of a positive definite matrix involves The log-determinant of a kernel matrix ap- the use of Cholesky decomposition (Golub & Van Loan, pears in a variety of machine learning prob- 1996), which is employed in various applications of lems, ranging from determinantal point pro- statistical models such as kernel machines. However, cesses and generalized Markov random fields, the use of Cholesky decomposition for general dense 3 through to the training of Gaussian processes. matrices requires O(n ) operations, whilst also entail- 2 Exact calculation of this term is often in- ing memory requirements of O(n ). In view of this tractable when the size of the kernel matrix ex- computational bottleneck, various models requiring ceeds a few thousands. In the spirit of proba- the log-determinant for inference bypass the need to bilistic numerics, we reinterpret the problem of compute it altogether (Anitescu et al., 2012; Stein et al., computing the log-determinant as a Bayesian 2013; Cutajar et al., 2016; Filippone & Engler, 2015). inference problem. In particular, we com- Alternatively, several methods exploit sparsity and struc- bine prior knowledge in the form of bounds ture within the matrix itself to accelerate computations. from matrix theory and evidence derived from For example, sparsity in Gaussian Markov Random stochastic trace estimation to obtain proba- fields (GMRFs) arises from encoding conditional inde- bilistic estimates for the log-determinant and pendence assumptions that are readily available when its associated uncertainty within a given com- considering low-dimensional problems.
    [Show full text]