Entropic Trace Estimates for Log Determinants

Entropic Trace Estimates for Log Determinants

Entropic Trace Estimates for Log Determinants Jack Fitzsimons1, Diego Granziol1, Kurt Cutajar2 Michael Osborne1, Maurizio Filippone2, and Stephen Roberts1 1 Department of Engineering, University of Oxford, UK fjack,diego,mosb,[email protected] 2 Department of Data Science, EURECOM, France fkurt.cutajar,[email protected] Abstract. The scalable calculation of matrix determinants has been a bottleneck to the widespread application of many machine learning methods such as determinantal point processes, Gaussian processes, gen- eralised Markov random fields, graph models and many others. In this work, we estimate log determinants under the framework of maximum en- tropy, given information in the form of moment constraints from stochas- tic trace estimation. The estimates demonstrate a significant improve- ment on state-of-the-art alternative methods, as shown on a wide vari- ety of matrices from the SparseSuite Matrix Collection. By taking the example of a general Markov random field, we also demonstrate how this approach can significantly accelerate inference in large-scale learn- ing methods involving the log determinant. 1 Introduction Scalability is a key concern for machine learning in the big data era, whereby inference schemes are expected to yield optimal results within a constrained computational budget. Underlying these algorithms, linear algebraic operations with high computational complexity pose a significant bottleneck to scalability, and the log determinant of a matrix [5] falls firmly within this category of opera- tions. The canonical solution involving Cholesky decomposition [16] for a general n × n positive definite matrix, A, entails time complexity of O(n3) and storage requirements of O(n2), which is infeasible for large matrices. Consequently, this term greatly hinders widespread use of the learning models where it appears, which includes determinantal point processes [24], Gaussian processes [31], and graph problems [36]. The application of kernel machines to vector valued input data has gained considerable attention in recent years, enabling fast linear algebra techniques. Examples include Gaussian Markov random fields [32] and Kronecker-based alge- bra [33], while similar computational speed-ups may also be obtained for sparse matrices. Nonetheless, such structure can only be expected in selected applica- tions, thus limiting the widespread use of such techniques. In light of this computational constraint, several approximate inference schemes have been developed for estimating the log determinant of a matrix more ef- ficiently. Generalised approximation schemes frequently build upon iterative stochastic trace estimation techniques [4]. This includes polynomial approxi- mations such as Taylor and Chebyshev expansions [2, 20]. Recent developments shown to outperform the aforementioned approximations include estimating the trace using stochastic Lanczos quadrature [35], and a probabilistic numerics approach based on Gaussian process inference which incorporates bound infor- mation [12]. The latter technique is particularly significant as it introduces the possibility of quantifying the numerical uncertainty inherent to the approxima- tion. In this paper, we present an alternative probabilistic approximation of log determinants rooted in information theory, which exploits the relationship be- tween stochastic trace estimation and the moments of a matrix's eigenspectrum. These estimates are used as moment constraints on the probability distribution of eigenvalues. This is achieved by maximising the entropy of the probability density p(λ) given our moment constraints. In our inference scheme, we circum- vent the issue inherent to the Gaussian process approach [12], whereby positive probability mass may occur in the region of negative densities. In contrast, our proposed entropic approach implicitly encodes the constraint that densities are necessarily positive. Given equivalent moment information, we achieve compet- itive results on matrices obtained from the SuiteSparse Matrix Collection [11] which consistently outperform competing approximations to the log-determinant [12, 7]. The most significant contributions of this work are listed below.3 1. We develop a novel approximation to the log determinant of a matrix which relies on the principle of maximum entropy enhanced with moment con- straints derived from stochastic trace estimation. 2. We present the theory motivating the use of maximum entropy for solving this problem, along with insights on why we expect particularly significant improvements over competing techniques for large matrices. 3. We directly compare the performance of our entropic approach to other state- of-the-art approximations to the log-determinant. This evaluation covers real sparse matrices obtained from the SuiteSparse Matrix Collection [11]. 4. Finally, to showcase how the proposed approach may be applied in a practical scenario, we incorporate our approximation within the computation of the log-likelihood term of a Gaussian Markov random field, where we obtain a significant increase in speed. 1.1 Related Work The methodology presented in this work predominantly draws inspiration from the recently introduced probabilistic numerics approach to estimating the log de- terminant of a matrix [12]. In that work, the computation of the log determinant 3 Code for algorithms proposed in this paper are available at https://github.com/OxfordML/EntropicTraceEstimation is reinterpreted as a probabilistic estimation problem, whereby results obtained from budgeted computations are used to infer accurate estimates for the log de- terminant. In particular, within that proposed framework, the eigenvalues of a matrix A are modelled from noisy observations of Tr(Ak) obtained from stochas- tic trace estimation [4] using the Taylor approximation method. By modelling such noisy observations using a Gaussian process [31], Bayesian quadrature [29] can then be invoked for making predictions on the infinite series of the Taylor expansion, and in turn estimating the log determinant. Of particular interest is the uncertainty quantification inherent to this approach, which is a notable step forward in the direction of measuring the complete numerical uncertainty asso- ciated with approximating large-scale inference models. The estimates obtained using this Bayesian set-up may be further improved by considering known up- per and lower bounds on the value of the log determinant [5]. In this paper, we provide an alternative to this approach by interpreting the observed moments as being constraints on the probability distribution of eigenvalues underlying the computation of the log determinant. As we shall explore, our novel entropic formulation makes better calibrated prior assumptions than the previous work, and consequently yields superior performance. More traditional approaches to approximating the log determinant build upon iterative algorithms, and exploit the fact that the log determinant may be rewritten as the trace of the logarithm of the matrix. This features in both the Chebyshev expansion approximation [20], as well as the widely-used Tay- lor series approximation upon which the aforementioned probabilistic inference approaches are built. Recently, an approximation to the log determinant using stochastic Lanczos quadrature [35] has been shown to outperform the aforemen- tioned polynomial approaches, while also providing probabilistic error bounds. Finally, given that the logarithm of a matrix often appears multiplied by a vec- tor (for example the log likelihood term of a Gaussian process [31]), the spline approximation proposed in [10] may be used to accelerate computation. 2 Background In this section, we shall formally introduce the concepts underlying the proposed maximum entropy approach to approximating the log determinant. We start by describing stochastic trace estimation and demonstrate how this can be applied to estimating the trace term of matrix powers. Subsequently, we illustrate how the latter terms correspond to the raw moments of the matrix's eigenspectrum, and show how the log determinant may be inferred from the distribution of eigenvalues constrained by such moments. 2.1 Stochastic Trace Estimation Estimating the trace of implicit matrices is a central component of many ap- proaches to approximate the log determinant of a matrix. Stochastic trace esti- mation [4] builds a Monte Carlo estimate of the trace of a matrix A by repeatedly multiplying it by probing vectors z, m 1 X Tr(A) ≈ zT Az ; m i i i=1 T T such that the expectation of zizi is the identity, E[zizi ] = I. This can be readily T verified using the expectation of Tr(zi Azi) by exploiting the cyclical property of the trace operation. As such, many choices of how to sample the probing vectors have emerged. Possibly the most na¨ıve choice involves sampling from the columns of the identity matrix; however, due to poor expected sample vari- ance this is not widely used in the literature. Sampling from vectors on the unit hyper-sphere, and correspondingly sampling normal random vectors (Gaussian estimator), significantly reduces the sample variance, but more random bits are required to generate each sample. A major progression for stochastic trace esti- mation was the introduction of Hutchinson's method [21], which sampled each element as a Bernoulli random variable requiring only a linear number of ran- dom bits, while also reducing the sample variance even further. A more recent approach involves sampling from sets of mutually

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