MGGD Parameter Estimation on the Space of SPD Matrices Zois Boukouvalas 1 Jialun Zhou 2 Mark Fuge 1 Salem Said 2 1. Problem Statement Due to its simple parametric form, the family of multivari- ate generalized Gaussian distributions (MGGD) has been widely used for modeling vector-valued signals. Therefore, efficient estimation of its parameters is of significant inter- est for a number of machine learning tasks. The MGGD probability density functions are given by (Kotz, 1975) p(x; Σ,β,m)= η β m p 2 |Σ| 1 2 ×exp  - 1 2m β ( x ⊤ Σ −1 x ) β  , where x ∈ R p , η = Γ( p 2 ) π p 2 Γ( p 2β )2 p 2β , m> 0 is a scale param- eter, β> 0 is a shape parameter that controls the distribu- tion’s peakedness and spread, and Σ ∈ R p×p is a symmetric positive definite (spd) matrix, called the scatter matrix. For a random sample {x 1 , x 2 , ..., x N } of p-dimensional observation vectors, the computation of the maximum like- lihood (ML) estimates ˆ β, ˆ Σ, and ˆ m lies in solving the non- linear equation given by Σ = N  i=1 p u i + u 1−β i  i=j u β j x i x ⊤ i , (1) where u i = x ⊤ i Σ −1 x i . The method of moments (MoM) and ML estimation techniques have been proposed (Ver- doolaege & Scheunders, 2011; 2012; Bombrun et al., 2012; Sra & Hosseini, 2013; Pascal et al., 2013) for estimating the scatter matrix. However, their accuracy suffers when β be- comes large, making them unsuitable for many applications. Here, we present an effective algorithm on the space of spd matrices S p + —Riemannian-averaged fixed point algorithm (RA-FP)—that accurately estimates Σ for any β. 2. RA-FP Algorithm Boukouvalas et al. (2015) formulated (1) as a fixed point equation by defining the right hand side as a function on S p + 1 Department of Mechanical Engineering, University of Mary- land, College Park, Maryland, USA 2 Laboratoire IMS, Universit de Bordeaux, Bordeaux, France. Correspondence to: Zois Bouk- ouvalas <zoisb@umd.edu>. Copyright 2018 by the author(s). and used a FP algorithm Σ k+1 = f (Σ k ) for k =0, 1, 2,... to estimate ˆ Σ. The algorithm’s convergence requires f be contractive, which numerical experiments showed is not the case when β ≥ 2. By taking advantage of the Riemannian geometry of S p + , RA-FP overcomes this difficulty. Precisely speaking, given Σ k , the new estimate Σ k+1 is: Σ k+1 = Σ k # t k f (Σ k ), (2) where the right hand side of (2) denotes the Riemannian average with ratio t k between Σ k and Σ k+1 . Thus, RA- FP implements Riemannian averages of successive fixed point iterates, preventing them from diverging when β in- creases. For a full discussion of RA-FP, as well as it proof of convergence, we refer the reader to Boukouvalas et al. (2015). 3. Numerical Experiments To numerically verify RA-FP’s effectiveness, Fig. 1 shows the Frobenius norm of the difference between the estimated and original scatter matrices, when Σ and β are jointly estimated. 0.25 0.5 1 2 4 8 10 -4 10 -2 10 0 10 2 Norm Error MoM ML-FP RA-FP Figure 1. Scatter matrix estimation performance for different val- ues of the shape parameter when Σ and β have been jointly esti- mated. N = 10000. 4. Future Directions Future work will focus on high-dimensional cases, specifi- cally how to estimate MGGD parameters when dimension p increases. Providing globally convergent algorithms that also scale is non-trivial. However, one promising approach is averaged constant-step-size Riemannian stochastic gradi- ent descent—an MCMC method with a geometric mixing property that converges exponentially to the stationary dis- tribution. The online averaging (as in RA-FP) stabilises the Markov chain to a unique deterministic limit, which experi- mentally approximates the true parameter values well.