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A common assumption when sampling $p$-dimensional observations from $K$distinct group is the equality of the covariance matrices. In this paper, wepropose two penalized $M$-estimation approaches for the estimation of thecovariance or scatter matrices under the broader assumption that they maysimply be close to each other, and hence roughly deviate from some positivedefinite "center". The first approach begins by generating a pooled$M$-estimator of scatter based on all the data, followed by a penalised$M$-estimator of scatter for each group, with the penalty term chosen so thatthe individual scatter matrices are shrunk towards the pooled scatter matrix.In the second approach, we minimize the sum of the individual group$M$-estimation cost functions together with an additive joint penalty termwhich enforces some similarity between the individual scatter estimators, i.e.shrinkage towards a mutual center. In both approaches, we utilize the conceptof geodesic convexity to prove the existence and uniqueness of the penalizedsolution under general conditions. We consider three specific penalty functionsbased on the Euclidean, the Riemannian, and the Kullback-Leibler distances. Inthe second approach, the distance based penalties are shown to lead toestimators of the mutual center that are related to the arithmetic, theRiemannian and the harmonic means of positive definite matrices, respectively.A penalty based on an ellipticity measure is also considered which isparticularly useful for shape matrix estimators. Fixed point equations arederived for each penalty function and the benefits of the estimators areillustrated in regularized discriminant analysis problem.

Simultaneous penalized M-estimation of covariance matrices using geodesically convex optimization