arXiv:1403.5977v1 [stat.AP] 24 Mar 2014 1 Uniqueness of Maronna’s M -estimators of scatter Yacine Chitour, Romain Couillet, and Fr´ ed´ eric Pascal Abstract—In this paper, we prove the uniqueness of Maronna’s M-estimator of scatter [1] for N vector obser- vations c1,...,cN ∈ R m under a mild constraint of linear independence of any subset of m of these vectors. This entails in particular almost sure uniqueness for random vectors ci with a density as long as N >m. Further relations are established which demonstrate that a prop- erly normalized Tyler’s M-estimator of scatter [2] can be considered as a limit of Maronna’s M-estimator. These results find important implications in recent works on the large dimensional (random matrix) regime of robust M- estimation. I. I NTRODUCTION Subsequent to Huber’s introduction of robust statistics in [3], Maronna proposed in [1] a class of robust estimates for scatter matrices defined as the solution of an implicit equation. In [1], the existence and uniqueness of such a solution are proved, under conditions involving both the ratio m/N of the population size m and the sample size N , and the parametrization of the estimate. A sufficient practical condition for uniqueness is that m/N be small enough. This constraint therefore had little impact on the asymptotic study of these estimators in the regime N →∞ and m fixed. However, although this constraint was slightly relaxed in [4], [5], with the recent renewed interest in robust M -estimation under the random matrix regime N,m → ∞ with m/N → c ∞ ∈ (0, 1) [6], [7], [8], [9], Maronna as well as Kent and Tyler conditions have become too stringent and have led these recent works to produce alternative proofs of existence and uniqueness. While Maronna’s original results are valid for any (well-behaved) set of samples satisfying the condition on m/N , the results in e.g. [6] are expressed in probabilistic terms and are only valid for all large m, N . Based on the ideas from [10], [11], [12], the present article generalizes both results by showing that existence and uniqueness can be extended to all well-behaved set of samples and for any ratio m/N ∈ (0, 1). In addition, Chitour is with Laboratoire des Signaux et Syst` emes at Sup´ elec, 91192 Gif s/Yvette, France and Universit´ e Paris Sud, Orsay, France yacine.chitour@lss.supelec.fr. Couillet is with the Telecommunications Department at Sup´ elec romain.couillet@supelec.fr. Pascal is with the SONDRA laboratory at Sup´ elec frederic.pascal@supelec.fr. Couillet’s work is funded by ERC–MORE EC–120133. by a proper parametrization of the weight function, we prove that some sequences of Maronna’s M -estimators converge to Tyler’s distribution-free M -estimator of scat- ter [2]. The paper is organized as follows: Section II presents our main results, the proofs of which are provided in Section III. II. NOTATIONS AND STATEMENT OF THE RESULTS Let m, N be positive integers with c := m N ∈ (0, 1). We use M m (R) and Sym m to denote the vector space of m × m matrices with real entries and the linear subspace of M m (R) made of the symmetric matrices, respectively. We also use Sym + m and PSD m to denote the non trivial cones in M m (R) of the non negative symmetric ma- trices and of the symmetric positive definite matrices, respectively. Also, (·) T stands for the transpose, Tr(·) and det(·) for the trace and the determinant. On M m (R), we use the inner product defined by the Frobenius norm ‖A‖ =  Tr(AA T ). We also use ≤ to denote the partial order on Sym m and I m the m × m identity matrix. Functions of two non negative real variables (t, x) will be considered. If f is such a function, we use f t , f x , f tx , ... to denote (when defined) the partial derivatives of f with respect to t and/or x. Definition II.1 A family (c i ) 1≤i≤N of vectors in R m is admissible if (C1) for 1 ≤ i ≤ N , ‖c i ‖ =1; (C2) for every injective map L : {1, ··· ,m} → {1, ··· ,N }, the m vectors c L(1) , ··· ,c L(m) are linearly independent. To prove the main result of this paper, the following lemma is required. Lemma II.2 Let (c i ) 1≤i≤N be an admissible family of vectors in R m . Fix m vectors (say) c 1 , ··· ,c m which are then linearly independent by (C2). For m +1 ≤ l ≤ N , we can write c l = ∑ n j=1 γ lj c j . Then, γ lj =0 for every 1 ≤ j ≤ m and m +1 ≤ l ≤ N . Lemma II.2: If the conclusion is not true, then there exists an index l ≥ m +1 and (at most) m − 1 indices in {1, ··· ,m} such that the corresponding c i ’s are linearly dependent, contradicting (C2).