Geometric Means

By Chi-Kwong Li. 2003

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Citations in the corpus

Learning explanations that are hard to vary (2020, background): “3, gradient descent on the pooled loss 12θ >HA+Bθ is much faster than gradient descent on the geometrically averaged loss 12θ >HA∧Bθ.”
Shampoo: Preconditioned Stochastic Tensor Optimization (2018, background): “Ando et al. [3] proved a stronger result which does not require the PSD matrices to commute with each other, relying on a generalized notion of geometric mean, but for our purposes the simpler commuting case suﬃces.”
A general framework for extending means to higher orders (2006, unknown): “As mentioned in the introduction we pattern our approach after [16] and [1], an approach that has some relation to the process of compounding three means in three variables to obtain another such mean (see [8]).”
Operator log-convex functions and operator means (2009, background): “, An) for n ≥ 3 was rather recently introduced in [2] in a recursive way.”
The Riemannian mean and matrix inequalities related to the Ando-Hiai inequality and chaotic order (2012, unknown): “De ﬁ ne three sequences and { C n } ∞ n = 0 converge the same limit, and we de ﬁ ne it as ALM mean [3] (denoted by G alm ( A , B , C ) ), i.e., On the other hand, BMP mean is de ﬁ ned as follows: De ﬁ ne three sequences { A n } ∞ n = 0 , { B n } ∞ n = 0 and { C n } ∞ n = 0 by A 0 = A , B 0 = B , C…”
Introduction to Matrix Analysis and Applications (2014, background): “The geometric mean for more matrices is from the paper [9].”
Matrix power means and the Karcher mean (2012, background): “Ando, Li and Mathias [2] listed the ten properties (P1)–(P10) for the unweighted case ω = (1/n, . . . ,1/n) as properties that a geometric mean of n positive definite matrices should satisfy, and their mean, called the ALM geometric mean, possesses all of them.”
Monotonic properties of the least squares mean (2010, methodology): “The Ando-Li-Mathias paper was also important for listing, and deriving for their mean, ten desirable properties for extended geometric means g : Pn → P that one might anticipate from properties of the two-variable geometric mean, where P = Pm denotes the convex cone of m×m positive definite Hermitia”