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# an Inductive Mean?

Communicated by *Notices* Associate Editor Richard Levine

### Notions of means

The notion of means 10 is central to mathematics and statistics, and plays a key role in machine learning and data analytics. The three classical Pythagorean means of two positive reals and are the arithmetic (A), geometric (G), and harmonic (H) means, given respectively by

These Pythagorean means were originally geometrically studied to define proportions, and the harmonic mean led to a beautiful connection between mathematics and music. The Pythagorean means enjoy the following inequalities:

with equality if and only if These Pythagorean means belong to a broader parametric family of means, the power means . defined for We have . , and in the limits: , and , Power means are also called binomial, Minkowski, or Hölder means in the literature. .

There are many ways to define and axiomatize means with a rich literature 8. An important class of means are the quasi-arithmetic means induced by strictly increasing and differentiable real-valued functional generators :

1Quasi-arithmetic means satisfy the in-betweenness property of means: and are called so because , is the arithmetic mean on the of numbers. -representation

The power means are quasi-arithmetic means, obtained for the following continuous family of generators: ,

Power means are the only homogeneous quasi-arithmetic means, where a mean is said to be homogeneous when for any .

Quasi-arithmetic means can also be defined for means (i.e., -variable and more generally for calculating expected values of random variables ),10: We denote by the quasi-arithmetic expected value of a random variable induced by a strictly monotone and differentiable function For example, the geometric and harmonic expected values of . are defined by and respectively. The ordinary expectation is recovered for , : The quasi-arithmetic expected values satisfy a strong law of large numbers and a central limit theorem ( .10, Theorem 1): Let be independent and identically distributed (i.i.d.) with finite variance and derivative at Then we have .

as where , denotes a normal distribution of expectation and variance .

### Inductive means

An inductive mean is a mean defined as a limit of a convergence sequence of other means 15. The notion of inductive means defined as limits of sequences was pioneered independently by Lagrange and Gauss 7 who studied the following double sequence of iterations:

initialized with and We have .

where the homogeneous arithmetic-geometric mean (AGM) is obtained in the limit:

There is no closed-form formula for the AGM in terms of elementary functions as this induced mean is related to the complete elliptic integral of the first kind 7:

where is the elliptic integral. The fast quadratic convergence 11 of the AGM iterations makes it computationally attractive, and the AGM iterations have been used to numerically calculate digits of or approximate the perimeters of ellipses among others 7.

Some inductive means admit closed-form formulas: For example, the arithmetic-harmonic mean obtained as the limit of the double sequence

initialized with and converges to the geometric mean:

In general, inductive means defined as the limits of double sequences with respect to two smooth symmetric means and :

are proven to converge quadratically 11 to (order- convergence).

### Inductive means and matrix means

We have obtained so far three ways to get the geometric scalar mean between positive reals and :

- 1.
As an inductive mean with the arithmetic-harmonic double sequence: ,

- 2.
As a quasi-arithmetic mean obtained for the generator : and ,

- 3.
As the limit of power means: .

Let us now consider the geometric mean of two symmetric positive-definite (SPD) matrices and of size SPD matrices generalize positive reals. We shall investigate the three generalizations of the above approaches of the scalar geometric mean, and show that they yield different notions of matrix geometric means when . .

First, the AHM iterations can be extended to SPD matrices instead of reals:

where the matrix arithmetic mean is and the matrix harmonic mean is The AHM iterations initialized with . and yield in the limit the matrix arithmetic-harmonic mean ,314 (AHM):

Remarkably, the matrix AHM enjoys quadratic convergence to the following SPD matrix:

When and are positive reals, we recover When . the identity matrix, we get , the positive square root of SPD matrix , Thus the matrix AHM iterations provide a fast method in practice to numerically approximate matrix square roots by bypassing the matrix eigendecomposition. When matrices . and commute (i.e., we have ), The geometric mean . is proven to be the unique solution to the matrix Ricatti equation is invariant under inversion (i.e., , and satisfies the determinant property ), .

Let denote the set of symmetric positive-definite matrices. The matrix geometric mean can be interpreted using a Riemannian geometry 5 of the cone Equip : with the trace metric tensor, i.e., a collection of smoothly varying inner products for defined by

where and are matrices belonging to the vector space of symmetric matrices (i.e., and are geometrically vectors of the tangent plane of The geodesic length distance on the Riemannian manifold ). is

where

This Riemannian least squares mean is also called the Cartan, Kärcher, or Fréchet mean in the literature. More generally, the Riemannian geodesic

This Riemannian barycenter can be solved as

with

Second, let us consider the matrix geometric mean as the limit of matrix quasi-arithmetic power means which can be defined 13 as

where

Third, we can define matrix power means

Let

In general, we get the following closed-form expression 13 of this matrix power mean for

### Inductive means, circumcenters, and medians of several matrices

To extend these various binary matrix means of two matrices to matrix means of