26 research outputs found
Hopf algebras and the logarithm of the S-transform in free probability
Let k be a positive integer and let G_k denote the set of non-commutative
k-variable distributions \mu such that \mu (X_1) = ... = \mu (X_k) = 1. G_k is
a group under the operation of free multiplicative convolution. We identify G_k
as the group of characters of a certain Hopf algebra Y_k. Then, by using the
log map from characters to infinitesimal characters of Y_k, we introduce a
transform LS_{\mu} for distributions \mu in G_k. The main property of the
LS-transform is that it linearizes commuting products in G_k. For \mu in G_k,
the transform LS_{\mu} is a power series in k non-commuting indeterminates; its
coefficients can be computed from the coefficients of the R-transform of \mu by
using summations over chains in the lattices NC(n) of non-crossing partitions.
In the particular case k=1 one has that Y_1 is naturally isomorphic to the
Hopf algebra Sym of symmetric functions, and that the LS-transform is very
closely related to the logarithm of the S-transform of Voiculescu, by the
formula LS(z) = - z log S(z). In this case the group G_1 can be identified as
the group of characters of Sym, in such a way that the S-transform, its
reciprocal 1/S and its logarithm log S relate in a natural sense to the
sequences of complete, elementary and respectively power sum symmetric
functions.Comment: Version with minor revisions, to appear in Transactions of the
American Mathematical Society. 39 pages, no figure
S-AMP: Approximate Message Passing for General Matrix Ensembles
In this work we propose a novel iterative estimation algorithm for linear
observation systems called S-AMP whose fixed points are the stationary points
of the exact Gibbs free energy under a set of (first- and second-) moment
consistency constraints in the large system limit. S-AMP extends the
approximate message-passing (AMP) algorithm to general matrix ensembles. The
generalization is based on the S-transform (in free probability) of the
spectrum of the measurement matrix. Furthermore, we show that the optimality of
S-AMP follows directly from its design rather than from solving a separate
optimization problem as done for AMP.Comment: 5 pages, 1 figur
S-AMP for Non-linear Observation Models
Recently we extended Approximate message passing (AMP) algorithm to be able
to handle general invariant matrix ensembles. In this contribution we extend
our S-AMP approach to non-linear observation models. We obtain generalized AMP
(GAMP) algorithm as the special case when the measurement matrix has zero-mean
iid Gaussian entries. Our derivation is based upon 1) deriving expectation
propagation (EP) like algorithms from the stationary-points equations of the
Gibbs free energy under first- and second-moment constraints and 2) applying
additive free convolution in free probability theory to get low-complexity
updates for the second moment quantities.Comment: 6 page
New spectral relations between products and powers of isotropic random matrices
We show that the limiting eigenvalue density of the product of n identically
distributed random matrices from an isotropic unitary ensemble (IUE) is equal
to the eigenvalue density of n-th power of a single matrix from this ensemble,
in the limit when the size of the matrix tends to infinity. Using this
observation one can derive the limiting density of the product of n independent
identically distributed non-hermitian matrices with unitary invariant measures.
In this paper we discuss two examples: the product of n Girko-Ginibre matrices
and the product of n truncated unitary matrices. We also provide an evidence
that the result holds also for isotropic orthogonal ensembles (IOE).Comment: 8 pages, 3 figures (in version 2 we added a figure and discussion on
finite size effects for isotropic orthogonal ensemble
Cumulants, free cumulants and half-shuffles
Free cumulants were introduced as the proper analog of classical cumulants in
the theory of free probability. There is a mix of similarities and differences,
when one considers the two families of cumulants. Whereas the combinatorics of
classical cumulants is well expressed in terms of set partitions, the one of
free cumulants is described, and often introduced in terms of non-crossing set
partitions. The formal series approach to classical and free cumulants also
largely differ. It is the purpose of the present article to put forward a
different approach to these phenomena. Namely, we show that cumulants, whether
classical or free, can be understood in terms of the algebra and combinatorics
underlying commutative as well as non-commutative (half-)shuffles and
(half-)unshuffles. As a corollary, cumulants and free cumulants can be
characterized through linear fixed point equations. We study the exponential
solutions of these linear fixed point equations, which display well the
commutative, respectively non-commutative, character of classical, respectively
free, cumulants.Comment: updated and revised version; accepted for publication in PRS
The splitting process in free probability theory
Free cumulants were introduced by Speicher as a proper analog of classical
cumulants in Voiculescu's theory of free probability. The relation between free
moments and free cumulants is usually described in terms of Moebius calculus
over the lattice of non-crossing partitions. In this work we explore another
approach to free cumulants and to their combinatorial study using a
combinatorial Hopf algebra structure on the linear span of non-crossing
partitions. The generating series of free moments is seen as a character on
this Hopf algebra. It is characterized by solving a linear fixed point equation
that relates it to the generating series of free cumulants. These phenomena are
explained through a process similar to (though different from) the
arborification process familiar in the theory of dynamical systems, and
originating in Cayley's work