Page 14 - Contributed Paper Session (CPS) - Volume 4

CPS2101 Bertail Patrice et al.
will assume that G has a density with respect to the Lebesgue measure,
denoted by belonging to some regular space G (we will precise later the
regulatity assumption needed on G ensuring convergence of our estimators) .
The distribution of the r.v. v is entirely determined by the law of the pair (W,G):
we denote it by PW,g and pW,g its density . The semiparametric model is thus
entirely described by the set

.
In the semiparametric terminology W is the parameter of interest and g is
the nuisance parameter.
Remark : This model can be extended to noisy models. Although additive
noises are generally modelled in an additive way in most applications, it should
be noticed that multiplicative and poisson noise models may also be used to
take advantage of leading to nonnegative data vectors in the context of NMF.
Hence, as highlighted in [6], uniqueness of matrices (W,H) cannot be
guaranteed in absence of further assumptions on W and/or H’s distribution.
We now set the hypotheses which will be assumed throughout this paper and
ensuring the existence and the unicity of the representation in a
semiparametric framework.
H1 The matrix W is of full rank K, 1 ≤ K ≤ F.
H2 The columns of the matrix W are of unit (euclidian) norm: ∀k ∈ {1, ..., K},
2
||W.k|| = 1.
H3 The columns of the matrix W are sorted by lexicographic order of the
vectors (α1,k(W), ..., αF,k(W)).
H4 The span of the support of the distribution of the v’is isdenoted by
.
H5 The distribution G(dh) is such that for any
, where {λw = (λwf) : λ > 0}
F
for any w = (wf) ∈ R and supp(G) denotes the support of h’s distribution.
We will denote by W+ the set of matrices W ∈ MFK(R+) fulfilling assumptions
H1 − H3.
Theorem (Semi-parametric identifiability in NMF models) Let G be a set of
probability distributions on . Assume that all the distributions in G fulfill
assumptions H4 −H5. The family of distributions

is then identifiable.
Now, under assumption H1, the likelihood pW,g is given by [2]
pW,g(v) = vol(W )g(W v). (3)
−1
−1
In that case the likelihood of this semi-parametric model based on a
sample vn = (v1, ..., vn) of n independent copies of the random variable v, is
simply given by

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