A coloring is distinguishing (or symmetry breaking) if no non-identity automorphism preserves it. The threshold of a graph G, denoted by θ(G), the minimum number colors k so that every k-coloring G distinguishing. We generalize this concept to edge-coloring defining an alternative index θ′(G). consider θ′ for some families graphs and find its connection with edge-cycles group. Then we show θ′(G...

In [1], it was conjectured that the permanent of a P-lifting θ of a matrix θ of degree M is less than or equal to the M th power of the permanent perm(θ), i.e., perm(θ) 6 perm(θ) and, consequently, that the degree-M Bethe permanent permM,B(θ) of a matrix θ is less than or equal to the permanent perm(θ) of θ, i.e., permM,B(θ) 6 perm(θ). In this paper, we prove these related conjectures and show ...

Here we will consider the problem min f ( ) θ∈Θ θ where, for a given value of θ , we are not able to evaluate analytically or numerically, but must obtain (noisy) estimates of f ( f ( ) θ ) θ using simulation. We will assume for now that the set of possible solutions Θ is uncountably infinite. In particular, suppose that Θ is an interval [ θ θ, θ Θ ] of real numbers. One approach to solving the...

I will not describe the underlying assumptions in details. These are the usual sorts of assumptions one makes for parametric models, in order to be able to establish sensible results. See Page 11 of Chapter 3 of Wellner’s notes for a detailed description of the conditions involved. For a multidimensional parametric model {p(x, θ) : θ ∈ Θ ⊂ Rk}, the information matrix I(θ) is given by: I(θ) = Eθ...

Let X(n) be an observation sampled from a distribution Pθ(n) with unknown parameter θ, θ being vector in Banach space E (most often, high-dimensional of dimension d). We study the problem estimation f(θ) for functional f:E↦R some smoothness s>0 based on X(n)∼Pθ(n). Assuming that there exists estimator θˆn=θˆn(X(n)) such n(θˆ n−θ) is sufficiently close to mean zero Gaussian random E, we construc...

Proof. Let M = sup θ∈Θ max{{h(θ), |v(θ)|} and V ε = {θ : d(θ, L) ≤ ε}. Applying Taylor's expansion formula (Folland, 1990), we have v(θ t+1) = v(θ t) + γ n+1 v h (θ t+1) + R t+1 , t ≥ 0, which implies that t i=0 γ i+1 v h (θ i) = v(θ t+1) − v(θ 0) − t i=0 R i+1 ≥ −2M − t i=0 R i+1. Since t i=0 R i+1 converges (owing to Lemma A.2), t i=0 γ i+1 v h (θ i) also converges. Furthermore, v(θ t) = v(θ ...

P~ θ : V 7→ [0, 1], where ~ θ is an element of the m-dimensional probability simplex. Hence the probability assigned to a single term vj is defined as: P~ θ (vj) def = θ[j]. Also recall from the previous lecture that the Kullback–Leibler (KL) divergence between two probability distributions P~ θ and P~ θ′ , i.e. the expected log-likelihood ratio with respect to P~ θ, is defined as: D(P~ θ ‖P~ θ...