On the positive semi-definite property of similarity matrices

Nader, Rafic; Bretto, Alain; Mourad, Bassam; Abbas, Hassan

doi:10.1016/j.tcs.2018.06.052

Cited by 11 publications

(10 citation statements)

References 15 publications

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“…The first proof sees the cosine pairwise similarity matrix as written as the Hadamard product between two matrices, say

and

, respectively, defined as

The former is the Gram matrix between histograms, therefore it trivially satisfies Mercer’s condition [ 91 ]. The latter has been shown to be positive semi-definite in [ 92 ]. Therefore, thanks to Theorem 1, HCK is a valid kernel.…”

Section: On the Positive Definiteness Of The Proposed Kernelsmentioning

confidence: 99%

(Hyper)graph Kernels over Simplicial Complexes

Martino

2020

Entropy

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Graph kernels are one of the mainstream approaches when dealing with measuring similarity between graphs, especially for pattern recognition and machine learning tasks. In turn, graphs gained a lot of attention due to their modeling capabilities for several real-world phenomena ranging from bioinformatics to social network analysis. However, the attention has been recently moved towards hypergraphs, generalization of plain graphs where multi-way relations (other than pairwise relations) can be considered. In this paper, four (hyper)graph kernels are proposed and their efficiency and effectiveness are compared in a twofold fashion. First, by inferring the simplicial complexes on the top of underlying graphs and by performing a comparison among 18 benchmark datasets against state-of-the-art approaches; second, by facing a real-world case study (i.e., metabolic pathways classification) where input data are natively represented by hypergraphs. With this work, we aim at fostering the extension of graph kernels towards hypergraphs and, more in general, bridging the gap between structural pattern recognition and the domain of hypergraphs.

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“…The first proof sees the cosine pairwise similarity matrix as written as the Hadamard product between two matrices, say

and

, respectively, defined as

Section: On the Positive Definiteness Of The Proposed Kernelsmentioning

confidence: 99%

(Hyper)graph Kernels over Simplicial Complexes

Martino

2020

Entropy

View full text Add to dashboard Cite

show abstract

“…In the paper [7] published in this journal an interesting Conjecture is presented concerning the positive definiteness of some similarities very much related to Fuzzy Logic [11] and especially to the theory of indistinguishability operators [8]. This Conjecture is not true in its current form as will be stated in the next section but in Section 3 a reformulation leading to interesting consequences is stated and proved.…”

Section: Introductionmentioning

confidence: 99%

“…Definition 1.1. [7] Let E be a finite set and let P (E) be its power set. A similarity is a mapping s from P (E) × P (E) into R + such that a) s(X, Y ) = s(Y, X) for all X, Y ∈ P (E) b) s(X, Y ) ≤ s(X, X) for all X, Y ∈ P (E).…”

Section: Introductionmentioning

confidence: 99%

“…A similarity s gives rise to a matrix S = (s(A i , A j )) that is called a similarity matrix in [7].…”

Section: Introductionmentioning

confidence: 99%

“…

In [7] a conjecture relating the positive definiteness of a similarity with its transitivity with respect to the Lukasiewicz t-norm is made. In its current form, the conjecture is not true but from a modified version interesting consequences can be derived.

…”

mentioning

confidence: 99%

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