arXiv subjects

Jinko Kanno

Publications and source records attributed to Jinko Kanno.

4 recordsLinked to original sources

arXiv2018

On the Definiteness of Earth Mover's Distance and Its Relation to Set Intersection

Positive definite kernels are an important tool in machine learning that enable efficient solutions to otherwise difficult or intractable problems by implicitly linearizing the problem geometry. In this paper we develop a set-theoretic interpretation of the Earth Mover's Distance (EMD) and propose Earth Mover's Intersection (EMI), a positive definite analog to EMD for sets of different sizes. We provide conditions under which EMD or certain approximations to EMD are negative definite. We also present a positive-definite-preserving transformation that can be applied to any kernel and can also be used to derive positive definite EMD-based kernels and show that the Jaccard index is simply the result of this transformation. Finally, we evaluate kernels based on EMI and the proposed transformation versus EMD in various computer vision tasks and show that EMD is generally inferior even with indefinite kernel techniques.

cs.LG↗

arXiv2017

Classifying Unordered Feature Sets with Convolutional Deep Averaging Networks

Unordered feature sets are a nonstandard data structure that traditional neural networks are incapable of addressing in a principled manner. Providing a concatenation of features in an arbitrary order may lead to the learning of spurious patterns or biases that do not actually exist. Another complication is introduced if the number of features varies between each set. We propose convolutional deep averaging networks (CDANs) for classifying and learning representations of datasets whose instances comprise variable-size, unordered feature sets. CDANs are efficient, permutation-invariant, and capable of accepting sets of arbitrary size. We emphasize the importance of nonlinear feature embeddings for obtaining effective CDAN classifiers and illustrate their advantages in experiments versus linear embeddings and alternative permutation-invariant and -equivariant architectures.

cs.LG↗

arXiv2017

Vizing's 2-factor Conjecture Involving Toughness and Maximum Degree Conditions

Let $G$ be a simple graph, and let $Δ(G)$ and $χ'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $χ'(G)=Δ(G)$ or $Δ(G)+1$. We say $G$ is $Δ$-critical if $χ'(G)=Δ+1$ and $χ'(H)<χ'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $Δ$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $Δ$-critical graph. It was proved that if $Δ(G)\ge n/2$, then $G$ has a 2-factor; and that if $Δ(G)\ge 2n/3+12$, then $G$ has a hamiltonian cycle, and thus a 2-factor. It is well known that every 2-tough graph with at least three vertices has a 2-factor. We investigate the existence of a 2-factor in a $Δ$-critical graph under "moderate" given toughness and maximum degree conditions. In particular, we show that if $G$ is an $n$-vertex $Δ$-critical graph with toughness at least 3/2 and with maximum degree at least $n/3$, then $G$ has a 2-factor. In addition, we develop new techniques in proving the existence of 2-factors in graphs.

math.CO↗

arXiv2016

Oriented Book Embeddings

A graph $G$ has a $k$-page book embedding if $G$ can be embedded into a $k$-page book. The minimum $k$ such that $G$ has a $k$-page book embedding is the book thickness of $G$, denoted $bt(G)$. Most of the work on this subject has been done for unoriented graphs and oriented acyclic graphs (no directed cycles). In this work we discuss oriented graphs $\overrightarrow{D}$ containing directed cycles by using oriented book embeddings and oriented book thickness, $obt(\overrightarrow{D})$. To characterize $\overrightarrow{D}$ such that $obt(\overrightarrow{D}) = k$, we define the class $\mathcal{M}^k$ of $k$-page critical oriented graphs to be all oriented graphs $\overrightarrow{D}$ with $obt(\overrightarrow{D}) =k$, but for every proper oriented subgraph of $\overrightarrow{D}$, denoted $\overrightarrow{D}'$, we have that $obt(\overrightarrow{D}') < k$. Determining $\mathcal{M}^k$ for general $k$ is challenging; we narrow down the list of oriented graphs in $\mathcal{M}^k$ for small $k$. In this work we show complete lists for $\mathcal{M}^1$ and for $\mathcal{M}^2 \cap \mathcal{U}$, where $\mathcal{U}$ consists of all strictly dicyclic oriented graphs, that is, oriented graphs containing exactly one oriented cycle, which is a directed cycle. Keywords: book embedding, book thickness, oriented book embedding, oriented book thickness, directed cycle, critical graph

math.CO↗