The Experts below are selected from a list of 14397 Experts worldwide ranked by ideXlab platform

Ron M Roth - One of the best experts on this subject based on the ideXlab platform.

Ata Kaban - One of the best experts on this subject based on the ideXlab platform.

  • improved bounds on the Dot Product under random projection and random sign projection
    Knowledge Discovery and Data Mining, 2015
    Co-Authors: Ata Kaban
    Abstract:

    Dot Product is a key building block in a number of data mining algorithms from classification, regression, correlation clustering, to information retrieval and many others. When data is high dimensional, the use of random projections may serve as a universal dimensionality reduction method that provides both low distortion guarantees and computational savings. Yet, contrary to the optimal guarantees that are known on the preservation of the Euclidean distance cf. the Johnson-Lindenstrauss lemma, the existing guarantees on the Dot Product under random projection are loose and incomplete in the current data mining and machine learning literature. Some recent literature even suggested that the Dot Product may not be preserved when the angle between the original vectors is obtuse. In this paper we provide improved bounds on the Dot Product under random projection that matches the optimal bounds on the Euclidean distance. As a corollary, we elucidate the impact of the angle between the original vectors on the relative distortion of the Dot Product under random projection, and we show that the obtuse vs. acute angles behave symmetrically in the same way. In a further corollary we make a link to sign random projection, where we generalise earlier results. Numerical simulations confirm our theoretical results. Finally we give an application of our results to bounding the generalisation error of compressive linear classifiers under the margin loss.

  • KDD - Improved Bounds on the Dot Product under Random Projection and Random Sign Projection
    Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2015
    Co-Authors: Ata Kaban
    Abstract:

    Dot Product is a key building block in a number of data mining algorithms from classification, regression, correlation clustering, to information retrieval and many others. When data is high dimensional, the use of random projections may serve as a universal dimensionality reduction method that provides both low distortion guarantees and computational savings. Yet, contrary to the optimal guarantees that are known on the preservation of the Euclidean distance cf. the Johnson-Lindenstrauss lemma, the existing guarantees on the Dot Product under random projection are loose and incomplete in the current data mining and machine learning literature. Some recent literature even suggested that the Dot Product may not be preserved when the angle between the original vectors is obtuse. In this paper we provide improved bounds on the Dot Product under random projection that matches the optimal bounds on the Euclidean distance. As a corollary, we elucidate the impact of the angle between the original vectors on the relative distortion of the Dot Product under random projection, and we show that the obtuse vs. acute angles behave symmetrically in the same way. In a further corollary we make a link to sign random projection, where we generalise earlier results. Numerical simulations confirm our theoretical results. Finally we give an application of our results to bounding the generalisation error of compressive linear classifiers under the margin loss.

Qinyichen - One of the best experts on this subject based on the ideXlab platform.

  • Statistical inference on random Dot Product graphs
    Journal of Machine Learning Research, 2017
    Co-Authors: Athreyaavanti, E Fishkinddonniell, Tangminh, E Priebecarey, Parkyoungser, T Vogelsteinjoshua, Levinkeith, Lyzinskivince, Qinyichen
    Abstract:

    The random Dot Product graph (RDPG) is an independent-edge random graph that is analytically tractable and, simultaneously, either encompasses or can successfully approximate a wide range of random...

Tobias Müller - One of the best experts on this subject based on the ideXlab platform.

  • Sphere and Dot Product Representations of Graphs
    Discrete & Computational Geometry, 2012
    Co-Authors: Ross J. Kang, Tobias Müller
    Abstract:

    A graph G is a k-sphere graph if there are k-dimensional real vectors v 1,…,v n such that ij∈E(G) if and only if the distance between v i and v j is at most 1. A graph G is a k-Dot Product graph if there are k-dimensional real vectors v 1,…,v n such that ij∈E(G) if and only if the Dot Product of v i and v j is at least 1. By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-Dot Product graph are NP-hard for all k>1. In the former case, this proves a conjecture of Breu and Kirkpatrick (Comput. Geom. 9:3–24, 1998). In the latter, this answers a question of Fiduccia et al. (Discrete Math. 181:113–138, 1998). Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the implicit graph conjecture, we demonstrate that, for all k>1, there exist k-sphere graphs and k-Dot Product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (Efficient Graph Representations, 2003).

  • Dot Product Representations of Planar Graphs
    The Electronic Journal of Combinatorics, 2011
    Co-Authors: Ross J. Kang, László Lovász, Tobias Müller, Edward R. Scheinerman
    Abstract:

    A graph $G$ is a $k$-Dot Product graph if there exists a vector labelling $u: V(G) \to \mathbb{R}^k$ such that $u(i)^{T}u(j) \geq 1$ if and only if $ij \in E(G)$. Fiduccia, Scheinerman, Trenk and Zito [ Discrete Math. , 1998] asked whether every planar graph is a $3$-Dot Product graph. We show that the answer is "no". On the other hand, every planar graph is a $4$-Dot Product graph. We also answer the corresponding questions for planar graphs of prescribed girth and for outerplanar graphs.

  • sphere and Dot Product representations of graphs
    Symposium on Computational Geometry, 2011
    Co-Authors: Ross J. Kang, Tobias Müller
    Abstract:

    A graph G is a k-sphere graph if there are k-dimensional real vectors v1,..., vn such that ij ∈ E(G) if and only if the distance between vi and vj is at most 1. A graph G is a k-Dot Product graph if there are k-dimensional real vectors v1,...,vn such that ij ∈ E(G) if and only if the Dot Product of vi and vj is at least 1. By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-Dot Product graph are NP-hard for all k>1. In the former case, this proves a conjecture of Breu and Kirkpatrick (1998). In the latter, this answers a question of Fiduccia, Scheinerman, Trenk and Zito (1998). Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the Implicit Graph Conjecture, we demonstrate that, for all k > 1, there exist k-sphere graphs and k-Dot Product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (2003).

  • Graph Drawing - Dot Product representations of planar graphs
    Graph Drawing, 2011
    Co-Authors: Ross J. Kang, Tobias Müller
    Abstract:

    A graph G on n vertices is a k-Dot Product graph if there are vectors u1,..., un ∈ Rk, one for each vertex of G, such that uiT uj ≥ 1 if and only if ij ∈ E(G). Fiduccia, Scheinerman, Trenk and Zito (1998) asked whether every planar graph is a 3-Dot Product graph. We show that the answer is "no". On the other hand, every planar graph is a 4-Dot Product graph.

  • Symposium on Computational Geometry - Sphere and Dot Product representations of graphs
    Proceedings of the 27th annual ACM symposium on Computational geometry - SoCG '11, 2011
    Co-Authors: Ross J. Kang, Tobias Müller
    Abstract:

    A graph G is a k-sphere graph if there are k-dimensional real vectors v1,..., vn such that ij ∈ E(G) if and only if the distance between vi and vj is at most 1. A graph G is a k-Dot Product graph if there are k-dimensional real vectors v1,...,vn such that ij ∈ E(G) if and only if the Dot Product of vi and vj is at least 1. By relating these two geometric graph constructions to oriented k-hyperplane arrangements, we prove that the problems of deciding, given a graph G, whether G is a k-sphere or a k-Dot Product graph are NP-hard for all k>1. In the former case, this proves a conjecture of Breu and Kirkpatrick (1998). In the latter, this answers a question of Fiduccia, Scheinerman, Trenk and Zito (1998). Furthermore, motivated by the question of whether these two recognition problems are in NP, as well as by the Implicit Graph Conjecture, we demonstrate that, for all k > 1, there exist k-sphere graphs and k-Dot Product graphs such that each representation in k-dimensional real vectors needs at least an exponential number of bits to be stored in the memory of a computer. On the other hand, we show that exponentially many bits are always enough. This resolves a question of Spinrad (2003).

Erik Jan Van Leeuwen - One of the best experts on this subject based on the ideXlab platform.

  • What Graphs are 2-Dot Product Graphs?⋆
    arXiv: Combinatorics, 2015
    Co-Authors: Matthew Johnson, Daniël Paulusma, Erik Jan Van Leeuwen
    Abstract:

    Let $d \geq 1$ be an integer. From a set of $d$-dimensional vectors, we obtain a $d$-Dot Product graph by letting each vector ${\bf a}^u$ correspond to a vertex $u$ and by adding an edge between two vertices $u$ and $v$ if and only if their Dot Product ${\bf a}^{u} \cDot {\bf a}^{v} \geq t$, for some fixed, positive threshold~$t$. Dot Product graphs can be used to model social networks. Recognizing a $d$-Dot Product graph is known to be \NP-hard for all fixed $d\geq 2$. To understand the position of $d$-Dot Product graphs in the landscape of graph classes, we consider the case $d=2$, and investigate how $2$-Dot Product graphs relate to a number of other known graph classes.

  • What Graphs are 2-Dot Product Graphs?
    Electronic Notes in Discrete Mathematics, 2015
    Co-Authors: Matthew Johnson, Erik Jan Van Leeuwen, Daniël Paulusma
    Abstract:

    Abstract From a set of d-dimensional vectors for some integer d ≥ 1 , we obtain a d-Dot Product graph by letting each vector a u correspond to a vertex u and by adding an edge between two vertices u and v if and only if their Dot Product a u ⋅ a v ≥ t , for some fixed, positive threshold t. Dot Product graphs can be used to model social networks. To understand the position of d-Dot Product graphs in the landscape of graph classes, we consider the case d = 2 , and investigate how 2-Dot Product graphs relate to a number of other known graph classes.

  • Algorithms for diversity and clustering in social networks through Dot Product graphs
    Social Networks, 2015
    Co-Authors: Matthew Johnson, Daniël Paulusma, Erik Jan Van Leeuwen
    Abstract:

    Abstract In this paper, we investigate a graph-theoretical model of social networks. The Dot Product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d -dimensional vector a v represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if a u · a v ≥ t , for some fixed, positive threshold t . The resulting graph is called a d-Dot Product graph . We consider diversity and clustering in social networks by using a d -Dot Product graph model for the network. Diversity is considered through the size of the largest independent set of the graph, and clustering through the size of the largest clique. We present both positive and negative results on the potential of this model. We obtain a tight result for the diversity problem, namely that it is polynomial-time solvable for d  = 2, but NP -hard for d  ≥ 3. We show that the clustering problem is polynomial-time solvable for d  = 2. To our knowledge, these results are also the first on the computational complexity of combinatorial optimization problems on Dot Product graphs. We also give new insights into the structure of Dot Product graphs. We also consider the situation when two individuals u and v are connected if and only if their preferences are not antithetical, that is, if and only if a u · a v ≥ 0 , and the situation when two individuals u and v are connected if and only if their preferences are neither antithetical nor “orthogonal”, that is, if and only if a u · a v > 0 . For these two cases we prove that the diversity problem is polynomial-time solvable for any fixed d and that the clustering problem is polynomial-time solvable for d  ≤ 3.

  • algorithms to measure diversity and clustering in social networks through Dot Product graphs
    International Symposium on Algorithms and Computation, 2013
    Co-Authors: Matthew Johnson, Daniël Paulusma, Erik Jan Van Leeuwen
    Abstract:

    Social networks are often analyzed through a graph model of the network. The Dot Product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d-dimensional vector a v represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if a u · a v ≥ t, for some fixed, positive threshold t. The resulting graph is called a d-Dot Product graph..

  • ISAAC - Algorithms to Measure Diversity and Clustering in Social Networks through Dot Product Graphs
    Algorithms and Computation, 2013
    Co-Authors: Matthew Johnson, Daniël Paulusma, Erik Jan Van Leeuwen
    Abstract:

    Social networks are often analyzed through a graph model of the network. The Dot Product model assumes that two individuals are connected in the social network if their attributes or opinions are similar. In the model, a d-dimensional vector a v represents the extent to which individual v has each of a set of d attributes or opinions. Then two individuals u and v are assumed to be friends, that is, they are connected in the graph model, if and only if a u · a v ≥ t, for some fixed, positive threshold t. The resulting graph is called a d-Dot Product graph..