The Experts below are selected from a list of 36033 Experts worldwide ranked by ideXlab platform

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

  • populations of models experimental designs and coverage of parameter space by latin hypercube and orthogonal sampling
    arXiv: Methodology, 2015
    Co-Authors: Kevin Burrage, Pamela Burrage, Diane Donovan, Bevan Thompson
    Abstract:

    In this paper we have used simulations to make a conjecture about the coverage of a $t$ Dimensional Subspace of a $d$ Dimensional parameter space of size $n$ when performing $k$ trials of Latin Hypercube sampling. This takes the form $P(k,n,d,t)=1-e^{-k/n^{t-1}}$. We suggest that this coverage formula is independent of $d$ and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the $t$ Dimensional Subspace at the sub-block size level.

  • populations of models experimental designs and coverage of parameter space by latin hypercube and orthogonal sampling
    International Conference on Conceptual Structures, 2015
    Co-Authors: Kevin Burrage, Pamela Burrage, Diane Donovan, Bevan Thompson
    Abstract:

    In this paper we have used simulations to make a conjecture about the coverage of a t-Dimensional Subspace of a d-Dimensional parameter space of size n when performing k trials of Latin Hypercube sampling. This takes the form P(k,n,d,t) = 1 - e^(-k/n^(t-1)). We suggest that this coverage formula is independent of d and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the t-Dimensional Subspace at the sub-block size level. These ideas have particular relevance when attempting to perform uncertainty quantification and sensitivity analyses.

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

  • the johnson lindenstrauss lemma almost characterizes hilbert space but not quite
    Discrete and Computational Geometry, 2010
    Co-Authors: William B Johnson, Assaf Naor
    Abstract:

    Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x 1,…,x n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear Subspace of dimension O(log n), such that ‖x i −x j ‖≤‖L(x i )−L(x j )‖≤O(1)⋅‖x i −x j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-Dimensional Subspace of X embeds into Hilbert space with distortion $2^{2^{O(\log^{*}n)}}$. On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-Dimensional Subspace E n ⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

  • the johnson lindenstrauss lemma almost characterizes hilbert space but not quite
    Symposium on Discrete Algorithms, 2009
    Co-Authors: William B Johnson, Assaf Naor
    Abstract:

    Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J--L lemma, in short) in the sense that for any integer n and any x1,...,xn e X there exists a linear mapping L: X → F, where F ⊆ X is a linear Subspace of dimension O(log n), such that ||xi - xj|| ≤ ||L(xi) - L(xj)|| ≤ O(1) · ||xi - xj|| for all i, j e {1,..., n). We show that this implies that X is almost Euclidean in the following sense: Every n-Dimensional Subspace of X embeds into Hilbert space with distortion [EQUATION]. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n there exists an n-Dimensional Subspace En ⊆ Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

  • the johnson lindenstrauss lemma almost characterizes hilbert space but not quite
    arXiv: Functional Analysis, 2008
    Co-Authors: William B Johnson, Assaf Naor
    Abstract:

    Let $X$ be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer $n$ and any $x_1,\ldots,x_n\in X$ there exists a linear mapping $L:X\to F$, where $F\subseteq X$ is a linear Subspace of dimension $O(\log n)$, such that $\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$ for all $i,j\in \{1,\ldots, n\}$. We show that this implies that $X$ is almost Euclidean in the following sense: Every $n$-Dimensional Subspace of $X$ embeds into Hilbert space with distortion $2^{2^{O(\log^*n)}}$. On the other hand, we show that there exists a normed space $Y$ which satisfies the J-L lemma, but for every $n$ there exists an $n$-Dimensional Subspace $E_n\subseteq Y$ whose Euclidean distortion is at least $2^{\Omega(\alpha(n))}$, where $\alpha$ is the inverse Ackermann function.

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

  • discovering a sparse set of pairwise discriminating features in high Dimensional data
    Bioinformatics, 2020
    Co-Authors: Samuel Melton, Sharad Ramanathan
    Abstract:

    MOTIVATION Recent technological advances produce a wealth of high Dimensional descriptions of biological processes, yet extracting meaningful insight and mechanistic understanding from these data remains challenging. For example, in developmental biology, the dynamics of differentiation can now be mapped quantitatively using single cell RNA-sequencing, yet it is difficult to infer molecular regulators of developmental transitions. Here we show that discovering informative features in the data is crucial for statistical analysis as well as making experimental predictions. RESULTS We identify features based on their ability to discriminate between clusters of the data points. We define a class of problems in which linear separability of clusters is hidden in a low Dimensional space. We propose an unsupervised method to identify the subset of features that define a low Dimensional Subspace in which clustering can be conducted. This is achieved by averaging over discriminators trained on an ensemble of proposed cluster configurations. We then apply our method to single cell RNA-seq data from mouse gastrulation, and identify 27 key transcription factors (out of 409 total), 18 of which are known to define cell states through their expression levels. In this inferred Subspace, we find clear signatures of known cell types that eluded classification prior to discovery of the correct low Dimensional Subspace. AVAILABILITY https://github.com/smelton/SMD. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

  • discovering a sparse set of pairwise discriminating features in high Dimensional data
    arXiv: Machine Learning, 2019
    Co-Authors: Samuel Melton, Sharad Ramanathan
    Abstract:

    Extracting an understanding of the underlying system from high Dimensional data is a growing problem in science. Discovering informative and meaningful features is crucial for clustering, classification, and low Dimensional data embedding. Here we propose to construct features based on their ability to discriminate between clusters of the data points. We define a class of problems in which linear separability of clusters is hidden in a low Dimensional space. We propose an unsupervised method to identify the subset of features that define a low Dimensional Subspace in which clustering can be conducted. This is achieved by averaging over discriminators trained on an ensemble of proposed cluster configurations. We then apply our method to single cell RNA-seq data from mouse gastrulation, and identify 27 key transcription factors (out of 409 total), 18 of which are known to define cell states through their expression levels. In this inferred Subspace, we find clear signatures of known cell types that eluded classification prior to discovery of the correct low Dimensional Subspace.

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

  • populations of models experimental designs and coverage of parameter space by latin hypercube and orthogonal sampling
    arXiv: Methodology, 2015
    Co-Authors: Kevin Burrage, Pamela Burrage, Diane Donovan, Bevan Thompson
    Abstract:

    In this paper we have used simulations to make a conjecture about the coverage of a $t$ Dimensional Subspace of a $d$ Dimensional parameter space of size $n$ when performing $k$ trials of Latin Hypercube sampling. This takes the form $P(k,n,d,t)=1-e^{-k/n^{t-1}}$. We suggest that this coverage formula is independent of $d$ and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the $t$ Dimensional Subspace at the sub-block size level.

  • populations of models experimental designs and coverage of parameter space by latin hypercube and orthogonal sampling
    International Conference on Conceptual Structures, 2015
    Co-Authors: Kevin Burrage, Pamela Burrage, Diane Donovan, Bevan Thompson
    Abstract:

    In this paper we have used simulations to make a conjecture about the coverage of a t-Dimensional Subspace of a d-Dimensional parameter space of size n when performing k trials of Latin Hypercube sampling. This takes the form P(k,n,d,t) = 1 - e^(-k/n^(t-1)). We suggest that this coverage formula is independent of d and this allows us to make connections between building Populations of Models and Experimental Designs. We also show that Orthogonal sampling is superior to Latin Hypercube sampling in terms of allowing a more uniform coverage of the t-Dimensional Subspace at the sub-block size level. These ideas have particular relevance when attempting to perform uncertainty quantification and sensitivity analyses.

William B Johnson - One of the best experts on this subject based on the ideXlab platform.

  • the johnson lindenstrauss lemma almost characterizes hilbert space but not quite
    Discrete and Computational Geometry, 2010
    Co-Authors: William B Johnson, Assaf Naor
    Abstract:

    Let X be a normed space that satisfies the Johnson–Lindenstrauss lemma (J–L lemma, in short) in the sense that for any integer n and any x 1,…,x n ∈X, there exists a linear mapping L:X→F, where F⊆X is a linear Subspace of dimension O(log n), such that ‖x i −x j ‖≤‖L(x i )−L(x j )‖≤O(1)⋅‖x i −x j ‖ for all i,j∈{1,…,n}. We show that this implies that X is almost Euclidean in the following sense: Every n-Dimensional Subspace of X embeds into Hilbert space with distortion $2^{2^{O(\log^{*}n)}}$. On the other hand, we show that there exists a normed space Y which satisfies the J–L lemma, but for every n, there exists an n-Dimensional Subspace E n ⊆Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

  • the johnson lindenstrauss lemma almost characterizes hilbert space but not quite
    Symposium on Discrete Algorithms, 2009
    Co-Authors: William B Johnson, Assaf Naor
    Abstract:

    Let X be a normed space that satisfies the Johnson-Lindenstrauss lemma (J--L lemma, in short) in the sense that for any integer n and any x1,...,xn e X there exists a linear mapping L: X → F, where F ⊆ X is a linear Subspace of dimension O(log n), such that ||xi - xj|| ≤ ||L(xi) - L(xj)|| ≤ O(1) · ||xi - xj|| for all i, j e {1,..., n). We show that this implies that X is almost Euclidean in the following sense: Every n-Dimensional Subspace of X embeds into Hilbert space with distortion [EQUATION]. On the other hand, we show that there exists a normed space Y which satisfies the J-L lemma, but for every n there exists an n-Dimensional Subspace En ⊆ Y whose Euclidean distortion is at least 2Ω(α(n)), where α is the inverse Ackermann function.

  • the johnson lindenstrauss lemma almost characterizes hilbert space but not quite
    arXiv: Functional Analysis, 2008
    Co-Authors: William B Johnson, Assaf Naor
    Abstract:

    Let $X$ be a normed space that satisfies the Johnson-Lindenstrauss lemma (J-L lemma, in short) in the sense that for any integer $n$ and any $x_1,\ldots,x_n\in X$ there exists a linear mapping $L:X\to F$, where $F\subseteq X$ is a linear Subspace of dimension $O(\log n)$, such that $\|x_i-x_j\|\le\|L(x_i)-L(x_j)\|\le O(1)\cdot\|x_i-x_j\|$ for all $i,j\in \{1,\ldots, n\}$. We show that this implies that $X$ is almost Euclidean in the following sense: Every $n$-Dimensional Subspace of $X$ embeds into Hilbert space with distortion $2^{2^{O(\log^*n)}}$. On the other hand, we show that there exists a normed space $Y$ which satisfies the J-L lemma, but for every $n$ there exists an $n$-Dimensional Subspace $E_n\subseteq Y$ whose Euclidean distortion is at least $2^{\Omega(\alpha(n))}$, where $\alpha$ is the inverse Ackermann function.