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A Concentration Theorem for Projections

A Concentration Theorem for Projections,Sanjoy Dasgupta,Daniel Hsu,Nakul Verma

A Concentration Theorem for Projections   (Citations: 5)
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Suppose the random vector X ∈ RD has mean zero and finite second moments. We show that there is a pre- cise sense in which almost all linear projections of X into Rd (for d < D) look like a scale-mixture of spherical Gaussians—specifically, a mixture of distributions N(0,�2Id) where thevalues follow the same distribution as k Xk / √ D. The extent of this effect depends upon the ratio of d to D, and upon a particular coefficient of eccentricity of X's distribution.
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    • ...assuming the distribution of X is unimodal, as proved in [13], the smaller k is, the more the distribution of RX will be like a spherical Gaussian, with a covariance matrix (� U ) which is diagonal...
    • ...have no essential difference from U. By [13], when some zeromean data X is randomly projected into a lower dimensional space, the distribution of projected data U is like a spherical Gaussian whose covariance matrix is a diagonal matrix...

    Yingpeng Sanget al. Effective Reconstruction of Data Perturbed by Random Projections

    • ...Proof: In [18], Dasgupta et al. have shown that almost all p linear projections behave like a scale-...
    • ...Specifically, in this case, the main theorem of [18] reads app roximately like the following: For any ball B ∈ R p and for almost all W ,...
    • ...Interestingly, such an order O(p 2 /d) appears also in [18] (e.g...

    Hyun Sung Changet al. Informative Sensing

    • ...Although PCA analysis is widely used in a variety of applications, random projections have recently emerged as a powerful method for dimensionality reduction that offers many benefits over PCA for data sets that do not follow a multivariate Gaussian distribution [5]...
    • ...This value of K was chosen using the model selection techniques described in section II-D.3 (Fig. 7). We performed experiments with 2 [1; 5] and 2 [1; 50] and obtained error rates that are comparable to those reported above and in Table I...

    S. Ruiz-Correaet al. A Bayesian hierarchical model for classifying craniofacial malformatio...

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