From Low- to High-Dimensional Moments Without Magic

We aim to compute the first few moments of a high-dimensional random vector from the first few moments of a number of its low-dimensional projections. To this end, we identify algebraic conditions on the set of low-dimensional projectors that yield explicit reconstruction formulas. We also provide a computational framework, with which suitable projectors can be derived by solving an optimization problem. Finally, we show that randomized projections permit approximate recovery.

[1]  M. Gräf.  Efficient Algorithms for the Computation of Optimal Quadrature Points on Riemannian Manifolds , 2013 .

[2]  J. A. Cuesta-Albertos,et al.  A Sharp Form of the Cramér–Wold Theorem , 2007 .

[3]  R. Calderbank,et al.  Robust dimension reduction, fusion frames, and Grassmannian packings , 2007, 0709.2340.

[4]  A. James.  Distributions of Matrix Variates and Latent Roots Derived from Normal Samples , 1964 .

[5]  Christine Bachoc,et al.  Codes and designs in Grassmannian spaces , 2004, Discret. Math..

[6]  A. Heppes.  On the determination of probability distributions of more dimensions by their projections , 1956 .

[7]  A. Rényi,et al.  On projections of probability distributions , 1952 .

[8]  R. Vershynin.  How Close is the Sample Covariance Matrix to the Actual Covariance Matrix? , 2010, 1004.3484.

[9]  P. Casazza,et al.  Fusion frames and distributed processing , 2006, math/0605374.

[10]  Claudio Procesi,et al.  The invariant theory of ? matrices , 2017 .

[11]  Thomas Ransford,et al.  When is a probability measure determined by infinitely many projections , 1997 .

[12]  Claudio Procesi,et al.  The invariant theory of n × n matrices , 1976 .

[13]  Y. Chikuse.  Statistics on special manifolds , 2003 .

[14]  Marie-Françoise Roy,et al.  Real algebraic geometry , 1992 .

[15]  M. Ehler,et al.  Cubatures and designs in unions of Grassmannians , 2014 .

[16]  A. G. Constantine,et al.  Generalized Jacobi Polynomials as Spherical Functions of the Grassmann Manifold , 1974 .

[17]  Christine Bachoc,et al.  Tight p-fusion frames , 2012, ArXiv.

[18]  Ronald E. Kutz.  Cohen-Macaulay rings and ideal theory in rings of invariants of algebraic groups , 1974 .

[19]  P. Harpe,et al.  Cubature Formulas, Geometrical Designs, Reproducing Kernels, and Markov Operators , 2005, math/0502312.

[20]  Christine Bachoc,et al.  Designs in Grassmannian Spaces and Lattices , 2002 .

[21]  Bernhard G. Bodmann,et al.  Random fusion frames are nearly equiangular and tight , 2013, 1303.5816.

[22]  Stanislav Minsker.  On Some Extensions of Bernstein's Inequality for Self-adjoint Operators , 2011, 1112.5448.

[23]  R. H. Hardin,et al.  A Group-Theoretic Framework for the Construction of Packings in Grassmannian Spaces , 1999, math/0208002.

[24]  Massimo Piccardi,et al.  Robust Dimensionality Reduction for Human Action Recognition , 2010, 2010 International Conference on Digital Image Computing: Techniques and Applications.

[25]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.

[26]  R. Vershynin,et al.  Partial estimation of covariance matrices , 2010, 1008.1716.

[27]  Martin Ehler,et al.  Phase retrieval using random cubatures and fusion frames of positive semidefinite matrices , 2015, 1505.05003.

[28]  R. Vershynin.  Spectral norm of products of random and deterministic matrices , 2008, 0812.2432.

[29]  Donald St. P. Richards,et al.  Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions , 1987 .