Learning Trigonometric Polynomials from Random Samples and Exponential Inequalities for Eigenvalues of Random Matrices

Motivated by problems arising in random sampling of trigonometric poly- nomials, we derive exponential inequalities for the operator norm of the dif- ference between the sample second moment matrix n 1 UU and its expec- tation where U is a complex random nD matrix with independent rows. These results immediately imply deviation inequalities for the largest (small- est) eigenvalues of the sample second moment matrix, which in turn lead to results on the condition number of the sample second moment matrix. We also show that trigonometric polynomials in several variables can be learned from constDlnD random samples.

[1]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1958 .

[2]  John Riordan,et al.  Introduction to Combinatorial Analysis , 1959 .

[3]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[4]  N. J. A. Sloane,et al.  The On-Line Encyclopedia of Integer Sequences , 2003, Electron. J. Comb..

[5]  Jon A. Wellner,et al.  Weak Convergence and Empirical Processes: With Applications to Statistics , 1996 .

[6]  Felipe Cucker,et al.  On the mathematical foundations of learning , 2001 .

[7]  Gilles Blanchard,et al.  Statistical properties of Kernel Prinicipal Component Analysis , 2019 .

[8]  S. Smale,et al.  Shannon sampling and function reconstruction from point values , 2004 .

[9]  Ññøøøññøø Blockin.  Random Sampling of Multivariate Trigonometric Polynomials , 2004 .

[10]  H. Rauhut.  Random Sampling of Sparse Trigonometric Polynomials , 2005, math/0512642.

[11]  S. Mendelson,et al.  On singular values of matrices with independent rows , 2006 .

[12]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[13]  Gilles Blanchard,et al.  Statistical properties of kernel principal component analysis , 2007, Machine Learning.

[14]  Gábor Lugosi,et al.  Prediction, learning, and games , 2006 .

[15]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[16]  M. Rudelson,et al.  Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements , 2006, 2006 40th Annual Conference on Information Sciences and Systems.

[17]  S. Smale,et al.  Learning Theory Estimates via Integral Operators and Their Approximations , 2007 .

[18]  Holger Rauhut,et al.  Random Sampling of Sparse Trigonometric Polynomials, II. Orthogonal Matching Pursuit versus Basis Pursuit , 2008, Found. Comput. Math..

[19]  Holger Rauhut.  Stability Results for Random Sampling of Sparse Trigonometric Polynomials , 2008, IEEE Transactions on Information Theory.