In this paper we look at isometry properties of random matrices. During the last decade these properties gained a lot attention in a field called compressed sensing in first place due to their initial use in [7, 8]. Namely, in [7, 8] these quantities were used as a critical tool in providing a rigorous analysis of l1 optimization’s ability to solve an under-determined system of linear equations...

For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by `1-minimization under the optimal condition m ≥ c s ln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes...

Recently,manyworks have focused on the characterization of nonlinear dimensionality reductionmethods obtained by quantizing linear embeddings, e.g. to reach fast processing time, efficient data compression procedures, novel geometry-preserving embeddings or to estimate the information/bits stored in this reduced data representation. In this work, we prove that many linear maps known to respect ...

— In this work, we analyze modulated sampling schemes, such as the Nyquist Folding Receiver, which are highly efficient, readily implementable, non-uniform sampling schemes that allows for the blind estimation of a narrow-band signal's spectral content and location in a wide-band environment. This non-uniform sampling, achieved by narrow-band modulation of the RF instantaneous sample rate, resu...

We prove that every isometry between the unit spheres of 2-dimensional Banach spaces extends to a linear spaces. This resolves famous Tingley's problem in class

In this chapter, we propose a pixel adjustment procedure for digital watermarking methods using uniform color space. Many digital watermarking methods use uniform color space due to the convenience of controlling image quality (Yoshiura & Echizen, 2006). However, there is the problem that watermarked RGB integer pixel values in such methods are not quite corresponding to the watermarked real-nu...

It is well known that the initialization of weights in deep neural networks can have a dramatic impact on learning speed. For example, ensuring the mean squared singular value of a network’s input-output Jacobian isO(1) is essential for avoiding the exponential vanishing or explosion of gradients. The stronger condition that all singular values of the Jacobian concentrate near 1 is a property k...