A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion

Low-rank matrix completion (LRMC) problems arise in a wide variety of applications. Previous theory mainly provides conditions for completion under missing-at-random samplings. This paper studies deterministic conditions for completion. An incomplete $d \times N$ matrix is finitely rank-$r$ completable if there are at most finitely many rank-$r$ matrices that agree with all its observed entries. Finite completability is the tipping point in LRMC, as a few additional samples of a finitely completable matrix guarantee its unique completability. The main contribution of this paper is a deterministic sampling condition for finite completability. We use this to also derive deterministic sampling conditions for unique completability that can be efficiently verified. We also show that under uniform random sampling schemes, these conditions are satisfied with high probability if $O(\max\{r,\log d\})$ entries per column are observed. These findings have several implications on LRMC regarding lower bounds, sample and computational complexity, the role of coherence, adaptive settings and the validation of any completion algorithm. We complement our theoretical results with experiments that support our findings and motivate future analysis of uncharted sampling regimes.Comment: This update corrects an error in version 2 of this paper, where we erroneously assumed that columns with more than r+1 observed entries would yield multiple independent constraint

On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality

The search for sharp constants for inequalities of the type Littlewood's 4/3 and Bohnenblust-Hille, besides its pure mathematical interest, has shown unexpected applications in many different fields, such as Analytic Number Theory, Quantum Information Theory, or (for instance) in deep results on the $n$-dimensional Bohr radius. The recent estimates obtained for the multilinear Bohnenblust-Hille inequality (in the case of real scalars) have been recently used, as a crucial step, by A. Montanaro in order to solve problems in the theory of quantum XOR games. Here, among other results, we obtain new upper bounds for the Bohnenblust-Hille constants in the case of complex scalars. For bilinear forms, we obtain the optimal constants of variants of Littlewood's 4/3 inequality (in the case of real scalars) when the exponent 4/3 is replaced by any $r\geq4/3.$ As a consequence of our estimates we show that the optimal constants for the real case are always strictly greater than the constants for the complex case

The Discrete Markus-Yamabe Problem for Symmetric Planar Polynomial Maps

We probe deeper into the Discrete Markus-Yamabe Question for polynomial planar maps and into the normal form for those maps which answer this question in the affirmative. Furthermore, in a symmetric context, we show that the only nonlinear equivariant polynomial maps providing an affirmative answer to the Discrete Markus-Yamabe Question are those possessing Z2 as their group of symmetries. We use this to establish two new tools which give information about the spectrum of a planar polynomial map

Global Saddles for Planar Maps

We study the dynamics of planar diffeomorphisms having a unique fixed point that is a hyperbolic local saddle. We obtain sufficient conditions under which the fixed point is a global saddle. We also address the special case of $D_2$-symmetric maps, for which we obtain a similar result for $C^1$ homeomorphisms. Some applications to differential equations are also given