Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and q sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given dataset. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, q sensitivity, has certain

Almost all three dimensional manifolds admit canonical metrics with constant sectional curvature. In this paper we proposed a new algorithm pipeline to compute such canonical metrics for hyperbolic 3-manifolds with high genus boundary surfaces. The computation is based on the discrete curvature flow for 3-manifolds, where the metric is deformed in an angle-preserving fashion until the curvature becomes uniform inside the volume and vanishes on the boundary. We also proposed algorithms to visualize the canonical metric by realizing the volume in the hyperbolic space ³, both in single period and in multiple periods. The proposed methods could not only facilitate the theoretical study of 3-manifold topology and geometry using computers, but also have great potentials in volumetric parameterizations, 3D shape comparison, volumetric biomedical image analysis and etc.

In this paper, we will systematically study who indeed are the hard lattice cases in LLL reduction. The “hard” cases here mean for their special geometric structures, with a comparatively high “failure probability” that LLL can not solve SVP even by using a powerful relaxation factor. We define the perfect lattice as the “Beauty”, which is given by basis of vectors of the same length with the mutual angles of any two vectors to be exactly 60°. Simultaneously the “Beasts” lattice is defined as the lattice close to the Beauty lattice. There is a relatively high probability (e.g. 15.0% in 3 dimensions) that our “Beasts” bases can withstand the exact-arithmetic LLL reduction (relaxation factors δ close to 1), comparing to the probability (corresponding <0.01%) when apply same LLL on random bases from TU Darmstadt SVP Challenge. Our theoretical proof gives us a direct explanation of this phenomenon. Moreover, we give rational Beauty bases of 3 and 8 dimensions, an irrational Beauty bases of general high dimensions. We also give a general way to construct Beasts lattice bases from the Beauty ones. Experimental results show the Beasts bases derived from Beauty can withstand LLL reduction by a stable probability even for high dimensions. Our work in a way gives a simple and direct way to explain how to build a hard lattice in LLL reduction.

We study motion of a charged particle confined to a Dirichlet layer of a fixed width placed into a homogeneous magnetic field. If the layer is planar and the field is perpendicular to it, the spectrum consists of infinitely degenerate eigenvalues. We consider translationally invariant geometric perturbations and derive several sufficient conditions under which a magnetic transport is possible, that is, the spectrum, in its entirety or a part of it, becomes absolutely continuous.

Multivariate Cryptography, as one of the main candidates for establishing post-quantum cryptosystems, provides strong, efficient and well-understood digital signature schemes such as UOV, Rainbow, and Gui. While Gui provides very short signatures, it is, for efficiency reasons, restricted to very small finite fields, which makes it hard to scale it to higher levels of security and leads to large key sizes. In this paper we propose a signature scheme called HMFEv (“Hidden Medium Field Equations”), which can be seen as a multivariate version of HFEv. We obtain our scheme by applying the Vinegar Variation to the MultiHFE encryption scheme of Chen et al. We show both theoretically and by experiments that our new scheme is secure against direct and Rank attacks. In contrast to other schemes of the HFE family such as Gui, HMFEv can be defined over arbitrary base fields and therefore is much more efficient in terms of both performance and memory requirements. Our scheme is therefore a good candidate for the upcoming standardization of post-quantum signature schemes.