Dimensionality reduction for k-distance applied to persistent homology

Given a set P of n points and a constant k, we are interested in computing the persistent homology of the Čech filtration of P for the k-distance, and investigate the effectiveness of dimensionality reduction for this problem, answering an open question of Sheehy (The persistent homology of distance functions under random projection. In Cheng, Devillers (eds), 30th Annual Symposium on Computational Geometry, SOCG’14, Kyoto, Japan, June 08–11, p 328, ACM, 2014). We show that any linear transformation that preserves pairwise distances up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) multiplicative factor, must preserve the persistent homology of the Čech filtration up to a factor of $$(1-{\varepsilon })^{-1}$$ ( 1 - ε ) - 1 . Our results also show that the Vietoris-Rips and Delaunay filtrations for the k-distance, as well as the Čech filtration for the approximate k-distance of Buchet et al. [J Comput Geom, 58:70–96, 2016] are preserved up to a $$(1\pm {\varepsilon })$$ ( 1 ± ε ) factor. We also prove extensions of our main theorem, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional submanifold, obtaining embeddings having the dimension bounds of Lotz (Proc R Soc A Math Phys Eng Sci, 475(2230):20190081, 2019) and Clarkson (Tighter bounds for random projections of manifolds. In Teillaud (ed) Proceedings of the 24th ACM Symposium on Computational Geom- etry, College Park, MD, USA, June 9–11, pp 39–48, ACM, 2008) respectively. Our results also work in the terminal dimensionality reduction setting, where the distance of any point in the original ambient space, to any point in P, needs to be approximately preserved.

Propagation of Vortex Cosine-hyperbolic-gaussian Beams in Atmospheric Turbulence

In this paper, the propagation properties of a vortex cosh-Gaussian beam (vChGB) in turbulent atmosphere are investigated. Based on the extended Huygens–Fresnel diffraction integral and the Rytov method, the analytical expression for the average intensity of the vChGB propagating in the atmospheric turbulence is derived. The effects of the turbulent strength and the beam parameters on the intensity distribution and the beam spreading are illustrated numerically and analyzed in detail. It is shown that upon propagating, the incident vChGB keeps its initial hollow dark profile within a certain propagation distance, then the field loses gradually its central hole-intensity and transformed into a Gaussian–like beam for large propagation distance. The rising speed of the central peak is demonstrated to be faster when the constant strength turbulence or the wavelength are larger and the Gaussian width is smaller. The obtained results can be beneficial for applications in optical communications and remote sensing.

On Polyhedral Approximations of the Positive Semidefinite Cone

It is well known that state-of-the-art linear programming solvers are more efficient than their semidefinite programming counterparts and can scale to much larger problem sizes. This leads us to consider the question, how well can we approximate semidefinite programs with linear programs? In this paper, we prove lower bounds on the size of linear programs that approximate the positive semidefinite cone. Let D be the set of n × n positive semidefinite matrices of trace equal to one. We prove two results on the hardness of approximating D with polytopes. We show that if 0 < ε < 1and A is an arbitrary matrix of trace equal to one, any polytope P such that (1-ε) (D-A) ⊂ P ⊂ D-A must have linear programming extension complexity at least [Formula: see text], where c > 0 is a constant that depends on ε. Second, we show that any polytope P such that D ⊂ P and such that the Gaussian width of P is at most twice the Gaussian width of D must have extension complexity at least [Formula: see text]. Our bounds are both superpolynomial in n and demonstrate that there is no generic way of approximating semidefinite programs with compact linear programs. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.

Persistent homology for low-complexity models

We show that recent results on randomized dimension reduction schemes that exploit structural properties of data can be applied in the context of persistent homology. In the spirit of compressed sensing, the dimension reduction is determined by the Gaussian width of a structure associated with the dataset, rather than its size, and such a reduction can be computed efficiently. We further relate the Gaussian width to the doubling dimension of a finite metric space, which appears in the study of the complexity of other methods for approximating persistent homology. We can, therefore, literally replace the ambient dimension by an intrinsic notion of dimension related to the structure of the data.

Dimension-Free Error Bounds from Random Projections

Learning from high dimensional data is challenging in general – however, often the data is not truly high dimensional in the sense that it may have some hidden low complexity geometry. We give new, user-friendly PAC-bounds that are able to take advantage of such benign geometry to reduce dimensional-dependence of error-guarantees in settings where such dependence is known to be essential in general. This is achieved by employing random projection as an analytic tool, and exploiting its structure-preserving compression ability. We introduce an auxiliary function class that operates on reduced dimensional inputs, and a new complexity term, as the distortion of the loss under random projections. The latter is a hypothesis-dependent data-complexity, whose analytic estimates turn out to recover various regularisation schemes in parametric models, and a notion of intrinsic dimension, as quantified by the Gaussian width of the input support in the case of the nearest neighbour rule. If there is benign geometry present, then the bounds become tighter, otherwise they recover the original dimension-dependent bounds.

Relationship between the line width of the atomic and molecular ISM in M33

ABSTRACT We investigate how the spectral properties of atomic (H i) and molecular (H2) gas, traced by CO(2−1) , are related in M33 on 80 pc scales. We find the H i and CO(2−1) velocity at peak intensity to be highly correlated, consistent with previous studies. By stacking spectra aligned to the velocity of H i peak intensity, we find that the CO line width (σHWHM = 4.6 ± 0.9 ${\rm km\, s^{-1}}$ ; σHWHM is the effective Gaussian width) is consistently smaller than the H i line width (σHWHM = 6.6 ± 0.1 ${\rm km\, s^{-1}}$), with a ratio of ∼0.7, in agreement with Druard et al. The ratio of the line widths remains less than unity when the data are smoothed to a coarser spatial resolution. In other nearby galaxies, this line width ratio is close to unity which has been used as evidence for a thick, diffuse molecular disc that is distinct from the thin molecular disc dominated by molecular clouds. The smaller line width ratio found here suggests that M33 has a marginal thick molecular disc. From modelling individual lines of sight, we recover a strong correlation between H i and CO line widths when only the H i located closest to the CO component is considered. The median line width ratio of the line-of-sight line widths is 0.56 ± 0.01. There is substantial scatter in the H i –CO(2−1) line width relation, larger than the uncertainties, that results from regional variations on &lt;500 pc scales, and there is no significant trend in the line widths, or their ratios, with galactocentric radius. These regional line width variations may be a useful probe of changes in the local cloud environment or the evolutionary state of molecular clouds.