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1. Manifold Curvature from Covariance Analysis

   Javier Alvarez-Vizoso, Michael Kirby ( June 2018 , 2018 IEEE Statistical Signal Processing Workshop (SSP 2018))

   Principal component analysis of cylindrical neighborhoods is proposed to study the local geometry of embedded Riemannian manifolds. At every generic point and scale, a highdimensional cylinder orthogonal to the tangent space at the point cuts out a path-connected patch whose point-set distribution in ambient space encodes the intrinsic and extrinsic curvature. The covariance matrix of the points from that neighborhood has eigenvectors whose scale limit tends to the Frenet-Serret frame for curves, and to what we call the Ricci-Weingarten principal directions for submanifolds. More importantly, the limit of differences and products of eigenvalues can be used to recover curvature information at the point. The formula for hypersurfaces in terms of principal curvatures is particularly simple and plays a crucial role in the study of higher-codimension cases. 

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2. DISCRETE-TO-CONTINUUM LIMITS OF LONG-RANGE ELECTRICAL INTERACTIONS IN NANOSTRUCTURES

   Jha, Prashant ; Breitzman, Timothy ; Dayal, Kaushik ( January 2023 , Archive for rational mechanics and analysis)

   We consider electrostatic interactions in two classes of nanostructures embedded in a three dimensional space: (1) helical nanotubes, and (2) thin films with uniform bending (i.e., constant mean curvature). Starting from the atomic scale with a discrete distribution of dipoles, we obtain the continuum limit of the electrostatic energy; the continuum energy depends on the geometric parameters that define the nanostructure, such as the pitch and twist of the helical nanotubes and the curvature of the thin film. We find that the limiting energy is local in nature. This can be rationalized by noticing that the decay of the dipole kernel is sufficiently fast when the lattice sums run over one and two dimensions, and is also consistent with prior work on dimension reduction of continuum micromagnetic bodies to the thin film limit. However, an interesting contrast between the discrete-to-continuum approach and the continuum dimension reduction approaches is that the limit energy in the latter depends only on the normal component of the dipole field, whereas in the discrete-to-continuum approach, both tangential and normal components of the dipole field contribute to the limit energy. 

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3. Surface Networks

   Kostrikov, Ilya ; Jiang, Zhongshi ; Panozzo, Daniele ; Zorin, Denis ; Bruna, Joan ( June 2018 , IEEE Conference on Computer Vision and Pattern Recognition)

   We study data-driven representations for three- dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curva- ture directions — this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models Surface Networks (SN). We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs. 

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4. Gradient Dynamics of Shallow Univariate ReLU networks

   Williams, Francis ; Trager, Matthew ; Silva, Claudio ; Zorin, Denis ; Panozzo, Daniele ; Bruna, Joan ( January 2019 , Advances in neural information processing systems)

   We present a theoretical and empirical study of the gradient dynamics of overparam- eterized shallow ReLU networks with one-dimensional input, solving least-squares interpolation. We show that the gradient dynamics of such networks are determined by the gradient flow in a non-redundant parameterization of the network function. We examine the principal qualitative features of this gradient flow. In particular, we determine conditions for two learning regimes: kernel and adaptive, which depend both on the relative magnitude of initialization of weights in different layers and the asymptotic behavior of initialization coefficients in the limit of large network widths. We show that learning in the kernel regime yields smooth interpolants, minimizing curvature, and reduces to cubic splines for uniform initializations. Learning in the adaptive regime favors instead linear splines, where knots cluster adaptively at the sample points. 

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5. Redshift space three-point correlation function of IGM at z < 0.48

   https://doi.org/10.1093/mnras/stab3308  

   Maitra, Soumak ; Srianand, Raghunathan ; Gaikwad, Prakash ; Khandai, Nishikanta ( November 2021 , Monthly Notices of the Royal Astronomical Society)

   Ly α forest decomposed into Voigt profile components allows us to study clustering properties of the intergalactic-medium and its dependence on various physical quantities. Here, we report the first detections of probability excess of low-z (i.e z < 0.48) Ly α absorber triplets over redshift-space scale of r∥ ≤ 8 pMpc (Mpc in physical units) with maximum amplitude of $8.76^{+1.96}_{-1.65}$ at a longitudinal separation of 1–2 pMpc. We measure non-zero three-point correlation ($\zeta = 4.76^{+1.98}_{-1.67}$) only at this scale with reduced three-point correlation Q = $0.95^{+0.39}_{-0.38}$. The measured ζ shows an increasing trend with increasing minimum H i column density (NH i) threshold while Q does not show any NH i dependence. About 88 per cent of the triplets contributing to ζ (at z ≤ 0.2) have nearby galaxies (whose distribution is known to be complete for ∼0.1L* at z < 0.1 and for ∼L* at z ∼ 0.25 within 20 arcsec to the quasar sightlines) within velocity separation of 500 km s−1 and median impact parameter of 405 pkpc. The measured impact parameters are consistent with majority of the identified triplets not originating from individual galaxies but tracing the underlying galaxy distribution. Frequency of occurrence of Broad-Ly α absorbers (b > 40 km s−1) in triplets (∼85 per cent) is factor ∼3 higher than that found among the full sample (∼32 per cent). Using four different cosmological simulations, we quantify the effect of peculiar velocities and feedback and show that most of the observed trends are broadly reproduced. However, ζ at small scales (r∥ < 1 pMpc) and its b-dependence found in simulations are inconsistent with observations. This could either be related to the failure of these simulations to reproduce the observed b and NH i distributions for NH i > 1014 cm−2 self-consistently or to the wide spread of signal-to-noise ratio in the observed data.

    

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