arXiv research

This work is an endeavor to develop a deep learning methodology for automated anatomical labeling of a given region of interest (ROI) in brain computed tomography (CT) scans. We combine both local and global context to obtain a representation of the ROI. We then use Relation Networks (RNs) to predict the corresponding …

We construct new explicit proper biharmonic functions on the $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $H^2\times\rn$ and $S^2\times\rn$.

In this paper we derive necessary and sufficient conditions for a smooth surface in Rn+1 to admit a local 1-quasiconformal parameterization by a domain in Rn (n >= 3). We then apply these conditions to specific hypersurfaces such as cylinders, paraboloids, and ellipsoids. As a consequence, we show that the classical Li…

For any positive natural number $r\in\N^+$ we construct new explicit proper $r$-harmonic functions on the celebrated $3$-dimensional Thurston geometries $\Sol$, $\Nil$, $\SL2$, $\H^2\times\rn$ and $\s^2\times\rn$.

Driven by climatic processes, wind power generation is inherently variable. Long-term simulated wind power time series are therefore an essential component for understanding the temporal availability of wind power and its integration into future renewable energy systems. In the recent past, mainly power curve based mod…

The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivi…

We provide a measure based topology for certain unions of C2 rectifiable submanifolds of mixed dimensions in Rn. In this topology lower dimensional sets remain in the limit as measures when higher dimensional sets collapse down to them. For example a decreasing sequence of spheres may have a limit consisting of just a …

Given a positive function F on Sn which satisfies a convexity condition, we introduce the r-th anisotropic mean curvature Mr for hypersurfaces in Rn+1 which is a generalization of the usual r-th mean curvature Hr. We get integral formulas of Minkowski type for compact hypersurfaces in Rn+1. We give some new characteriz…

Zero-sum games such as chess and poker are, abstractly, functions that evaluate pairs of agents, for example labeling them `winner' and `loser'. If the game is approximately transitive, then self-play generates sequences of agents of increasing strength. However, nontransitive games, such as rock-paper-scissors, can ex…

These are notes on the theory of supermanifolds and integration on them, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.

We give a sufficient condition for a metric (homology) manifold to be locally bi-Lipschitz equivalent to an open subset in $\rn$. The condition is a Sobolev condition for a measurable coframe of flat 1-forms. In combination with an earlier work of D. Sullivan, our methods also yield an analytic characterization for smo…

These are notes on the theory of super Riemann surfaces and their moduli spaces, aiming to collect results that are useful for a better understanding of superstring perturbation theory in the RNS formalism.

Paper presents quantum algorithms for pricing financial derivatives using complex models.

A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…

An ODE variational calculation shows that an image principle curvature ratio factor can raise the lower bound, 2(Image Area), on energy of a harmonic map of a surface into Rn. In certain situations, including all radially symmetry harmonic maps, equality is achieved.

Efficient Winograd convolution for INT8 networks using RNS.

The paper shows that g-convex functions on manifolds are sparse.

Let M be a random (alpha n) x n matrix of rank r<<n, and assume that a uniformly random subset E of its entries is observed. We describe an efficient algorithm that reconstructs M from |E| = O(rn) observed entries with relative root mean square error RMSE <= C(rn/|E|)^0.5 . Further, if r=O(1), M can be reconstructed ex…

We construct sequences of pseudo-Anosov mapping classes whose dilatations behave asymptotically like the inverse of the Euler characteristic of the surface they are defined on. These sequences are used to show that if the genus, g, and punctures, n, of a surface are related by a rational ray g=rn then the minimal dilat…

The notion of $L^p$-distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the Laplace-Beltrami operator. The techniques are based on a-priori estimates for ellipt…

Our principal goal is to study the Prescribed Curvature Tensor problem in locally conformally flat manifolds. The solution to this problem is given explicitly for the special cases of the tensor R, including a case where the metric g is complete on Rn. Similar problems are considered for locally conformally flat manifo…

It is shown that existence of a global solution to a particular nonlinear system of second order partial differential equations on a complete connected Riemannian manifold has topological and geometric implications and that in the domain of positivity of such solution its reciprocal is the radial function of only one o…

Classifies SL(n) covariant matrix-valued valuations on Lp-spaces.

The study evaluates different parameter selection methods for Gaussian process interpolation.

Framework for transitioning financial models from risk-neutral to real-world measure.

Willmore flow preserves low energy surfaces to planes.

Double Machine Learning estimators are asymptotically inadmissible under structure-agnostic models.

An L-space link is a link in $S^3$ on which all sufficiently large integral surgeries are L-spaces. We prove that for m, n relatively prime, the r-component cable link $K_{rm,rn}$ is an L-space link if and only if K is an L-space knot and $n/m \geq 2g(K)-1$. We also compute HFL-minus and HFL-hat of an L-space cable lin…

Relational Networks (RN) as introduced by Santoro et al. (2017) have demonstrated strong relational reasoning capabilities with a rather shallow architecture. Its single-layer design, however, only considers pairs of information objects, making it unsuitable for problems requiring reasoning across a higher number of fa…

We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) …

GROUSE (Grassmannian Rank-One Update Subspace Estimation) is an incremental algorithm for identifying a subspace of Rn from a sequence of vectors in this subspace, where only a subset of components of each vector is revealed at each iteration. Recent analysis has shown that GROUSE converges locally at an expected linea…

For a congruence of straight lines defined by a hypersurface in $R^{n+1}, n \geq 1,$ and a field of reflected directions created by a point source we define the notion of intensity in a tangent direction and introduce elementary symmetric functions $S_m, m=1, 2,...,n,$ of {\it principal intensities}. The problem of exi…

Rotation intertwining maps from the set of convex bodies in Rn into itself that are continuous linear operators with respect to Minkowski and Blaschke addition are investigated. The main focus is on Blaschke-Minkowski homomorphisms. We show that such maps are represented by a spherical convolution operator. An applicat…

Let M be a compact manifold, possibly with boundary. We show that the group of homeomorphisms of M has the automatic continuity property: any homomorphism from Homeo(M) to any separable group is necessarily continuous. This answers a question of C. Rosendal. If N is a submanifold of M, the group of homeomorphisms of M …

We present new families of continuous piecewise linear (CPWL) functions in Rn having a number of affine pieces growing exponentially in $n$. We show that these functions can be seen as the high-dimensional generalization of the triangle wave function used by Telgarsky in 2016. We prove that they can be computed by ReLU…

Critical nets in $\mathbb{R}^k$ (sometimes called geodesic nets) are embedded graph with the property that their embedding is a critical point of the total (edge) length functional and under the constraint that certain 1-valent vertices (leaves) have a fixed position. In contrast to what happens on generic manifolds, w…

The paper examines stability of Minkowski inequalities in warped product spaces.

We build polyhedral complexes in Rn that coincide with dyadic grids with different orientations, while keeping uniform lower bounds (depending only on n) on the flatness of the added polyhedrons including their subfaces in all dimensions. After the definitions and first properties of compact Euclidean polyhedrons and c…

Inspired by the adaptation phenomenon of neuronal firing, we propose the regularity normalization (RN) as an unsupervised attention mechanism (UAM) which computes the statistical regularity in the implicit space of neural networks under the Minimum Description Length (MDL) principle. Treating the neural network optimiz…

In this paper, we consider matrix completion from non-uniformly sampled entries including fully observed and partially observed columns. Specifically, we assume that a small number of columns are randomly selected and fully observed, and each remaining column is partially observed with uniform sampling. To recover the …

The vulnerability to adversarial attacks has been a critical issue for deep neural networks. Addressing this issue requires a reliable way to evaluate the robustness of a network. Recently, several methods have been developed to compute $\textit{robustness quantification}$ for neural networks, namely, certified lower b…

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph …

The first time that the connection between isometric immersion of surfaces and solutions of the Dirac equation appeared in the literature was in the seminal paper of Thomas Friedrich in 1998. In consequence of that, several authors contributed to this topic hereafter, by obtaining the spinorial representation of Spin m…

CUR matrix decomposition is a randomized algorithm that can efficiently compute the low rank approximation for a given rectangle matrix. One limitation with the existing CUR algorithms is that they require an access to the full matrix A for computing U. In this work, we aim to alleviate this limitation. In particular, …

We study the problem of exact completion for $m \times n$ sized matrix of rank $r$ with the adaptive sampling method. We introduce a relation of the exact completion problem with the sparsest vector of column and row spaces (which we call \textit{sparsity-number} here). Using this relation, we propose matrix completion…

Proves Thom's conjecture for parabolic flows on Hilbert spaces.

We investigate graph neural networks for multi-relational data.

Modeling business cycles via collective risk fluctuations in economic agents' risk space.