arXiv research

New approach to nonuniform learnability using measure theory.

New method shows Hessian estimator from random samples converges to true Hessian on complex manifolds.

New property identifies arithmetic lattices from nonuniform lattices.

This paper studies the covolumes of nonuniform arithmetic lattices in PU(n, 1). We determine the smallest covolume nonuniform arithmetic lattices for each n, the number of minimal covolume lattices for each n, and study the growth of the minimal covolume as n varies. In particular, there is a unique lattice (up to conj…

Improves statistical inference using machine learning predictions with imputed data.

LA-VDM accelerates VDM using landmarks to improve data analysis.

The Besson-Courtois-Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices.

Nonuniform tubular neighborhoods of curves in Euclidean n-space are studied by using weighted distance functions and generalizing the normal exponential map. Different notions of injectivity radii are introduced to investigate singular but injective exponential maps. A generalization of the thickness formula is obtaine…

Improved estimation for imbalanced data using log odds correction and optimal sampling.

New learning algorithm for real analytic functions without gradient descent.

New geometric theory explains nonuniform origami responses.

For any n>1 we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group Sp(n,1). We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the icosian ring in the uniform case for all n>1.

Hierarchical GANs reduce anomaly detection costs.

Let $Γ$ be a nonuniform lattice acting on real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of $Γ$ in SO(n,1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the repr…

If Gamma is a nonuniform, irreducible lattice in a semisimple Lie group whose real rank is greater than 1, we show Gamma contains a subgroup that is isomorphic to a nonuniform, irreducible lattice in either SL(3,R), SL(3,C), or a direct product SL(2,R)^m x SL(2,C)^n$, with m + n > 1. (In geometric terms, this can be in…

The paper explores the dynamics of composite symplectic Dehn twists with nonuniform hyperbolicity.

Deep networks adapt to function regularity and data distribution.

Optimal multiscale learning of linear operators

We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6.

We study upper bounds for the torsion in homology of nonuniform arithmetic lattices. Together with recent results of Calegari-Venkatesh, this can be used to obtain upper bounds on K2 of the ring of integers of totally imaginary fields.

New lattices in higher dimensions have dense surface subgroups.

In this paper, we consider the problem of column subset selection. We present a novel analysis of the spectral norm reconstruction for a simple randomized algorithm and establish a new bound that depends explicitly on the sampling probabilities. The sampling dependent error bound (i) allows us to better understand the …

In this paper we answer the following question: what is the infinitesimal generator of the diffusion process defined by a kernel that is normalized such that it is bi-stochastic with respect to a specified measure? More precisely, under the assumption that data is sampled from a Riemannian manifold we determine how the…

Perfect clustering achieved in hypergraphs with enough interactions.

New summary measures reveal geometric structure in weighted measures on manifolds.

We determine the number of cusps of minimal Picard modular surfaces. The proof also counts cusps of other Picard modular surfaces of arithmetic interest. Consequently, for each N > 0 there are finitely many commensurability classes of nonuniform arithmetic lattices in SU(2, 1) that contain an N-cusped surface. We also …

The existing approaches to intrinsic dimension estimation usually are not reliable when the data are nonlinearly embedded in the high dimensional space. In this work, we show that the explicit accounting to geometric properties of unknown support leads to the polynomial correction to the standard maximum likelihood est…

ERDMD discovers sparse, nonuniformly timed DMD models from chaotic attractors.

In this paper, we consider the numerical pricing of financial derivatives using Radial Basis Function generated Finite Differences in space. Such discretization methods have the advantage of not requiring Cartesian grids. Instead, the nodes can be placed with higher density in areas where there is a need for higher acc…

We prove Zimmer's conjecture for $C^2$ actions by finite-index subgroups of $\mathrm{SL}(m,\mathbb{Z})$ provided $m>3$. The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in $\mathrm{SL}(m,\mathbb{R})$ but new ideas are needed to overcome the lack of compactn…

We report on results concerning a partially aggregated Stock Flow Consistent (SFC) macroeconomic model in the stationary state where the sectors of banks and firms are aggregated, the sector of households is dis-aggregated, and the probability density function (pdf) of the wealth of households is exogenous, constrained…

Framework for designing nonlinearities in neural networks with slope constraints.

T-Rex uses EM to fit robust factor models in noisy data.

This paper optimizes subsampling for large datasets using Poisson distribution.

Let $Γ$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $Γ$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this vo…

We study the classification of smooth toroidal compactifications of nonuniform ball quotients in the sense of Kodaira and Enriques. Moreover, several results concerning the Riemannian and complex algebraic geometry of these spaces are given. In particular we show that there are compact complex surfaces which admit Riem…

If $Γ$ is any nonuniform lattice in the group ${\rm PU}(2,1)$, let $\overlineΓ$ be the quotient of $Γ$ obtained by filling the cusps of $Γ$ (i.e. killing the center of parabolic subgroups). Assuming that such a lattice $Γ$ has positive first Betti number, we prove that for any sufficiently deep subgroup of finite index…

This paper proposes BAT to balance accuracy and robustness in adversarial training.

Social sensing has emerged as a new sensing paradigm where humans (or devices on their behalf) collectively report measurements about the physical world. This paper focuses on a quality-cost-aware task allocation problem in multi-attribute social sensing applications. The goal is to identify a task allocation strategy …

Estimates functions on unknown manifolds using multiscale regression.

Stochastic optimization is key to efficient inversion in PDE-constrained optimization. Using 'simultaneous shots', or random superposition of source terms, works very well in simple acquisition geometries where all sources see all receivers, but this rarely occurs in practice. We develop an approach that interpolates d…

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virt…

We construct two infinite families of ball quotient compactifications birational to bielliptic surfaces. For each family, the volume spectrum of the associated noncompact finite volume ball quotient surfaces is the set of all positive integral multiples of $\frac{8}{3}π^{2}$, i.e., they attain all possible volumes of c…

In this paper we investigate the effectiveness of Alternating Direction Implicit (ADI) time discretization schemes in the numerical solution of the three-dimensional Heston-Hull-White partial differential equation, which is semidiscretized by applying finite difference schemes on nonuniform spatial grids. We consider t…

As the size and complexity of models and datasets grow, so does the need for communication-efficient variants of stochastic gradient descent that can be deployed to perform parallel model training. One popular communication-compression method for data-parallel SGD is QSGD (Alistarh et al., 2017), which quantizes and en…

Optimal subset selection for hypothesis testing with penalties.

Simultaneous reproduction of all financial stylized facts is so difficult that most existing stochastic process-based and agent-based models are unable to achieve the goal. In this study, by extending the decision-making structure of Minority Game, we propose a novel agent-based model called "Speculation Game," for a b…

We propose to optimize the activation functions of a deep neural network by adding a corresponding functional regularization to the cost function. We justify the use of a second-order total-variation criterion. This allows us to derive a general representer theorem for deep neural networks that makes a direct connectio…