arXiv research

Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.

We study the one parameter family of genus 2 Riemann surfaces defined by the orbit of the L-shaped translation surface tiled by three squares under the Teichmüller geodesic flow. These surfaces are real algebraic curves with three real components. We are interested in describing these surfaces by their period matrices.…

New tractable density models from squaring neural networks.

Exponential family extensions of principal component analysis (EPCA) have received a considerable amount of attention in recent years, demonstrating the growing need for basic modeling tools that do not assume the squared loss or Gaussian distribution. We extend the EPCA model toolbox by presenting the first exponentia…

We consider adaptive system identification problems with convex constraints and propose a family of regularized Least-Mean-Square (LMS) algorithms. We show that with a properly selected regularization parameter the regularized LMS provably dominates its conventional counterpart in terms of mean square deviations. We es…

The study finds arithmetic groups often in square-tiled surface monodromies.

The square-peg problem is solved using configuration spaces and multijet transversality.

Computes Steenrod squares on Khovanov homology for knots up to 11 crossings.

Synthesizes robust estimators for domain adaptation.

The family of temporal difference (TD) methods span a spectrum from computationally frugal linear methods like TD(λ) to data efficient least squares methods. Least square methods make the best use of available data directly computing the TD solution and thus do not require tuning a typically highly sensitive learning r…

This paper reviews SDR methods for multivariate response regression.

Study of $n$-cylinder surfaces to calculate Masur-Veech volumes.

New puzzles from geometry and topology.

We prove a transversality "lifting property" for compactified configuration spaces as an application of the multijet transversality theorem: the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclid…

New groups derived from square configurations have right-angled and HNN structures.

For every spatial embedding of each graph in the Petersen family, it is known that the sum of the linking numbers over all of the constituent 2-component links is congruent to 1 modulo 2. In this paper, we give an integral lift of this formula in terms of the square of the linking number and the second coefficient of t…

Origami graphs' Euler characteristics grow as origami complexity increases.

The paper is on the vanishing topology of singular Milnor fibres of holomorphic families of arbitrary square, symmetric and skew-symmetric matrices with sufficiently many parameters. We define vanishing cycles on such fibres, prove an extended form of the Damon-Pike $μ=τ$ conjecture about the families of a special type…

SNEPPPs use squared neural networks to efficiently model Poisson point processes.

The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic Klein-Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An i…

In this paper, we revisit the analyses of Antonie Stern (1925) and Hans Lewy (1977) devoted to the construction of spherical harmonics with two or three nodal domains. Our method yields sharp quantitative results and a better understanding of the occurrence of bifurcations in the families of nodal sets.This paper is a …

This paper focuses on the graphs in the Petersen family, the set of minor minimal intrinsically linked graphs. We prove there is a relationship between algebraic linking of an embedding and knotting in an embedding. We also present a more explicit relationship for the graph $K_{3,3,1}$ between knotting and linking, whi…

We show how the families Seiberg-Witten invariants of a family of smooth $4$-manifolds can be recovered from the families Bauer-Furuta invariant via a cohomological formula. We use this formula to deduce several properties of the families Seiberg-Witten invariants. We give a formula for the Steenrod squares of the fami…

We study cubical sets without degeneracies, which we call square sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a square set C has an infinite family of associated square sets J^i(C), i=1,2,..., which we call James complexes. There are mock bundle pr…

We show that for every integer $b\geq 3$, there exists a link in a $b$-bridge position with respect to a critical bridge sphere. In fact, for each $b$, we construct an infinite family of links which we call square whose bridge spheres are critical.

Let $S$ be an $n$-punctured sphere, with $n \geq 3$. We prove that $\binom{n}{3}$ is the maximum size of a family of pairwise non-homotopic simple arcs on $S$ joining a fixed pair of distinct punctures of $S$ and pairwise intersecting at most twice. On the way, we show that a square annular diagram $A$ has a corner on …

We present a new description of the genus 3 Arnoux--Yoccoz translation surface in terms of its Delaunay polygons and show that, up to affine equivalence, it belongs to two families of surfaces whose isometry groups include the dihedral group of the square.

If F is a family of mod 2 flat k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp.

Earlier work with Robert Gompf and Abigail Thompson classified, via a natural slope indexed by the rationals, all two-component links which contain the square knot and from which $(S^1 \times S^2) \# (S^1 \times S^2)$ can be obtained by surgery. It was argued that a certain family $L_n$ of such links probably contradic…

We have shown that the Beltrami Theorem in Riemannian geometry is still true for square metrics if the dimension $n\ge 3$, namely, an $n(\ge 3)$-dimensional square metric is locally projectively flat if and only if it is of scalar flag curvature. In this paper, we go on with the study of the Beltrami Theorem for a larg…

New elastic metrics for surface shape analysis.

This paper calculates the derivative of surface holonomy for non-abelian gerbes.

Unified multi-view learning framework using OPLS with regularization and deep extensions.

Using Gauge theoretical techniques employed by Lisca for 2-bridge knots and by Greene-Jabuka for 3-stranded pretzel knots, we show that no member of the family of Montesinos knots M(0;[m_1+1,n_1+2],[m_2+1,n_2+2],q), with certain restrictions on m_i, n_i, and q, can be (smoothly) slice. Our techniques use Donaldson's di…

We combine recent developments on weakly symmetric pseudo--riemannian nilmanifolds with with geometric methods for construction of unitary representations on square integrable Dolbeault cohomology spaces. This runs parallel to construction of discrete series representations on spaces of square integrable harmonic forms…

Unified analysis of reweighted least-squares algorithms for linear models.

We develop Square Root Graphical Models (SQR), a novel class of parametric graphical models that provides multivariate generalizations of univariate exponential family distributions. Previous multivariate graphical models [Yang et al. 2015] did not allow positive dependencies for the exponential and Poisson generalizat…

New method tracks time-varying parameters in data.

If there are any 2-component counterexamples to the Generalized Property R Conjecture, a least genus component of all such counterexamples cannot be a fibered knot. Furthermore, the monodromy of a fibered component of any such counterexample has unexpected restrictions. The simplest plausible counterexample to the Gene…

New CH covariance class improves spatial statistics by balancing differentiability and tail behavior.

Explicitly constructed 3XOR instances hard for Sum-of-Squares hierarchy.

We establish $L^{\infty}$ and $L^2$ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $\ell^1$ and $\ell^0$ controls on their inner and outer parameters. With the squared ReLU ridge function, we show th…

Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations …

Unified CCA methods for large-scale data with fast SGD algorithms.

Dual-sPLS improves feature selection and prediction in high-dimensional data.

This paper establishes a statistical versus computational trade-off for solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis} (Sparse PCA) problem, and the family of {\em Sum-of-Squares} (SoS, aka Lasserre/…

This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where $q\le p^*$, and $1/p+1/p^{*}=1$, we discuss properties of a width-independent ca…

We present an efficient second-order algorithm with $\tilde{O}(\frac{1}η\sqrt{T})$ regret for the bandit online multiclass problem. The regret bound holds simultaneously with respect to a family of loss functions parameterized by $η$, for a range of $η$ restricted by the norm of the competitor. The family of loss funct…