arXiv research

We study the Fox coloring invariants of rational knots. We express the propagation of the colors down the twists of these knots and ultimately the determinant of them with the help of finite increasing sequences whose terms of even order are even and whose terms of odd order are odd.

The study finds smooth structures on specific 4-manifolds with even fundamental groups.

This short note resolves the most important part of the PS braid conjecture while introducing the first known examples of knots and links with odd torsion of order 9, 27, 81, and 25 in their even Khovanov homology.

The $L^p$-cosine transform of an even, continuous function $f\in C_e(\Sn)$ is defined by: $$H(x)=\int_{\Sn}|\ip{x}ξ|^pf(ξ) dξ,\quad x\in {\R}^n.$$ It is shown that if $p$ is not an even integer then all partial derivatives of even order of $H(x)$ up to order $p+1$ (including $p+1$ if $p$ is an odd integer) exist and ar…

In this paper, we prove a Morse index theorem for the index form of even order linear Hamiltonian systems on the closed interval with reasonable self-adjoint boundary conditions. The highest order term is assumed to be nondegenerate.

The Hopf Lemma for second order elliptic operators is proved to hold in domains with $C^{1,α}$, and even less regular, boundaries. It need not hold for $C^1$ boundaries. Corresponding results are proved for second order parabolic operators.

On conformal manifolds of even dimension $n\geq 4$ we construct a family of new conformally invariant differential complexes. Each bundle in each of these complexes appears either in the de Rham complex or in its dual. Each of the new complexes is elliptic if the signature is Riemannian. We also construct gauge compani…

State-of-the-art methods in convex and non-convex optimization employ higher-order derivative information, either implicitly or explicitly. We explore the limitations of higher-order optimization and prove that even for convex optimization, a polynomial dependence on the approximation guarantee and higher-order smoothn…

Researchers construct irreducible 4-manifolds with specific properties.

New algorithm finds approximate stationary points in non-convex optimization.

Study proves higher-order conformal forms don't exist in odd dimensions.

The topological significance of the spectral Atiyah-Patodi-Singer eta-invariant is investigated under the parity conditions of P. Gilkey. We show that twice the fractional part of the invariant is computed by the linking pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin duality implies the…

Transformers can approximate Newton's method for logistic regression.

Algorithm learns causal structures from low-order conditional independencies.

Researchers confirm a conjecture about metrics on a specific Teichmüller space.

We investigate the orderability properties of fundamental groups of 3-dimensional manifolds. Many 3-manifold groups support left-invariant orderings, including all compact P^2-irreducible manifolds with positive first Betti number. For seven of the eight geometries (excluding hyperbolic) we are able to characterize whi…

In the paper we consider the theory of elliptic operators acting in subspaces defined by pseudodifferential projections. This theory on closed manifolds is connected with the theory of boundary value problems for operators violating Atiyah-Bott condition. We prove an index formula for elliptic operators in subspaces de…

Factorization machines (FMs) are a supervised learning approach that can use second-order feature combinations even when the data is very high-dimensional. Unfortunately, despite increasing interest in FMs, there exists to date no efficient training algorithm for higher-order FMs (HOFMs). In this paper, we present the …

Data whitening and second order optimization harm generalization by reducing access to dataset information.

Careful tuning of the learning rate, or even schedules thereof, can be crucial to effective neural net training. There has been much recent interest in gradient-based meta-optimization, where one tunes hyperparameters, or even learns an optimizer, in order to minimize the expected loss when the training procedure is un…

Topological degrees of continuous mappings between manifolds of even dimension are studied in terms of index theory of pseudo-differential operators. The index formalism of non-commutative geometry is used to derive analytic integral formulas for the index of a 0:th order pseudo-differential operator twisted by a Hölde…

In this paper, we investigate the popular deep learning optimization routine, Adam, from the perspective of statistical moments. While Adam is an adaptive lower-order moment based (of the stochastic gradient) method, we propose an extension namely, HAdam, which uses higher order moments of the stochastic gradient. Our …

This paper presents the first use of graph neural networks (GNNs) for higher-order proof search and demonstrates that GNNs can improve upon state-of-the-art results in this domain. Interactive, higher-order theorem provers allow for the formalization of most mathematical theories and have been shown to pose a significa…

An elliptic theory is constructed for operators acting in subspaces defined via odd pseudodifferential projections. Subspaces of this type arise as Calderon subspaces for first order elliptic differential operators on manifolds with boundary, or as spectral subspaces for self-adjoint elliptic differential operators of …

For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators $\hatψ_{1}$ are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting wi…

First-order ANIL learns shared representations even with overparametrization.

Local search heuristics for non-convex optimizations are popular in applied machine learning. However, in general it is hard to guarantee that such algorithms even converge to a local minimum, due to the existence of complicated saddle point structures in high dimensions. Many functions have degenerate saddle points su…

Efficient algorithm for contextual bandits with first-order guarantees.

Defining the $m$-th stratum of a closed subset of an $n$ dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least $n-m$ linearly independent directions, we establish that the $m$-th stratum is second-order rectifiable of dimension $m$ and a Borel set. This was known for co…

New conservation laws found for polyharmonic maps in critical dimension.

Paper proposes efficient SHAP computation methods.

For an even dimensional, compact, conformal manifold without boundary we construct a conformally invariant differential operator of order the dimension of the manifold. In the conformally flat case, this operator coincides with the critical {\sf GJMS} operator of Graham-Jenne-Mason-Sparling. We use the Wodzicki residue…

Study finds existence of $Q$-curvature metrics on even-dimensional manifolds with conical singularities.

New algorithms optimize constrained problems faster, avoiding full set optimization.

New method identifies causal direction with latent confounders.

We prove that the quantum SO(3)-invariant of an arbitrary 3-manifold $M$ is always an algebraic integer, if the order of the quantum parameter is co-prime with the order of the torsion part of $H_1(M,\BZ)$. An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-…

While recent continual learning methods largely alleviate the catastrophic problem on toy-sized datasets, some issues remain to be tackled to apply them to real-world problem domains. First, a continual learning model should effectively handle catastrophic forgetting and be efficient to train even with a large number o…

Paper proposes methods to learn DAGs from partial orderings.

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.

Improved GAN performance using higher-order Wasserstein moments.

Let $Σ_g (g>1)$ be a closed surface embedded in $S^3$. If a group $G$ can acts on the pair $(S^3, Σ_g)$, then we call such a group action on $Σ_g$ extendable over $S^3$. In this paper we show that the maximum order of extendable cyclic group actions is $4g+4$ when $g$ is even and $4g-4$ when $g$ is odd; the maximum ord…

An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the logarithm of a defining function homogeneous of degree 2, and an invariant procedur…

We show that the resulting manifold by $r$-surgery on the hyperbolic twist knot $K_m, \, m \ge 2$, has left-orderable fundamental group if the slope $r$ satisfies the condition $r \in (-4,2m)$ if $m$ is even, and $r \in [0,4] \cup (\frac{4m}ω+4, 2m+4)$ if $m$ is odd, where $ω>1$ is the unique real solution of the equat…

It is known that the order of a finite group of diffeomorphisms of a 3-dimensional handlebody of genus g > 1 is bounded by the linear polynomial 12(g-1), and that the order of a finite group of diffeomorphisms of a 4-dimensional handlebody (or equivalently, of its boundary 3-manifold), faithful on the fundamental group…

In order to be useful, visualizations need to be interpretable. This paper uses a user-based approach to combine and assess quality measures in order to better model user preferences. Results show that cluster separability measures are outperformed by a neighborhood conservation measure, even though the former are usua…

Using more than 6.7 billions of trades, we explore how the tick-by-tick dynamics of limit order books depends on the aggregate actions of large investment funds on a much larger (quarterly) timescale. In particular, we find that the well-established long memory of market order signs is markedly weaker when large invest…

We show that the statistics of spreads in real order books is characterized by an intrinsic asymmetry due to discreteness effects for even or odd values of the spread. An analysis of data from the NYSE order book points out that traders' strategies contribute to this asymmetry. We also investigate this phenomenon in th…

A new algorithm for optimizing huge-scale black-box problems with reduced memory usage.