arXiv research

Study Euler characteristics and loop lengths in hyperbolic 3-manifolds.

Local gauge freedom in relativistic quantum mechanics is derived from a measurement principle for space and time. For the Dirac equation, one obtains local U(2,2) gauge transformations acting on the spinor index of the wave functions. This local U(2,2) symmetry allows a unified description of electrodynamics and genera…

A new property fixes look-ahead bias in backtesting and trading pipelines.

Derives log-corrections in AdS4/CFT3 using supergravity localization.

In this paper, we explore degrees of freedom in deep sigmoidal neural networks. We show that the degrees of freedom in these models is related to the expected optimism, which is the expected difference between test error and training error. We provide an efficient Monte-Carlo method to estimate the degrees of freedom f…

We introduce two numerical conjugacy invariants for dynamical systems -- the complexity and weak complexity indices -- which are well-suited for the study of "completely integrable" Hamiltonian systems. These invariants can be seen as "slow entropies", they describe the polynomial growth rate of the number of balls (fo…

We give the first example of systolic freedom over torsion coefficients. The phenomenon is a bit unexpected (contrary to a conjecture of Gromov's) and more delicate than systolic freedom over the integers.

Study gauge freedoms in elastic wave equations and Dirichlet-to-Neumann map.

The derivation of statistical properties for Partial Least Squares regression can be a challenging task. The reason is that the construction of latent components from the predictor variables also depends on the response variable. While this typically leads to good performance and interpretable models in practice, it ma…

Measuring supernova neutrinos removes spacetime's conformal freedom.

Measures neural network complexity via effective degrees of freedom.

Classifies solutions in multisymplectic field theories using geometric gauge freedom.

This paper investigates the model degrees of freedom in k-means clustering. An extension of Stein's lemma provides an expression for the effective degrees of freedom in the k-means model. Approximating the degrees of freedom in practice requires simplifications of this expression, however empirical studies evince the a…

Regularization aims to improve prediction performance of a given statistical modeling approach by moving to a second approach which achieves worse training error but is expected to have fewer degrees of freedom, i.e., better agreement between training and prediction error. We show here, however, that this expected beha…

A new distribution family extends the $\alpha$-stable distribution with a degree of freedom parameter.

This note presents an operational measure of fat-tailedness for univariate probability distributions, in $[0,1]$ where 0 is maximally thin-tailed (Gaussian) and 1 is maximally fat-tailed. Among others,1) it helps assess the sample size needed to establish a comparative $n$ needed for statistical significance, 2) allows…

A neural network with a single hidden layer can't represent certain multivariable functions.

Unified finetuning of all quantization degrees of freedom achieves state-of-the-art 4-bit quantization.

A new formalism simplifies SO(3) Yang-Mills theory connections.

We re-examine classical mechanics with both commuting and anticommuting degrees of freedom. We do this by defining the phase dynamics of a general Lagrangian system as an implicit differential equation in the spirit of Tulczyjew. Rather than parametrising our basic degrees of freedom by a specified Grassmann algebra, w…

Fewer degrees of freedom can train deep networks, showing a sharp phase transition.

The paper examines parallel one forms on Riemannian and Finslerian manifolds.

The paper argues for using more degrees of freedom in empirical financial analysis to improve conclusions.

We discuss normal forms and symplectic invariants of parabolic orbits and cuspidal tori in integrable Hamiltonian systems with two degrees of freedom. Such singularities appear in many integrable systems in geometry and mathematical physics and can be considered as the simplest example of degenerate singularities. We a…

EPGP surrogate outperforms finite elements in solving wave equations.

A central question in modern machine learning and imaging sciences is to quantify the number of effective parameters of vastly over-parameterized models. The degrees of freedom is a mathematically convenient way to define this number of parameters. Its computation and properties are well understood when dealing with di…

John Lott has computed an integer-valued signature for the orbit space of a compact orientable $(4k+1)$ manifold with a semi-free $S^1$-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on…

Geometrically describes pseudo-gauge freedom in relativistic hydrodynamics.

Physical systems differring in their microscopic details often display strikingly similar behaviour when probed at macroscopic scales. Those universal properties, largely determining their physical characteristics, are revealed by the powerful renormalization group (RG) procedure, which systematically retains "slow" de…

Directly simulates squared Bessel processes efficiently.

The center of mass in General Relativity is hard to define due to coordinate freedom.

Deep neural networks reduce loan portfolio risk.

A new method joins two arcs with a degree of freedom.

We developed a perturbation model for affine gravity theories.

Given a pair of integers m and n such that 1 < m < n, we show that every n-dimensional manifold admits metrics of arbitrarily small total volume, and possessing the following property: every m-dimensional submanifold of less than unit m-volume is necessarily torsion in homology. This result is different from the case o…

Researchers explore gauge freedom in entropies of $q$-Gaussian measures.

Developed a new thresholding method that connects soft and hard thresholding.

We prove the simultaneous (k,n-k)-systolic freedom, for a pair of adjacent integers k smaller than n/2, of a simply connected n-manifold X. Our construction, related to recent results of I. Babenko, is concentrated in a neighborhood of suitable k-dimensional submanifolds of X. We employ calibration by differential form…

Develops unisolvent weights for Nédélec second family finite elements in 2D.

In this paper we are investigating variational homogeneous second order differential equations by considering the questions of how many different variational principles exist for a given spray. We focus our attention on h(2)-variationality; that is, the regular Lagrange function is homogeneous of degree two in the dire…

We simplify supergravity in 10D using geometric insights.

For an integrable Hamiltonian with $d\ (d\geq 2)$ degrees of freedom, we show the conditions on perturbations, for which invariant tori can be destructed.

This paper studies schemes to de-bias the Lasso in a linear model $y=Xβ+ε$ where the goal is to construct confidence intervals for $a_0^Tβ$ in a direction $a_0$, where $X$ has iid $N(0,Σ)$ rows. We show that previously analyzed propositions to de-bias the Lasso require a modification in order to enjoy efficiency in a f…

Transformers reduce redundancy by focusing on invariant relational quantities.

AI agents improve forecast combination in empirical economics.

Paper develops methods for estimating and simulating a Student-t Lévy regression model.

A method to automatically choose feature dimensions in linear attention for better approximation quality.

Choosing appropriate architectures and regularization strategies for deep networks is crucial to good predictive performance. To shed light on this problem, we analyze the analogous problem of constructing useful priors on compositions of functions. Specifically, we study the deep Gaussian process, a type of infinitely…