arXiv research

Study fractal and regular geometry in deep neural networks.

In this paper we investigate how the volume of hyperbolic manifolds increases under the process of removing a curve, that is, Dehn drilling. If the curve we remove is a geodesic we are able to show that for a certain family of manifolds the volume increase is bounded above by $π\cdot l$ where $l$ is the length of the g…

Study increasing profits in a flexible financial market model.

Online reinforcement learning agents are currently able to process an increasing amount of data by converting it into a higher order value functions. This expansion of the information collected from the environment increases the agent's state space enabling it to scale up to a more complex problems but also increases t…

We propose a training method for deep neural network (DNN)-based source enhancement to increase objective sound quality assessment (OSQA) scores such as the perceptual evaluation of speech quality (PESQ). In many conventional studies, DNNs have been used as a mapping function to estimate time-frequency masks and traine…

We provide several new depth-based separation results for feed-forward neural networks, proving that various types of simple and natural functions can be better approximated using deeper networks than shallower ones, even if the shallower networks are much larger. This includes indicators of balls and ellipses; non-lin…

For a knot K, let b_n(K) be the minimum length of an n-stranded braid representative of K. Examples of knots exist for which b_n(K) is a non-increasing function. We investigate the behavior of b_n(K). We develop bounds on the function in terms of the genus of K, with stronger results for homogeneous knots and braid pos…

We study the average price impact of a single trade executed in the NYSE. After appropriate averaging and rescaling, the data for the 1000 most highly capitalized stocks collapse onto a single function, giving average price shift as a function of trade size. This function increases as a power that is the order of 1/2 f…

With a new deprivation (or poverty) function, in this paper, we theoretically study the changes in poverty with respect to the `global' mean and variance of the income distribution using Indian survey data. We show that when the income obeys a log-normal distribution, a rising mean income generally indicates a reductio…

Probabilistic mixture models have been widely used for different machine learning and pattern recognition tasks such as clustering, dimensionality reduction, and classification. In this paper, we focus on trying to solve the most common challenges related to supervised learning algorithms by using mixture probability d…

New findings on hidden symmetries in ReLU networks.

Study on critical points in random neural networks, revealing three regimes based on activation function.

Gaussian processes (GP) are attractive building blocks for many probabilistic models. Their drawbacks, however, are the rapidly increasing inference time and memory requirement alongside increasing data. The problem can be alleviated with compactly supported (CS) covariance functions, which produce sparse covariance ma…

Gaussian prior and likelihood improve bandit learning performance.

Productions functions map the inputs of a firm or a productive system onto its outputs. This article expounds generalizations of the production function that include state variables, organizational structures and increasing returns to scale. These extensions are needed in order to explain the regularities of the empiri…

When the parameters are independently and identically distributed (initialized) neural networks exhibit undesirable properties that emerge as the number of layers increases, e.g. a vanishing dependency on the input and a concentration on restrictive families of functions including constant functions. We consider parame…

Monte Carlo (MC) techniques are often used to estimate integrals of a multivariate function using randomly generated samples of the function. In light of the increasing interest in uncertainty quantification and robust design applications in aerospace engineering, the calculation of expected values of such functions (e…

A quaternionic contact (qc) heat equation and the corresponding qc energy functional are introduced. It is shown that the qc energy functional is monotone non-increasing along the qc heat equation on a compact qc manifold provided certain positivity conditions are satisfied.

We show that a variety of modern deep learning tasks exhibit a "double-descent" phenomenon where, as we increase model size, performance first gets worse and then gets better. Moreover, we show that double descent occurs not just as a function of model size, but also as a function of the number of training epochs. We u…

A diffusion model estimates data manifold dimension by tracking likelihood increases.

The three-state agent-based 2D model of financial markets in the version proposed by Giulia Iori in 2002 has been herein extended. We have introduced the increase of herding behaviour by modelling the altering trust of an agent in his nearest neighbours. The trust increases if the neighbour has foreseen the price chang…

We are interested in the maximum value achieved by the systole function over all complete finite area hyperbolic surfaces of a given signature $(g,n)$. This maximum is shown to be strictly increasing in terms of the number of cusps for small values of $n$. We also show that this function is greater than a function that…

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), …

A novel neural network approach for optimization problems.

In order to push the performance on realistic computer vision tasks, the number of classes in modern benchmark datasets has significantly increased in recent years. This increase in the number of classes comes along with increased ambiguity between the class labels, raising the question if top-1 error is the right perf…

Study of loss functions for learning to defer, proving consistency.

We introduce a faithful representation of the heavy tail multivariate distribution of asset returns, as parsimonous as the Gaussian framework. Using calculation techniques of functional integration and Feynman diagrams borrowed from particle physics, we characterize precisely, through its cumulants of high order, the d…

The standard asset pricing models (the CCAPM and the Epstein-Zin non-expected utility model) counterintuitively predict that equilibrium asset prices can rise if the representative agent's risk aversion increases. If the income effect, which implies enhanced saving as a result of an increase in risk aversion, dominates…

Mathematically, a homothetic function is a function of the form $f({\bf x})=F(h(x_1,...,x_n))$, where $h$ is a homogeneous function of any degree $d\ne 0$ and $F$ is a monotonically increasing function. In economics homothetic functions are production functions whose marginal technical rate of substitution is homogeneo…

The study shows that limited liability can make banks more stable by choosing less risky assets.

Quasi-Monte Carlo speeds up option Greeks calculation on GPUs.

The main purpose of this note is to construct two functionals of the positive solutions to the conjugate heat equation associated to the metrics evolving by the conformal Ricci flow on closed manifolds. We show that they are nondecreasing by calculating the explicit evolution formulas of these functionals. For the entr…

Expressive efficiency refers to the relation between two architectures A and B, whereby any function realized by B could be replicated by A, but there exists functions realized by A, which cannot be replicated by B unless its size grows significantly larger. For example, it is known that deep networks are exponentially…

In this paper, we propose a stochastic optimization method that adaptively controls the sample size used in the computation of gradient approximations. Unlike other variance reduction techniques that either require additional storage or the regular computation of full gradients, the proposed method reduces variance by …

All DeFi markets are essentially CFMMs with increasing invariants.

Bernard et al. (2015) study an optimal insurance design problem where an individual's preference is of the rank-dependent utility (RDU) type, and show that in general an optimal contract covers both large and small losses. However, their contracts suffer from a problem of moral hazard for paying more compensation for a…

Method bounds tail probabilities of continuous RVs.

LM optimization outperforms other methods in deep learning tasks but at high computational cost.

Predictions are issued on the basis of certain information. If the forecasting mechanisms are correctly specified, a larger amount of available information should lead to better forecasts. For point forecasts, we show how the effect of increasing the information set can be quantified by using strictly consistent scorin…

Global minima found for multidimensional scaling with penalties.

We study an optimal liquidation problem under the ambiguity with respect to price impact parameters. Our main results show that the value function and the optimal trading strategy can be characterized by the solution to a semi-linear PDE with superlinear gradient, monotone generator and singular terminal value. We also…

BAxUS optimizes high-dimensional functions adaptively, avoiding performance degradation and failure.

Neural networks cannot approximate certain functions in Sobolev spaces, leading to unbounded parameter growth.

Proposes an exponentially increasing step-size for faster parameter estimation in statistical models.

Bayesian free energy remains bounded for deep ReLU networks in overparametrized cases.

Two new criteria help understand the advantage of deep neural networks.

Paper proposes LANN to measure model complexity of neural networks with curve activation functions.

Based on the tick-by-tick price changes of the companies from the U.S. and from the German stock markets over the period 1998-99 we reanalyse several characteristics established by the Boston Group for the U.S. market in the period 1994-95, which serves to verify their space and time-translational invariance. By increa…