arXiv research

Improved root-finding method for smooth functions.

In this paper, we find a condition under which a Finsler space with Kropina change of mth-root metric is projectively related to a mth-root metric and also we find a condition under which this Kropina transformed mth-root metric is locally dually flat. Moreover we find the condition for its Projective flatness.

Algorithm removes leaves to find root in uniform trees.

In this paper, we characterize locally dually flat generalized m-th root Finsler metrics. Then we find a condition under which a generalized m-th root metric is projectively related to a m-th root metric. Finally, we prove that if a generalized m-th root metric is conformal to a m-th root metric, then both of them redu…

Study finds root vertex in large networks with high probability.

Many independent studies on stocks and futures contracts have established that market impact is proportional to the square-root of the executed volume. Is market impact quantitatively similar for option markets as well? In order to answer this question, we have analyzed the impact of a large proprietary data set of opt…

In this paper, we consider Kropina change of $m$-th root Finsler metrics. We find necessary and sufficient condition under which the Kropina change of an $m$-th root Finsler metric be locally dually flat. Then we prove that the Kropina change of an $m$-th root Finsler metric is locally projectively flat if and only if …

Root-finding methods improve efficiency of conformal prediction sets.

In this paper, we prove that every m-th root metric with isotropic mean Berwald curvature reduces to a weakly Berwald metric. Then we show that an m-th root metric with isotropic mean Landsberg curvature is a weakly Landsberg metric. We find necessary and sufficient condition under which conformal $β$-change of an m-th…

Margalit and Schleimer constructed nontrivial roots of the Dehn twist about a nonseparating curve. We prove that the conjugacy classes of roots of the Dehn twist about a nonseparating curve correspond to the conjugacy classes of periodic maps with certain conditions. Futhermore, we give data set which determine the con…

Improved bounds for Black-Scholes volatility lead to faster root-finding.

We consider numerical schemes for root finding of noisy responses through generalizing the Probabilistic Bisection Algorithm (PBA) to the more practical context where the sampling distribution is unknown and location-dependent. As in standard PBA, we rely on a knowledge state for the approximate posterior of the root l…

Enhanced ROOT-SGD optimizes stochastic optimization with diminishing stepsizes.

Study finds price impact follows a 'double' square-root law, suggesting mechanical origin.

We investigate the behavior of the Shanghai Stock Exchange Composite (SSEC) index for the period from 1990:12 to 2007:06 using an unconstrained two-regime threshold autoregressive (TAR) model with an unit root developed by Caner and Hansen. The method allows us to simultaneously consider non-stationarity and nonlineari…

The notion of market impact is subtle and sometimes misinterpreted. Here we argue that impact should not be misconstrued as volatility. In particular, the so-called ``square-root impact law'', which states that impact grows as the square-root of traded volume, has nothing to do with price diffusion, i.e. that typical p…

The study of random positive 3-strand braids reveals patterns in the roots of their Alexander polynomials.

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$…

The probabilistic bisection algorithm (PBA) solves a class of stochastic root-finding problems in one dimension by successively updating a prior belief on the location of the root based on noisy responses to queries at chosen points. The responses indicate the direction of the root from the queried point, and are incor…

Develops new algorithms for solving root-finding problems in large-scale settings.

The paper finds formulas for a specific invariant order 7.

Optimizes AMM markets with a new framework reducing complex optimization to simpler root finding.

Develops efficient method for nonconvex problems using Regula Falsi.

Probabilistic Bisection Algorithm performs root finding based on knowledge acquired from noisy oracle responses. We consider the generalized PBA setting (G-PBA) where the statistical distribution of the oracle is unknown and location-dependent, so that model inference and Bayesian knowledge updating must be performed s…

New method solves root-finding problems with faster convergence.

We study $k$-GenEV, the problem of finding the top $k$ generalized eigenvectors, and $k$-CCA, the problem of finding the top $k$ vectors in canonical-correlation analysis. We propose algorithms $\mathtt{LazyEV}$ and $\mathtt{LazyCCA}$ to solve the two problems with running times linearly dependent on the input size and…

We find two different families of $Sp(2,R)$ symmetric $G_2$ structures in seven dimensions. These are $G_2$ structures with $G_2$ being the split real form of the simple exceptional complex Lie group $G_2$. The first family has $τ_2\equiv 0$, while the second family has $τ_1\equivτ_2\equiv 0$. The families are differen…

Study finds Deep Taylor Decomposition is unreliable for explaining neural networks.

Study of Hitchin map on specific Higgs bundles.

Deep neural networks (DNNs) are shown to be promising solutions in many challenging artificial intelligence tasks. However, it is very hard to figure out whether the low precision of a DNN model is an inevitable result, or caused by defects. This paper aims at addressing this challenging problem. We find that the inter…

Method estimates noise variance in Gaussian process regression.

We find explicit bases for naturally defined lattices over a ring of algebraic integers in the SO(3) TQFT-modules of surfaces at roots of unity of odd prime order. Some applications relating quantum invariants to classical 3-manifold topology are given.

This study applies old and new generations of panel unit root tests to test the validity of long-run real interest rate parity (RIP) hypothesis for ten Central and Eastern European Countries (CEECs) with respect to the Euro area and an average of the CEECs' real interest rates, respectively. When the panel unit root te…

MSLs use parallelizable root-finding for efficient ODE and PDE solutions.

Algorithm generates realistic metaorders from public trade data.

The recent increase in the scale and complexity of software systems has introduced new challenges to the time series monitoring and anomaly detection process. A major drawback of existing anomaly detection methods is that they lack contextual information to help stakeholders identify the cause of anomalies. This proble…

Feature selection is important for modeling high-dimensional data, where the number of variables can be much larger than the sample size. In this paper, we develop a support detection and root finding procedure to learn the high dimensional sparse generalized linear models and denote this method by GSDAR. Based on the …

Study tail risk in high-frequency finance using $L_1$-regularized regression.

New method identifies causal order without sparsity assumptions.

Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time…

New findings show the Gilmer-Masbaum map isn't always one-to-one.

We confirm the square-root law of market impact on Apple Inc. using a large dataset.

The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.

We compute the noncommutative de Rham cohomology for the finite-dimensional q-deformed coordinate ring $C_q[SL_2]$ at odd roots of unity and with its standard 4-dimensional differential structure. We find that $H^1$ and $H^3$ have three additional modes beyond the generic $q$-case where they are 1-dimensional, while $H…

Two-root Riemannian manifolds have no odd-dimensional examples.

New definition of patient-specific root causes of disease using counterfactuals.

A new line search rule improves support recovery in high-dimensional data.

Uniqueness of quasi-roots explored in right-angled Artin groups.