Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics in dimension n≥3, namely, an n(≥3)-dimensional square metric is locall…
In this paper, we study an important class of Finsler metrics--square metrics. We give two expressions of such metrics in terms of a Riemannian metric and a 1-form. We show that Einstein square metrics can be classified up to the classification of Einstein Riemannian metrics.
Investigates hypersurfaces in Finsler spaces with generalized square metrics.
problem Classifying and understanding hypersurfaces in Finsler spaces with generalized square metrics.
method Examined the generalized square metric F(x,y) = (α(x,y) + β(x,y))^(n+1)/(α^n(x,y)) and its application to Finslerian hypersurfaces.
result Established the classification and existence of first, second, and third kind of hyperplanes in the Finsler manifold.
Square metrics F=α(α+β)2 are a special class of Finsler metrics. It is the rate kind of metric category to be of excellent geometrical properties. In this paper, we discuss the so-called singular square metrics F=α(bα+β)2. A characterization for such metrics to be of vanishing Douglas curvature is p…
In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of Finsler metrics of Einstein-reversibility are always Einstein metrics. In parti…
Study Ricci curvature of homogeneous Finsler spaces with specific metrics.
problem Curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Derived explicit formulae for Ricci curvature and found conditions for vanishing S-curvature. result Spaces with vanishing S-curvature and negative Ricci curvature are Riemannian. Solutions to Strominger system found for square of Kähler class.
problem Finding solutions to Strominger system with specific balanced classes.
method Deforming Calabi-Yau and Hermitian-Yang-Mills metrics.
result Classes that are squares of Kähler metrics admit solutions.
The chapter reviews metrics for comparing curves, focusing on quotient elastic and square root velocity metrics.
problem Comparing and analyzing shapes of curves.
method Construction and theoretical properties of quotient elastic metrics, special case of square root velocity metric, numerical approaches for estimation.
result Simplified expression for the square root velocity metric distance.
We have shown that the Beltrami Theorem in Riemannian geometry is still true for square metrics if the dimension n≥3, namely, an n(≥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…
Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
The paper calculates S-curvature for specific Finsler metrics.
problem Calculating curvature properties of homogeneous Finsler spaces with (α,β)-metrics. method Proved existence of invariant vector fields and derived explicit formula for S-curvature. result Explicit formula for S-curvature of (α,β)-metrics is established. Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.
problem Proving non-degeneracy of critical points for a manifold's squared norm of second fundamental form.
method Generic Riemannian metric and conformal class restriction.
result Squared norm of the second fundamental form is a Morse function with non-degenerate critical points.
Researchers find unique metrics solving complex PDEs for constant scalar curvature.
problem Finding metrics with constant scalar curvature in complex manifolds.
method Proving existence and uniqueness of smooth functions f that solve a fourth-order nonlinear PDE related to the Calabi functional. result Critical metrics minimize the Calabi functional and have constant Chern scalar curvature.
The paper establishes a uniform Lipschitz bound on the square root of the systole function in Teichmüller space.
problem Uniform Lipschitz bounds on geometric functions in Teichmüller space.
method Injectivity radius analysis and Lipschitz bounds on systole function.
result Uniform Lipschitz constant for the square root of the systole function on Teichmüller space.
Sharp estimates on 2-step nilpotent Lie groups' metrics and cones.
problem Estimating asymptotic metrics in 2-step nilpotent Lie groups.
method Developed a novel technique to perturb rectifiable curves.
result Every 2-step nilpotent Riemannian Lie group is at bounded distance from its asymptotic cone.
In light of the power problems of statistical tests and undisciplined use of alpha-based statistics to compare models, this paper proposes a unified set of distance-based performance metrics, derived as the square root of the sum of squared alphas and squared standard errors. The Bayesian investor views model performan…
SRNF framework extends surface distance to Lipschitz surfaces.
problem Defining a distance metric for unparametrized surfaces.
method Square Root Normal Fields (SRNF) and Wasserstein Fisher Rao (WFR) metric.
result SRNF distance on Lipschitz surfaces is equivalent to WFR metric.
This paper analyzes sampling from heavy-tailed distributions using discretized Itô diffusions.
problem Sampling from heavy-tailed distributions with finite variance.
method Mean-square analysis of discretized Itô diffusions with weighted Poincaré inequalities.
result Explicit iteration complexity for obtaining samples close to target distributions in Wasserstein-2 metric.
Paper proposes a chi-square test for distance correlation.
problem Testing distance correlation is computationally expensive.
method Proposes a chi-square test for distance correlation, non-parametric, fast, applicable to various metrics.
result Chi-square test exhibits similar power to permutation test and can be valid and universally consistent for testing independence.
We develop efficient algorithms to estimate the stability of Ordinary Least Squares regression results.
problem Measuring the stability of regression conclusions in low dimensions.
method Efficient algorithms for estimating the minimum number of samples that need to be removed to change a regression conclusion.
result We can estimate stability up to a factor of 3 better than the greedy heuristic and certify stability even for dropping a majority of samples.
Squared families are a new model class derived from linear transformations, offering convenient properties and universal approximation.
problem Developing a new class of probability models that are easier to handle and have useful properties.
method Introducing squared families as families of probability densities obtained by squaring a linear transformation of a statistic, and showing their properties and applications.
result Squared families have convenient properties and can approximate target densities well.
We study distributed learning with the least squares regularization scheme in a reproducing kernel Hilbert space (RKHS). By a divide-and-conquer approach, the algorithm partitions a data set into disjoint data subsets, applies the least squares regularization scheme to each data subset to produce an output function, an…
Study of n-cylinder surfaces to calculate Masur-Veech volumes.
problem Calculating Masur-Veech volumes for hyperbolic surfaces.
method Combinatorial approach using metric ribbon graphs and plane trees.
result Found generating function for n-cylinder contributions. Paper classifies Randers metrics based on Ricci curvature properties.
problem Investigating isotropic projective Ricci curvature in Randers metrics.
method Classification of Randers metrics based on isotropic projective Ricci curvature properties.
result Randers metric of isotropic projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.
The study classifies harmonic cubic polynomials in up to 4 dimensions.
problem Describing harmonic cubic polynomials with specific Hessian properties.
method Construction and classification in all dimensions; techniques for inequivalence determination.
result Classification of solutions in dimensions up to 4.
Embedding complex objects as vectors in low dimensional spaces is a longstanding problem in machine learning. We propose in this work an extension of that approach, which consists in embedding objects as elliptical probability distributions, namely distributions whose densities have elliptical level sets. We endow thes…
The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. Through every point in such a metric there is a geodesic that saturates the length condition, and saturating geodesics …
This paper proposes a new method to prevent backtesting overfitting in trading strategies.
problem Preventing misleading results in backtesting of trading strategies.
method Covariance-Penalty Correction approach to reduce risk metrics based on the number of parameters and data used.
result Covariance-Penalties are effective in avoiding backtesting overfitting, with Total Least Squares outperforming Ordinary Least Squares.
In this paper, we study generalized Douglas-Weyl (α,β)-metrics. Suppose that an regular (α,β)-metric F is not of Randers type. We prove that F is a generalized Douglas-Weyl metric with vanishing S-curvature if and only if it is a Berwald metric. Moreover by ignoring the regularity, if F is not a Berwald met…
The paper studies metrics on manifolds with scalar curvature properties.
problem Finding metrics with specific scalar curvature properties.
method Analyzing the squared L2-norm of the scalar curvature over constant volume metrics. result Critical points of the functional correspond to Einstein or scalar flat metrics.
In this short note, we prove that the space of all admissible piecewise linear metrics parameterized by length square on a triangulated manifolds is a convex cone. We further study Regge's Einstein-Hilbert action and give a much more reasonable definition of discrete Einstein metric than our former version in \cite{G}.…
The most important aspect of any classifier is its error rate, because this quantifies its predictive capacity. Thus, the accuracy of error estimation is critical. Error estimation is problematic in small-sample classifier design because the error must be estimated using the same data from which the classifier has been…
LMC algorithm converges to target in Chi-squared and Renyi divergence.
problem Sampling from target distribution using LMC with strong dissipativity and smoothness conditions.
method LMC algorithm with strong dissipativity and first-order smoothness, initialized with Gaussian.
result LMC reaches ε-neighborhood of target in Chi-squared and Renyi divergence in O(λ²dε⁻¹) steps.
Develops Green operators for quantum fields on low-regularity spacetimes.
problem Describes quantum fields on low-regularity spacetimes with C1,1 metrics. method Shows well-posedness of wave equation, constructs Green operators, defines symplectic form.
result Provides a locally covariant description of quantum fields.
Study improves least squares estimation for heavy-tailed errors.
problem Improving least squares estimation under heteroscedastic and heavy-tailed errors.
method Analyzes the rate of convergence of least squares estimator under bounded conditional variance and finitely many moments of errors.
result Upper bounds on rates of convergence of LSE for heavy-tailed errors are found.
Study of circle homeomorphisms with square summable diamond shears.
problem Characterizing circle homeomorphisms with specific summability properties.
method Analysis of homeomorphisms in modular coordinates and comparison to Weil-Petersson class.
result Sharp results comparing new class to Weil-Petersson class and Hölder classes.
We bound the value of the Casson invariant of any integral homology 3-sphere M by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\T$, of the element of $\T$ associated to any Heegaard splitting of M. We construct examples which show this bound is asymptotica…
We study the systolic area (defined as the ratio of the area over the square of the systole) of the 2-sphere endowed with a smooth riemannian metric as a function of this metric. This function, bounded from below by a positive constant over the space of metrics, have the standard metric g_0 for critic point, althoug…
Unified treatment of elastic metrics for curves in any dimension.
problem Defining metrics on spaces of Euclidean curves for statistical analysis.
method Developing a unified approach to elastic metrics, extending results on existence of solutions and algorithms for computing distances and geodesics.
result Unified treatment of elastic metrics for all parameter choices, extending previous work.
Extended metric defined on Siegel-Jacobi space using invariant forms.
problem Defining a metric on the extended Siegel-Jacobi upper half space.
method Matrix embedding, pre-Iwasawa decomposition, invariant forms, sum of squares of forms.
result Invariant metric on the extended Siegel-Jacobi upper half space is derived.
We derive a mapping between MSE and CCC, revealing counterintuitive insights.
problem Missing mapping between mean square error and concordance correlation coefficient.
method Derive mathematical formula connecting MSE and CCC, analyze graphical implications.
result Formula uncovers counterintuitive insights and precise range for CCC given MSE.
Least Squares Estimators are suboptimal for 5D convex functions.
problem Suboptimality of Least Squares Estimators in estimating multidimensional convex functions.
method Analysis of natural subclasses of convex functions in random and fixed design settings.
result Risk of LSE is n−2/d while minimax risk is n−4/(d+4) for d≥5. The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric…
Proposes a variational NNCC formulation for infinite dimensions.
problem Optimization and gradient flows in infinite-dimensional settings.
method Variational formulation of NNCC on c-convex domains.
result Wasserstein spaces inherit NNCC from their base space.
We study a singular Hermitian metric of a vector bundle. First, we prove the sheaf of locally square integrable holomorphic sections of a vector bundle with a singular Hermitian metric, which is a higher rank analogy of a multiplier ideal sheaf, is coherent under some assumptions. Second, we prove a Nadel-Nakano type v…
We consider the conformal class of the Riemannian product g0+g, where g0 is the constant curvature metric on Sm and g is a metric of constant scalar curvature on some closed manifold. We show that the number of metrics of constant scalar curvature in the conformal class grows at least linearly with respect…
Improved estimator for least squares using random projections achieves smaller error.
problem Improving the accuracy of least squares solutions for large-scale problems.
method James-Stein estimator applied to Gaussian sketching of least squares problems.
result Upper and lower bounds match when SNR is small and data matrix is well-conditioned.