The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form …
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48 results for “UNIT”
Magnetic unit vector fieldsmath.DG
Characterizes magnetic unit vector fields on Lie groups.
problem Classifying magnetic unit vector fields on Lie groups.
method Characterization through critical points of Landau Hall and Dirichlet energy functionals.
result Classification of all magnetic left invariant unit vector fields on 3-dimensional Lie groups.
In this paper we propose and investigate a novel nonlinear unit, called unit, for deep neural networks. The proposed unit receives signals from several projections of a subset of units in the layer below and computes a normalized norm. We notice two interesting interpretations of the unit. First…
Randomly chosen primary hidden units and derived secondary units reduce neural network complexity.
problem Large number of hidden units in neural networks.
method Introducing primary and secondary hidden units with random weights for primary units and derived weights for secondary units.
result Significant reduction in the number of hidden units without compromising accuracy.
Study examines how business units can benefit from group cohesion under regulatory constraints.
problem Regulatory constraints limit business units' ability to form a single cohesive group.
method Defined and analyzed cohesive risk measures to minimize capital costs.
result Cohesive risk measures allow groups to achieve minimal capital costs without altering individual liabilities.
Study examines dependence properties of Bayesian neural network units in finite-width networks.
problem Understanding dependence properties of hidden units in practical finite-width Bayesian neural networks.
method Theoretical analysis and empirical evaluation of depth and width impacts.
result Hidden units in finite-width Bayesian neural networks are dependent, contrary to the infinite-width limit assumption.
Smooth groupoid algebras are H-unital, with implications for algebraic and homological properties.
problem Understanding the structure of convolution algebras on Lie groupoids.
method Analyzing smooth functions and invariant subsets to prove H-unitality.
result H-unitality of groupoid algebras and their quotients, leading to excision properties.
We study deep Bayesian neural networks with Gaussian priors, revealing heavy-tailed unit activations.
problem Characterizing regularization effects in deep Bayesian neural networks.
method Investigation of deep Bayesian neural networks with Gaussian weight priors and ReLU-like nonlinearities.
result The prior distribution on units becomes increasingly heavy-tailed with depth, influencing activation patterns.
WEST compresses word embeddings and softmax layers for memory efficiency.
problem Memory constraints in large vocabulary models.
method WEST encodes words with sequences of sub-units, improving compression without performance loss.
result WEST achieves significant compression without sacrificing performance.
Wasserstein t-SNEcs.LG
Wasserstein t-SNE embeds hierarchical datasets considering within-unit distributions.
problem Exploring hierarchical datasets where units are compared based on means of sample distributions.
method Uses Wasserstein distance metric for 2D embeddings of units, approximating Gaussian distributions for efficiency.
result Demonstrates effective embedding of hierarchical datasets, uncovering meaningful structure.
Study on hidden units in finite Bayesian neural networks and their tail properties.
problem Understanding the behavior of hidden units in finite Bayesian neural networks.
method Introduced a generalized Weibull-tail property to describe hidden units tails.
result Unit priors become heavier-tailed going deeper, providing insights into finite Bayesian neural networks.
New characterization of Calabi torus in unit sphere found.
problem Rigidity of closed minimally immersed Legendrian submanifolds in unit sphere.
method Maximum principle and Simons' type integral inequality.
result New characterization of Calabi torus in unit sphere.
Study on fractional curvature flow on unit sphere, extending previous work.
problem Fractional Nirenberg problem on unit sphere.
method Fractional conformal curvature flow on unit sphere.
result Perturbation result for fractional Nirenberg problem with .
We present a new equation with respect to a unit vector field on Riemannian manifold such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to some classes of unit vector fields. We introduce a class of covariantly normal unit vector fields and pro…
Bayesian units improve speech recognition with minimal parameters.
problem Improving speech recognition models with fewer parameters.
method Derived Bayesian recurrent units integrated into deep learning frameworks.
result Adding Bayesian units improves speech recognition performance.
Let Σbe a k-dimensional minimal surface in the unit ball B^n which meets the unit sphere orthogonally. We show that the area of Σis bounded from below by the volume of the unit ball in R^k. This answers a question posed by R. Schoen.
Minimal vector fields on oscillator groups studied, with specific conditions for minimality.
problem Characterizing minimal left-invariant unit vector fields on oscillator groups.
method Analyzing structure constants and harmonic maps into the unit tangent bundle.
result Minimal vector fields defined by specific conditions on oscillator groups.
New neural network units resist adversarial attacks effectively.
problem Adversarial attacks on neural networks that misclassify inputs.
method Introduced RBFI units with non-linear structure.
result RBFI units maintain high accuracy in adversarial attacks.
Neural Power Unitscs.LG
Neural Power Unit (NPU) learns arbitrary power functions on real numbers.
problem Neural Networks struggle with generalizing beyond seen data and arithmetic operations.
method Introduces Neural Power Unit (NPU) that operates on real numbers and learns arbitrary power functions.
result NPU outperforms competitors in accuracy and sparsity on arithmetic datasets and discovers governing equations from data.
Study calculates first -widths of unit disk.
problem Computing first -widths of the unit disk.
method Regularity result for integral 1-varifolds on compact 2-manifolds with convex boundary, applied to unit disk.
result Computed first -widths for .
ReLU units can 'die' in neural networks, causing slower convergence.
problem ReLU units sometimes produce near-zero outputs during training.
method Simulation and statistical analysis of a simplified ReLU unit model.
result Activation probability decreases as training progresses, leading to slower convergence.
In a seminal paper Abadie, Diamond, and Hainmueller [2010] (ADH), see also Abadie and Gardeazabal [2003], Abadie et al. [2014], develop the synthetic control procedure for estimating the effect of a treatment, in the presence of a single treated unit and a number of control units, with pre-treatment outcomes observed f…
Bayesian SHMM discovers acoustic units from unlabeled speech.
problem Discovering language-specific acoustic units from unlabeled speech.
method Bayesian Subspace Hidden Markov Model (SHMM) trained on labeled data to find new acoustic units on target language.
result Significantly outperforms previous HMM-based systems and compares favorably with Variational Auto Encoder-HMM.
We construct homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere with arbitrarily small k-dilation for each k greater than (m + 1)/2. We prove that homotopically non-trivial maps from the unit m-sphere to the unit (m-1)-sphere cannot have arbitrarily small k-dilation for k less than or equal …
What can you draw?math.MG
Can certain shapes be drawn with a pencil and eraser?
problem Characterizing which planar sets can be drawn with a pencil and eraser.
method Analyzes the properties of sets drawable with a pencil and eraser, using open and closed unit disks.
result Drawability cannot be characterized by local obstructions.
We present a probabilistic variant of the recently introduced maxout unit. The success of deep neural networks utilizing maxout can partly be attributed to favorable performance under dropout, when compared to rectified linear units. It however also depends on the fact that each maxout unit performs a pooling operation…
We study unit horizontal bundles associated with Riemannian submersions. First we investigate metric properties of an arbitrary unit horizontal bundle equipped with a Riemannian metric of the Cheeger-Gromoll type. Next we examine it from the Gromov-Hausdorff convergence theory point of view, and we state a collapse the…
Foreign exchange markets show that currency units (= accounting or nominal price units) are variables. Technical and economic progress evidence that the consumer baskets (= purchasing power units or real price units) are also variables. In contrast, all physical measurement units are constants and either defined in the…
Bayesian SHMM models speech units from unannotated speech.
problem Discover latent speech units from unannotated data.
method Bayesian Subspace Hidden Markov Model (SHMM).
result System performs well on synthesis quality, lower bitrate.
New equations connect unit Killing vectors to initial data.
problem Characterizing initial data for Einstein vacuum with unit Killing vectors.
method Developed new equations (uKID) by eliminating scaling and using propagation identity.
result Found equations that are finite type and characterize unit normalized Killing vectors.
New minimal surfaces found in ball with boundary constraints.
problem Finding minimal surfaces with boundary conditions.
method Equivariant differential geometry approach.
result A family of free boundary minimal surfaces in the unit ball.
Researchers determine the Thurston unit ball for a family of -chained links and find conditions for fibered faces.
problem Determining the Thurston unit ball and conditions for fibered faces in a family of -chained links.
method Analyzing the family of -chained links , proving the Thurston unit ball is an -dimensional cocube for , and finding conditions for fibered faces.
result The Thurston unit ball for is an -dimensional cocube for and provides at least one fibered face for any .
Synthetic control method improves policy evaluation in high-dimensional settings.
problem Evaluating the impact of new policies in large-scale applications.
method Two-phase approach: nearest neighbor matching followed by supervised learning.
result The method successfully improves estimate accuracy in large-scale experiments.
The paper addresses selection bias in conformal prediction for focal units.
problem Selection bias in marginally valid conformal prediction intervals for focal units.
method A general framework for constructing selection-conditional coverage prediction sets.
result Efficient methods for various selection rules with exact finite-sample coverage.
Study shows how certain foliations in unit tangent bundles behave.
problem Characterizing behavior of foliations in unit tangent bundles.
method Analyzing intersections and properties of foliations.
result Certain partially hyperbolic diffeomorphisms are collapsed Anosov flows.
A method for making machine learning units-equivariant using dimensional analysis.
problem Ensuring machine learning models respect dimensional consistency.
method Constructing dimensionless inputs and applying equivariant machine learning methods.
result Improved accuracy in tasks requiring dimensional consistency.
The paper finds the Finsler structure of Apollonian weak metric on unit disc.
problem Understanding the Finsler structure of Apollonian weak metric on the unit disc.
method Analyzing the deformation of hyperbolic Poincaré metric by a closed 1-form.
result The Apollonian weak-Finsler structure has bounded below -curvature and flag curvature satisfying .
A remarkable characteristic of overparameterized deep neural networks (DNNs) is that their accuracy does not degrade when the network's width is increased. Recent evidence suggests that developing compressible representations is key for adjusting the complexity of large networks to the learning task at hand. However, t…
Study inverse curvature flows for capillary hypersurfaces in a unit ball.
problem Understanding the behavior of capillary hypersurfaces under inverse curvature flows.
method Investigate inverse curvature flows for strictly convex, capillary hypersurfaces in the unit Euclidean ball.
result Establish existence and convergence results for inverse curvature flows.
Full description of totally geodesic unit vector fields on 2-dimensional Riemannian manifoldsmath.DG
We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local imbedding of the manifold into its unit tangent bundle with the Sasaki metric.
Unsupervised methods detect faults in industrial systems with limited data.
problem Challenges in training data-driven fault detection models with scarce or unavailable data.
method Proposes five approaches: baseline, incremental learning, fleet comparison, and UFAN.
result UFAN outperforms other methods in detecting faults in dissimilar units.
We establish upper bounds for the minimal number of hidden units for which a binary stochastic feedforward network with sigmoid activation probabilities and a single hidden layer is a universal approximator of Markov kernels. We show that each possible probabilistic assignment of the states of output units, given t…
Study constructs disks with curved boundaries in a 3D ball.
problem Constructing non-planar free boundary disks in a unit ball.
method Infinite family of non-planar disks with non-positive Gaussian curvature.
result Constructs disks with curved boundaries in a unit ball.
New eigenvalue problem for free boundary minimal surfaces in a ball.
problem Eigenvalue problem for free boundary minimal surfaces in a unit ball.
method New eigenvalue problem associated with the Jacobi operator.
result Eigenfunctions of the unit normal are associated with eigenvalue -2.
Study -dim hypersurfaces with constant mean curvature in unit spheres.
problem Characterize complete hypersurfaces with constant mean curvature in spheres.
method Analyze scalar curvature and provide a new proof.
result Give a lower bound of scalar curvature.
We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…
Characterizes special curves on surface tangent bundles.
problem Understanding curves on surface tangent bundles.
method Characterization of Legendre and slant curves.
result Characterizations for N-Legendre and N-slant curves.
The unit ball is characterized by a Kähler-Einstein potential.
problem Characterizing the unit ball in complex geometry.
method Using a global potential function of the Kähler-Einstein metric.
result A compact Kähler manifold with an ample canonical bundle is the unit ball if it has a specific potential function.