For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…
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48 results for “floating body”
Study floating bodies in various space forms using a new approach.
problem Investigate floating bodies in different space forms.
method Develop a unified approach to study floating bodies in Euclidean space, Euclidean unit sphere, and hyperbolic space.
result Establish a relation between the derivative of the volume of the floating body and the floating area.
Study floating bodies of polytopes, linking volume to flags.
problem Understanding floating bodies of polytopes in various spaces.
method Introducing flag simplices to connect metric and combinatorial structures.
result Weighted volume depends on complete flags of polytopes.
New methods prove floating areas on spheres and hyperbolic spaces.
problem Existence of floating areas on curved spaces.
method Weighted floating bodies and polytopal approximation.
result New asymptotic approximation results on curved spaces.
Study spherical convex bodies using -floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced -floating areas and curvature entropy for spherical convex bodies.
result Established isoperimetric inequalities and dual isoperimetric inequalities.
The floating body approach to affine surface area is adapted to a holomorphic context providing an alternate approach to Fefferman's invariant hypersurface measure.
The paper proves a conjecture about the shape of floating bodies.
problem The shape of bodies of flotation and buoyancy.
method Modern differential geometry techniques.
result If a body of flotation is homothetic to a body of buoyancy, it must be an ellipse.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing relative surface areas, proving invariance and inequalities, and using geometric interpretations.
result Established inequalities and a new notion of entropy for ball-convex bodies.
Study on curvature measures in non-Euclidean spaces linked to Euclidean geometry.
problem Investigating curvature measures in spherical, hyperbolic, and de Sitter spaces.
method Establishing a unifying framework for curvature measures in real-analytic spaces of constant curvature.
result Floating bodies and duality in non-Euclidean spaces are connected to curvature measures in Euclidean space.
Introduces new weighted floating functions and affine surface areas.
problem Developing new mathematical concepts for convex bodies.
method Introducing weighted floating functions and weighted functional affine surface areas.
result New relations to traditional and classical affine surface areas.
The paper generalizes the second Pappus-Guldin theorem for calculating volumes of bodies.
problem Calculating the volume of a body cut into perpendicular slices.
method Using a generalized formula and properties of centroids and floating bodies.
result A curve with centroid property exists for convex bodies, leading to simpler volume calculations.
New illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.
problem Characterizing properties of ball-convex shapes.
method Introducing illumination bodies and weighted illumination bodies, proving convexity, and establishing surface area measures.
result Illumination bodies are convex and provide surface area measures for ball-convex shapes.
Method trains sparse neural networks without sacrificing accuracy.
problem Training sparse neural networks limits model size.
method Updates sparse network topology during training.
result Requires fewer FLOPs to achieve accuracy.
Hamiltonian method applied to floating barrier options pricing.
problem Pricing of floating barrier options.
method Hamiltonian approach in quantum mechanics applied to barrier options.
result Analytical expressions for pricing kernel and option price derived.
Floating Gossip improves continuous machine learning in a decentralized manner.
problem Continuous learning in infrastructure-less environments.
method Mean field approach to analyze the impact of communication and computing in Floating Gossip.
result Floating Gossip can effectively incorporate large amounts of data into machine learning models.
Hybrid BFP-FP improves DNN training accuracy with 8.5x higher throughput.
problem Limited dynamic range of fixed-point arithmetic for DNN training convergence.
method Introducing HBFP, a hybrid BFP-FP approach.
result HBFP matches floating point's accuracy while delivering up to 8.5x higher throughput.
We establish a connection between capillary floating in neutral equilibrium and the billiard ball problem. This allows us to reduce the question of floating in neutral equilibrium at any orientation with a prescribed contact angle for infinite homogeneous cylinders to a question about billiard caustics for their orthog…
No floating point, no multiplications, no problem! Training efficient networks for resource-constrained devices.
problem Designing efficient neural networks for resource-constrained devices without floating-point operations.
method Discretizing both in-network non-linearities and network weights to avoid floating-point and multiplication operations.
result Training networks without floating-point operations can achieve comparable performance to those using floating-point operations, with less memory usage.
Paper examines floating exercise boundaries for American options in time-inhomogeneous models.
problem Floating exercise boundaries in time-inhomogeneous models with negative interest rates or yields.
method Semi-analytical approach for pricing American options.
result Specialized pricing methodologies are required for models with floating exercise boundaries.
Researchers find floating point errors can mislead neural network verifiers.
problem Floating point arithmetic inaccuracies mislead neural network verifiers.
method Efficiently searches inputs and constructs neural network architectures to exploit verification errors.
result Floating point errors can systematically mislead neural network verifiers.
Floating geodesic planes in Hitchin manifolds have fractal closures with non-integer dimensions.
problem Rigidity of geodesic planes in Hitchin manifolds.
method Constructing a specific surface group and analyzing its action on the Hitchin manifold.
result Existence of floating geodesic planes in Hitchin manifolds with fractal closures.
Unique floating and buoyancy surfaces identify convex polytopes.
problem Identifying convex polytopes from their flotation and buoyancy surfaces.
method Proving uniqueness of surfaces for polytopes with uniform or prescribed density.
result Floating and buoyancy surfaces uniquely determine convex polytopes.
We compare the results of our earlier paper on the floating in neutral equilibrium at arbitrary orientation in the sense of Finn-Young with the literature on its counterpart in the sense of Archimedes. We add a few remarks of personal and social-historical character.
AdaptivFloat improves deep learning inference accuracy at low precision.
problem Low precision quantization issues in deep learning inference.
method Dynamic floating-point representation with adaptive clipping.
result Consistently higher inference accuracy at low precision compared to other methods.
Training deep neural networks with 8-bit floating point numbers is now possible and more efficient.
problem Challenges in training DNNs with reduced precision, especially for gradient computations.
method Introduction of chunk-based accumulation and floating point stochastic rounding to reduce arithmetic precision to 16 bits.
result Successful training of DNNs using 8-bit floating point numbers, maintaining accuracy on various models and datasets.
Constructs minimal immersions with singularities.
problem Minimal immersions with singularities in metric spaces.
method Constructs minimal immersions with catenoidal necks or floating disks converging to a singular point.
result Constructs minimal immersions with singularities.
Paper explores low-precision arithmetic for neural networks, improving efficiency.
problem Training neural networks with high precision and energy efficiency.
method 12-bit fixed-point, 12-bit floating-point, local scaling, Power-of-Two arithmetic.
result 7-bit Power-of-Two arithmetic achieves minimal loss in accuracy with reduced computation.
Paper proposes training deep neural networks with 8-bit floating point precision.
problem Challenges in training deep neural networks at 8-bit precision due to higher precision and dynamic range requirements.
method Proposes a method to train deep neural networks using 8-bit floating point for weights, activations, errors, and gradients. Introduces an enhanced loss scaling method and stochastic rounding technique.
result Demonstrates state-of-the-art accuracy across multiple datasets and workloads compared to full precision baseline.
Binarized CNNs improve GPU inference efficiency on resource-constrained devices.
problem Efficient inference on resource-constrained devices for image classification.
method Binarization of weights and computations in CNNs, implemented on GPUs.
result 7.4X speedup with 4.4% accuracy loss on embedded GPU platforms.
IntSGD compresses SGD gradients without floats, converging as SGD.
problem Efficiently compressing stochastic gradients in distributed SGD.
method Adaptive integer compression of gradients, estimating scaling adaptively.
result IntSGD matches SGD's iteration complexity for convex and non-convex functions.
Unveiling phase transitions with machine learningcond-mat.str-el
Machine learning identifies phase transitions in condensed matter physics.
problem Classifying phase transitions in condensed matter physics.
method Unsupervised and supervised machine learning techniques applied to the Ising model.
result Machine learning can detect multiple phases and regions within the paramagnetic phase.
Hamel's Equations and Geometric Mechanics of Constrained and Floating Multibody and Space Systemsmath.DS
Geometric mechanics approach to constrained and floating multibody systems using Hamel's equations.
problem Analytical mechanics of constrained and floating multibody systems.
method Geometric approach using bundle structures and connections, with Hamel's equations as a universal non-holonomic formulation.
result Achieved intrinsic splitting and inertial decoupling of reduced Euler-Lagrange equations.
The paper studies self-Bäcklund curves in centroaffine geometry using elliptic functions.
problem Understanding self-Bäcklund curves in centroaffine geometry.
method Description of general properties and detailed analysis using elliptic functions.
result Provides a detailed description of self-Bäcklund centroaffine curves in terms of elliptic functions.
Paper proposes a deep learning strategy for efficient FC management.
problem Inaccurate and conservative FC dimensioning approaches in realistic settings.
method Uses a Convolutional Neural Network (CNN) to dynamically modulate FC resources.
result Achieves a maximum rejection rate of 3% and 37.5% resource savings.
Proposes standards for evaluating online machine learning methods in evolving data streams.
problem Difficulty in evaluating online machine learning methods under realistic conditions.
method Proposes comprehensive evaluation standards, performance measures, and evaluation strategies.
result Provides a new Python framework (float) for modular integration of libraries and custom code.
Custom narrow-precision representations boost DNN inference speed by 7.6x with minimal accuracy loss.
problem Improving computational efficiency of deep neural networks.
method Exploring and utilizing unconventional narrow-precision floating-point representations for DNN weights and activations.
result Average speedup of 7.6x with less than 1% accuracy loss.
Posits improve DNN training efficiency on edge devices.
problem Training deep neural networks with low-precision formats.
method Used posits (5-8 bit) for DNN training compared to floating point.
result 16-bit posits outperform 16-bit floating point for end-to-end training.
StatQAT optimizes quantization for deep networks, reducing computational cost and memory usage.
problem Optimal quantization parameters selection for deep neural networks with diverse data distributions.
method Statistical error analysis framework for uniform and floating-point quantization, iterative and analytic quantizers designed for arbitrary and Gaussian-like distributions.
result Improved accuracy and stability in training low-precision neural networks.
Cheetah framework optimizes DNNs for edge devices using low-precision formats.
problem Reducing DNN model size for edge devices while maintaining accuracy.
method Mixed low-precision hardware and software co-design framework using posit and other formats.
result 16-bit posits outperform 16-bit floating point in training, and [5..8]-bit posits improve inference performance.
Quantization scheme improves inference efficiency for deep learning models.
problem Limited computational resources and energy constraints in deploying deep learning models.
method Proposes a quantization scheme to reduce precision and improve efficiency without sacrificing accuracy.
result End-to-end post-quantization accuracies comparable to reference model achieved using a single inference batch calibration.
We present a numerical approach for solving the free boundary problem for the Black-Scholes equation for pricing American style of floating strike Asian options. A fixed domain transformation of the free boundary problem into a parabolic equation defined on a fixed spatial domain is performed. As a result a nonlinear t…
Flexpoint improves deep learning training efficiency by using adaptive 16-bit format.
problem Training deep neural networks in low bit-width formats is challenging.
method Flexpoint uses a shared exponent dynamically adjusted to minimize overflows and maximize dynamic range.
result 16-bit Flexpoint tensors closely match 32-bit floating point in training deep networks without tuning.
BFLOAT16 achieves SOTA results in deep learning training without hyper-parameter tuning.
problem Ensuring deep learning training achieves SOTA results without hyper-parameter tuning.
method Implemented a method to emulate BFLOAT16 operations in various frameworks.
result BFLOAT16 achieves SOTA results in deep learning training without hyper-parameter tuning.
The study analyzes convergence of adaptive optimizers under low-precision training.
problem Understanding why low-precision training remains effective for large models.
method Developed a theoretical framework for analyzing convergence of adaptive optimizers under floating-point quantization.
result Adaptive optimizers retain convergence rates close to full-precision methods under logarithmic mantissa scaling.
G-Net constructs binary neural networks with high accuracy using randomized binary embeddings.
problem Creating high-accuracy binary neural networks with theoretical guarantees.
method Proposes a novel floating-point G-Net family with randomized binary embeddings and theoretical accuracy guarantees.
result Empirically, G-Net achieves almost 30% higher accuracy on CIFAR-10 compared to prior HDC models.
A new approach uses deep learning to manage vehicular content efficiently.
problem Managing content replication and caching in vehicular networks efficiently.
method Data-driven, centralized approach using a Convolutional Neural Network (CNN).
result Effective strategies derived to modulate FC operation in space and adapt to mobility changes.
The Orlicz-Petty bodiesmath.MG
The paper studies properties of Orlicz-Petty bodies and related geometric areas.
problem Properties of Orlicz-Petty bodies and related geometric areas.
method Established properties through the existence and uniform boundedness of Orlicz-Petty bodies.
result Geominimal surface areas are continuous under certain conditions on convex bodies.
New material groupoid theory subdivides non-uniform bodies into smoothly uniform parts and isolated points.
problem Lack of differentiability in material bodies leads to non-uniformity.
method Introducing material groupoid and material distribution to study non-uniform bodies rigorously.
result Material bodies can be subdivided into smoothly uniform parts and isolated points.