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.
Proves inequalities for convex hypersurfaces in Euclidean space.
problem Geometric inequalities for hypersurfaces in Euclidean space.
method Proves two weighted geometric inequalities involving area, volume, and mean curvature.
result Examples of convex surfaces with ratios less than the round sphere.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
Asymptotic results for weighted floating bodies are established and used to obtain new proofs for the existence of floating areas on the sphere and in hyperbolic space and to establish the existence of floating areas in Hilbert geometries. Results on weighted best and random approximation and the new approach to floati…
Enhances linear regression with Kalman filter for loss minimization.
problem Minimizing loss in linear regression models.
method Integrates Kalman filter and SGD for optimal weight updates.
result Develops optimal linear regression equation with minimum area under curve.
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 Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
In this paper, we introduce a definition of λ-hypersurfaces of weighted volume-preserving mean curvature flow in Euclidean space. We prove that λ-hypersurfaces are critical points of the weighted area functional for the weighted volume-preserving variations. Furthermore, we classify complete λ-hypersurfaces with …
Let M be a weighted manifold with boundary ∂M, i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in $\p…
We use a new approach that we call unification to prove that standard weighted double bubbles in n-dimensional Euclidean space minimize immiscible fluid surface energy, that is, surface area weighted by constants. The result is new for weighted area, and also gives the simplest known proof to date of the (unit weight…
Extends illumination bodies to non-Euclidean spaces and proves their volume derivative defines surface area.
problem Defining surface area in non-Euclidean geometries.
method Generalizes illumination bodies to Riemannian spaces of constant curvature and projective Finsler geometries, proving their volume derivative defines surface area.
result Derivative of volume of illumination bodies defines surface area in non-Euclidean geometries.
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…
Hyperplanes, hyperspheres and hypercylinders in Rn with suitable densities are proved to be weighted minimizing by a calibration argument. Also calibration method is used to prove a weighted minimal hypersurface is weighted area-minimizing locally.
Since n-dimensional λ-hypersurfaces in the Euclidean space Rn+1 are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of complete λ-hypersurfaces. We give a gap theorem of complete λ-hypersurfaces with po…
In this paper we study the functional $\SW_{λ_1,λ_2}$, which is the the sum of the Willmore energy, λ1-weighted surface area, and λ2-weighted volume, for surfaces immersed in R3. This coincides with the Helfrich functional with zero `spontaneous curvature'. Our main result is a complete classification of all …
Study shows how certain metrics can be split into warped products.
problem Understanding conditions under which metrics can be split as warped products.
method Investigating warped area-minimizing hypersurfaces and spectral Ricci/scalar curvature bounds.
result Metrics can be locally split as warped products under specific curvature conditions.
Study proves no minimal surfaces can be contained in certain half-spaces or cones.
problem Prohibiting minimal surfaces from certain geometric configurations.
method Analyzes weighted minimal surfaces in R3 with height-dependent weights. result No proper surfaces can be contained in specific half-spaces or cones.
The paper extends rigidity results for λ-self-expanders to hyperplanes, spheres, and cylinders.
problem Characterizing λ-self-expanders as hyperplanes, spheres, and cylinders. method Extending results on self-expanders to λ-self-expanders, proving rigidity results. result Characterizes hyperplanes, spheres, and cylinders as λ-self-expanders. Paper finds relations between Willmore-type energies, weighted areas, and vertical potential energies for cylindrical critical points.
problem Tackles relations between three types of energy functions for cylindrical critical points.
method Uses differential equations and critical point analysis for Willmore-type energies and weighted areas.
result Generating curves coincide for Willmore-type energies and weighted areas, and similar results hold for Willmore-type energies and vertical potential energies.
Adapts AUM to identify ambiguous tasks in crowdsourced learning, improving generalization.
problem Discerning ambiguous tasks in crowdsourced labels to prevent mislabeling.
method Introduces Weighted Areas Under the Margin (WAUM) to average AUMs weighted by task-specific scores.
result Improves generalization performance by discarding ambiguous tasks.
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.
Paper proves Harnack inequality for f-mean curvature flow.
problem Proving Harnack inequality for f-mean curvature flow. method Gradient flow of the weighed area functional with measure density function e−f. result Proves Li-Yau-Hamilton type Harnack estimate.
One reflection suffices for orthogonal weights, reducing GPU usage.
problem Efficiently computing orthogonal weight matrices without high GPU utilization.
method Use an auxiliary neural network to compute one reflection instead of many.
result One reflection is sufficient for orthogonal weights, improving GPU utilization.
We study stable smooth solutions to the isoperimetric type problem for a Gaussian weight on Euclidean Space. That is, we study hypersurfaces Σn⊂Rn+1 that are second order stable critical points of compact variations that minimize Gaussian weighted area and preserve Gaussian weighted volume. We sho…
The paper proves weighted monotonicity theorems in various spaces and applies them to minimal surfaces.
problem Proving weighted monotonicity theorems in different spaces.
method Proving weighted monotonicity theorems for functions proportional to the metric tensor in Riemannian manifolds.
result Weighted monotonicity theorems in hyperbolic space imply unweighted theorems, leading to bounds on minimal surface areas.
Study on Lp affine surface areas and their inequalities for convex bodies.
problem Understanding weighted Lp affine surface areas in convex bodies. method Investigating valuations, isoperimetric inequalities, and connections to f divergences. result Established isoperimetric inequalities for weighted Lp affine surface areas. The paper proves Calabi-Bernstein type results for minimal and maximal surfaces in 3D and 3D-L spacetime.
problem Characterizing minimal and maximal surfaces in 3D and 3D-L spacetime.
method Analyzing surfaces with specific properties and using geometric and functional methods.
result Calabi-Bernstein type results for critical points of a weighted area functional in R3 and L3. The paper proves smoothness of almost-minimizers' boundaries near the free boundary.
problem Minimizing degenerate area functionals with weighted boundary conditions.
method Epsilon-regularity theorem applied to almost-minimizers.
result Almost-minimizers' boundaries are C1,γ0-smooth, orthogonal to the boundary Ω. Proves a similar inequality to a conjecture about hyperbolic space hypersurfaces.
problem Proving a conjecture about weighted Alexandrov-Fenchel inequalities for hyperbolic space hypersurfaces.
method Analyzes horospherically convex hypersurfaces in hyperbolic space.
result Proves a similar inequality to the conjectured one, provides a counterexample when applicable.
Simple proof for graph curvature in Gauss space leads to new theorems.
problem Proving curvature bounds for graphs in Gauss space.
method Simple proof using weighted area estimates.
result Bernstein type theorems for self-shrinkers and graphic hypersurfaces.
Paper proposes a method for predicting future passenger flow in urban transportation.
problem Predicting future passenger flow in urban transportation development.
method Multi-view localized correlation learning method with adaptive-weight.
result Our method achieves excellent performance compared with other baselines.
Paper proves a sharp weighted Isoperimetric inequality for substatic manifolds.
problem Proving geometric results for substatic Riemannian manifolds.
method Comparison theory based on a newly discovered conformal connection.
result Sharp, weighted Isoperimetric inequality quantifying boundary minimization.
The study connects minimal and maximal surfaces in 3D and 3-L space.
problem Describing correspondences between minimal and maximal surfaces in different spaces.
method Weierstrass representation and asymptotic analysis.
result Established criteria for singularity types and moduli spaces.
Stable capillary surfaces in weighted balls are disks.
problem Finding the shape of isoperimetric regions in weighted balls.
method Stability analysis and Hsiang symmetrization.
result Interior boundaries of isoperimetric regions in weighted balls are disks.
LUTNet optimizes FPGA neural network accelerators by leveraging LUTs for inference, achieving significant area savings.
problem Redundancy in deep neural networks and inefficient use of FPGA resources.
method End-to-end hardware-software framework using LUTs to implement any K-input Boolean operation for inference.
result Significant area savings and comparable accuracy compared to state-of-the-art binarized neural networks.
Predicting neural network accuracy from weights without testing.
problem Predicting neural network performance based on weights alone.
method Used simple statistics of weights to rank neural networks' performance.
result Simple predictors can rank networks' performance with high accuracy (R2 score > 0.98).
Bit-slice sparsity improves ReRAM-based DNN acceleration.
problem Limited ADC power and area constraints in ReRAM-based DNN accelerators.
method Proposed bit-slice L1 algorithm to induce sparsity during training.
result 2x sparsity improvement compared to previous methods.
New pseudometrics defined on knot spaces based on curve thickness and length.
problem Rigidity and non-degeneracy of knot spaces under isotopies.
method Swept-area pseudometrics on ropelength-filtered knot spaces.
result Proved non-degeneracy on polygonal strata and exact distance formulas.
A new model Weighted-SVD improves recommendation accuracy by adjusting latent factor weights.
problem Current Matrix Factorization models assume equal weights for all latent factors, which may not be accurate.
method Integrates linear regression with SVD to allow different weights for latent factors.
result The Weighted-SVD model outperforms other models in RMSE metrics on multiple datasets.
New model for detecting communities in weighted bipartite networks.
problem Lack of models for weighted bipartite networks.
method Introducing Bipartite Distribution-Free model and its extension.
result Spectral algorithms for consistent estimation of node labels.
We prove that on a Riemannian manifold, a smooth differential form has a primitive with a given (functional) upper bound provided the necessary weighted isoperimetric inequalities implied by Stokes are satisfied. We apply this to prove a comparison predicted by Gromov between the cofilling function and the filling area…
Random projection improves deep learning performance on high-dimensional data.
problem Training deep neural networks on high-dimensional data is infeasible.
method Prepending the network with an input layer initialized with random projection matrices.
result Neural networks with RP layers achieve competitive or improved performance on high-dimensional datasets.
Paper proposes a training framework for deploying DNNs on analog NVM crossbars, achieving significant efficiency gains.
problem Challenges in deploying deep learning models on microcontrollers due to limited compute and memory resources.
method Developed a training algorithm to eliminate tuning and propose unipolar-weighted matrices to reduce crossbar area and simplify hardware.
result Achieved up to 92.91% accuracy with 2-bit crossbars and up to 45% energy reduction.
SoftAdapt dynamically adjusts loss weights for multi-part functions.
problem Slow convergence and poor weight selection for multi-part loss functions.
method SoftAdapt dynamically changes weights based on live performance statistics.
result Improved convergence and better weight selection for multi-part loss functions.
We prove a generalization of Hsiung-Minkowski formulas for closed submanifolds in semi-Riemannian manifolds with constant curvature. As a corollary, we obtain volume and area upper bounds for k-convex hypersurfaces in terms of a weighted total k-th mean curvature of the hypersurface. We also obtain some Alexandrov-type…
Researchers find continuous solutions to minimizers in weighted least gradient problems.
problem Existence and regularity of minimizers to weighted least gradient problems.
method Constructing continuous solutions using Sternberg-Williams-Ziemer technique extended to inhomogeneous variations.
result Continuous solutions constructed for minimizers in any dimension n≥2, with level sets being minimal surfaces in a conformal metric.
Proposes a new method to initialize neural networks by estimating global curvature of weights.
problem Improving the initialization of neural networks for better training and convergence.
method Estimates the global curvature of weights across layers using the Hessian matrix norm.
result The proposed method helps in more rigorously initializing weights, leading to better performance.
The paper develops a spectral theory for hypergraphs with edge-dependent vertex weights using random walks.
problem Lack of spectral theory for hypergraphs with edge-dependent vertex weights.
method Random walks on hypergraphs with edge-dependent vertex weights, deriving a random walk-based hypergraph Laplacian.
result Random walks on hypergraphs with edge-dependent vertex weights can capture higher-order relationships in data.
OverQ increases model accuracy by handling outliers in neural networks with minimal hardware changes.
problem Handling outliers in neural network weights and activations for low-precision quantization.
method Overwrite quantization (OverQ) that opportunistically increases bitwidth for activation outliers.
result OverQ can handle over 90% of outliers and achieve +5% ImageNet Top-1 accuracy on a quantized ResNet-50 at 4 bits.