In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.
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The paper proves inequalities for hypersurfaces in weighted manifolds.
problem Willmore-type inequalities for closed hypersurfaces in weighted manifolds.
method Analyzes weighted manifolds with nonnegative Bakry-Émery Ricci curvature, proving sharp inequalities and characterizing equality cases.
result Derives sharp Willmore-type and Willmore-like inequalities in steady and shrinking gradient Ricci solitons.
New method detects overlapping communities in weighted graphs without pure nodes assumption.
problem Detect overlapping communities in weighted graphs without making pure nodes assumption.
method Convex optimization-based approach for weighted graphs.
result Success on artificial and real-world datasets.
Upper bounds for eigenvalues on submanifolds in weighted manifolds.
problem Eigenvalue bounds for submanifolds in weighted Riemannian manifolds.
method Proving upper bounds for divergence-type operators and Steklov problems on submanifolds.
result Reilly-type upper bounds for eigenvalues.
Paper proves new theorems about curvature in weighted manifolds.
problem Understanding curvature in weighted manifolds.
method Proved spectral comparison and splitting theorems for infinity-Bakry-Emery Ricci curvature.
result Results extend existing theorems and provide new supplements.
In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Lapl…
Proves Laplacian and Lichnerowicz Laplacian are sectorial in weighted Hölder spaces.
problem Analyzing sectoriality of Laplacian and Lichnerowicz Laplacian on asymptotically hyperbolic spaces.
method Proves sectoriality in weighted Hölder spaces using asymptotically hyperbolic metrics.
result Analytic semigroups apply, yielding well-posedness results for parabolic evolution equations.
Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylindersmath.DG
The study characterizes hypersurfaces in weighted cylinders and generalizes confinement properties.
problem Characterizing hypersurfaces in weighted Riemannian products.
method Analyzing parabolic hypersurfaces with boundary in weighted cylinders.
result Generalized confinement properties of hypersurfaces in weighted cylinders.
In study of eigenvalue problems, a classical problem is the Stekloff eigenvalue problem. There are many estimates of the first non- zero Stekloff eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997)". In this paper…
New findings show tool-augmented models can recall unlimited facts, outperforming purely memorized models.
problem Limitations of purely memorized models in recalling large amounts of factual information.
method Demonstrated the benefits of in-tool learning (external retrieval) over in-weight learning (memorization) for factual recall.
result Proved that tool-use enables unbounded factual recall via a simple and efficient circuit construction.
We study stability properties of -minimal hypersurfaces isometrically immersed in weighted manifolds with non-negative Bakry-Emery Ricci curvature under volume growth conditions. Moreover, exploiting a weighted version of a finiteness result and the adaptation to this setting of Li-Tam theory, we investigate the top…
The paper studies quaternionic Monge-Ampère equations in weighted energy classes.
problem Characterizing the finite energy range of quaternionic Monge-Ampère operator.
method Proving well-definedness and fine property of the operator in weighted energy classes.
result Explicit characterization of the finite energy range of quaternionic Monge-Ampère operator.
Unified framework for measuring concentration in weighted networks considering both weight distributions and network structure.
problem Traditional indices neglect the topology of relationships among network elements.
method Develops a family of topology-aware concentration indices that jointly account for weight distributions and network structure.
result The proposed indices preserve key properties and allow concentration to be evaluated across different dimensions of dependence.
Kähler information manifolds for signal filters in weighted Hardy spaces are explored.
problem Developing a geometric framework for signal processing filters in weighted Hardy spaces.
method Introducing weighted Hardy spaces and smooth transformations of transfer functions, demonstrating the Kähler manifold structure.
result The Riemannian geometry of weighted Hardy norms for transfer functions forms a Kähler manifold.
We study singular del Pezzo surfaces that are quasi-smooth and well-formed weighted hypersurfaces. We give an algorithm how to classify all of them.
The paper explores inequalities for strongly-convex sets in weighted Riemannian manifolds.
problem Investigating dilation type inequalities on weighted Riemannian manifolds.
method Introducing dilation profile and comparing it with model space under lower weighted Ricci curvature bounds.
result Showed several functional inequalities related to various entropies.
This paper presents a supervised learning algorithm, namely, the Synaptic Efficacy Function with Meta-neuron based learning algorithm (SEF-M) for a spiking neural network with a time-varying weight model. For a given pattern, SEF-M uses the learning algorithm derived from meta-neuron based learning algorithm to determi…
The Penrose theorem and Hawking's topology theorem are extended to weighted spacetimes.
problem Extending Penrose's singularity theorem and Hawking's topology theorem to weighted spacetimes.
method Using weighted null energy condition and synthetic dimension to generalize the theorems.
result Generalized versions of the Penrose and Hawking theorems hold under a weighted null energy condition.
Paper proves Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
problem Proving equivalence between Brunn-Minkowski inequality and curvature dimension condition.
method Analyzes weighted Riemannian manifolds, proving equivalence without optimal transport or differential structure.
result Brunn-Minkowski inequality and curvature dimension condition are equivalent in weighted Riemannian manifolds.
Paper explores exact recovery of communities in weighted graphs using Gaussian and exponential distributions.
problem Exact recovery of communities in weighted graphs with Gaussian and exponential distributions.
method Introduces a new semi-metric to describe conditions for exact recovery and analyzes conditions for both complete and incomplete graphs.
result Necessary and sufficient conditions for exact recovery are asymptotically tight and applicable to both complete and incomplete graphs.
New cyclicity measures defined in weighted Besov spaces, with stability and geometric analysis.
problem Characterizing cyclicity in weighted Besov spaces.
method Defining cyclicity indices based on potential theory and capacity, studying stability under perturbations, and linking zero set structure to cyclicity.
result Novel invariants and conditions for cyclicity in various function spaces.
We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev space.
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.
New criterion improves predictive evaluation in weighted inference scenarios.
problem Improving predictive evaluation in scenarios with different likelihoods for estimation and evaluation.
method Developed the posterior covariance information criterion (PCIC) to handle weighted likelihood inference.
result PCIC is asymptotically unbiased for quasi-Bayesian generalization error in weighted inference.
New method estimates Nishimori temperature for node classification in weighted graphs.
problem Estimating Nishimori temperature for Bayesian inference.
method Spectral method using eigenvalues of Bethe Hessian matrix.
result Spectral method outperforms existing approaches in node classification.
Cremona transformations of weighted projective planes, Zariski pairs, and rational cuspidal curvesmath.AG
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of curves in weighted projective planes with special properties. We explain how to co…
Weighted Classification Cascades for Optimizing Discovery Significance in the HiggsML Challengestat.ML
We introduce a minorization-maximization approach to optimizing common measures of discovery significance in high energy physics. The approach alternates between solving a weighted binary classification problem and updating class weights in a simple, closed-form manner. Moreover, an argument based on convex duality sho…
The paper proves geometric inequalities for hypersurfaces in weighted manifolds.
problem Geometric inequalities for hypersurfaces in weighted manifolds.
method Noncompact smooth metric measure spaces with nonnegative Bakry-Émery Ricci curvature.
result Sharp geometric inequalities for the boundary of open sets in weighted manifolds.
Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R^d. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges …
The focus in machine learning has branched beyond training classifiers on a single task to investigating how previously acquired knowledge in a source domain can be leveraged to facilitate learning in a related target domain, known as inductive transfer learning. Three active lines of research have independently explor…
The paper studies the Heath-Jarrow-Morton-Musiela equation of the bond market. The equation is analyzed in weighted spaces of functions defined on . Sufficient conditions for local and global existence are obtained . For equation with the linear diffusion term the conditions for global existence are close …
This paper evaluates heuristics and hyperparameters in weight-sharing NAS methods.
problem Improving the performance of weight-sharing NAS methods.
method Systematic evaluation of heuristics and hyperparameters in weight-sharing NAS algorithms.
result Some heuristics negatively impact super-net and stand-alone performance correlation.
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with b…
New method adapts neural networks without losing prior knowledge.
problem Understanding and enabling flexible adaptation of neural networks.
method Differential geometry framework, functionally invariant paths (FIP).
result Achieves comparable state-of-the-art performance on continual learning and sparsification tasks.
Lower weight Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4math.AT
In this paper, we investigate the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on R^4. In the case of formal Hamiltonian vector fields on R^2, we computed the relative Gel'fand-Kalinin-Fuks cohomology groups of weight <20 in the paper by Mikami-Nakae-Kodama. The main strategy…
New algorithm detects communities in weighted networks, improving on binary ones.
problem Few methods exist for detecting communities in weighted networks.
method Pseudo-likelihood approach for weighted stochastic block model.
result The method is consistent and works well for both homogeneous and heterogeneous networks.
In this paper we investigate codimension one Fano distributions on Fano manifolds with Picard number one. We classify Fano distributions of maximal index on complete intersections in weighted projective spaces, Fano contact manifolds, Grassmannians of lines and their linear sections, and describe their moduli spaces. A…
Deep reinforcement learning requires a heavy price in terms of sample efficiency and overparameterization in the neural networks used for function approximation. In this work, we use tensor factorization in order to learn more compact representation for reinforcement learning policies. We show empirically that in the l…
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for -harmonic functions in manifolds with curvature conditions.
result Established a quantitative second order Sobolev estimate for -harmonic functions.
New duality found linking neural network weights and activities for better generalization.
problem Understanding and improving neural network generalization.
method Activity-weight duality mapping between neural network layers.
result Generalization loss can be decomposed into geometric factors of sharpness and weight standard deviation.
Interpreting neural networks is a crucial and challenging task in machine learning. In this paper, we develop a novel framework for detecting statistical interactions captured by a feedforward multilayer neural network by directly interpreting its learned weights. Depending on the desired interactions, our method can a…
New method prevents deep learning forgetting past by remembering key examples.
problem Catastrophic forgetting in continual learning.
method Functional regularisation using Gaussian Process formulation.
result Achieves state-of-the-art performance on benchmarks.
MLDS dataset reveals hidden model behavior via weight-space analysis.
problem Neural networks' opacity makes them hard to evaluate.
method Presented MLDS dataset of trained neural networks.
result Weight-space analysis reveals meaningful divergence with small changes in training data.
Classifies surfaces of section for Seifert fibrations.
problem Classifying surfaces of section for Seifert fibrations.
method Discussing branched coverings and relating surfaces of section to algebraic curves.
result Relates surfaces of section to algebraic curves in weighted complex projective planes.
We derive the Simons' type equation for -minimal hypersurfaces in weighted Riemannian manifolds and apply it to obtain a pinching theorem for closed -minimal hypersurfaces immersed in the product manifold with . Also we classify closed -minimal h…
We find new examples of compact Spin(7)-manifolds using a construction of Joyce. The essential ingredient in Joyce's construction is a Calabi-Yau 4-orbifold with particular singularities admitting an antiholomorphic involution, which fixes the singularities. We search the class of well-formed quasismooth hypersurfaces …
The chiral de Rham complex of Malikov, Schechtman, and Vaintrob, is a sheaf of differential graded vertex algebras that exists on any smooth manifold , and contains the ordinary de Rham complex at weight zero. Given a closed 3-form on , we construct the twisted chiral de Rham differential , which coincid…
In "Isomonodromy aspects of the tt* equations of Cecotti and Vafa I. Stokes data" (arxiv:1209.2045) we described all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpret…