Graphs with stronger curvature grow faster.
problem Understanding volume growth on graphs with various curvatures.
method Examined inner-outer and Ricci-Ollivier curvatures to relate them to volume growth.
result Graphs with stronger inner-outer curvature growth have faster volume growth.
Paper proposes a structured approach to improve active learning performance.
problem Improving active learning performance in non-linear feature spaces.
method Approximates version space to a structured hypersphere, divides AL sampling into outer and inner volumes, and uses sequential optimization to globally optimize the kernel space.
result Proves guarantees for AL performance in version space and proposes a new algorithm called Volume-based AL (VAL).
Harish-Chandra's volume formula shows that the volume of a flag manifold G/T, where the measure is induced by an invariant inner product on the Lie algebra of G, is determined up to a scalar by the algebraic properties of G. This article explains how to deduce Harish-Chandra's formula from Weyl's law by utilizing…
In a noncommutative torus, effect of perturbation by inner derivation on the associated quantum stochastic process and geometric parameters like volume and scalar curvature have been studied. Cohomological calculations show that the above perturbation produces new spectral triples. Also for the Weyl C^*-algebra, the La…
Convex hypersurfaces evolve to spheres under a specific flow.
problem Volume preserving nonhomogeneous mean curvature flow of convex hypersurfaces.
method Monotonicity of isoperimetric ratio, inner and outer radius control, maximum principle arguments.
result Closed convex hypersurfaces converge to round spheres.
Study on renormalized volume of minimal submanifolds in Poincare-Einstein manifolds.
problem Analyzing minimal submanifolds in Poincare-Einstein manifolds.
method Deriving formulae for first and second variations of renormalized volume, proving asymptotic descriptions, and deriving inner-product relationships.
result Existence of asymptotic description and L2-inner-product relationship for specific cases. Study of non-homogeneous mean curvature flow in hyperbolic space converging to a geodesic sphere.
problem Volume/area preserving curvature flow of convex hypersurfaces in hyperbolic space.
method Generic positive, increasing mean curvature velocity; preserving convexity by horospheres; maximum principle; exponential convergence analysis.
result Exponential convergence to a geodesic sphere.
A new FV-ADI method calibrates SLV models efficiently.
problem Calibrating SLV models to their underlying local volatility models.
method Finite volume - Alternating Direction Implicit (ADI) approach for solving 1D and 2D forward Kolmogorov equations.
result The proposed method efficiently calibrates SLV models without requiring PDE transformations and conserves numerical mass.
New mathematical surfaces without boundaries found.
problem Existence of nonlocal free boundary minimal surfaces.
method Fractional perimeter critical points with invariant boundary.
result Existence of nonlocal free boundary minimal surfaces without boundaries.
Revises Gauss's Lemma using metrical distortion and differential slip.
problem Revising Gauss's Lemma in Riemannian geometry.
method Defining metrical distortion and differential slip, showing their geometric implications.
result Geodesically radial volume and length preservation properties.
New method avoids surface self-collision in geometric optimization.
problem Avoiding self-collision in surface optimization.
method Developed a numerical framework using tangent-point energy and fractional Sobolev inner product.
result Successfully accelerated collision avoidance scheme for triangle meshes.
Novel analysis of neural networks using geometric algebra and convex optimization.
problem Understanding the inner workings of deep neural networks.
method Geometric (Clifford) algebra and convex optimization.
result Optimal weights are given by the wedge product of training samples.
For any manifold M, the direct sum TM \oplus T*M carries a natural inner product given by the pairing of vectors and covectors. Differential forms on M may be viewed as spinors for the corresponding Clifford bundle, and in particular there is a notion of \emph{pure spinor}. In this paper, we study pure spinors and Dira…
Study higher rank inner products and their tilings to describe tori degenerations.
problem Understanding metric degenerations of tori.
method Introduce higher rank inner products and their tilings, use to describe degenerations.
result Describe metric degenerations of polarized tori and Hausdorff limits of tilings.
Characterizes quandles with abelian inner automorphisms.
problem Understanding quandles with specific automorphism properties.
method Generalizes previous work to construct new quandles.
result Homogeneous quandles with abelian inner automorphisms are abelian extensions of trivial quandles.
New spectral functionals for Dirac operators with inner fluctuations computed.
problem Spectral functionals and Dirac operators with inner fluctuations.
method Extension of spectral functionals for Dirac operators with inner fluctuations.
result Computed spectral Einstein functional for Dirac operator with inner fluctuations on even-dimensional spin manifolds.
A new algorithm tackles bilevel optimization with multiple inner minima.
problem Challenges in bilevel optimization with multiple inner minima.
method Reformulated as constrained optimization, solved via primal-dual bilevel optimization (PDBO) algorithm.
result First non-asymptotic convergence guarantee for bilevel optimization with multiple inner minima.
Deep learning detects glaucoma from raw OCT volumes, outperforming classical methods.
problem Glaucoma diagnosis from OCT measurements using traditional features.
method 3D Convolutional Neural Network (CNN) for unsegmented OCT volumes.
result Deep learning achieved higher accuracy (AUC 0.94) compared to logistic regression (AUC 0.89).
This paper constructs quandles with abelian inner automorphism groups from graphs, proving their homogeneity.
problem Finding quandles with specific automorphism properties.
method Starting from simple graphs, the paper constructs quandles with abelian inner automorphism groups and proves their homogeneity.
result Homogeneous quandles with abelian inner automorphism groups are constructed from vertex-transitive graphs.
The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.
problem Finding the domain with the largest first eigenvalue for a given volume and boundary conditions.
method Shape optimization for the first eigenvalue of the p-Laplace operator in hyperbolic space.
result The concentric annular region maximizes the first eigenvalue among multiply-connected domains.
Groups with specific properties have vanishing ℓ2-Betti numbers.
problem Understanding ℓ2-Betti numbers for certain groups. method Introduced cheap 1-rebuilding property and used structure theorem of Tucker-Drob.
result First ℓ2-Betti numbers vanish for specified groups. The author reviews his results on locally compact homogeneous spaces with inner metric, in particular, homogeneous manifolds with inner metric. The latter are isometric to homogeneous (sub-)Finslerian manifolds; under some additional conditions they are isometric to homogeneous (sub)-Riemannian manifolds. The class Ω…
Researchers prove inner product recovery is impossible in latent space models.
problem Recovering inner products in latent space models with random geometric graphs.
method Rate-distortion theory applied to Gaussian or spherical latent locations.
result Impossible to recover inner products if dimensionality exceeds nh(p), matching positive results' conditions. Paper proposes a new method to optimize feature coordinates for better image classification.
problem Improving feature extraction for better machine learning classification.
method Mutual-energy inner product optimization method.
result The method enhances low-frequency features and suppresses high-frequency noise, leading to better classification results.
We classify homotopes of classical symmetric spaces (studied in Part I of this work). Our classification uses the fibered structure of homotopes: they are fibered as symmetric spaces, with flat fibers, over a non-degenerate base; the base spaces correspond to inner ideals in Jordan pairs. Using that inner ideals in cla…
The paper challenges the belief that more inner iterations at test time improve performance in implicit deep learning.
problem The performance improvement of implicit deep learning models with increased inner iterations at test time.
method Theoretical analysis of a simple setting, validation on implicit deep learning problems.
result Overparametrization plays a key role; increasing the number of iterations at test time does not improve performance for overparametrized networks.
Estimates inner products between nonparametric distributions using Fourier basis.
problem Estimating inner products between two nonparametric distributions.
method Proposes estimators for inner products and induced norms, proves mean squared error bounds and minimax lower bounds.
result Proposed estimators are rate-optimal over Fourier ellipsoids.
WIPS optimizes inner product weights to approximate various similarities.
problem Learning high-quality node representations and accurate similarities.
method Weighted inner product similarity (WIPS) with adjustable weights.
result WIPS can approximate arbitrary general similarities including positive definite and indefinite kernels.
Study of Gaussian distributions using entropic Gromov-Wasserstein and inner product Gromov-Wasserstein.
problem Optimal transportation between Gaussian distributions with different dimensions.
method Entropic Gromov-Wasserstein and inner product Gromov-Wasserstein, with closed-form expressions and von Neumann's trace inequality.
result Closed-form expressions for the entropic IGW and its unbalanced variant between Gaussian distributions.
We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body K∈Rn and its polar body K∘. The Mahler conjecture asserts that the Mahler volume v(K) is minimiz…
ANIL adapts only a subset of parameters, reducing computational cost.
problem Efficiently adapt model parameters in meta-learning.
method Adapts only a small subset of parameters in the inner loop of ANIL.
result Theoretical convergence and computational complexity analysis for ANIL.
We propose a quantization based approach for fast approximate Maximum Inner Product Search (MIPS). Each database vector is quantized in multiple subspaces via a set of codebooks, learned directly by minimizing the inner product quantization error. Then, the inner product of a query to a database vector is approximated …
Study bounds the index of minimal submanifolds using energy measures and Yang-Mills-Higgs equations.
problem Bounding the index of codimension 2 minimal submanifolds.
method Second inner variation of energy, convergence of energy measures, and stress-energy tensors.
result Bound the Morse index of the submanifold by the index of critical points.
Let D− and D+ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as t→+∞ for the number of common perpendiculars of length at most t from D− to D+, count…
Each market has its singular characteristic. Its inner structure is directly responsible for the observed distributions of returns though this fact is widely overlooked. Big orders lead to doubling the tails. The behavior of a market maker with many or few ``friends'' who can reliably loan money or stock to him is quit…
Study on singularities of specific polynomial functions.
problem Characterizing the topology of singularities of mixed functions.
method Introduced inner non-degenerate mixed functions and used Newton boundary to characterize links.
result Links of singularities can be completely characterized under certain conditions.
Develops implicit MAML for efficient few-shot learning.
problem Efficient few-shot learning with limited data.
method Implicit differentiation for inner loop optimization.
result Agrees with inner loop optimizer choice and handles many gradient steps.
Study on kernel regression risk in high dimensions using Pinsker bound.
problem Kernel regression risk in high-dimensional inner product spaces.
method Investigation of Pinsker bound for kernel regression on sphere Sd with sample size n=αdγ(1+od(1)). result Exact minimax risk and Pinsker constant identified for kernel regression.
We point out that the Homfly polynomial (that is to say, Ocneanu's trace functional) contains two polynomial-valued inner products on the Hecke algebra representation of Artin's braid group. These bear a close connection to the Morton-Franks-Williams inequality. In these structures, the sets of positive, respectively n…
The paper solves the Andreadakis problem for specific groups using inner automorphisms.
problem Solving the Andreadakis problem for specific groups.
method Generalizing tools from [Dar19b] to study subgroups of IAn, focusing on the behavior of the Andreadakis problem with inner automorphisms.
result The Andreadakis equality holds for the pure braid group and the mapping class group of the n-punctured sphere.
We study one extremal problem on the product of power of generalized inner radii of non-overlapping domains in Cn.
New metrics measure knots' shapes without changing their orientation.
problem Measuring knots without considering their orientation.
method Defined Möbius invariant metrics on knot space.
result Found conditions for Möbius invariant weighted inner products.
New SVRG and SARAH schemes reduce tuning effort for variance reduction.
problem Optimal performance of SVRG and SARAH requires tuning of parameters.
method Introduces Barzilai-Borwein step sizes, averaging, and adaptive inner loop length.
result Improves SVRG, SARAH, and BB variants' convergence rates and performance.
A new encoding framework predicts brain activity from visual stimuli and intrinsic brain connections.
problem Traditional encoding models ignore brain inner states, limiting their performance in natural image identification.
method Proposes a novel encoding framework combining external stimuli and brain inner states, using a forward encoding model and an inner state model.
result The framework achieves better performance on natural image identification from fMRI responses than traditional models.
Convex learning for diverse invariances in semi-inner-product space.
problem Efficiently learning invariant representations for a wide range of invariances.
method Developed a convex representation learning algorithm for generalized invariances modeled as semi-norms, introducing Euclidean embeddings for kernel representers in a semi-inner-product space.
result Accurate invariant representations learned efficiently and effectively, validated by experiments.
We present the first provably sublinear time algorithm for approximate \emph{Maximum Inner Product Search} (MIPS). Our proposal is also the first hashing algorithm for searching with (un-normalized) inner product as the underlying similarity measure. Finding hashing schemes for MIPS was considered hard. We formally sho…
A new method for efficient nested Monte Carlo simulations in financial modeling.
problem Computational challenges in nested stochastic modeling for financial risk assessment.
method Sample recycling approach to speed up inner loop estimations.
result Significantly more efficient than traditional techniques.
Study on naked singularities without symmetry, forming incomplete future null infinity and singular inner Cauchy horizon.
problem Formation of naked singularities in Einstein-scalar field system without symmetry assumptions.
method Employing four-type differences and scale-invariant weighted norms to control geometry.
result Global naked singularity structure with incomplete future null infinity and singular inner Cauchy horizon.