Magnitude homology reveals that graphs can have torsion subgroups.
problem Understanding torsion in magnitude homology of graphs.
method Analysis of magnitude homology defined by Hepworth and Willerton.
result Torsion of any prime order can appear in graphs' magnitude homology.
Hepworth, Willerton, Leinster and Shulman introduced the magnitude homology groups for enriched categories, in particular, for metric spaces. The purpose of this paper is to describe the magnitude homology group of a metric space in terms of order complexes of posets. In a metric space, an interval (the set of points b…
Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.
problem Defining magnitude for non-finite metric spaces with measures.
method Integrals over geodesics, using counting and weight measures.
result Magnitude agrees with finite spaces' magnitude and volume under specific conditions.
In this paper we establish the existence of periodic orbits belonging to any σ-atoroidal free homotopy class for Hamiltonian systems in the twisted disc bundle, provided that the compactly supported time-dependent Hamiltonian function is sufficiently large over the zero section and the magnitude of the weakly exact $…
The issue of computing (co)homology generators of a cell complex is gaining a pivotal role in various branches of science. While this issue can be rigorously solved in polynomial time, it is still overly demanding for large scale problems. Drawing inspiration from low-frequency electrodynamics, this paper presents a ph…
A new method clusters complex networks using topological and geometric structure.
problem Clustering complex networks with intricate topology.
method Centroid-based clustering strategy using Wasserstein distance and barycenter for persistence barcodes.
result Demonstrated effectiveness on simulated and real-world networks.
This research quantifies neural networks using magnitude, a topological invariant.
problem Understanding the generalization capabilities of neural networks.
method Using a novel topological invariant called magnitude to study neural network representations.
result Magnitude dimension is theoretically connected to generalisation error and can predict it.
Magnitude of manifolds linked to Riesz energies and beta functions.
problem Magnitude invariant and its geometric significance.
method Relating magnitude invariant to Brylinski's beta function and pseudodifferential analysis.
result Precise relation between magnitude invariant and beta function for closed manifolds.
Lookahead pruning extends single-layer optimization to multi-layer, outperforming magnitude-based pruning.
problem Pruning neural networks to reduce computational cost and memory usage.
method Developed a multi-layer optimization approach extending the single-layer optimization of magnitude-based pruning.
result Consistently outperforms magnitude-based pruning on various networks, especially in high sparsity.
Magnitude is not continuous but may be stable for most finite metric spaces.
problem Stability of magnitude invariant in finite metric spaces.
method Investigates the continuity properties of magnitude with respect to Gromov-Hausdorff topology.
result Magnitude is nowhere continuous but may be generically continuous.
Magnitude of Euclidean domains predicts Willmore energy in odd dimensions.
problem Magnitude function of compact domains in odd dimensions.
method Asymptotic expansion of magnitude function at infinity.
result Magnitude function determines Willmore energy of boundary in odd dimensions.
Are expansions and recessions more likely to end as their magnitude increases? In this paper we apply parametric hazard models to investigate this issue in a sample of 16 countries from 1881 to 2000. For the total sample we find evidence of positive magnitude dependence for recessions, while for expansions we are not a…
Magnitude study on manifolds using fractional Laplacian.
problem Magnitude invariant of compact metric spaces via fractional Laplacian.
method Semiclassical analysis of nonlocal boundary value problem related to fractional Laplacian.
result Asymptotic expansion of magnitude in terms of curvature invariants.
Magnitude of geometric shapes studied for smooth manifolds, revealing spectral geometry insights.
problem Understanding the geometric significance of Leinster's magnitude for smooth manifolds.
method Investigation of magnitude function for various distance functions, including submanifolds and Riemannian manifolds, with asymptotic analysis in the limit.
result Magnitude function is well-defined and meromorphically continued for large distances, revealing volume, surface area, and curvature integrals.
This paper introduces new invariants for time series analysis.
problem Analyzing the diversity and invariants of time series data.
method Introduces new invariants derived from the continuity of magnitude and maximum diversity.
result Demonstrates improved performance in machine learning experiments with real-world data.
A new method models financial returns by separating sign and magnitude, improving forecasting accuracy.
problem Capturing nonlinear predictability in financial return dynamics.
method Decomposes returns into sign and magnitude components, using a joint distribution model.
result Significantly outperforms traditional linear models in forecasting U.S. stock market returns.
Magnitude is a real-valued invariant of metric spaces, analogous to the Euler characteristic of topological spaces and the cardinality of sets. The definition of magnitude is a special case of a general categorical definition that clarifies the analogies between various cardinality-like invariants in mathematics. Altho…
Novel metric space magnitude and weighting vectors improve machine learning tasks.
problem Improving machine learning algorithms using novel metric space concepts.
method Metric space magnitude and weighting vectors for better machine learning.
result The weighting vector effectively detects boundaries and improves classic machine learning tasks.
New measures quantify diversity of latent representations using metric space magnitude.
problem Evaluating the diversity of latent representations in machine learning models.
method Developed magnitude-based measures for latent representations, stable under data perturbations.
result Demonstrated superior performance across various domains and tasks.
Previous studies indicate that nonlinear properties of Gaussian time series with long-range correlations, ui, can be detected and quantified by studying the correlations in the magnitude series ∣ui∣, i.e., the ``volatility''. However, the origin for this empirical observation still remains unclear, and the exact …
Magnitude-based features capture interactions between different entities in multispecies spatial data.
problem Capturing interactions between different entities in multispecies spatial data.
method Developing magnitude-based features for multispecies spatial data.
result Identifies distinct neighbourhood types and spatial heterogeneity.
A new pruning criterion reduces model size and improves performance.
problem Overparameterized neural networks are computationally and memory intensive, leading to overfitting.
method Introduces a magnitude and uncertainty (M&U) pruning criterion inspired by statistical Wald test.
result Our M&U pruning criterion leads to more compressed models with less loss in predictive power.
SPADE-S improves time series forecasting accuracy for low-magnitude and sparse data.
problem Challenges in forecasting time series with strong heterogeneity in magnitude and sparsity.
method SPADE-S is a robust forecasting architecture that reduces biases and improves overall prediction accuracy.
result SPADE-S outperforms existing state-of-the-art approaches across diverse use cases, improving forecast accuracy by up to 15%.
In this paper we define the magnitude of metric spaces using measures rather than finite subsets as had been done previously and show that this agrees with earlier work with Leinster in arXiv:0908.1582. An explicit formula for the magnitude of an n-sphere with its intrinsic metric is given. For an arbitrary homogeneous…
Accelerated magnetic resonance (MR) scan acquisition with compressed sensing (CS) and parallel imaging is a powerful method to reduce MR imaging scan time. However, many reconstruction algorithms have high computational costs. To address this, we investigate deep residual learning networks to remove aliasing artifacts …
New pruning method retains model expressiveness for NLP tasks.
problem Pruning large pretrained transformer models for real-world deployment.
method Mixture Gaussian Prior Pruning (MGPP) algorithm.
result MGPP outperforms existing pruning methods in high sparsity settings.
Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.
problem Understanding the relationship between parameter magnitudes and Hessian curvature in deep learning models.
method Developed a matrix-free algorithm based on sketched SVDs to measure similarity between parameter masks and Hessian eigenspaces.
result Top Hessian eigenvectors tend to be concentrated around larger parameters, indicating a connection between parameter magnitudes and loss curvature.
This paper proposes an approach to the joint modeling of the short-time Fourier transform magnitude and phase spectrograms with a deep generative model. We assume that the magnitude follows a Gaussian distribution and the phase follows a von Mises distribution. To improve the consistency of the phase values in the time…
Active inference implemented for high-dimensional tasks shows efficient exploration and improved sample efficiency.
problem Achieving efficient exploration and learning in complex, uncertain environments.
method Active inference framework applied to high-dimensional tasks, with Bayesian evidence maximization.
result Order of magnitude increase in sample efficiency over model-free baselines.
A new energy-efficient pruning method for federated learning.
problem Energy inefficiency in gradient sparsification for federated learning.
method Formalized energy-constrained projection problem and proposed Cost-Weighted Magnitude Pruning (CWMP).
result CWMP optimally balances performance and energy efficiency in federated learning.
This paper considers magnitude, asymptotics and duration of drawdowns for some Lévy processes. First, we revisit some existing results on the magnitude of drawdowns for spectrally negative Lévy processes using an approximation approach. For any spectrally negative Lévy process whose scale functions are well-behaved at …
New method identifies whether equity return predictability is due to magnitude shrinkage or directional reversal.
problem Determining the nature of equity return predictability (directional reversal vs magnitude shrinkage).
method Developed the Fourier-Residue Identity (FRI) to decompose return autocorrelation into sign and magnitude channels.
result The lag-1 autocorrelation in SPY is driven entirely by magnitude shrinkage, not directional reversal.
Distributed training of massive machine learning models, in particular deep neural networks, via Stochastic Gradient Descent (SGD) is becoming commonplace. Several families of communication-reduction methods, such as quantization, large-batch methods, and gradient sparsification, have been proposed. To date, gradient s…
Cosine similarity can force points to grow in magnitude, causing convergence issues.
problem Cosine similarity loss can lead to convergence issues in deep learning.
method Analyzing under-explored settings and proposing cut-initialization.
result Cosine similarity optimization forces points to grow in magnitude, leading to convergence issues.
We address noisy Euclidean distances in high dimensions, estimating noise levels and correcting distances.
problem Distorted pairwise Euclidean distances due to heteroskedastic noise.
method Developed a hyperparameter-free approach to jointly estimate noise magnitudes and correct distances.
result Our method provides accurate noise magnitude estimates and corrected distances in high-dimensional settings.
Extends Khovanov homology spectral sequence using Heegaard Floer homology.
problem Relating Khovanov homology and Heegaard Floer homology of branched double covers.
method Involutive Heegaard Floer homology, bordered Floer homology, surgery exact triangle.
result Establishes spectral sequence connecting Khovanov homology and Heegaard Floer homology.
Predicting absolute magnitude of fluctuations of price, even if their sign remains unknown, is important for risk analysis and for option prices. In the present work, we display our predictions about absolute magnitude of daily fluctuations of the Dow Jones Industrials Average (DJIA), utilizing the original theory of c…
New homological action on sutured instanton homology defined.
problem Detecting link splitting in knot homology.
method Defining a homological action on sutured instanton Floer homology.
result Instanton knot homology detects link splitting for two-component links.
Order-flow entropy predicts price magnitude without directionality.
problem Predicting price magnitude in financial markets.
method Real-time order-flow entropy computed from a 15-state Markov transition matrix.
result Order-flow entropy predicts the magnitude of intraday returns with high accuracy.
The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.
problem Constructing rational homology 3-spheres that bound rational homology 4-balls.
method Exploring plumbed 3-manifolds and using rational homology circles.
result Infinite families of rational homology 3-spheres that bound rational homology 4-balls.
We propose a novel SPARsity and Clustering (SPARC) regularizer, which is a modified version of the previous octagonal shrinkage and clustering algorithm for regression (OSCAR), where, the proposed regularizer consists of a K-sparse constraint and a pair-wise ℓ∞ norm restricted on the K largest componen…
Study on knot concordance and homology cobordism using Heegaard Floer homology.
problem Knot concordance and homology cobordism.
method Heegaard Floer homology.
result Recent results in knot concordance and homology cobordism.
Maps quandle homology to relative group homology.
problem Understanding the relationship between quandle and group homology.
method Introducing a chain map and constructing quandle cocycles.
result Relates quandle homology to relative group homology through triangulations.
Homological stability aids in computing group homology.
problem Computing homology of families of groups.
method Proving homological stability theorems and computing stable homology.
result Computation of Higman-Thompson groups' homology.
Study ribbon homology concordances using link Floer homology.
problem Understanding ribbon homology concordances and their effects on link Floer homology.
method Combining results from Daemi, Lidman, Vela-Vick, Wong, and Zemke, using link Floer homology and torsion submodules.
result Ribbon homology concordances induce split injections on HFL−. New symplectic annular Khovanov homology connects knot theory to Floer homology.
problem Understanding the relationship between knot theory and Floer homology.
method Introducing a new version of symplectic annular Khovanov homology and establishing spectral sequences.
result Established spectral sequences linking different knot homologies.
Grid homology confirms the Upsilon invariant in knot theory.
problem Verifying the equivalence of Upsilon invariants in knot theory.
method Reconstructed Upsilon invariant using grid homology and proved equivalence.
result Upsilon invariants in knot Floer and grid homology are equivalent.
In this paper, we address the problem of reconstructing a time-domain signal (or a phase spectrogram) solely from a magnitude spectrogram. Since magnitude spectrograms do not contain phase information, we must restore or infer phase information to reconstruct a time-domain signal. One widely used approach for dealing w…