arXiv research

Magnitude homology is a bigraded homology theory for finite graphs defined by Hepworth and Willerton, categorifying the power series invariant known as magnitude which was introduced by Leinster. We analyze the structure and implications of torsion in magnitude homology. We show that any finitely generated abelian grou…

Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.

This research quantifies neural networks using magnitude, a topological invariant.

Magnitude of manifolds linked to Riesz energies and beta functions.

Magnitude is not continuous but may be stable for most finite metric spaces.

Magnitude of Euclidean domains predicts Willmore energy 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.

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…

Magnitude of geometric shapes studied for smooth manifolds, revealing spectral geometry insights.

This paper introduces new invariants for time series analysis.

A new method models financial returns by separating sign and magnitude, improving forecasting accuracy.

Magnitude-based pruning is one of the simplest methods for pruning neural networks. Despite its simplicity, magnitude-based pruning and its variants demonstrated remarkable performances for pruning modern architectures. Based on the observation that magnitude-based pruning indeed minimizes the Frobenius distortion of a…

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.

New measures quantify diversity of latent representations using metric space magnitude.

Previous studies indicate that nonlinear properties of Gaussian time series with long-range correlations, $u_i$, can be detected and quantified by studying the correlations in the magnitude series $|u_i|$, 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.

SPADE-S improves time series forecasting accuracy for low-magnitude and sparse data.

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…

Neural networks have achieved dramatic improvements in recent years and depict the state-of-the-art methods for many real-world tasks nowadays. One drawback is, however, that many of these models are overparameterized, which makes them both computationally and memory intensive. Furthermore, overparameterization can als…

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.

Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.

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…

A new energy-efficient pruning method for 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.

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.

We address noisy Euclidean distances in high dimensions, estimating noise levels and correcting distances.

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…

Order-flow entropy predicts price magnitude without directionality.

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 $\ell_{\infty}$ norm restricted on the $K$ largest componen…

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…

New algorithm robust to outliers in Bayesian Optimization.

We study involuntary micro-movements of the eye for biometric identification. While prior studies extract lower-frequency macro-movements from the output of video-based eye-tracking systems and engineer explicit features of these macro-movements, we develop a deep convolutional architecture that processes the raw eye-t…

Pruning method removes less important features in linear models.

The thesis explores IMP, a process that identifies winning tickets in DNNs, and its universality.

Hollow-tree Super resolves feature importance in large datasets.

Defines SETR to measure carbon transition risk for investors.

We investigate statistical properties of daily international market indices of seven countries, and high-frequency $S&P500$ and KOSDAQ data, by using the detrended fluctuation method and the surrogate test. We have found that the returns of international stock market indices of seven countries follow a universal power-…

A new approach to the understanding of complex behavior of financial markets index using tools from thermodynamics and statistical physics is developed. Physical complexity, a magnitude rooted in Kolmogorov-Chaitin theory is applied to binary sequences built up from real time series of financial markets indexes. The st…

Low rank matrix factorisation is often used in recommender systems as a way of extracting latent features. When dealing with large and sparse datasets, traditional recommendation algorithms face the problem of acquiring large, unrestrained, fluctuating values over predictions especially for users/items with very few co…

A new approach to the understanding of the complex behavior of financial markets index using tools from thermodynamics and statistical physics is developed. Physical complexity, a magnitude rooted in the Kolmogorov-Chaitin theory is applied to binary sequences built up from real time series of financial markets indices…

Softmax is found ineffective for NL block, leading to improved performance.

Asymmetries in volatility spillovers are highly relevant to risk valuation and portfolio diversification strategies in financial markets. Yet, the large literature studying information transmission mechanisms ignores the fact that bad and good volatility may spill over at different magnitudes. This paper fills this gap…

Neural model outperforms ETAS in forecasting Central Apennines earthquakes.