Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine lea…
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48 results for “Persistence Landscape”
Considering the creation of persistence landscape on a parametrized curve and structure of sampling, there exists a random process for which a finite mixture model of persistence landscape (FMMPL) can provide a better description for a given dataset. In this paper, a nonparametric approach for computing integrated mean…
In topological data analysis, persistent homology is used to study the "shape of data". Persistent homology computations are completely characterized by a set of intervals called a bar code. It is often said that the long intervals represent the "topological signal" and the short intervals represent "noise". We give ev…
Study the landscape of Lipschitz functions between manifolds using persistent homology.
problem Understanding the structure of homotopy paths between maps with high Lipschitz constants.
method Using persistent homology to analyze the landscape of Lipschitz functions between manifolds.
result First results on the persistence of higher-dimensional cycles in function spaces.
We explore the evolution of daily returns of four major US stock market indices during the technology crash of 2000, and the financial crisis of 2007-2009. Our methodology is based on topological data analysis (TDA). We use persistence homology to detect and quantify topological patterns that appear in multidimensional…
Persistent Neuronscs.LG
Persistent neurons improve neural network optimization by leveraging previous solutions.
problem Improving neural network optimization under different initialization and data distributions.
method Persistent neurons use information from previous converged solutions to explore new landscapes and avoid local minima.
result Persistent neurons converge to more optimal solutions and improve model performance under various initializations.
Study uses topological signatures to quantify financial market complexity.
problem Capturing temporal organization beyond volatility measures.
method Null validated topological approach using norm of persistence landscapes.
result Persistence landscape norms reveal dynamical structure during market stress.
This paper presents approximate confidence intervals for each function of parameters in a Banach space based on a bootstrap algorithm. We apply kernel density approach to estimate the persistence landscape. In addition, we evaluate the quality distribution function estimator of random variables using integrated mean sq…
We propose PLLay, a novel topological layer for general deep learning models based on persistence landscapes, in which we can efficiently exploit the underlying topological features of the input data structure. In this work, we show differentiability with respect to layer inputs, for a general persistent homology with …
TDA-based portfolios show better risk-adjusted returns than classical methods.
problem Traditional portfolio selection methods fail to capture complex asset dynamics.
method Topological Data Analysis (TDA) using persistence landscapes to quantify portfolio risk.
result TDA-based portfolios outperform classical models in excess mean return and financial ratios.
A new method compares persistent cycles in topological data.
problem Comparing persistent homology representations of two spaces.
method Direct comparison of individual persistent cycles based on persistence intervals and spatial placement.
result Demonstrated the effectiveness of the method in topological inference.
Unified pipeline classifies time series using complex networks and persistent homology.
problem Classifying univariate time series using various graph constructions and metrics.
method Time series to graph, graph to dissimilarity matrix, filtration to persistence diagrams, vectorization to features.
result Persistence-based features are robust to noise and optimal graph type depends on signal structure.
Monotonic Linear Interpolation property in neural networks persists despite non-convexity.
problem Understanding the geometric properties of neural network loss landscapes.
method Tools from differential geometry to analyze the monotonicity of neural network weights.
result Sufficient conditions for the Monotonic Linear Interpolation property under mean squared error.
In recent years there has been noticeable interest in the study of the "shape of data". Among the many ways a "shape" could be defined, topology is the most general one, as it describes an object in terms of its connectivity structure: connected components (topological features of dimension 0), cycles (features of dime…
Neural networks' optimization dynamics are confined to a single basin despite connected basins in the loss landscape.
problem Neural networks' optimization dynamics are confined to a single basin despite connected basins in the loss landscape.
method Identifying entropic barriers arising from the interplay between curvature variations along low-loss paths and noise in optimization dynamics.
result Curvature-induced entropic forces bias noisy dynamics back toward the endpoints, explaining the confinement and connectivity of solutions.
Persistence Spheres: a Bi-continuous Linear Representation of Measures for Partial Optimal Transportstat.ML
Improved persistence spheres map measures to functions, stable under partial transport.
problem Representing and comparing measures in topological machine learning.
method Persistence spheres map measures to continuous functions on the sphere, stable under 1-Wasserstein partial transport.
result Persistence spheres provide a stable, parameter-free representation of measures, improving upon existing methods.
PHINN: A generative model for rare-event time series using persistent homology
problem Generating rare events in time series
method Flow-matching framework with dynamic Betti curves and persistence landscape loss
result Outperforms statistical and diffusion baselines in topological fidelity and tail coverage
Stochasticity helps to navigate rough landscapes: comparing gradient-descent-based algorithms in the phase retrieval problemcond-mat.dis-nn
Gradient descent variants improve phase retrieval accuracy.
problem Phase retrieval problem in high-dimensional spaces.
method Gradient descent, stochastic gradient descent, Langevin algorithm, dynamical mean-field theory.
result Stochastic variants of gradient descent achieve better generalization in phase retrieval.
Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persist…
Mathematical pipeline identifies structural homology of knotted proteins.
problem Quantification and classification of protein structures, especially knotted proteins, require noise-free and complete data.
method Developed a geometric framework using persistent homology to analyze protein structures.
result Persistent homology accurately represents structural homology of knotted proteins and identifies geometric features of protein entanglement.
Paper estimates neural network size needed for topology learning.
problem Estimating the smallest neural network size for topology learning.
method Using algebraic topology and Lie theory, the paper introduces a procedure based on persistent homology to determine the required dimension.
result The derived dimension is the smallest capable of capturing the topology of the data manifold.
New method uses topological data analysis to study stock market crashes.
problem Characterizing and predicting stock market crashes.
method Topological data analysis, persistence landscape, dynamic time series analysis.
result Demonstrates effectiveness of new method for Flash Crash characterization and prediction.
Despite the phenomenal success of deep neural networks in a broad range of learning tasks, there is a lack of theory to understand the way they work. In particular, Convolutional Neural Networks (CNNs) are known to perform much better than Fully-Connected Networks (FCNs) on spatially structured data: the architectural …
TDA improves cryptocurrency portfolio management.
problem Traditional methods fail to manage cryptocurrencies effectively.
method Topological Data Analysis (TDA) for identifying investment opportunities.
result TDA-based portfolio management outperforms traditional methods.
Approaches for approximating persistent homology for large datasets.
problem Inability to compute persistent homology for large datasets.
method Multiple subsampling framework for statistical approximation of persistent homology.
result Derivation of finite sample convergence rates for empirical means of persistent homology.
Proposes deep graph persistence to address neural persistence issues in deep learning.
problem Variance of weights and lack of spatial structure in deep neural networks impact neural persistence.
method Extends neural persistence to the whole network, considering interactions between layers.
result Deep graph persistence alleviates variance-related issues and captures persistent paths through the network.
Reviews recent findings on neural network landscapes.
problem Non-convexity of loss functions causing bad landscapes.
method Rigorous geometric analysis and empirical exploration.
result Wide neural nets may have sub-optimal local minima.
Paper proves -means clustering works on persistence diagrams.
problem Complex geometry of persistence diagram space.
method Proves convergence of -means on persistence diagram space.
result Performance of -means on persistence diagrams and measures is superior.
Formula for interleaving distance of rectangle persistence modules.
problem Calculating distances between rectangle persistence modules.
method Formulas based on rectangle geometry, extended to decomposable modules.
result Closed formulas for interleaving and bottleneck distances.
This paper demonstrates the flaws of co-persistence theory proposed by Bollerslev and Engle (1993) which cause the theory can hardly be applied. With the introduction of the half-life of decay coefficient as the measure of the persistence, and both the weak definition of persistence and co-persistence in variance, this…
Optimizes wavelets for graph classification using spectral wavelet signatures and persistence diagrams.
problem Graph classification with geometric properties encoded in persistence diagrams.
method Optimizes spectral wavelets for graph datasets to capture best-suited features for classification.
result Competitive performance in graph classification problems compared to other persistence-based architectures.
New method for analyzing multiparameter persistence modules from smooth functions.
problem Analyzing multiparameter persistence modules from smooth functions.
method Generalized Morse theory applied to cobordism and Cerf theory.
result Complete description of persistence modules as direct sums of indecomposables.
Develops robust persistence diagrams using kernel methods.
problem Persistence diagrams are sensitive to data perturbations.
method Constructs robust persistence diagrams from superlevel filtrations of robust density estimators using reproducing kernels.
result Robust persistence diagrams are consistent estimators in bottleneck distance.
The paper uses TDA to select stocks for a sparse portfolio, improving performance across market scenarios.
problem Sparse portfolio selection in financial markets.
method Topological data analysis (TDA) for clustering stock price movements.
result The TDA-based clustering strategy significantly enhances sparse portfolio performance.
This paper interprets critical scales in persistent homology for compact metric spaces.
problem Understanding critical scales in persistent homology for general compact metric spaces.
method Analyzing local minima of the distance function and their impact on persistence.
result Each decrease in zero-dimensional persistence and increase in one-dimensional persistence is induced by local minima of the distance function.
LPINNs solve complex PDEs by reformulating PINNs on Lagrangian frame, reducing training complexity.
problem Complexity in training PINNs, especially for convection-diffusion equations.
method Propose LPINNs, a Lagrangian reformulation of PINNs, with two branches solving state variables and characteristics curves.
result Loss landscapes of LPINNs are less sensitive to problem complexity compared to traditional PINNs.
MuRiT efficiently computes multi-parameter persistence barcodes.
problem Efficient computation of multi-parameter persistent homology.
method Vietoris-Rips transformation to reduce multi-parameter to single-parameter computation.
result MuRiT computes pathwise persistence barcodes for multi-filtered flag complexes.
Persistent homology can recognize knotting in curves.
problem Recognizing knotting in curves
method Compute one-dimensional persistent homology, extract cycle representatives, and assign a hypergraph curvature-based score.
result Systematic differences between knotted and unknotted structures are revealed.
This article analyzes the relationship between co-persistence and hedging which indicates co-persistence ratio is just the long-term hedging ratio. The new method of exhaustive search algorithm for deriving co-persistence ratio is derived in the article. And we also develop a new hedging strategy of combining co-persis…
Despite the obvious similarities between the metrics used in topological data analysis and those of optimal transport, an optimal-transport based formalism to study persistence diagrams and similar topological descriptors has yet to come. In this article, by considering the space of persistence diagrams as a space of d…
The study examines when MAML's objective has a benign landscape.
problem Understanding when MAML's objective landscape is benign.
method Analyzing the landscape of MAML objective on LQR tasks.
result The benign landscape of the MAML objective depends on task similarities.
Persistence diagrams are important descriptors in Topological Data Analysis. Due to the nonlinearity of the space of persistence diagrams equipped with their {\em diagram distances}, most of the recent attempts at using persistence diagrams in machine learning have been done through kernel methods, i.e., embeddings of …
Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field. In this paper, we study topological persistence in general metric spaces, with a …
The choice of the control frequency of a system has a relevant impact on the ability of reinforcement learning algorithms to learn a highly performing policy. In this paper, we introduce the notion of action persistence that consists in the repetition of an action for a fixed number of decision steps, having the effect…
Given a compact geodesic space we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their preci…
We introduce several geometric notions, including the width of a homology class, to the theory of persistent homology. These ideas provide geometric interpretations of persistence diagrams. Indeed, we give quantitative and geometric descriptions of the "life span" or "persistence" of a homology class. As a case study, …
This article addresses persistent tangles. These are tangles whose presence in a knot diagram forces that diagram to be knotted. We provide new methods for constructing persistent tangles. Our techniques rely mainly on the existence of non-trivial colorings for the tangles in question. Our main result in this article i…
Black holes offer insights into machine learning's loss landscapes.
problem Understanding the loss landscape in machine learning.
method Comparing machine learning loss landscapes to black hole entropy.
result Black holes provide an infinite family of potential landscapes with known minima.