Persistently trained EBMs generate images and estimate complex densities.
problem Challenges in ML learning for energy-based models, especially non-convergence of MCMC.
method Introduce diffusion data, learn a joint EBM through persistent training with enhanced sampling.
result First simultaneous achievement of stability, post-training image generation, and superior out-of-distribution detection for image data.
The paper introduces a new ODE approach to improve Wasserstein GANs.
problem Improving Wasserstein GANs for better training results.
method Derives an ODE representing the gradient flow of Wasserstein-1 loss and proposes a new model W1-FE.
result W1-FE outperforms WGAN in training experiments across various dimensions.
Persistent homology reveals a topological signature of grokking in neural networks.
problem Understanding how neural networks learn and generalize from modular arithmetic tasks.
method Persistent homology on point clouds derived from embedding matrices of models trained on modular arithmetic.
result A sharp increase in first homology persistence indicates grokking, with a dominant long-lived topological feature and structured secondary features.
Unreduced PDs can perform similarly to reduced PDs in machine learning tasks.
problem Ignoring much of the information in persistence diagrams in machine learning pipelines.
method Developed methods to generate topological feature vectors from unreduced boundary matrices.
result Unreduced PDs can perform on par with, and sometimes outperform, fully-reduced PDs in machine learning tasks.
When an online learning algorithm is used to estimate the unknown parameters of a model, the signals interacting with the parameter estimates should not decay too quickly for the optimal values to be discovered correctly. This requirement is referred to as persistency of excitation, and it arises in various contexts, s…
FCM clustering adapts to persistence diagrams for topological data analysis.
problem Integrating topological data into machine learning workflows.
method Adapting Fuzzy c-Means to persistence diagrams.
result FCM clustering captures topological structure without additional processing.
Persistent entropy detects phase transitions in complex systems.
problem Detecting phase transitions in complex systems.
method Established a general theorem for persistent entropy to reliably detect phase transitions, introduced operational framework for finite-time computations.
result Persistent entropy exhibits an asymptotically non-vanishing gap across phases, robust numerical signatures across experiments.
New method improves neural network robustness to adversarial attacks.
problem Improving adversarial robustness of neural networks.
method Inspired by adaptive control theory, the approach uses persistency of excitation to constrain gradient descent updates.
result Networks trained with the PoE-motivated learning rate schedule are significantly more robust to adversarial attacks.
While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In this work, we propose neural persistence, a complexity measure for ne…
Topology applied to real world data using persistent homology has started to find applications within machine learning, including deep learning. We present a differentiable topology layer that computes persistent homology based on level set filtrations and edge-based filtrations. We present three novel applications: th…
The paper tackles long-term treatment effects with persistent confounders using sequential short-term outcomes.
problem Estimating long-term treatment effects with persistent unmeasured confounders.
method Exploiting the sequential structure of short-term outcomes, the paper develops three novel identification strategies and corresponding estimators.
result The proposed methods outperform existing approaches in handling persistent confounders.
Study forecasts U.S. bond index using deep learning, finding persistence is key.
problem Forecasting U.S. aggregate bond index with deep learning methods.
method Constructed a stationary but maximally persistent representation of the bond index, evaluated using MLPs and CNNs.
result Deep learning models outperform traditional methods in short-horizon forecasting of bond indices.
Paper develops a generative model using Wasserstein-2 loss.
problem Creating realistic data samples from limited data.
method Uses a distribution-dependent ODE with a gradient flow for W2 loss.
result The method converges to the true data distribution exponentially.
PES method reduces bias in gradient estimation for unrolled graphs.
problem High variance and bias in gradient estimation for unrolled computation graphs.
method Divide graph into unrolls, apply ES update, accumulate correction terms.
result PES provides unbiased, low-variance gradient estimates.
New method uses DTW to evaluate neural network forecasts of geomagnetic indices.
problem Evaluation metrics fail to capture persistence behavior in neural network forecasts.
method Dynamic Time Warping (DTW) to measure time series similarity, training neural networks to remove persistence.
result DTW reveals persistence behavior in neural network forecasts, confirming visual inspection.
We outline a detection method for adversarial inputs to deep neural networks. By viewing neural network computations as graphs upon which information flows from input space to out- put distribution, we compare the differences in graphs induced by different inputs. Specifically, by applying persistent homology to these …
It is well known that, for most datasets, the use of large-size minibatches for Stochastic Gradient Descent (SGD) typically leads to slow convergence and poor generalization. On the other hand, large minibatches are of great practical interest as they allow for a better exploitation of modern GPUs. Previous literature …
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.
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…
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.
Paper proves k-means clustering works on persistence diagrams.
problem Complex geometry of persistence diagram space.
method Proves convergence of k-means on persistence diagram space. result Performance of k-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…
Central to robot exploration and mapping is the task of persistent localization in environmental fields characterized by spatially correlated measurements. This paper presents a Gaussian process localization (GP-Localize) algorithm that, in contrast to existing works, can exploit the spatially correlated field measurem…
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.
This paper evaluates fractal dimension and persistent homology for neural network generalization.
problem Bounding and predicting the generalization gap of neural networks.
method Empirical evaluation of fractal dimension and persistent homology as generalization measures.
result Fractal dimension and persistent homology fail to predict generalization of models trained from poor initializations.
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.
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.
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.
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 …
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…
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 X we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of X to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their preci…
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.
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.
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…
Enhances graph embeddings by preserving graph topology.
problem Node2vec struggles to recreate the topology of input graphs.
method Introduces a topological loss term to Node2vec, aligning the persistence diagram of the embedding to that of the input graph.
result Reconstructs both geometry and topology of input graphs.
Study cosmic structures using Topological Data Analysis and Persistence Energy.
problem Investigate cosmic web evolution in ΛCDM cosmologies. method Apply LITE method to embed persistence diagrams into vector spaces and analyze cosmic structures.
result Discover a correlation between Persistence Energy and redshift values.
New lattice path method for statistical inference of persistent diagrams.
problem Statistical inference on persistent diagrams.
method Lattice path representation and combinatorial enumerations.
result Topological changes observed in spike proteins of COVID-19 virus.
The paper examines how long-memory dynamics, rough-volatility, and persistence affect equity volatility forecasting.
problem The study investigates how long-memory dynamics, rough-volatility, and persistence impact equity volatility forecasting.
method The paper combines semiparametric long-memory estimation, rough-volatility diagnostics, and structured forecasting regressions.
result Persistence measures improve out-of-sample volatility forecasts, particularly during periods of elevated market volatility and in volatility-managed portfolio applications.
This review explores TDA and TDL beyond persistent homology.
problem Limitations of persistent homology in capturing topological invariants and homotopic evolution.
method Spectral representations, sheaf theory, Mayer topology, interaction topology, differential topology, geometric topology.
result Review of topological tools for various data types.
Revises SWK for persistence diagrams using Figalli-Gigli distance.
problem Efficiently embedding persistence diagrams in a Hilbert space.
method Directly use Figalli-Gigli distance to build a positive definite kernel.
result SFGK shares properties with SWK and performs similarly on benchmarks.