TopoFisher optimizes trainable filtrations, vectorizations, and compressors in persistent homology to maximize Fisher information, yielding higher information than fixed cosmological summaries and approaching neural baselines with far fewer parameters while generalizing better under simulator shifts
Interactive visualization of 2-d persistence modules
7 Pith papers cite this work. Polarity classification is still indexing.
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abstract
The goal of this work is to extend the standard persistent homology pipeline for exploratory data analysis to the 2-D persistence setting, in a practical, computationally efficient way. To this end, we introduce RIVET, a software tool for the visualization of 2-D persistence modules, and present mathematical foundations for this tool. RIVET provides an interactive visualization of the barcodes of 1-D affine slices of a 2-D persistence module $M$. It also computes and visualizes the dimension of each vector space in $M$ and the bigraded Betti numbers of $M$. At the heart of our computational approach is a novel data structure based on planar line arrangements, on which we can perform fast queries to find the barcode of any slice of $M$. We present an efficient algorithm for constructing this data structure and establish bounds on its complexity.
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Constructs a computable 3-parameter Delaunay trifiltration for bifunction point clouds with O(|X|^⌈(d+1)/2⌉+1) size, an O(|X|^⌈d/2⌉+2) algorithm, and experiments on thousands of R³ points.
Interleaving distance on single- and multi-parameter persistence modules equals a Galois-edit distance, yielding a new proof of bottleneck stability.
Galois connections provide a new language that unifies interleavings and matchings in persistent homology and yields a simpler proof of bottleneck stability.
New algorithms compute Hom spaces for poset representations in O(n^4 (thick(Y) + thick(Omega^1 Y))^2) time using a uniqueness result for lifts, plus a classical O(n^3 thick(Y)^3) method, both improving on O(n^6) and strengthening AIDA for multiparameter persistence.
A new functor calculus for posets yields necessary and sufficient conditions for n-parameter multipersistence modules to have projective dimension at most n-1 or n-2.
Develops invariants for persistence modules over posets P by restricting to order-embedded finite-representation-type subposets X, decomposing into indecomposables, and generalizing signed barcodes via homological methods.
citing papers explorer
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TopoFisher: Learning Topological Summary Statistics by Maximizing Fisher Information
TopoFisher optimizes trainable filtrations, vectorizations, and compressors in persistent homology to maximize Fisher information, yielding higher information than fixed cosmological summaries and approaching neural baselines with far fewer parameters while generalizing better under simulator shifts
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Bifunction and Interlevel Delaunay Trifiltrations
Constructs a computable 3-parameter Delaunay trifiltration for bifunction point clouds with O(|X|^⌈(d+1)/2⌉+1) size, an O(|X|^⌈d/2⌉+2) algorithm, and experiments on thousands of R³ points.
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Interleaving Distance as a Galois-Edit Distance
Interleaving distance on single- and multi-parameter persistence modules equals a Galois-edit distance, yielding a new proof of bottleneck stability.
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Galois Connections in Persistent Homology
Galois connections provide a new language that unifies interleavings and matchings in persistent homology and yields a simpler proof of bottleneck stability.
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Computing Homomorphisms of Poset Representations with Applications to Multiparameter Persistence
New algorithms compute Hom spaces for poset representations in O(n^4 (thick(Y) + thick(Omega^1 Y))^2) time using a uniqueness result for lifts, plus a classical O(n^3 thick(Y)^3) method, both improving on O(n^6) and strengthening AIDA for multiparameter persistence.
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Cross effects for functors from posets
A new functor calculus for posets yields necessary and sufficient conditions for n-parameter multipersistence modules to have projective dimension at most n-1 or n-2.
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Invariants of persistence modules defined by order-embeddings
Develops invariants for persistence modules over posets P by restricting to order-embedded finite-representation-type subposets X, decomposing into indecomposables, and generalizing signed barcodes via homological methods.