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The Ring of Algebraic Functions on Persistence Bar Codes

Aaron Adcock, Erik Carlsson, Gunnar Carlsson

arXiv:1304.0530v1math.RA

TL;DR

Barcodes are useful for organizing topological data, but their interval-based representation motivates a coherent algebra of functions that treats zero-length intervals as zero modules. Using algebraic geometry and invariant theory, the paper identifies such coordinates and generators; these coordinates can organize barcode collections and databases of sampled shapes.

  • Problem

    The paper seeks algebraic functions on barcodes that are unchanged by adjoining zero-length intervals, which correspond to zero modules.

    Source: §1 · p. 24

  • Method

    Using algebraic geometry and invariant theory, the paper constructs coordinates and algebraic functions on the space of barcodes.

    Source: §1 · p. 2, §4.3 · p. 41

  • Results

    The paper identifies a family of coordinates on sampled shapes and describes their algebraic structure, including a basis for the invariant subring.

    Source: §3 · p. 27, §4.3 · p. 41

  • Takeaways & Limitations

    These coordinates can organize barcode collections and databases whose members produce barcodes.

    Source: §4.3 · p. 41

    A positive-length-only barcode space would split into components by interval count and would not capture the intended topology as short intervals converge to zero.

    Source: §2 · p. 4

Abstract

from arXiv · show

We study the ring of algebraic functions on the space of persistence barcodes, with applications to pattern recognition.

1 Introduction

The introduction frames barcodes as useful topological data whose interval-based form makes numerical interpretation difficult. It develops an algebraic description of barcode functions that is insensitive to adjoining zero-length intervals.

1 Introduction

  • Barcodes encode features by intervals whose endpoints mark their birth and death, but their interval-based form makes them difficult to use as numerical inputs to standard algorithms.The dimension k specifies the feature type, such as a cluster or loop.Source: §1 · p. 1
  • An n-interval barcode is represented by endpoint pairs, with interval order discarded through the n-fold symmetric product of R2.The symmetric group acts by permuting the interval coordinates.Source: §1 · p. 2
  • The symmetric products form an inverse system of graded coordinate rings whose inverse limit is denoted A[Sp∞(R2)].The cited construction describes functions on barcodes with varying interval counts.Source: §1 · p. 2
  • Because zero-length intervals represent zero modules, the target algebra consists of functions unchanged when any zero-length interval is adjoined to a barcode.The paper identifies this subring and describes its algebra generators for shape-database analysis.Source: §1 · p. 3

2 The Ind-scheme B

This section constructs the barcode space by assembling symmetric products of intervals and identifying barcodes that differ by zero-length intervals. It then defines algebraic functions on that space and characterizes a finite-stage subring.

2 The Ind-scheme B

  • Barcodes with n intervals are modeled by Spn(I), and adjoining [0,0] supplies the directed-system identification used to form the barcode space.I consists of endpoint pairs (x,y) with x ≤ y.Source: §2 · p. 3
  • Functions on the direct limit are compatible sequences, then restricted to those assigning equal values when a zero-length interval is replaced or removed.The resulting quotient set B identifies barcodes related by adjoining a zero-length interval.Source: §2 · pp. 3–4
  • Using only positive-length intervals would split barcodes by their positive interval count and would not capture the intended topology as short intervals approach zero.The section notes that a short interval can converge to the barcode without that interval.Source: §2 · p. 4
  • The paper studies algebraic functions because all functions, and even all continuous functions, are too complex to describe completely.It treats B as a colimit of algebraic varieties.Source: §2 · p. 4
  • At finite stage, Dn consists of polynomial functions whose restriction to each diagonal yi = xi is independent of xi.The stated characterization is that the corresponding derivative vanishes on each diagonal.Source: §2 · pp. 4–5

3 The ring of algebraic functions on B

The section characterizes the algebra through monomial conditions, symmetric-group invariants, and multisymmetric functions. Its main structural result is a free generating family for the inverse-limit algebra.

3 The ring of algebraic functions on B

  • The admissible monomials defining the finite-stage algebra are specified by the exponent condition that ai > 0 implies bi > 0.This is the basis condition stated for Dn.Source: §3 · p. 5
  • At finite stage, Dn is spanned by monomials in which a positive ξ_i exponent requires a positive η_i exponent.The condition follows from the diagonal-independence characterization after changing coordinates to ξ_i = x_i + y_i and η_i = y_i − x_i.Source: §3 · p. 5
  • The Sn-invariant subring has a basis formed by orbit sums of representatives of the admissible monomials.The action preserves the monomial basis, and the proposition gives the orbit sums as a k-basis.Source: §3 · p. 6
  • D embeds into the ring Λ2 of multisymmetric functions, whose inverse-limit power sums freely generate a polynomial algebra.The section uses this embedding to identify the structure of D.Source: §3 · p. 7
  • D is freely generated as a subalgebra of Λ2 by the elements pa+1,b − pa,b+1.The proof strategy establishes containment and absence of relations, then compares Hilbert series.Source: §3 · pp. 8–9

4.1 Digits Example

The digits example combines persistent-homology features with an SVM to illustrate classification, not to outperform existing algorithms. Directional sweeps add spatial information to barcode features.

4.1 Digits Example

  • The MNIST example demonstrates combining topological features with existing machine-learning methods, since topology alone cannot distinguish all digit classes.Source: §4.1 · p. 10

4.1.1 Topological Methods

  • Thresholded images are swept across rows and columns to add spatial information; using both Betti 0 and Betti 1 barcodes across four sweeps reveals further digit differences.The paper notes that spatial location can distinguish the loops of 6 and 9.Source: §4.1.1 · p. 10

4.1.2 Feature Selection

  • Four selected features applied to four sweeps and both Betti dimensions produce a 32-feature vector, with two features favoring longer bars.The selected polynomial exponents weight small bars or endpoints differently relative to large ones.Source: §4.1.2 · p. 12

4.1.3 Classification Results

  • The polynomial-kernel classifier has a 7.9% error, and the experiment is presented as an illustration rather than an attempt to outperform existing algorithms.Figure 5 shows typical problem digits.Source: §4.1.3 · p. 15
  • Common errors include loopless 5/2 pairs, style-dependent 7/3 and 4/9 pairs, and digits whose loops were added or removed.The paper relates some confusions to shared topology and handwriting variation.Source: §4.1.3 · p. 15

4.2 Hepatic Lesion Classification

For hepatic lesions, the paper constructs barcode features from intensity and border-based filtrations and evaluates SVM classification. Its reported comparison with the matching metric is close across most categories, with each method slightly better on some subsets.

4.2 Hepatic Lesion Classification

  • The 132-lesion CT dataset has nine diagnoses and highly variable lesion sizes; the analysis focuses on cysts, metastases, and hemangiomas because the data are unbalanced.Source: §4.2 · p. 16

4.2.1 Topological Methods

  • Border-distance filtrations use two-dimensional homology approximated by seven one-dimensional slices per lesion, with both Betti 0 and Betti 1 barcodes; this particularly improves results for hemangiomas.The border filtration is used because the lesions have no rotational orientation; intensity can also be filtered in either direction.Source: §4.2.1 · pp. 16–17

4.2.2 Feature Selection

  • Four barcode features, including long-bar and shorter-bar-sensitive features, are averaged over bars to reduce effects of lesion-size variation, yielding 224 features per lesion.Source: §4.2.2 · pp. 17–18

4.2.3 Classification Results

  • The SVM uses a Gaussian kernel with an exponential sweep to select γ, and classification accuracy is measured by leave-one-out cross-validation.Tables report results for 1D and 2D filtrations, lesion types, and lesion sizes.Source: §4.2.3 · p. 18
  • Topological features and the matching metric are virtually the same on the reported HeCM subset, and their results are close in most categories, with each slightly outperforming the other on some subsets.Source: §4.2.3 · pp. 18–19

4.3 Discussion

The discussion highlights the classification value of topological features and of combining topology with geometry, especially for hepatic lesions. It also presents algebraic coordinates as a way to organize barcode collections and databases of sampled shapes.

4.3 Discussion

  • Topological features show classifying power on real-world datasets and can be combined with traditional machine-learning methods.The examples use off-the-shelf algorithms to add classification features to the machine-learning toolbox.Source: §4.3 · p. 19
  • Combining topology with geometry is integral to classification in both datasets, with hepatic lesions illustrating potential gains.Source: §4.3 · p. 20
  • Algebraic coordinates organize barcodes and sampled shapes for databases, while motivating study of their relationship to barcode distances and multidimensional persistence.The discussion names bottleneck and Wasserstein distances and suggests analogous coordinates for multidimensional persistence modules.Source: §4.3 · p. 20

Persistence Bar Codes

The supplied passages identify the work as an arXiv preprint dated 2 April 2013 and name its authors.

Persistence Bar Codes

1 Introduction

The introduction motivates numerical, algebraic functions on barcodes and focuses on those unchanged by adjoining a zero-length interval. It proposes identifying this subring, describing its structure, and giving explicit algebra generators.

1 Introduction

  • Because barcodes are collections of intervals rather than numerical quantities, the introduction calls for numerical quantities that can feed standard machine-learning and cluster-analysis methods.It notes prior barcode applications to image patches and distinguishing handdrawn letters.Source: §1 · p. 22
  • A barcode with n intervals is represented by endpoint pairs, but its intervals are unordered, leading to a symmetric-product model.The coordinate ring for this model consists of multisymmetric polynomials, whose algebra generators can satisfy nontrivial relations.Source: §1 · p. 23
  • The coordinate rings form an inverse system whose inverse limit is A[Sp∞(R2)], an algebra freely generated by a minimal set of algebra generators.Source: §1 · p. 23
  • The construction identifies adjoining the zero interval [0, 0] with leaving the barcode unchanged, motivating the target subring's invariance requirement.Source: §1 · pp. 23–24
  • The paper seeks the algebraic functions that take the same value on a barcode and on that barcode with any zero-length interval adjoined.It aims to identify this subring, describe its structure, and give explicit generators for analyzing shape databases.Source: §1 · p. 24

2 The Ind-scheme B

The section models barcodes as symmetric products of intervals and builds a space in which all zero-length intervals are treated as equivalent. It then defines algebraic functions on that space and gives an algebraic condition for functions to ignore zero intervals.

2 The Ind-scheme B

  • Barcodes with n intervals are modeled by Spn(I), the symmetric product of intervals, and assembled into a directed system.Intervals are represented by pairs (x, y) with x ≤ y.Source: §2 · p. 24
  • Compatible functions on the barcode system must treat every zero-length interval as equivalent to no interval, regardless of its endpoint.Length-zero intervals correspond to zero vector spaces in the persistence-vector-space parametrization.Source: §2 · pp. 24–25
  • The space allows short intervals to approach a barcode limit, reflecting that small input perturbations can change interval lengths and add short intervals.The cited stability theorem for persistence diagrams is in.Source: §2 · p. 25
  • Since the full function ring is too large and even continuous functions are too complex to describe, the section defines algebraic functions using B as a colimit of algebraic varieties.Source: §2 · p. 25
  • Dn consists of functions whose restriction to each diagonal yi = xi is independent of xi, equivalently satisfying the induced derivative condition there.The section characterizes this algebraically using the vanishing of the corresponding partial derivative.Source: §2 · pp. 25–26

3 The ring of algebraic functions on B

The section develops the invariant-ring structure of the algebraic functions and identifies generators and bases for its finite-stage rings.

3 The ring of algebraic functions on B

  • A k-basis of Dn consists of monomials in which any positive exponent of ξ_i is paired with a positive exponent of η_i.Membership in Dn is characterized by this condition on monomials.Source: §3 · pp. 26–27
  • The orbit sums σ_α form a k-basis of the Sn-invariant subring of Dn.The symmetric-group action preserves the monomial basis, so orbit representatives yield the invariant basis.Source: §3 · p. 27
  • The inverse system of rings Dn has inverse limit D, embedded in the ring Λ2 of 2-multisymmetric functions.The construction uses restriction maps that discard variables beyond the first m pairs.Source: §3 · pp. 27–28
  • Theorem 1 states that D is freely generated, as a subalgebra of Λ2, by the elements p_a,b with b ≥ 1.The proof compares Hilbert series to establish that the generated polynomial subalgebra is all of D.Source: §3 · p. 29
  • A monomial basis for D is counted through orbit representatives and a generating-function recursion to establish its Hilbert series.The proof compares that series with the Hilbert series of k[∆].Source: §3 · pp. 29–30

4 Machine Learning on B with examples

The section illustrates barcode features in digit and hepatic-lesion classification, emphasizing spatial or geometric information alongside topology and standard machine-learning methods.

4.1.1 Topological Methods

  • Directional sweeps of thresholded digit images add spatial information: the loop in a 9 and a 6 appears in different locations, and four sweeps with Betti 0 and 1 reveal further differences.The method treats pixels as graph vertices and connects adjacent pixels, including diagonals.Source: §4.1.1 · p. 31, Figure 1 · p. 32, Figure 2 · p. 32

4.1.2 Feature Selection

  • Four selected invariant features across four sweeps and two Betti dimensions form a 32-feature vector; some features emphasize longer bars, while others account for all bars and endpoints.The features are visualized with MDS and a matrix spectrum.Source: §4.1.2 · p. 33, Figure 3 · p. 34, Figure 4 · p. 35

4.1.3 Classification Results

  • The polynomial-kernel SVM had a 7.9% error rate on the digit example, with common confusions including loopless 5s and 2s.Other frequent errors involved handwriting style or topological changes such as adding or removing a loop.Source: §4.1.3 · p. 36, Figure 5 · p. 36

4.2.1 Topological Methods

  • For lesions, boundary-distance filtrations add geometric information to pixel-intensity filtrations; seven slices per lesion use both Betti 0 and Betti 1 barcodes.The two-dimensional filtration is approximated with one-dimensional barcode slices because it is computationally intensive; the setup yields 56 barcodes per lesion.Source: §4.2.1 · pp. 37–38, Figure 6 · p. 38

4.2.2 Feature Selection

  • Lesion features cover both long and short bars, average values across bars to reduce lesion-size effects, and produce 224 features per lesion.The 224 features come from 56 barcodes and four features per barcode.Source: §4.2.2 · pp. 38–39

4.2.3 Classification Results

  • For the HeCM subset, topological features and the barcode matching metric produced virtually the same result; across most categories, each slightly outperformed the other on some subsets.The comparison is reported relative to results in.Source: §4.2.3 · pp. 39–40

4.3 Discussion

  • The examples demonstrate real-world classification with off-the-shelf learners and highlight topology–geometry combinations; the identified algebraic coordinates can organize barcode datasets.The discussion also proposes studying relationships with barcode metrics and extending analogous coordinates to multidimensional persistence modules.Source: §4.3 · pp. 40–41
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