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Bridging persistence of discrete and continuous 1D maps: Total variation, permutation equivalence, and diagonal affine transformations

Matthew Blake, Firas A. Khasawneh

arXiv:2610.31391v1math.ATcs.CG

TL;DR

The paper extends the equality between total persistence and total variation beyond generic continuous functions on compact intervals, and examines how critical-value order and function transformations relate to persistence diagrams. It concludes that the critical-value sequence determines persistence pairings, while computations are available for small cases and general derivative or antiderivative relationships do not hold.

  • Problem

    Prior work equates total persistence and total variation for tame, separating continuous functions on compact intervals, while broader function settings and predictable diagram changes remain under examination.

    Source: §1 · p. 2, §4.6 · p. 15

  • Method

    The paper derives a critical-value formula for total variation and persistence, studies persistence diagrams through ordinal critical-value sequences, and analyzes function transformations.

    Source: §1 · p. 2, §4.6 · p. 15, §5.1 · p. 17

  • Results

    The persistence diagram is determined by the critical-value sequence, and the paper characterizes diagonal affine effects but finds no general relationship with derivatives or antiderivatives.

    Source: §1 · p. 2, §7 · p. 34

  • Takeaways & Limitations

    The same persistence tools and results can apply to continuous functions and discrete time series, with critical-value ordering sufficient to determine ordinal pairings.

    Source: §7 · p. 34

    The computed permutation formulas cover generic interval functions with finitely many critical points only for n ≤ 6; no general formula is known for larger n.

    Source: §5.2 · p. 23

Abstract

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Fundamental to the field of topological signal processing (TSP) is the \textit{persistence diagram} (PD), or barcode, of a signal, which gives noise-resistant measures of topologically significant features as distinct from noise. Existing literature proves that the total persistence computed from these diagrams is equal to the total variation, for certain continuous signals on compact domains. We provide a closed formula for this shared quantity, and generalize to all continuous and discrete functions on bounded domains, and to certain functions on non-compact 1D manifolds. Extending the identification of continuous and discrete persistence, we investigate the similarities of functions with the same PDs, and prove that, for any $n$, the persistence diagram can be calculated as a function of the sequence of critical values; we compute these formulas for $n \leq 6$. Finally, we examine which function transformations cause predictable changes in the PD of the transformed function, characterizing affine and linear transformations.

7 Conclusions and Further Directions (34)

The listed material covers generalizations of the equivalence domain, computed persistence diagrams, diagonal affine transformations, and direct computation.

Generalizing Equivalence Domain

  • Appendix B is organized around formula derivation, non-separating and discrete functions, circle functions, non-tame functions, and open or unbounded domains.Source: §7 · p. 1

Persistence Diagram Computed Examples; Diagonal Affine Transformation Cases

  • The listed later material includes computed persistence-diagram examples and cases for diagonal affine transformations.Source: §7 · p. 1

Direct Computation

  • The direct-computation material lists persistence-diagram preimage illustrations and computed-PD flowcharts.Source: §7 · p. 1

1 Introduction

For one-dimensional signals, persistence diagrams pair local minima and maxima to distinguish larger-scale variation from finer details often attributable to noise. This paper extends total-persistence and total-variation results, studies how critical-value order determines diagrams, and characterizes selected transformations.

1 Introduction

  • For one-dimensional signals, sublevel persistent homology identifies underlying shapes and distinguishes them from noise, while their characterization is simpler than for higher-dimensional point clouds.The passage contrasts 1D signals with point-cloud filtrations such as Čech and Vietoris–Rips complexes [7, p.27-28].Source: §1 · p. 2
  • Persistence diagrams pair local minima and maxima by the elder rule, with persistence given by the absolute difference between paired values and total persistence their sum.The pairings are plotted by birth and death time; smaller differences often correspond to finer details or noise.Source: §1 · p. 2
  • Biswas et al. equate total persistence and total variation for tame, separating continuous functions on compact intervals, while this paper evaluates non-separating functions directly.The earlier result applies to functions whose critical points have unique critical values; the direct approach avoids approximation by separating functions.Source: §1 · p. 2
  • The paper gives a weighted alternating-sum formula for total variation and persistence, extending it to infinite critical points, discrete time series, and certain noncompact 1D manifolds.It presents the same tools as applicable to persistence of discrete or continuous signals.Source: §1 · p. 2
  • The Permutation Theorem says a function’s persistence diagram is determined by its critical-value sequence, and the paper computes the associated permutation formulas through n = 6.It also identifies two diagram-preserving operations and conjectures they connect any permutations with the same diagram.Source: §1 · p. 2
  • The paper characterizes diagram changes under diagonal affine transformations, finds conditions for effects of adding functions, and reports no general relationship for diagrams of functions and their derivatives or antiderivatives.The no-general-relationship result also covers discrete analogs.Source: §1 · p. 2

2 Contributions

The contributions extend total variation and persistence formulas across broader function classes, relate diagrams to critical-value order, and characterize selected diagram-preserving operations and transformations.

2 Contributions

  • The total-variation and total-persistence formula extends to continuous or discrete functions on bounded domains and certain piecewise C1, absolutely convergent functions on unbounded domains with two asymptotes.In each case, the quantities are given explicitly as functions of critical values.Source: §2 · p. 3
  • A persistence diagram is completely determined by the relative ordering of a function’s critical-value sequence.Source: §2 · p. 3
  • For small n ≤ 6, the paper computes persistence-diagram formulas from ordinal sequences of local-extremum values, but gives no general formula for larger n.Source: §2 · p. 3
  • Reversal and persistence transposition preserve persistence diagrams, and the paper conjectures that sequences of these operations connect all functions with the same diagram.Source: §2 · p. 3
  • The effects of diagonal affine transformations on a signal’s persistence diagram are completely characterized.Source: §2 · p. 3
  • Adding two functions has no general rule for its effect on persistence diagrams, though the paper identifies specific conditions under which the effect can be characterized.Source: §2 · p. 3
  • There is no meaningful general relationship between a function’s persistence diagram and those of its difference or derivative, or its summation or integral.The paper supports this by showing that any two functions can be connected through these operations and diagram-preserving function changes.Source: §2 · p. 3

3 Domains, Level Sets, and Persistence

The section defines the domains and critical points considered, then uses level sets and persistence diagrams to describe how topological features change across function values. Its framework applies to discrete and continuous maps, with persistence tracked through connected components and loops.

3.1 Discrete and Continuous Maps

  • Discrete maps have disconnected domains, typically {1, . . . , n}, while continuous maps satisfy the usual real-analysis definition of continuity.Source: §3.1 · p. 3

3.2 Domains, Genericity, and Critical Points

  • The paper considers bounded functions on finite intervals or circles and treats critical points as extrema, excluding inflection points.For discrete functions, critical points are defined using differences and include domain endpoints.Source: §3.2 · pp. 3–4

3.3 Level sets

  • Between critical values, level-set topology is unchanged, so characterizing sublevel and superlevel sets requires examining the sets at critical values.The framework assumes a finite interval unless otherwise indicated; the circle has no boundary points.Source: §3.2 · p. 3, §3.3 · pp. 4–5

3.4 Persistence Diagrams

  • Persistence diagrams record feature births and deaths: ordinary persistence tracks connected components, while relative persistence tracks loops in superlevel sets.The elder rule lets the older feature persist when features merge; on an interval, the first maximum creates no loop, so loop counts are one below the number of superlevel components.Source: §3.3 · p. 5, §3.4 · pp. 5–6

4 Total Variation and Total Persistence

This section relates total variation to total persistence through critical-value formulas and extends their equality across circular, non-separating, and finite discrete domains. It also clarifies existence and equality conditions for non-tame functions, including extensions to interval closures and unbounded domains.

4.1 Computation and Comparison

  • The total variation is a weighted alternating sum of critical values, and the critical-value sequence determines the total persistence and entire persistence diagram.When the first critical point is a local minimum, the alternating sum is nonnegative.Source: §4.1 · p. 8

4.2 Formula Derivation

  • For continuous functions with finitely many isolated critical points, counting critical-value contributions shows that total variation equals total persistence.The derivation uses monotonicity between consecutive critical points and alternating local minima and maxima.Source: §4.2 · pp. 9–10

4.3 Circular Domains

  • On the circle, total variation equals total persistence, and the persistence diagram is symmetric because every critical value supplies both a birth and a death.Unlike the interval case, circular domains have no boundary-point births that create diagram asymmetry.Source: §4.3 · p. 10

4.4 Non-Separating Functions

  • For continuous non-separating functions with finitely many critical points and horizontal segments, total variation still equals total persistence, including diagrams with repeated points.Equal-valued components may be assigned seniority arbitrarily without changing the diagram; horizontal segments can be shrunk to points for the proof.Source: §4.4 · pp. 11–13

4.5 Discrete Functions

  • For a finite discrete function, linear interpolation preserves total variation and persistence, yielding equality between its total variation and total persistence.Only local extrema contribute persistence, and their pairing is identical for the discrete function and its interpolation.Source: §4.5 · pp. 13–14

4.6 Non-Tame Functions

  • A continuous function on a compact domain can have infinitely many oscillations yet finite total variation when the contributions of its critical points form a convergent sum.The passage identifies this finite-variation condition but does not establish total-persistence equality for non-tame functions.Source: §4.6 · pp. 14–15
  • On open or clopen intervals, endpoint limits allow evaluation by continuous extension to the closure, preserving equality of total variation and total persistence.Both quantities are defined only when the function has limits at all endpoints.Source: §4.6 · p. 15
  • For absolutely continuous functions, total persistence equals total variation, including on unbounded domains when the function approaches a limit at infinity.On the real line or a half-line, these limits allow total variation to be computed.Source: §4.6 · pp. 16–17
  • The Cantor function shows absolute continuity is not necessary for equality: its total persistence and total variation are both one despite its lack of absolute continuity.The passage cites [1, sec. 4.5.3, p. 136].Source: §4.6 · p. 16
  • For continuous functions, bounded variation is necessary and sufficient for total variation and total persistence to exist, while absolute continuity is required for the convenient formula.Absolute continuity is sufficient, but not necessary, for total variation and total persistence to exist.Source: §4.6 · p. 16

5 Permutation Equivalence

The section studies how critical-value orderings and their permutations relate to persistence pairings and diagrams, including computed permutation sequences through n = 6 and transformations connecting discrete signals with the same diagram. It shows that persistence transpositions and reversals preserve diagrams, while the converse remains a conjecture and a non-persistence transposition alone need not change a diagram.

5.1 Permutation Theorem

  • Having the same persistence diagram is not sufficient to conclude that two functions have the same ordering of critical values.The section explicitly states that the converse of its sufficient condition is false.Source: §5.1 · p. 19
  • Under the section’s sufficient condition, the persistence diagrams of f and F have matching pairings by ordinal index; corresponding critical points also have the same local-extremum type.The result applies to continuous functions on an interval because their persistence diagrams agree with those of the discrete sequences of their critical points.Source: §5.1 · pp. 17, 18, 20

5.2 Persistence Diagram Computation

  • For a generic interval function, the section characterizes ordinary and relative diagram pairings with a map from each birth level to its paired level, using permutations of interior extrema values.The formulas are reversed when the first critical point is a local maximum, by the duality of ordinary and relative persistence.Source: §5.2 · pp. 21–22
  • The essential class is the same for every function on the interval: (y1, yn).The section therefore focuses its computation on the ordinary and relative parts of the diagram.Source: §5.2 · p. 22
  • Formulas for πA and πB have been computed through n ≤ 6 for generic discrete or continuous interval functions with finitely many critical points; no general formula is known.Further cases n ≥ 7 and extensions to circles or other domains are identified as future research.Source: §5.2 · p. 23

5.3 Persistence Diagram Preimage

  • For the persistence transposition τ_i, swapping f(x_i) and f(x_{i+2}) preserves the persistence diagram when x_{i+1} is a local maximum below x_{i-1} and x_{i+3}, or a local minimum above both.Source: §5.3 · pp. 23–24
  • A finite sequence of persistence transpositions and reversals connects functions with identical persistence diagrams.This is a sufficient condition; the converse is stated as Conjecture 1.Source: §5.3 · p. 24
  • A non-persistence transposition does not by itself imply different persistence diagrams: the passage gives a pair separated by such a swap whose diagrams are equal.The example swaps τ*4.Source: §5.3 · p. 25

6 Signal Transformations

This section examines how specified transformations affect persistence diagrams and how they connect discrete and continuous functions. It characterizes diagonal affine transformations and describes connections through summation, differences, persistence-diagram operations, and calculus-related operations.

6.1 Diagonal Affine Transformation Theorem

  • Horizontal translations, scalings, and reflections preserve total variation, total persistence, and the persistence diagram; vertical shifts preserve the totals but shift the diagram diagonally.Positive vertical scaling scales all three, while vertical reflection preserves the totals and reflects ordinary and relative points through the origin and essential points across the anti-diagonal.Source: §6.1 · p. 26

6.2 Function Addition

  • In general, the persistence diagram of f + g bears no necessary resemblance to either input diagram, because addition can change the functions’ local extrema.When g is constant, a scalar multiple of f, or has sufficiently small range, the diagram can be related through the corresponding affine transformation or stability.Source: §6.2 · p. 26
  • For a function f, with F the restriction of f to its local extrema, Dgm(f + g) consists of points (A + g(F^-1(A)), B + g(F^-1(B))) for each (A, B) in Dgm(f).Source: §6.2 · p. 27

6.3 Integration and Differentiation

  • Neither summation nor differences preserves the equivalence relation of having identical persistence diagrams: examples show changed diagram points and, for differences, a changed point count.The running sums of functions with identical diagrams can have different diagrams.Source: §6.3 · pp. 29, 31
  • No nontrivial invariant remains under persistence-diagram replacement together with summation or differences, since sequences of these operations can connect any two discrete signals.The section frames this as the absence of a nontrivial equivalence relation invariant under the operations.Source: §6.3 · pp. 27, 31

6.3.1 Calculus Connection Theorem

  • For any two discrete integer-valued functions, a sequence of the theorem’s listed operations can transform one into the other.The preceding section identifies the connecting operations as summation, difference, persistence-diagram replacement, and its inverse.Source: §6.3.1 · p. 31, §6.3 · p. 31
  • In the construction, inserting points that respect monotonicity creates no new local extrema and leaves each function’s persistence diagram unchanged.Source: §6.3.1 · p. 33
  • Discrete real-valued functions can be transformed into one another by a sequence of specified operations.The corollary establishes this existence result for functions with domains I_n and real-valued outputs.Source: §6.3.1 · p. 33
  • Continuous functions with finitely many critical points can likewise be connected by a sequence of specified operations.The proof reduces the continuous case to discrete functions defined by their critical values, which retain the original persistence diagrams.Source: §6.3.1 · p. 34
  • Future work could seek meaningful invariants by restricting operation sequences, but merely bounding their length cannot yield an invariant because equivalence classes require transitivity.Source: §6.3.1 · p. 34

7 Conclusions and Further Directions

The conclusions frame total persistence as the finite total variation apportioned among feature pairs, with larger-persistence pairs corresponding to large-scale signal features. The appendices support the generalization beyond generic functions and summarize results about critical values and persistence diagrams.

7 Conclusions and Further Directions

  • Large-persistence pairs correspond to large-scale signal features, while pairs with less persistence account for less total variation and are more likely noise or negligible.The total variation and total persistence provide a shared quantity for interpreting how variation is divided among persistence pairs.Source: §7 · p. 34
  • The total-variation/total-persistence equivalence extends beyond generic continuous functions, indicating that the relation is not merely an artifact of restrictive function or domain assumptions.The conclusions also extend the equivalence to discrete signals or time series, where many tools can be used with equal facility as in the continuous setting.Source: §7 · p. 34
  • The persistence diagram can be computed from the sequence of critical values, and sequences with the same relative ordering pair the same ordinal critical values.The conclusions also describe signal transformations as unifying the continuous and discrete settings.Source: §7 · p. 34

A.1 Negation Lemmas

  • The appendix states that negation reverses filtration order and swaps minima with maxima, relating ordinary persistence of a function to relative persistence of its negation.It identifies the discrete-case statements as Belton’s [2, Theorem 3.14].Source: §A.1 · p. 35

A.2 Other Lemmas

  • For piecewise C1 functions, bounded variation implies absolute continuity; local-extrema accumulation points are nowhere dense where the function has no constant subinterval.The appendix also states that a function on an open interval whose endpoint limit does not exist cannot have bounded variation.Source: §A.2 · pp. 36–38

B.1 Formula Derivation Proof

  • The formula derivation establishes equality of total variation and total persistence for generic continuous functions and argues that the persistence diagram remains identical when tied critical values leave component seniority ambiguous.The diagram is independent of the strict total ordering used to break ties among critical values.Source: §B.1 · pp. 38, 40

B.2 Non-Separating Functions

This section extends the equality of total variation and total persistence to additional domain and function classes, and establishes persistence-diagram invariance under non-generic choices and flat-segment collapse. It also presents critical-value formulas and permutation-based persistence-diagram calculations for discrete functions, including cases with up to six critical points.

B.2 Non-Separating Functions

  • For continuous interval functions with finitely many isolated critical points and horizontal segments, the persistence diagram is independent of persistence ordering at any level.The result follows by collapsing flat intervals, which preserves the diagram, and applying invariance under swaps of equal-level critical points.Source: §B.2 · p. 41
  • For non-generic continuous interval functions with finitely many critical points and horizontal segments, total variation equals total persistence and is given by a weighted sum of critical values.Collapsing flat intervals preserves both quantities because horizontal segments have zero variation.Source: §B.2 · pp. 41–42

B.3 Discrete Functions

  • A finite discrete function and its linear interpolation have equal total variation and total persistence, and the same holds for interpolations that remain strictly monotonic between critical points.The discrete total variation and total persistence are given by the same weighted critical-value formula as for continuous interval functions.Source: §B.3 · pp. 42–43

B.4 Functions on the Circle

  • On the circle, continuous functions with finitely many isolated critical points satisfy equality of total variation and total persistence, computed as an alternating sum of critical values.For functions with horizontal segments, collapsing flat intervals preserves both quantities; the equality also extends to discrete circle functions with finitely many isolated critical points and horizontal segments.Source: §B.4 · pp. 43–44

B.5 Non-Tame Functions

  • For absolutely continuous functions on a compact interval, total persistence equals total variation and can be computed as a weighted sum of critical values.The result covers functions with infinitely many local extrema as well as those with finitely many, after shrinking horizontal sections without changing either quantity.Source: §B.5 · p. 44
  • The Cantor function is not absolutely continuous yet has total variation and total persistence of 1, showing absolute continuity is not necessary for their equality.The preceding argument also concludes tPer(f) = tVar(f).Source: §B.5 · pp. 46–47

B.6 Open and Unbounded Domains

  • For bounded absolutely continuous functions on open intervals with finite endpoint limits, extending by those limits preserves total variation and the persistence diagram.The argument also obtains tPer(f) = tVar(f) for this setting.Source: §B.6 · p. 47
  • For piecewise C1 functions of bounded variation on the real line, persistence diagrams arise as limits over expanding finite intervals; the same construction applies to half-lines.The proof uses truncations and monotone perturbations that preserve total variation and total persistence.Source: §B.6 · pp. 48–50
  • Total persistence equals total variation for absolutely continuous signals on bounded domains and piecewise C1 signals of bounded variation on unbounded domains.The theorem states that the shared quantity has the expression given in the passage.Source: §B.6 · p. 50

C Persistence Diagram Computed Examples

  • For discrete functions, persistence diagrams are computed from critical-value sequences using permutations πA and πB, with the formulas depending on whether the first critical point is a minimum or maximum.The formulas cover all functions with n critical points; for n ≥ 7, enumeration grows on the order of O((n/2 − 1)!).Source: C Persistence Diagram Computed Examples · pp. 50–51, §12 · p. 52

12 In particular, we can show

  • For n=4, assignments to y(2) and y(3) exchange x2 and x3 according to whether x1 is a local minimum or maximum, while πA and πB are both identity maps.For n=5 with x1 a local minimum, πA is trivial, whereas πB is nontrivial and can take one of two values.Source: C Persistence Diagram Computed Examples · p. 52

D Diagonal Affine Transformation Cases

The Diagonal Affine Transformation Theorem specifies how affine changes affect total variation, total persistence, and persistence diagrams. The direct-computation discussion relates persistence diagrams to sequences of local extrema and illustrates functions sharing a diagram.

D Diagonal Affine Transformation Cases

  • Horizontal shifts, scaling, and reflection preserve total variation, total persistence, and the persistence diagram.The same invariance holds for partial positive horizontal scaling.Source: D Diagonal Affine Transformation Cases · pp. 53–54
  • Vertical shifts leave total variation and total persistence invariant while translating the persistence diagram diagonally; positive vertical scaling multiplies all three by its factor.Under vertical reflection, total variation and total persistence remain invariant, while ordinary and relative diagram points reflect through the origin and the essential point maps to (−y(n), −y(1)).Source: D Diagonal Affine Transformation Cases · pp. 53, 55
  • The transformation rules also apply to rotations, reflections, vertical shifts, and vertical scaling of circle signals, and to affine transformations of discrete functions.For circle functions with finitely many isolated critical points, the construction in Proposition 8.7 preserves the diagram and total variation.Source: D Diagonal Affine Transformation Cases · p. 56

E Direct Computation

  • Functions with the same sequence of local extrema have the same persistence diagram, including when changes between extrema preserve monotonicity.Figure 24 illustrates two functions with the same local extrema and identical persistence diagrams.Source: E Direct Computation · p. 56, Figure 24 · p. 57

E.1 PD Preimage Illustrations

  • The preimage illustrations show connecting persistence transpositions and identify black-and-blue functions as reversals of one another.A dotted line with τ*_i denotes a swap that is not a persistence transposition.Source: §E.1 · p. 56

E.2 Computed PD Flowcharts

This section gives case-by-case flowcharts for computing persistence diagrams from generic interval functions with five or six local extrema, using ordered critical values and branch conditions.

E.2 Computed PD Flowcharts

  • For five local extrema with ordered levels y(1) < y(2) < y(3) < y(4) < y(5), the diagram’s ordinary and relative pairs include the alternatives listed in the flowchart.Both listed alternatives include the relative pair (y(3), f(x3)) in Rel(f), while their ordinary pairs differ.Source: §E.2 · pp. 57–58
  • The six-extrema flowchart branches on comparisons between function values and on whether selected values equal y(3).Its listed conditions include comparisons at x1 and x2, x4 and x2, and x3 and x5.Source: §E.2 · pp. 59–60
  • The six-extrema legend identifies the interior local minima and maxima used in the flowchart.The flowchart also distinguishes a local minimum from a local maximum using whether f(x1) is below or above f(x2).Source: §E.2 · pp. 59–60, Table 2 · p. 60
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