Monotone Path Filtration

• Monotone path filtration is a method that restricts multiparameter persistence modules along continuous, coordinate-wise nondecreasing paths for one-parameter barcode analysis.
• It generalizes straight-line slices, allowing detection of subtle topological differences by flexibly traversing parameter spaces and enhancing discriminatory power.
• The approach enables efficient computation by discretizing paths and applying standard persistence algorithms, with stability guaranteed through interleaving distance bounds.

Monotone path filtration is a construction in multiparameter persistent homology that restricts a $d$-parameter persistence module or multifiltration along a continuous path in $\mathbb{R}^d$ whose coordinate functions are nondecreasing, thereby producing a one-parameter filtration to which standard barcode algorithms apply (Sun et al., 31 Jul 2025). In the formulation of Sun–Corbet–Marr, it generalizes the straight-line slices used by the classical matching distance, and it supports comparison of two data sets by taking a supremum of bottleneck distances over monotone paths. Because multiparameter persistence modules can capture more topological differences across data instances than a single parameter, replacing straight-line slices by arbitrary monotone paths enlarges the family of one-parameter restrictions available for analysis (Sun et al., 31 Jul 2025).

1. Formal setting in multiparameter persistence

Let $K$ be a field. A $d$-parameter persistence module is a functor

$M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$

where $\mathbb{R}^d$ is ordered coordinate-wise, so $p \le q$ iff $p_i \le q_i$ for all $i$, and each structure map

$M_{p,q} : M(p) \to M(q)$

is a linear injection whenever $\mathbb{R}^d$0 (Sun et al., 31 Jul 2025). Often $\mathbb{R}^d$1 arises as homology of a multifiltration

$\mathbb{R}^d$2

A monotone path in $\mathbb{R}^d$3 is a continuous map

$\mathbb{R}^d$4

whose coordinate functions are nondecreasing, meaning that for all $\mathbb{R}^d$5 and all $\mathbb{R}^d$6,

$\mathbb{R}^d$7

Given a multifiltration $\mathbb{R}^d$8 or a module $\mathbb{R}^d$9 indexed by $K$0, such a path induces a one-parameter filtration

$K$1

with structure maps $K$2, or equivalently a one-parameter persistence module

$K$3

Concretely, the induced sublevel at time $K$4 is written

$K$5

Passing to homology, or starting from $K$6 directly, one applies a standard one-parameter barcode algorithm to a sequence of the form

$K$7

2. From straight-line slices to arbitrary monotone paths

The classical matching distance compares two $K$8-parameter modules $K$9 by looking only at straight lines $d$0 of positive slope in parameter space. Writing

$d$1

one restricts $d$2 to $d$3 and takes a weighted bottleneck distance

$d$4

(Sun et al., 31 Jul 2025).

Monotone path filtration replaces each such slice by an arbitrary piecewise-linear monotone path $d$5. This enlarges the family of restrictions under consideration, and in particular

$d$6

This inequality records a structural inclusion: straight-line slices are special cases of monotone paths. A plausible implication is that features invisible on diagonal or near-diagonal slices may become detectable when the restriction is allowed to change direction while remaining coordinate-wise nondecreasing.

The distinction is consequential in multiparameter settings. Straight-line slicing imposes a single joint progression through all parameters, whereas a monotone path can traverse one coordinate range first and another later. This preserves the partial-order compatibility of the original filtration while relaxing the geometric rigidity of straight slices.

3. Path-based distances and stability

For any two modules $d$7 and any monotone path $d$8, the bottleneck distance between the restricted barcodes is controlled by the interleaving distance on the full module: $d$9 (Sun et al., 31 Jul 2025). Each fixed path therefore yields a stable descriptor.

The path-based bottleneck distance is defined by

$M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$0

Because slices $M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$1 are special monotone paths, one has

$M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$2

Moreover, by the usual matching-versus-interleaving stability,

$M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$3

These inequalities situate monotone path filtration between two familiar quantities. It is at least as discriminating as the matching distance, in the sense encoded by the supremum over a larger family of restrictions, while remaining bounded above by interleaving distance. A common misconception is that enlarging the class of admissible restrictions necessarily compromises robustness; the stated bound shows that this is not the case for the bottleneck distance computed along any monotone path.

4. Computation and software realization

The basic computational procedure is to discretize a monotone path $M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$4 by a partition

$M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$5

evaluate the filtration or module at the points $M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$6, and then run a standard one-parameter persistence reduction on the resulting sequence (Sun et al., 31 Jul 2025). In the pseudocode given for a fixed path, the steps are:

1. compute $M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$7;
2. extract the complex or vector space at each $M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$8;
3. return the barcode of the sequence $M : (\mathbb{R}^d, \le) \to \mathrm{Vect}_K$9.

The implementation notes specify that extract_complex builds the simplicial or cell complex at parameter $\mathbb{R}^d$0, or directly reads off the boundary matrix of $\mathbb{R}^d$1, and that PERSISTENCE_BARCODE runs a standard one-parameter reduction such as matrix reduction in $\mathbb{R}^d$2 time, with $\mathbb{R}^d$3 the number of simplices (Sun et al., 31 Jul 2025).

If the path is subdivided into $\mathbb{R}^d$4 steps and each step differs by $\mathbb{R}^d$5 simplices, one can update the boundary matrix incrementally in $\mathbb{R}^d$6 per step rather than rebuilding from scratch. In practice one uses streaming updates or vineyard algorithms to exploit the fact that $\mathbb{R}^d$7 is close to $\mathbb{R}^d$8. The overall cost is roughly $\mathbb{R}^d$9 in the worst case, but typical implementations using PHAT, Ripser-like optimizations, or CGAL for geometry achieve far better performance (Sun et al., 31 Jul 2025).

The software interface described for this framework includes the following components.

Component	Role
ModuleLoader	reads a multifiltration, for example a bifiltration from rhomboid tiling
Path	stores breakpoints $p \le q$0 in $p \le q$1 and a parameterization $p \le q$2
restrict_to_path(M, Path)	builds the one-parameter filtration $p \le q$3
compute_barcode(filtration)	invokes a PH library such as PHAT, Ripser, or CGAL
query_distance(M,N,Path; metric="bottleneck" or "wasserstein")	restricts both modules, computes barcodes, and returns diagram difference

A Python-style usage example loads two bifiltrations, defines a monotone path in $p \le q$4 with breakpoints $p \le q$5, $p \le q$6, $p \le q$7, and computes the bottleneck distance along that path (Sun et al., 31 Jul 2025). This emphasizes that the construction is not merely existential; it is designed to interoperate with established persistent homology software.

5. Illustrative behavior in $p \le q$8

Two examples clarify what becomes visible when the restriction is allowed to follow non-straight monotone paths (Sun et al., 31 Jul 2025).

In the first example, two point clouds each form a noisy circle, but one has a small indentation in the $p \le q$9-direction. Along the diagonal path $p_i \le q_i$0, the two $p_i \le q_i$1 barcodes look nearly identical, each exhibiting a single long feature. Along the L-shaped path

$p_i \le q_i$2

the indentation becomes visible in the first segment, yielding an extra short bar in $p_i \le q_i$3. Hence $p_i \le q_i$4 is larger.

In the second example, two grayscale images are viewed as a superlevel-sublevel bifiltration, and one image is slightly brighter in the top half. If $p_i \le q_i$5 tracks first intensity and then a scale parameter, the barcode difference is small. If a path first traverses intensity up to the midpoint and then turns to traverse scale, one captures the regional brightness difference as an early birth of a connected-component feature, producing a larger distance.

These examples are limited in scope, but they make precise a central point: straight-line slices can miss differences that become apparent when the order of parameter traversal is varied without violating monotonicity. This suggests a use case for data sets whose distinguishing structure is aligned with non-diagonal directions in parameter space.

6. Distinct usage of path-filtration in modal logic

The expression “path-filtration” also appears in modal logic with a different technical meaning. Gasquet introduces a path-based filtration method for an unravelled canonical pointed model

$p_i \le q_i$6

where each world $p_i \le q_i$7 has exactly one shortest $p_i \le q_i$8-path

$p_i \le q_i$9

and $i$0 is the length of that path (Gasquet, 15 Jul 2025).

For a modal formula $i$1 of depth $i$2, the construction defines sets of relevant subformulas $i$3, local restrictions

$i$4

and a path-vector

$i$5

Two worlds are equivalent when they have the same depth and the same path-vector: $i$6 The quotient classes are path-types, and truncation to depth $i$7 yields a finite path-filtration $i$8.

In this setting, monotonicity refers to two facts. First, enlarging the set of relevant subformulas makes the induced equivalence finer: $i$9 Second, the map $M_{p,q} : M(p) \to M(q)$0 is monotone with respect to generated submodels: if $M_{p,q} : M(p) \to M(q)$1 is a generated submodel of $M_{p,q} : M(p) \to M(q)$2, then $M_{p,q} : M(p) \to M(q)$3 embeds as a generated submodel into $M_{p,q} : M(p) \to M(q)$4 (Gasquet, 15 Jul 2025).

The correctness theorem states that for $M_{p,q} : M(p) \to M(q)$5 and $M_{p,q} : M(p) \to M(q)$6 with $M_{p,q} : M(p) \to M(q)$7,

$M_{p,q} : M(p) \to M(q)$8

The associated finite model property gives

$M_{p,q} : M(p) \to M(q)$9

and the satisfiability problem for any quasi-dense extension of $\mathbb{R}^d$00 lies in NEXPTIME, while validity lies in co-NEXPTIME (Gasquet, 15 Jul 2025).

This modal-logic construction is terminologically related but mathematically separate from monotone path filtration in persistent homology. The former records the history of formula satisfaction along canonical paths in a Kripke model; the latter restricts a multiparameter filtration along coordinate-wise nondecreasing paths in $\mathbb{R}^d$01. The shared terminology reflects a common emphasis on preserving pathwise information that standard quotienting or straight-line slicing would otherwise discard.