Persistent Homology Convolutions

Updated 6 July 2026

Persistent homology convolutions are methods that integrate convolution operations with persistent homology to capture local motifs and yield robust topological features.
They address limitations of traditional persistence by applying convolution at various stages—such as pre-processing, local window extraction, lattice operations, or stabilization—across diverse settings.
These constructions are validated theoretically and empirically in applications from image-based motif detection to histopathology, lattice-structured data, and sheaf-theoretic frameworks.

Persistent homology convolutions designate several constructions that combine convolutional operators with persistent homology, but the locus of convolution differs across the literature. Depending on context, convolution is applied to the input signal before persistence is computed, to overlapping local windows through persistent homology-based feature extraction, to lattice-indexed invariants of multiparameter persistence modules through meet/join operations, to PH-derived real-valued functions on input or parameter space as a stabilizing average, or to sheaf-theoretic realizations of persistence objects as a thickening operator (Solomon et al., 2022, Pothagoni et al., 18 Jul 2025, Riess et al., 2020, Bendich et al., 2015, Kashiwara et al., 2017). Across these formulations, the common objective is to make topological information compatible with locality, translation, order structure, or stability requirements that ordinary persistent homology or ordinary convolution alone do not directly address.

1. Scope and terminological landscape

The literature does not use a single standardized meaning for the phrase. A common misconception is that persistent homology convolutions always mean convolving persistence diagrams themselves. The principal lines of work summarized here do not adopt that interpretation. Instead, they treat convolution as motif detection before persistence, as a local topological receptive-field mechanism, as an order-theoretic operator on multiparameter persistence data, as a smoothing device on unstable PH-derived outputs, or as a categorical operation in sheaf-theoretic persistence (Solomon et al., 2022, Pothagoni et al., 18 Jul 2025, Riess et al., 2020, Bendich et al., 2015, Kashiwara et al., 2017).

Construction	Role of convolution	Representative source
Convolutional persistence	Convolve data with filters before computing persistence	(Solomon et al., 2022)
Persistent Homology Convolutions	Compute PH on local windows and combine via a kernel	(Pothagoni et al., 18 Jul 2025)
Lattice-theoretic convolutions	Replace translations with meet/join on a finite lattice	(Riess et al., 2020)
Stabilizing unstable PH output	Convolve a PH-derived function on input/parameter space	(Bendich et al., 2015)
Microlocal sheaf convolution	Use $F\star G = Rs_!(F\boxtimes G)$ as a thickening operator	(Kashiwara et al., 2017)

Two recurrent themes organize these approaches. First, several papers argue that ordinary persistence can be too coarse when the relevant information is local or motif-dependent. Second, several papers argue that ordinary convolution may be algebraically mismatched to persistence data, either because topology should be extracted locally rather than globally, or because multiparameter persistence is indexed by a poset or lattice rather than by a translation group.

2. Convolutional persistence as motif detection

In "Convolutional Persistence Transforms" (Solomon et al., 2022), convolution is applied to data defined over simplicial or cubical complexes before persistent homology is computed. The central idea is to interpret a filter as a local motif. For an image $f:P\to \mathbb{R}$ on a rectangle $P \subset \mathbb{Z}^d$ , with filter $g:B\to \mathbb{R}$ on another rectangle $B\subset \mathbb{Z}^d$ and stride vector $k$ , the convolution is

$(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$

For multi-channel data $f:P\to \mathbb{R}^n$ , $g:B\to \mathbb{R}^n$ , the convolution becomes

$(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$

The scalar response field is then extended by the lower- $f:P\to \mathbb{R}$ 0 rule to a cubical complex on the output grid, and persistence is computed on the resulting filtration.

The simplicial analogue replaces image patches by a vertex signal on a connected simplicial complex. If $f:P\to \mathbb{R}$ 1 encodes vertex features, $f:P\to \mathbb{R}$ 2 is an arbitrary fixed $f:P\to \mathbb{R}$ 3 matrix, and $f:P\to \mathbb{R}$ 4 is a $f:P\to \mathbb{R}$ 5 weight vector, then the convolved signal is the scalar vertex function associated to $f:P\to \mathbb{R}$ 6. Persistent homology is computed on the lower- $f:P\to \mathbb{R}$ 7 extension of that scalar function over the original simplicial complex. This produces a transform viewpoint: for a family of filters, one obtains a family of persistence diagrams indexed by filters.

The paper’s conceptual claim is that the persistence diagram of the resulting convolution describes the way the motif is distributed across the simplicial complex. This is not merely smoothing. The convolved scalar field is a motif-response field, and persistent homology measures the topology of regions where the motif response is strong or weak. The trivial $f:P\to \mathbb{R}$ 8 filter recovers ordinary persistence, so convolutional persistence strictly generalizes standard persistence.

The theoretical contribution is unusually strong. The paper proves that, generically speaking, for any two labeled complexes one can find some filter for which they produce different persistence diagrams, so that the collection of all possible convolutional persistence diagrams is an injective invariant. This is established by showing convolutional persistence to be a special case of the Persistent Homology Transform. The paper also proves explicit stability bounds; for example, in the image setting,

$f:P\to \mathbb{R}$ 9

A further practical point is computational: with stride $P \subset \mathbb{Z}^d$ 0, persistence complexity becomes

$P \subset \mathbb{Z}^d$ 1

and more generally, for $P \subset \mathbb{Z}^d$ 2 filters and stride-product $P \subset \mathbb{Z}^d$ 3, the cost is

$P \subset \mathbb{Z}^d$ 4

The experiments use UCI digits, MNIST, Chinese handwritten digits, Devanagari characters, and a Kuramoto–Sivashinsky PDE parameter-estimation task. The reported empirical message is that convolutional persistence substantially outperforms ordinary persistence, even when one uses random filters and vectorizes the resulting diagrams by recording only their total persistences (Solomon et al., 2022).

3. Local persistent homology convolutions for histopathology

"Classification of Histopathology Slides with Persistence Homology Convolutions" (Pothagoni et al., 18 Jul 2025) introduces Persistent Homology Convolutions (PHCs) as a convolution-like operator that replaces local intensity-based feature extraction with local topological feature extraction. The motivation is domain-specific: in histopathology, topology is an important descriptor because disease-relevant abnormalities include cell and nucleus size variation, multinucleation, disorganization of tissue, and changes in the arrangement of cellular voids. The paper argues that previous methods used global topological summaries computed over an entire image, whereas PHC is designed to retain both locality and the convolutional notion of translation invariance of topological features.

The formalism begins with an $P \subset \mathbb{Z}^d$ 5 grayscale image $P \subset \mathbb{Z}^d$ 6, where

$P \subset \mathbb{Z}^d$ 7

For $P \subset \mathbb{Z}^d$ 8, the map

$P \subset \mathbb{Z}^d$ 9

translates the $g:B\to \mathbb{R}$ 0-coordinate of $g:B\to \mathbb{R}$ 1 to the origin and restricts to the $g:B\to \mathbb{R}$ 2 window $g:B\to \mathbb{R}$ 3. PHC then slides a local window over the image, builds a filtration on each local subimage, computes persistent homology in that receptive field, vectorizes the resulting persistence diagram, and combines the local topological vectors via a kernel. In the implementation emphasized in the paper, $g:B\to \mathbb{R}$ 4 is the extended sublevel-set filtration of the height function on the $g:B\to \mathbb{R}$ 5 subimage after thresholding; persistence is computed primarily in dimension $g:B\to \mathbb{R}$ 6; and the vectorization is a persistence image.

The local topological pipeline is highly specific. Images are converted from RGB to grayscale, resized from $g:B\to \mathbb{R}$ 7 to $g:B\to \mathbb{R}$ 8, thresholded, and then morphologically conditioned. The text contains an inconsistency: one part refers to “image erosion,” while another describes applying dilation once with a $g:B\to \mathbb{R}$ 9 kernel. The thresholded pixels are inserted into a simplex tree, along with edges between neighboring nonzero pixels, producing the adjacency complex of the image. The filtration is extended persistence of the height function $B\subset \mathbb{Z}^d$ 0, and the paper restricts attention to

$B\subset \mathbb{Z}^d$ 1

explicitly excluding summaries generated when $B\subset \mathbb{Z}^d$ 2. Each local diagram is vectorized as a persistence image $B\subset \mathbb{Z}^d$ 3, with $B\subset \mathbb{Z}^d$ 4 and $B\subset \mathbb{Z}^d$ 5 fixed for reported experiments.

In experiments, $B\subset \mathbb{Z}^d$ 6 and $B\subset \mathbb{Z}^d$ 7, so windows are $B\subset \mathbb{Z}^d$ 8. For a $B\subset \mathbb{Z}^d$ 9 image, the output is

$k$ 0

that is, 256 local persistence images, one per spatial locality. The persistent homology computation itself is not learned. The learnable part is the kernel $k$ 1, described as “a linear operator (an arbitrary $k$ 2 matrix) whose entries are optimized via backpropagation during model training.” The authors explicitly note that most of the PHC data generation is a separate preprocessing step applied on the image data outside of the training cycle except for optimizing the weights of $k$ 3. PHC is therefore not end-to-end differentiable in the reported implementation.

The experimental setting uses the Osteosarcoma histopathology dataset from TCIA: 1144 RGB images, originally $k$ 4, with 3 classes—non-tumorous (47%), necrotic tumor (23%), and viable tumor (30%). After balancing by resampling and augmentation, the experimental dataset contains 1143 images. The study compares thresholded image data, global persistent homology, local persistent homology via PHC, images plus global PH, and images plus PHC. For the best accuracy-maximizing models, Height PHC attains accuracy $k$ 5, precision $k$ 6, sensitivity $k$ 7, and specificity $k$ 8, while Images + Height PHC attains accuracy $k$ 9, precision $(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 0, sensitivity $(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 1, and specificity $(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 2. Height PHC also compares favorably in runtime: for a sample of 100 images, average per-image computation times were $(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 3 sec for Height PHC and $(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 4 sec for Global Height PH. Across 1000 model initializations, Height PHC yields

$(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 5

compared with

$(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 6

for Image Data and

$(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 7

for Images + Height PHC. The paper interprets these results as evidence that PHC preserves where topological features occur and is less sensitive to hyperparameters (Pothagoni et al., 18 Jul 2025).

4. Lattice-theoretic convolutions on multiparameter persistence modules

"Multidimensional Persistence Module Classification via Lattice-Theoretic Convolutions" (Riess et al., 2020) addresses a different object: not raw images, but multiparameter persistence modules themselves. The starting point is that persistent homology naturally produces data indexed by an ordered parameter set. In the one-parameter case, a filtration such as $(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 8 yields a persistence module

$(f \ast g)(v) = \sum_{p \in B} g(p)f(p+v \odot k).$ 9

and the total order on $f:P\to \mathbb{R}^n$ 0 permits interval decomposition and barcode summaries. In the multiparameter case, the paper considers a bifiltration

$f:P\to \mathbb{R}^n$ 1

which gives a $f:P\to \mathbb{R}^n$ 2-parameter persistence module

$f:P\to \mathbb{R}^n$ 3

Because the indexing category is now the poset $f:P\to \mathbb{R}^n$ 4 with product order, there is “no complete compact barcode-like characterization” of such modules.

The paper therefore uses computable incomplete invariants. For a fixed homological degree $f:P\to \mathbb{R}^n$ 5, it encodes the module by the Hilbert function

$f:P\to \mathbb{R}^n$ 6

together with the multigraded Betti numbers

$f:P\to \mathbb{R}^n$ 7

This yields four channels over a discretized parameter grid: one Hilbert function plus $f:P\to \mathbb{R}^n$ 8. The conceptual claim is that ordinary convolutions on these arrays are not obviously the natural operators, because standard convolution assumes an abelian group such as $f:P\to \mathbb{R}^n$ 9, while the natural algebraic structure of the indexing set is a lattice.

The proposed alternative is lattice-theoretic signal processing. For signals $g:B\to \mathbb{R}^n$ 0 on a lattice $g:B\to \mathbb{R}^n$ 1, the paper defines

$g:B\to \mathbb{R}^n$ 2

For the finite lattice

$g:B\to \mathbb{R}^n$ 3

meet and join are computed coordinatewise: $g:B\to \mathbb{R}^n$ 4 The multichannel meet-convolution and join-convolution layers are

$g:B\to \mathbb{R}^n$ 5

and

$g:B\to \mathbb{R}^n$ 6

The actual layer used in experiments combines both directions as

$g:B\to \mathbb{R}^n$ 7

with

$g:B\to \mathbb{R}^n$ 8

A key technical issue is receptive-field design. For meet-convolution the appropriate neutral element is the maximum $g:B\to \mathbb{R}^n$ 9, while for join-convolution it is the minimum $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 0. The paper remarks that kernels supported only on a small contiguous block near one corner can produce degenerate receptive fields; for join-convolution, many neurons may see only themselves. The authors therefore hypothesize that appropriate lattice kernels should be supported on evenly spaced sublattices including both $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 1 and $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 2, and in experiments they use an evenly spaced $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 3 grid of lattice points.

The end-to-end pipeline uses a subset of Princeton ModelNet containing 10 object classes. For each CAD model, 3000 vertices are sampled to obtain a point cloud in $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 4. A bifiltered simplicial complex is built using a geometric scale parameter and the codensity function

$(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 5

with $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 6. Using RIVET, the authors compute degree- $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 7 multiparameter persistent homology on a discrete $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 8 grid, forming a $(f \ast g)(v) = \sum_{p \in B} g(p)\cdot f(p+v \odot k).$ 9 tensor. The lattice-convolutional CNN and a standard CNN baseline each have three convolutional layers followed by two fully connected layers, with hidden convolutional layers of 16 channels, a final convolutional layer of 8 channels, $f:P\to \mathbb{R}$ 00 max-pooling after the first two convolutional layers, a fully connected hidden layer of 32 features, a final softmax with cross-entropy loss, and the Adam optimizer with learning rate

$f:P\to \mathbb{R}$ 01

for 300 epochs. The main empirical result is modest: the lattice convolutional network “slightly underperforms” the standard convolutional classifier. The paper presents this not as superiority, but as a proof of concept for an order-aware convolutional formalism for multiparameter persistence modules (Riess et al., 2020).

5. Convolution as stabilization in input or parameter space

"Stabilizing the unstable output of persistent homology computations" (Bendich et al., 2015) uses the word convolution in yet another sense. Here convolution does not act on images, on persistence modules, or on diagrams. Instead, some PH-derived quantity of interest is viewed as a real-valued function on the space of inputs or parameters, and convolution with a kernel is used to make that quantity Lipschitz-stable.

The formal setup is as follows. Let $f:P\to \mathbb{R}$ 02 denote all real parameters entering a persistent homology pipeline $f:P\to \mathbb{R}$ 03, and let $f:P\to \mathbb{R}$ 04 denote the induced output map. One then chooses a real-valued summary $f:P\to \mathbb{R}$ 05, such as the persistence of the class born by a specific simplex, an indicator that a representative cycle intersects a chosen region, or the longest persistence after density thresholding. This defines

$f:P\to \mathbb{R}$ 06

The function $f:P\to \mathbb{R}$ 07 may be discontinuous but measurable. The stabilized quantity is

$f:P\to \mathbb{R}$ 08

where

$f:P\to \mathbb{R}$ 09

The paper discusses triangular, Epanechnikov, and Gaussian kernels.

The core theorems state that smoothing by convolution yields Lipschitz control under mild assumptions. If

$f:P\to \mathbb{R}$ 10

and $f:P\to \mathbb{R}$ 11 is $f:P\to \mathbb{R}$ 12-Lipschitz, then $f:P\to \mathbb{R}$ 13 is $f:P\to \mathbb{R}$ 14-Lipschitz. If $f:P\to \mathbb{R}$ 15 locally and $f:P\to \mathbb{R}$ 16 is compactly supported and Lipschitz, then $f:P\to \mathbb{R}$ 17 is locally Lipschitz. If $f:P\to \mathbb{R}$ 18 and $f:P\to \mathbb{R}$ 19 satisfies an $f:P\to \mathbb{R}$ 20-translation bound, then $f:P\to \mathbb{R}$ 21 is globally Lipschitz. The conceptual interpretation is that exact localization of a generator or exact identification of a simplex may be unstable, while an average score under small perturbations is stable.

The method is computationally practical because convolution can be estimated by Monte Carlo: $f:P\to \mathbb{R}$ 22 when the $f:P\to \mathbb{R}$ 23 are sampled i.i.d. from $f:P\to \mathbb{R}$ 24. The operational prescription is therefore to perturb the input many times, rerun persistent homology, and average the unstable output. The paper illustrates this with localization of representative cycles in a double-annulus example, with a brain-imaging example in which the stabilized probability for the generator of the $f:P\to \mathbb{R}$ 25th longest bar to lie in a chosen ball is estimated as

$f:P\to \mathbb{R}$ 26

and with thresholding-based examples in which the persistence diagram itself is unstable with respect to preprocessing parameters. The paper is explicit that this framework complements stable diagram summaries by stabilizing information beyond the diagram, such as where a generator is located or which simplex created a feature (Bendich et al., 2015).

6. Sheaf-theoretic convolution and derived persistence

"Persistent homology and microlocal sheaf theory" (Kashiwara et al., 2017) develops a categorical notion of convolution on the derived category of sheaves on a finite-dimensional real vector space $f:P\to \mathbb{R}$ 27. This is not a convolution on persistence modules in the usual algebraic sense. Rather, it is a sheaf-theoretic convolution used to reinterpret persistence, stability, and higher-dimensional barcode-like structures.

The ambient category is

$f:P\to \mathbb{R}$ 28

and the convolution functor is

$f:P\to \mathbb{R}$ 29

where $f:P\to \mathbb{R}$ 30 is addition. The category becomes a commutative monoidal category with unit $f:P\to \mathbb{R}$ 31. A family of kernels $f:P\to \mathbb{R}$ 32 satisfies

$f:P\to \mathbb{R}$ 33

so convolution with $f:P\to \mathbb{R}$ 34 behaves like a metric thickening operator. This is the basis for a pseudo-distance.

For $f:P\to \mathbb{R}$ 35 and $f:P\to \mathbb{R}$ 36, the paper says that $f:P\to \mathbb{R}$ 37 and $f:P\to \mathbb{R}$ 38 are $f:P\to \mathbb{R}$ 39-isomorphic if there exist morphisms

$f:P\to \mathbb{R}$ 40

satisfying compatibility conditions through $f:P\to \mathbb{R}$ 41. The pseudo-distance is

$f:P\to \mathbb{R}$ 42

This is explicitly interleaving-like, but formulated in the derived sheaf category and expressed through convolution kernels $f:P\to \mathbb{R}$ 43 rather than shifts of persistence modules. The distance is symmetric, satisfies the triangle inequality, and can be finite even for sheaves concentrated in different cohomological degrees.

The main stability theorem states that for a locally compact space $f:P\to \mathbb{R}$ 44, continuous maps $f:P\to \mathbb{R}$ 45, and any $f:P\to \mathbb{R}$ 46,

$f:P\to \mathbb{R}$ 47

This is a sheaf-theoretic analogue of classical persistence stability. The broader microlocal framework introduces $f:P\to \mathbb{R}$ 48-sheaves, that is, sheaves on $f:P\to \mathbb{R}$ 49 whose microsupport lies in a cone determined by a closed convex proper cone $f:P\to \mathbb{R}$ 50 with nonempty interior. The paper proves an equivalence between $f:P\to \mathbb{R}$ 51-sheaves and sheaves on the $f:P\to \mathbb{R}$ 52-topology $f:P\to \mathbb{R}$ 53, approximation of constructible sheaves by piecewise-linear objects in the convolution pseudo-distance, and a higher-dimensional barcode analogue in which piecewise-linear $f:P\to \mathbb{R}$ 54-sheaves are constant on stratifications by $f:P\to \mathbb{R}$ 55-locally closed sets.

In dimension one, the framework recovers ordinary barcode theory. With $f:P\to \mathbb{R}$ 56 and $f:P\to \mathbb{R}$ 57, $f:P\to \mathbb{R}$ 58-locally closed sets are intervals $f:P\to \mathbb{R}$ 59, and any $f:P\to \mathbb{R}$ 60-constructible $f:P\to \mathbb{R}$ 61-sheaf decomposes uniquely as a locally finite direct sum of interval sheaves. In higher dimensions, the paper does not obtain a full barcode classification. Instead, it establishes a richer categorical structure in which higher-dimensional “barcodes” embed fully faithfully but are not essentially surjective. This suggests that convolution, in this setting, functions as a conceptual higher-dimensional and categorical analogue of thickening and interleaving in persistence (Kashiwara et al., 2017).