Inverse Problems in TDA

• Inverse problems in TDA are techniques that determine if persistent homology barcodes uniquely capture the geometry or topology of underlying data.
• They explore both surjectivity—ensuring every barcode has a realizable pre-image—and injectivity, where unique recovery up to equivalence is sought.
• Recent methods, including distributed persistence and topological regularization, provide stability and improved discrimination amidst noise and sampling challenges.

The inverse problem in topological data analysis (TDA) concerns the extent to which topological invariants—specifically, persistent homology barcodes or persistence diagrams—capture enough information to recover, or meaningfully constrain, the geometry or topology of the underlying data. Given a collection of barcodes arising from data (often point clouds or functions on topological spaces), the inverse problem asks whether there exists—uniquely or up to some equivalence—a geometric or topological object that realizes these invariants, and under what conditions this recovery is stable, computable, and discriminative. This problem is fundamentally tied to the surjectivity and injectivity properties of the “persistence map,” as well as practical inference in the presence of noise, sampling artifacts, and model ambiguity.

1. Formulation and Types of Inverse Problems

Inverse problems in persistent homology can be formulated in two primary directions: surjectivity (the existence of pre-images or right inverses) and injectivity (uniqueness or the existence of left inverses) of the persistence functor. For a field $k$, a one-parameter persistence module $M$ is a functor $M: (\mathbb{R},\leq) \to \mathrm{Vect}_k$. The structure theorem implies $M$ decomposes as a direct sum of interval modules, whose multiset of intervals $\{I_j\}$ constitutes the barcode $B(M)$ (Oudot et al., 2018).

• Surjectivity: Given an abstract barcode $B$, does there exist a filtered topological space $X$ such that persistent homology of $X$ yields $B$?
• Injectivity: If two spaces produce identical barcodes, are they equivalent in some geometric or topological sense? Under what conditions is this possible?

This dichotomy frames the essential question: can topological features observed in data meaningfully determine, or even tightly constrain, the underlying signal or structure?

2. Surjectivity: Realizability of Barcodes and Modules

Several constructive methods demonstrate that, for many classes of persistence modules, any barcode can be realized by some filtered CW-complex. The Moore-space construction shows that any pointwise-finite-dimensional (pfd) module with sufficient degree-0 coverage can be realized via persistent Moore spaces formed by gluing spheres and disks according to the interval structure of the barcode (Oudot et al., 2018). Functional surjectivity extends this to show that, for any pfd module $M$ and any $\epsilon > 0$, one can find real-valued functions $\gamma^M, \gamma^N$ on a fixed topological space $X$ such that their persistent homologies approximate any two given modules as closely as desired in the interleaving distance.

Algorithmic approaches such as point-cloud continuation and functional map optimization enable one, at least locally, to converge to point clouds or function parameters whose persistence barcode matches a given target, using e.g. Newton-type schemes or gradient descent (Oudot et al., 2018).

These surjectivity results highlight that the persistence map is onto for broad classes of modules, but do not guarantee uniqueness or interpretability of preimages.

3. Injectivity, Discriminativity, and Stability

Ordinary persistent homology is not injective: distinct metric spaces or subsets can yield identical barcodes. For instance, non-isometric triangles, all trees under Čech filtration, and functions differing by homeomorphism on the domain can share their barcodes (Oudot et al., 2018). Thus, persistence alone cannot reliably distinguish all underlying spaces.

However, enhanced invariants improve discriminativity. The persistent homology transform (PHT) records the persistent homology of height functions in all directions $v \in S^{d-1}$, and the Euler characteristic transform (ECT) applies Schapira's Radon inversion framework to recover subanalytic sets in $\mathbb{R}^d$ from ECT or PHT with sufficiently many directions. For planar graphs, $O(n^2)$ directions suffice for reconstruction; for general complexes, the number of required directions depends on angular visibility and the number of critical values per direction (Oudot et al., 2018).

Intrinsic analogues such as the intrinsic persistent homology transform (IPHT), which uses distance-to-point functions, achieve stability and near-generic injectivity for metric graphs, modulo automorphism obstructions. The persistence-distortion metric gives quantifiable bounds: $d_{PD}(X,Y) \leq 18\,d_{GH}(X,Y)$, where $d_{GH}$ is the Gromov-Hausdorff distance (Oudot et al., 2018).

4. Distributed Persistence and Bi-Lipschitz Inversion

Distributed persistence advances the inverse problem by associating to each point cloud $X$ the multiset of persistence diagrams of all (or suitably many) $k$-point subsets, denoted $D_{\mathrm{dist},k}(X)$. Equipped with the Hausdorff-bottleneck distance, the map from finite metric spaces to distributed persistence invariants is globally bi-Lipschitz with respect to the quasi-isometry metric. Specifically,

$$d_{\mathrm{HB}}(D_{\mathrm{dist},k}(X), D_{\mathrm{dist},k}(Y)) \leq d_{\mathrm{QI}}(X, Y)$$

and conversely,

$$d_{\mathrm{QI}}(X, Y) \leq 112\,k^2\,d_{\mathrm{HB}}(D_{\mathrm{dist},k}(X), D_{\mathrm{dist},k}(Y))$$

for Rips persistence, with the bounds improving for small $k$ and becoming loose as $k \to |X|$ (Solomon et al., 2021).

This construction interpolates between geometry and topology: for $k=2$, the invariant recovers all edge lengths; for larger $k$, more global shape information appears, but injectivity weakens as $k$ increases.

These results demonstrate that, for sufficiently rich collections of diagrams, the inverse problem has a quantitative stability property not found for full-diagram persistence alone.

5. Inverse Problems in Topological Inference from Sampled Data

For spaces such as manifolds with boundary sampled by finite point clouds, the inverse problem is to recover the topology (e.g., homology or homotopy type) of the underlying space. Probabilistic results show that if one samples $$n > \beta(\epsilon) (\ln \beta(\epsilon/2) + \ln(1/\gamma))$$ i.i.d. points on a $C^2$ manifold $M$ of positive reach, then the union of $\epsilon$-balls around the sample with $\epsilon < \min\{\tau(M), \tau(\partial M)\}/2$ deformation-retracts onto $M$ with probability at least $1-\gamma$, thus recovering its homological invariants (Wang et al., 2018). The reach $\tau(M)$, local chart conditions, and volume lower bounds are critical to guaranteeing the deformation retract.

This provides a rigorous, parameterized recipe for the inverse problem in manifold inference. The result is stronger than simple homology recovery; the union of balls is homotopy equivalent, and indeed a deformation retract, of the original manifold under appropriate conditions.

6. Ill-Posedness and Topological Regularization in Applications

The ill-posedness of inverse problems is acute in practical scenarios such as cryo-electron microscopy (cryo-EM), where reconstruction from noisy projections leads to non-unique density fits under standard least-squares or correlation-based objectives. For example, fitting various coarse-grained helix models to cryo-EM data may yield indistinguishable cross-correlation scores, but persistent homology breaks the degeneracy: only models with the correct topological fingerprint, as measured by $\beta_0$ and $\beta_1$ barcodes, match the denoised experimental data (Xia et al., 2014).

By imposing topological constraints via persistent homology, well-posedness is restored: among many optimizers, only those matching the persistent fingerprint of the data are selected. This shifts inference from purely geometric or correlation-based metrics to topologically regularized criteria, sensitive to connective and loop features invisible to classical methods.

7. Open Challenges and Future Research

Open problems include efficiently selecting minimal sets of directions for transforms such as PHT/ECT, extending intrinsically stable and injective transforms (e.g., IPHT) to higher-dimensional and stratified spaces, studying transforms based on alternative filtrations (e.g., Laplacian eigenfunctions), improving algorithmic scaling (seeking near-linear approaches for special classes), and establishing statistical theory (hypothesis testing, confidence intervals, sample complexity) for inverse recovery.

There is ongoing interest in using inverse problem tools for data explainability and model interpretation, guiding visualization and understanding of the shapes that underlie observed barcodes (Oudot et al., 2018). A major challenge remains in generalizing deformation-retract and injectivity results beyond positive-reach manifolds to settings with singularities or more complex stratifications (Wang et al., 2018).

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Key references: (Xia et al., 2014, Oudot et al., 2018, Wang et al., 2018, Solomon et al., 2021)