Topological Denoising Consistency (TDC)

• Topological Denoising Consistency is a framework ensuring that estimators reliably recover true topological features, such as Betti numbers, from noisy observations.
• It leverages manifold constructions (CkNN), kernel-based level-set estimators, and topology-aware deep models like TopoDiffusionNet to achieve consistent persistent homology recovery.
• TDC provides practical guidance for parameter tuning and error control by offering convergence guarantees and explicit loss formulations for robust topological data analysis.

Topological Denoising Consistency (TDC) refers to a suite of theoretical and algorithmic guarantees under which estimators recover the true topological invariants (homology, or persistent homology) of an underlying object from noisy or indirect observations. It is a key property in topological data analysis (TDA), where real-world data are often polluted by sampling noise or structured artifacts. TDC requires that an estimator, given increasingly many or more precise samples, reconstructs the correct Betti numbers or persistence diagrams with high probability, despite denoising and statistical fluctuation. TDC has been formalized and achieved in distinct frameworks: manifold-based constructions (notably the Continuous k-Nearest-Neighbors (CkNN) graph), kernel-based level-set estimators, and topology-aware learning models such as TopoDiffusionNet for image synthesis. Each offers rigorous results, parameter guidelines, and error bounds appropriate to the underlying noise model and data modality.

1. Formal Definitions and Measurement of TDC

TDC characterizes the reliability with which a procedure recovers the correct topological descriptors from noisy data. In image synthesis and generative modeling, TDC is defined as the degree to which intermediate denoised representations $\hat x_0^t$ retain the target Betti number $c = \beta_k(x_0)$ over all denoising steps. A scalar metric is: $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$ and equivalently by its indicator form,

$$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$

Maximizing TDC ensures that the estimated Betti number matches the ground truth across the entire process, implying “consistency” in both the statistical and topological sense (Gupta et al., 22 Oct 2024).

In the setting of manifold and level-set inference, TDC is formulated probabilistically: for a given estimator $\hat H_*$, one establishes high-probability convergence

$\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$

as sample size grows and smoothing parameters shrink (Bobrowski et al., 2014, Berry et al., 2016).

2. CkNN and Manifold Representation: Uniqueness and Consistency

CkNN achieves TDC in the context of point-cloud data sampled (possibly noisily) from a submanifold $M \subset \mathbb{R}^N$ via the following construction:

• For sample $x_i$, let $r_i$ be the Euclidean distance to its $k$th nearest neighbor.
• For continuous scale parameter $\delta > 0$, the unweighted CkNN graph $G_n(\delta)$ sets $W_{ij}=1$ iff $d(x_i,x_j) < \delta\,\sqrt{r_i r_j}$, and $0$ otherwise.
• The unnormalized graph Laplacian $L_n = D - W$ (where $D_{ii} = \sum_j W_{ij}$) is analyzed under random sampling $x_i \sim p$, where $p$ is a positive density on $M$.

Under precise smoothness and sampling density assumptions [(A1), (A2) in (Berry et al., 2016)], CkNN is shown to produce a graph Laplacian that converges (spectrally and pointwise) to the Laplace–Beltrami operator $\Delta_{\tilde g}$ of a conformally changed metric $\tilde g = p^{2/m}g$. This ensures the multiplicity of the zero-eigenvalue of $L_n$ recovers the correct number of connected components; thus the CkNN approach is the unique unweighted construction admitting topological (connected-component) denoising consistency under nonuniform sampling (Berry et al., 2016).

A central conjecture extends this to full persistent homology: for a suitable scaling $\delta_n \sim n^{-2/(m+6)}$, the Vietoris–Rips complex built from the CkNN graph satisfies

$$\mathbb{P}[\,H_k(\mathrm{VR}_{\delta_n}(X_n)) \cong H_k(M)\,] \to 1,$$

valid for all $k$ as $n \to \infty$, implying topological consistency under increasing sample size. This approach yields a unified homology estimator, in contrast to classical persistent homology, which detects features at varying scales.

3. Kernel-Based Level-Set Estimators and Homology Consistency

Kernel estimation methods achieve TDC for topology of level sets of density or regression functions. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a density or regression function. The kernel estimator $\hat f_h$ approximates $f$ locally via: $$\hat f_h(x) = \frac{1}{nC_K h^d} \sum_{i=1}^n K\left(\frac{x-X_i}{h}\right)$$ for some kernel $K$ and bandwidth $h > 0$. To robustly recover the homology of a level set $D_L = \{x : f(x) \geq L\}$, the estimator constructs empirical level-sets at $L_\pm = L \pm \varepsilon$, with $\varepsilon \sim h$, and uses the inclusion-induced map: $$i_* : H_k(\hat D_{L_+}) \to H_k(\hat D_{L_-}),\quad \hat H_k(L) := \mathrm{Im}(i_*).$$ This “image-of-inclusion” estimator suppresses spurious topological features that appear/disappear in a narrow band, greatly enhancing denoising reliability. Consistency theorems establish that, if $f$ is tame and $L$ is regular (i.e., excludes critical values of $f$), for $h\to 0,\,n h^d \gg \log n$,

$$\mathbb{P}[\,\hat H_*(L) \cong H_*(D_L)\,] \geq 1 - 6n \exp(-Cn h^d)$$

with $C>0$ (Bobrowski et al., 2014). Persistent homology barcodes are also recovered within $5\varepsilon$ bottleneck distance under similar conditions. These guarantees deliver full persistent-homology denoising consistency across a variety of real and simulated tasks.

4. Topology-Aware Deep Generative Models: TDC in TopoDiffusionNet

Topological Denoising Consistency is operationalized in deep generative frameworks by integrating persistent homology directly into the loss function. In TopoDiffusionNet, a topology-based loss $\mathcal{L}_{\mathrm{top}}$ is introduced:

• For each intermediate denoised sample $\hat x_0^t$, compute the persistence diagram of its super-level-set filtration.
• Given target $c$ (desired Betti number), partition persistent features into (i) top $c$ features to be preserved ($\mathcal{P}$), and (ii) others to be suppressed ($\mathcal{Q}$), using their lifetimes $\ell_p = |d_p - b_p|$ (where $(b_p,d_p)$ are birth/death of feature $p$).
• Define the loss: $$\mathcal{L}_{\mathrm{top}} = -\sum_{p \in \mathcal{P}} \ell_p^2 + \sum_{p \in \mathcal{Q}} \ell_p^2$$ which directly penalizes persistent noise and enforces the survival of only the correct number of topological features.

Empirical results on datasets including synthetic shapes, COCO-Animals, CREMI, and Google-Maps segmentation masks, demonstrate that enforcing this loss achieves TDC $\approx 0.93$ (indicator metric) across all denoising steps, compared to TDC $\approx 0.45$ for topology-agnostic baselines. The approach yields accuracy improvements (e.g., for 1-dimensional (hole) topology, TDN $0.8318 \pm 0.1159$ vs. ADM-T $0.5494 \pm 0.1386$ on the Google-Maps dataset). Ablation studies confirm that both preservation and denoising components are necessary for optimal TDC (Gupta et al., 22 Oct 2024).

5. Analytic Principles and Parameter Selection

The analytic foundation underlying TDC includes bias–variance tradeoffs, conformal metric changes, and spectral convergence theorems. For CkNN and graph-based methods:

• Pointwise bias: $O(\delta^2)$; variance: $O(n^{-1}\delta^{-(m+2)})$.
• Spectral convergence rate optimal at $\delta \sim n^{-2/(m+6)}$ for operator spectrum tasks.
• Local rescaling via $r_i \sim p(x_i)^{-1/m}$ ensures that low-density (“noise”) samples are increasingly isolated, rendering the graph robust to variable sampling or noise.
• For kernel approaches, bandwidth scheduling $h_n \sim c (\log n/n)^{1/d}$ balances bias and misclassification probability.

In TopoDiffusionNet, the scalar weight $\lambda$ for the topological loss is empirically tuned ($\lambda \sim 1\textrm{e}-5$ is optimal); both terms of the loss must be active. The persistence threshold capping (top-$c$ features) and persistence-diagram sorting are crucial for TDC attainment.

6. Practical Implications, Limitations, and Connections

TDC provides a principled framework for denoising in topological learning and inference:

• In data sampled from manifolds, TDC underpins the recovery of all Betti numbers with a single graph representation, eliminating the need to scan multiple scales.
• In function estimation (density or regression), TDC enables recovery of homology of level-sets or super-level-sets, with explicit non-asymptotic error bounds.
• In generative modeling, TDC yields explicit control over global topology, critical for robotics, segmentation, and scientific imaging.

A key practical consideration is the selection of scale parameters ($\delta$, $h$, etc.) and tuning of topological loss weights, which may be guided by theoretical rates but must be adapted to data characteristics. No finite-sample error bounds exist for certain TDC statements (e.g., the CkNN topological consistency conjecture for $k>0$ features on noncompact manifolds). Empirical validation and ablation remain essential in these settings.

TDC is closely related to the concepts of statistical consistency, persistent homology stability, and robust clustering under nonuniform sampling, offering a unified perspective across discrete, functional, and deep-learning paradigms.