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., 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 $c = \beta_k(x_0)$0, the unweighted CkNN graph $c = \beta_k(x_0)$1 sets $c = \beta_k(x_0)$2 iff $c = \beta_k(x_0)$3, and $c = \beta_k(x_0)$4 otherwise.
• The unnormalized graph Laplacian $c = \beta_k(x_0)$5 (where $c = \beta_k(x_0)$6) is analyzed under random sampling $c = \beta_k(x_0)$7, where $c = \beta_k(x_0)$8 is a positive density on $c = \beta_k(x_0)$9.

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 $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$0 of a conformally changed metric $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$1. This ensures the multiplicity of the zero-eigenvalue of $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$2 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 $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$3, the Vietoris–Rips complex built from the CkNN graph satisfies

$$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$4

valid for all $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$5 as $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$6, 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 $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$7 be a density or regression function. The kernel estimator $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$8 approximates $$\mathrm{TDC} = 1 - \frac{1}{T} \sum_{t=1}^T \frac{|\beta_k(\hat x_0^t) - c|}{\max(c,1)},$$9 locally via: $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$0 for some kernel $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$1 and bandwidth $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$2. To robustly recover the homology of a level set $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$3, the estimator constructs empirical level-sets at $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$4, with $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$5, and uses the inclusion-induced map: $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$6 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 $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$7 is tame and $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$8 is regular (i.e., excludes critical values of $$\mathrm{TDC} = \frac{1}{T} \sum_{t=1}^T \mathbf{1}\left[\beta_k(\hat x_0^t) = c\right].$$9), for $\hat H_*$0,

$\hat H_*$1

with $\hat H_*$2 (Bobrowski et al., 2014). Persistent homology barcodes are also recovered within $\hat H_*$3 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 $\hat H_*$4 is introduced:

• For each intermediate denoised sample $\hat H_*$5, compute the persistence diagram of its super-level-set filtration.
• Given target $\hat H_*$6 (desired Betti number), partition persistent features into (i) top $\hat H_*$7 features to be preserved ($\hat H_*$8), and (ii) others to be suppressed ($\hat H_*$9), using their lifetimes $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$0 (where $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$1 are birth/death of feature $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$2).
• Define the loss: $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$3 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 $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$4 (indicator metric) across all denoising steps, compared to TDC $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$5 for topology-agnostic baselines. The approach yields accuracy improvements (e.g., for 1-dimensional (hole) topology, TDN $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$6 vs. ADM-T $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$7 on the Google-Maps dataset). Ablation studies confirm that both preservation and denoising components are necessary for optimal TDC (Gupta et al., 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: $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$8; variance: $\mathbb{P}[\,\hat H_*(L)\simeq H_*(D_L)\,] \to 1$9.
• Spectral convergence rate optimal at $M \subset \mathbb{R}^N$0 for operator spectrum tasks.
• Local rescaling via $M \subset \mathbb{R}^N$1 ensures that low-density (“noise”) samples are increasingly isolated, rendering the graph robust to variable sampling or noise.
• For kernel approaches, bandwidth scheduling $M \subset \mathbb{R}^N$2 balances bias and misclassification probability.

In TopoDiffusionNet, the scalar weight $M \subset \mathbb{R}^N$3 for the topological loss is empirically tuned ($M \subset \mathbb{R}^N$4 is optimal); both terms of the loss must be active. The persistence threshold capping (top-$M \subset \mathbb{R}^N$5 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 ($M \subset \mathbb{R}^N$6, $M \subset \mathbb{R}^N$7, 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 $M \subset \mathbb{R}^N$8 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.